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ACCIDENT OR DESIGN? New Theories on the Unfinished Contrapunctus 14 in J. S. Bach’s The Art of Fugue BWV 1080
INDRA NICHOLAS MARTINDALE HUGHES
A thesis submitted in partial fulfilment of the requirements for the Degree of DOCTOR OF MUSICAL ARTS
University of Auckland, 2006
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Frontispiece
Deutsche Staatsbibliothek, Berlin
Mus. ms. P200/Beilage 3 page 5
The last page of Bach’s autograph manuscript of Contrapunctus 14
This reproduction is a colour postcard published by Carus, Stuttgart CV 40.331/90. It is included here, despite its reduced size, as the clearest available reproduction of the
last page.
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ABSTRACT
The literature about the unfinished ending of J S Bach’s The Art of Fugue (Die Kunst der Fuga) BWV 1080 is in universal agreement that the work remained unfinished at Bach’s death; some texts go a step further to say that it is unfinished because he died. After giving a series of performances of the work, the author became convinced that this latter view must be incorrect, and that Bach left the work unfinished deliberately. This thesis explores this idea in detail and, by presenting a number of new theories, suggests not only that Bach left the work unfinished deliberately as an invitation to the reader, student or performer to work out his or her own completion, but also that he left a number of clues, hidden to a greater or lesser extent, to indicate that that was his intention and to supply vital information about the content of the missing bars.
Divided into two parts, the thesis first considers some of the evidence contained within the manuscript itself, up to and including the final written bar, and then in the second part goes on to consider two essential aspects of the completion.
By way of introduction, the first chapter surveys the controversial area of Bach’s use of numbers in his music and draws attention to the number of the final bar, which can be interpreted as a clue to the fact that Bach expects the music to be continued.
Chapter Two invites a reconsideration of Christoph Wolff’s famous “Fragment X” theory, which suggests that the continuation of the final fugue was written on a separate, now lost, piece of paper. Many inconsistencies and details in the manuscript suggest strongly that Wolff’s theory is incorrect. As part of this theory, the author reports on his own examination of the original manuscript in Berlin.
Chapter Three, through a detailed study of the architecture of the final fugue, makes the bold claim that the author has definitively proved the exact number of bars required to complete the music in accordance with Bach’s intentions: this theory develops and refines the work of Gregory Butler in this area, and, to corroborate the theory, presents a possible interpretation of the unusual markings at the end of Bach’s score and of a significant correction made by Bach in his manuscript.
Finally, in Chapter Four, the question of the proposed inverted combination of all four fugue subjects is revisited – a combination that several writers have claimed to be impossible – and a new and convincing solution to this problem is presented and justified.
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Acknowledgements
I should like to thank my two Supervisors at the University of Auckland: Dr
Fiona McAlpine, who was principally responsible for the supervision of this thesis; and Dr John Wells. I am very grateful for their expertise and wisdom.
Thanks are due also to those who have shared their knowledge and opinions with me in private correspondence: Michael Ferguson, Lionel Rogg, Dr Bradley Lehman, Dr Yo Tomita of Queen’s University, Belfast, Dr John Bertalot, Ewald Demeyere of the Koninklijk Vlaams Muziekconservatorium, Antwerp and Nikolas Krawchenko.
Dr Helmut Hell of the Deutsche Staatsbibliothek in Berlin graciously granted permission for me to view Bach’s original manuscript of The Art of Fugue, enabling me to confirm my theories about the markings on Bach’s last page, about the handwriting characteristics, and about the significant alteration made by Bach in one place in the score of Contrapunctus 14. The staff at the Bach‐Archiv in Leipzig generously allowed me to take photocopies of some material that was not easily available elsewhere.
A number of people and organizations made financial contributions to assist me with my research visit to Berlin: they are listed here in alphabetical order with gratitude and appreciation. Margaret Barriball; Graeme Edwards; David Goodenough; Mrs C E Hughes; Wayne Hughes; Neil Ingram; Margo Knightbridge; Joan Pasche; John Sinclair; the University of Auckland Graduate Research Fund; the University of Auckland School of Music; Dr David Yu.
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Table of Contents ABSTRACT .................................................................................................................................................III
ACKNOWLEDGEMENTS .............................................................................................................................. IV TABLE OF CONTENTS .................................................................................................................................. V LIST OF FIGURES ........................................................................................................................................ VI LIST OF MUSICAL EXAMPLES ...................................................................................................................VII
PART ONE: BEFORE BAR 239 ................................................................................................................ 1 BACH’S USE OF GEMATRIA IN THE ART OF FUGUE.................................................................... 1
BACKGROUND............................................................................................................................................. 2 NUMBER ALPHABETS .................................................................................................................................. 3 EXAMPLES OF GEMATRIA IN BACH’S MUSIC ............................................................................................. 4
Smend’s theories.................................................................................................................................... 4 The number 43: The B Minor Mass Credo and the Organ Prelude Wir glauben all’ an einen Gott BWV 680................................................................................................................................................ 8 The number 3: The “St Anne” Fugue and the Organ Prelude Wachet auf, ruft uns die Stimme ....... 8 The numbers 14 and 41 (the “BACH numbers”): the Mass in B Minor; the Schübler Chorales and the Chorale Vor deinen Thron .............................................................................................................. 9
THE BACH NUMBERS AND THE ART OF FUGUE......................................................................................... 15 Herbert Kellner and the significance of the Well-Tempered tuning system ...................................... 15 Other occurrences of the number 14 .................................................................................................. 18 The last bar.......................................................................................................................................... 20
‘FRAGMENT X’ RECONSIDERED....................................................................................................... 22 THE SOURCES............................................................................................................................................ 22
The Autograph: P200.......................................................................................................................... 23 Beilage 3 .............................................................................................................................................. 23
CHRISTOPH WOLFF’S ARTICLE: THE ‘FRAGMENT X’ THEORY .................................................................. 27 PROBLEMS WITH WOLFF’S THEORY.......................................................................................................... 31
‘Fragment x’........................................................................................................................................ 31 “Two different pieces”........................................................................................................................ 33 The structure of Contrapunctus 14 ..................................................................................................... 34 The appearance of page 5 ................................................................................................................... 35
WHAT CATEGORY OF SCORE IS BEILAGE 3? .............................................................................................. 47 Examination of the manuscript ........................................................................................................... 57 More on the appearance and ruling of page 5 ................................................................................... 63
CONCLUSIONS ........................................................................................................................................... 70 PART TWO: AFTER BAR 239 ................................................................................................................ 75 THE BACH NUMBERS, PROPORTIONAL RATIOS AND THE MARKINGS ON THE LAST PAGE: THE LENGTH OF THE QUADRUPLE FUGUE CONCLUSIVELY ESTABLISHED..... 75
INTRODUCTION ......................................................................................................................................... 75 BACH’S USE OF GEMATRIA OR NUMEROLOGY: BRIEF OVERVIEW............................................................ 76 VARYING LENGTHS OF COMPLETIONS OF CONTRAPUNCTUS 14 ............................................................... 77
Gregory Butler’s article...................................................................................................................... 80 REFINEMENT OF BUTLER’S THEORY......................................................................................................... 84 EXPLANATION OF A “CORRECTION” IN THE MANUSCRIPT ........................................................................ 91 MARKINGS ON THE LAST PAGE OF THE AUTOGRAPH: A POSSIBLE NUMEROLOGICAL INTERPRETATION... 95
Gematria applied to the markings ...................................................................................................... 97 CORROBORATIVE EXAMPLES .................................................................................................................. 100
The Mass in B minor ......................................................................................................................... 100 The title page of The Well-Tempered Clavier .................................................................................. 102 The Musical Offering: “Quaerendo invenietis” and the Houdemann Canon BWV 1074 .............. 104
CONCLUSION........................................................................................................................................... 106
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QUESTIONS RAISED BY THE OBITUARY, AND NEW SOLUTIONS TO THE PROBLEM OF THE INVERTED COMBINATIONS .................................................................................................... 108
“This is the last work of the author…”............................................................................................. 113 “His last illness prevented him from completing…” ....................................................................... 114 “the next-to-last fugue” .................................................................................................................... 117 “the last one was to contain four themes…”.................................................................................... 119
THE RECTUS COMBINATION OF THE FOUR SUBJECTS ............................................................................ 121 “… was to contain four themes” ...................................................................................................... 121 Consecutives in Bach’s Music........................................................................................................... 124
THE INVERSUS COMBINATION OF THE FOUR SUBJECTS ......................................................................... 142 “…and to have been afterward inverted note for note in all four voices” ...................................... 142 Inverting the subjects ........................................................................................................................ 144 More on subject II: Bach’s sometimes irregular resolution of suspensions.................................... 147 Combining the four inverted subjects ............................................................................................... 153
AN OUTLINE OF A POSSIBLE COMPLETION .............................................................................................. 167 CONCLUSIONS ....................................................................................................................................... 171 APPENDIX: FACSIMILE OF MUS. MS. P200, BEILAGE 3............................................................ 174 BIBLIOGRAPHY ..................................................................................................................................... 180
List of Figures Fig. 1.1: number alphabet by Georg Philipp Harsdoerffer, Poetischer Trichter, Nürnberg 1648................. 6 Fig. 1.2: the last bar of Bach’s manuscript ................................................................................................... 20 Fig. 2.1: Beilage 3, page 5 ............................................................................................................................ 25 Fig. 2.2: Beilage 3, page 1 ............................................................................................................................ 30 Fig. 2.3: Contrapunctus 1, page 1 (autograph) ............................................................................................. 37 Fig. 2.4: Well-Tempered Clavier Book 1, Prelude I in C (autograph) ........................................................ 38 Fig. 2.5: Well-Tempered Clavier Book 1, Fugue in B minor (autograph) .................................................. 38 Fig. 2.6: Fantasia in D major BWV 906 (autograph)................................................................................... 38 Fig. 2.7: Fugue from Violin Sonata BWV 1001 (autograph) ...................................................................... 39 Fig. 2.8: Fugue from Violin Sonata BWV 1005 (autograph) .................................................................... 39 Fig. 2.9: Contrapunctus 1 (autograph), bars 19 & 20, tenor part ................................................................. 40 Fig. 2.10: Cantata BWV 61 .......................................................................................................................... 43 Fig. 2.11: Beilage 1, page 2 (autograph, final version of the Augmentation Canon).................................. 45 Fig. 2.12: 1751 edition, page 12, end of Contrapunctus 4 and Schmuckgraphik........................................ 46 Fig. 2.13: 1751 edition, end of Contrapunctus 8 and Schmuckgraphik incorporating “JSB” monogram.. 46 Fig. 2.14: Contrapunctus 14 (autograph), last two bars ............................................................................... 54 Fig. 2.15: Contrapunctus 14 (autograph), bar 165 ....................................................................................... 61 Fig. 3.1: page 61 of the printed edition: first page of the unfinished fugue ................................................ 83 Fig. 3.2: the structure of a completed Contrapunctus 14 ............................................................................. 90 Fig. 3.3: Contrapunctus 14 (autograph, page 2), showing the original bars 111 and 112 crossed out and
three replacement bars notated in the bottom margin ........................................................................ 92 Fig. 3.4: Beilage 3, page 5, showing “doodles” at the bottom corners of the page..................................... 96 Fig. 3.5: bottom left....................................................................................................................................... 96 Fig. 3.6: bottom right .................................................................................................................................... 96 Fig. 3.7: The manuscript of the “Patrem omnipotentem” chorus from the Mass in B Minor, showing the
number 84 written at the end of the score......................................................................................... 101 Fig. 3.8: The title page of the Well-Tempered Clavier .............................................................................. 103 Fig. 3.9: detail of the loops at the top of the page ...................................................................................... 103 Fig. 3.10: the drawing turned upside down, showing correspondence with Bach’s tuning system ......... 104 Fig. 3.11: Musical Offering, Canon No 9 “quaerendo invenietis” ............................................................ 105 Fig. 3.12: one possible realisation of Canon No. 9 .................................................................................... 105 Fig. 3.13: puzzle canon BWV 1074 ........................................................................................................... 106
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List of Musical Examples
Ex. 1.1: the last bar of the Christe eleison from the Mass in B minor............................................... 10 Ex. 1.2: the first four notes of the subject of Kyrie II ......................................................................... 12 Ex. 1.3: subject of Kyrie II .................................................................................................................. 12 Ex. 1.4: soprano entry of the subject, bar 40, Kyrie II....................................................................... 13 Ex. 1.5: the opening of Wachet auf, ruft uns die Stimme ................................................................... 14 Ex. 1.6: the first subject of Contrapunctus 14 .................................................................................... 17 Ex. 1.7: Contrapunctus 5, bar 41........................................................................................................ 19 Ex. 1.8: the third subject of Contrapunctus 14................................................................................... 19 Ex. 2.1: the combination of the four subjects as given by Nottebohm and Wolff .............................. 27 Ex. 3.1: Contrapunctus 14, bars 113 - 115......................................................................................... 85 Ex. 3.2: Contrapunctus 14, bars 192-194........................................................................................... 86 Ex. 3.3: bars 143-4, showing a hint of the BACH motif, and the notes D, A, C and F (=14) sounding in the fourth beat ................................................................................................................................. 91 Ex. 3.4: Contrapunctus 14 (autograph, page 2), showing the original music of bars 111 and 112 before Bach crossed them out ............................................................................................................. 93 Ex. 3.5: The same passage using treble and bass clefs ...................................................................... 93 Ex. 3.6: the three new bars (111, 112 and 113 in this example) that replaced bars 111 and 112 in the previous examples ......................................................................................................................... 93 Ex. 3.7: graphic representation of the markings ................................................................................ 97 Ex. 4.1: The three subjects of Contrapunctus 14.............................................................................. 108 Ex. 4.2 The triple combination at bar 233........................................................................................ 109 Ex. 4.3: The quadruple combination as given by Nottebohm .......................................................... 110 Ex. 4.4: bars 229-238, soprano voice, following the outline of the main Art of Fugue subject...... 110 Ex. 4.5: The quadruple combination as given by Nottebohm .......................................................... 121 Ex. 4.6: Tovey’s third and last rectus combination.......................................................................... 122 Ex. 4.7: Contrapunctus 3, bars 55 - 61............................................................................................. 123 Ex. 4.8: Chorale “Gott sei uns gnädig und barmherzig”, showing consecutive fifths between T and S in the last bar.................................................................................................................................. 125 Ex. 4.9: “Ein' feste Burg ist unser Gott”, showing consecutive 5ths between S and B in the last bar........................................................................................................................................................... 125 Ex. 4.10: Ricercar a 3 from the Musical Offering, bar 64............................................................... 126 Ex. 4.11: Contrapunctus 6, bar 15.................................................................................................... 126 Ex. 4.12: Contrapunctus 6, bar 18.................................................................................................... 126 Ex. 4.13: The subjects layered I-II-III-IV ......................................................................................... 128 Ex. 4.14: The five triple combinations from the end of Contrapunctus 8........................................ 130 Ex. 4.15: The three triple combinations from the end of Contrapunctus 11 ................................... 131 Ex. 4.16: The consecutives created by layering IV above II ............................................................ 132 Ex. 4.17: Tovey’s solution to the problem of consecutives .............................................................. 132 Ex. 4.18: The subjects layered with II above IV: the consecutives vanish ...................................... 133 Ex. 4.19: How the 14-note form of IV fits in the counterpoint ......................................................... 134 Ex. 4.20: The subjects layered I12-II-III-IV....................................................................................... 135 Ex. 4.21: The subjects layered II-I-IV-III ......................................................................................... 135 Ex. 4.22: The subjects layered II-I12-IV-III....................................................................................... 136 Ex. 4.23: Tovey’s first rectus combination ....................................................................................... 136 Ex. 4.24: Tovey’s first rectus combination (I12-III-II-IV) transposed to D minor ........................... 137 Ex. 4.25: Tovey’s third and final rectus combination: IV-II-III-I.................................................... 138 Ex. 4.26: II-IV-III-I............................................................................................................................ 139 Ex. 4.27: II-IV-III-I with conjectural continuation .......................................................................... 140 Ex. 4.28: II-IV-III-I with conjectural continuation, transposed into A minor ................................. 140 Ex. 4.29: Tovey’s second rectus combination II-III-IV-I ................................................................. 141 Ex. 4.30: II-III-IV-I using the 14-note form of IV............................................................................. 141 Ex. 4.31: the standard scale of inversion ......................................................................................... 145 Ex. 4.32: subject I and Iinv ................................................................................................................. 145 Ex. 4.33: subject II and IIinv .............................................................................................................. 145 Ex. 4.34: inversion scale for III as given by Tovey .......................................................................... 145 Ex. 4.35: First appearance of IIIinv as given by Bach in the alto at bar 213 ................................... 146
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Ex. 4.36: Second and final appearance of IIIinv as given by Bach in the bass at bar 222............... 146 Ex. 4.37: subject III and IIIinv............................................................................................................ 146 Ex. 4.38: subject IV and IVinv ............................................................................................................ 147 Ex. 4.39: WTC, Book 2, Prelude 1, bars 9 - 10 ................................................................................ 150 Ex. 4.40: Verlieh’ uns Frieden gnädiglich (Riemenschneider No. 91)............................................ 150 Ex. 4.41: WTC, Book 2, Prelude 11.................................................................................................. 151 Ex. 4.42: WTC, Book 2, Prelude 11 bars 15-16 ............................................................................... 151 Ex. 4.43: Contrapunctus 10, bars 20-22........................................................................................... 152 Ex. 4.44: Iinv-IIinv-IIIinv-IVinv............................................................................................................... 153 Ex. 4.45: IVinv-IIIinv-Iinv-IIinv............................................................................................................... 154 Ex. 4.46: Tovey’s first inverted combination.................................................................................... 155 Ex. 4.47: IIIinv- IIinv- IVinv- Iinv............................................................................................................ 156 Ex. 4.48: IIIinv- IIinv- IVinv- Iinv-12 ........................................................................................................ 157 Ex. 4.49: Tovey’s second inversus combination............................................................................... 158 Ex. 4.50: IVinv-IIIinv- IIinv- Iinv............................................................................................................. 158 Ex. 4.51: IVinv-IIIinv- IIinv- Iin-12 .......................................................................................................... 159 Ex. 4.52: The four subjects inverted semitone for semitone............................................................. 160 Ex. 4.53: Michael Ferguson’s inverted quadruple combination ..................................................... 161 Ex. 4.54: Ferguson’s inversion layered in the order IVinv-IIIinv-IIinv-Iinv.......................................... 162 Ex. 4.55: The inversion layered in the order IVinv-IIIinv-IIinv-Iinv-12, with the 14-note form of IV ..... 163 Ex. 4.56: The solution to the inversus quadruple combination, as proposed by Indra Hughes ..... 164 Ex. 4.57: Three-part Invention No 12, bars 3 and 4 ........................................................................ 165 Ex. 4.58: The combination complete with suggested non-thematic material .................................. 167
PART ONE: BEFORE BAR 239
Chapter One
BACH’S USE OF GEMATRIA IN THE ART OF FUGUE
There is no certainty…that Bach used any eighteenth-century number-alphabet forms, even assuming that he knew about them.
Ruth Tatlow1
Mathematics is the heart and soul of music … Without question the bar, the rhythm, the proportion of the parts of a musical work and so on must all be measured … Notes and other signs are only tools in music, the heart and soul is the good proportion of melody and harmony. It is ridiculous to say that mathematics is not the heart and soul of music.
Johann Mattheson2
The legitimacy and soundness of such procedures is nowadays established: Baroque gematria is a historical fact. No serious and competent musicologist could claim controversy here anymore.
Herbert Kellner3
Gematria is “a system by which hidden truths and meanings are
discovered within words. Each letter of an alphabet corresponds to a number.
Numerical values of words are totalled up and then these words are said to
correspond with other words sharing the same numerical value.”4 The word
gematria is often used in a loosely interchangeable manner with the word
‘numerology’: “divination by numbers; the study of the esoteric meaning of
numbers”.5
1 Tatlow, R. (1991) Bach and the riddle of the number alphabet Cambridge, Cambridge University Press, page 127 2 Mattheson, J. (1743) "Neu eröffnete musikalische Bibliothek" 2: 54 3 Kellner, H. A. (2000) "Johann Sebastian Bach and 'Die Kunst der Fuga'" The Diapason 91(3:1084): 13 4 Hefner, A. G. (1997-2004) Gematria <http://www.agnosticwitch.catcara.com/divindex-part1.htm>, (web site) Accessed 4 June 2005. The derivation of the word is from Rabbinical Hebrew gēmatriyā, adj.: Greek γεωμετρί (geometry). 5 Sykes, J. B., Ed. (1982) Concise Oxford Dictionary, Oxford, Oxford University Press
2
Perhaps no other area of Bach studies has generated such controversy as
the claim that much of Bach’s music is founded on, or at least contains,
hidden numerological or gematric structures or meanings. Opinion is sharply
divided between those who delight in hunting out such numbers in Bach’s
music and who believe that he put them there intentionally, and those who
are sceptics and who believe that the numbers are either not there or are at
best coincidences. Chief among the former was Friedrich Smend (1893‐1980);
one of the leading sceptics is Ruth Tatlow.6
In this Chapter I give a brief overview of the background to gematria and
discuss some examples of Bach’s use of numbers in his music, in particular
the number 14. I then consider appearances of this number in The Art of Fugue
and suggest that it brings to bear on my theory that Bach left the work
unfinished deliberately.
Background
The Babylonian king Sargon II, in the 8th century BC, is believed to have
been the first to use gematria when building the wall of Khorsabad exactly
16,283 cubits long, because that was the numerical value of his name. In
Jewish mysticism this is a traditional system of associating numbers with
Hebrew letters for the purpose of discovering hidden meanings in words.
This is accomplished by systematically associating letters with numbers and
then finding other words with similar numbers. These latter words are
regarded as comments on the original words. The Hebrews used gematria for
divination (a means of communicating with the supernatural world,
principally to foretell the future); and the ancient Greeks used gematria in
dream interpretation. Gematria carried over into early Christianity which
helped make the dove a representation of Jesus: the Greek word for dove,
6 Tatlow, op. cit.
3
peristera, equals 801 as do the Greek letters in alpha and omega, which
represent the Beginning and the End. It was the Kabbalists, however, who
seriously studied gematria and developed it into an art form. The Kabbalists
of the 13th century believed that the Old Testament was written in a hidden
code inspired by God. They used gematria as one of the chief means by
which to decipher this code. As an example, gematria shows that the first
verse of the Bible, (Genesis 1:1) In the beginning God created the heaven and the
earth, is full of multiple references to the number 7 – the number of
completeness and principally the number of the 7 days of creation. Entire
verses were numerically added up and interpreted in such a fashion. The
13th century German Kabbalistic scholar, Eleazar of Worms, did extensive
gematric commentaries on the Bible.
Number alphabets
Ruth Tatlow lists 33 different number alphabets and their variants in
Appendix 1 of her book. The most commonly used number alphabet was a
Latin ‘natural order’ alphabet which Tatlow designates as “Latin natural‐
order: variant 1”.
A B C D E F G H
1 2 3 4 5 6 7 8
I / J K L M N O P Q
9 10 11 12 13 14 15 16
R S T U / V W X Y Z
17 18 19 20 21 22 23 24
Note that the letter J was not included – or, on the rare occasions that it
was included, it shared the same number as I. The Roman letter I was used
up to the 18th century to represent both the vowel sound i (as in ‘sit’) and the
4
consonant dzh (as in ‘jump’). Roman numerals ending with i came to be
written with an elongated final letter, as in “Kynge Henry the viij”. During
the 17th century, i came to be used for the vowel, and the elongated version j
for the consonant, with the elongated upper‐case form J being introduced.
Some however objected to this departure from Classical Latin. For example,
when Captain Cook drew his chart of New Zealand he signed it “by Lieut. J.
Cook, Commander of the Endeavour Bark 1770”; but when the map was
published in Sydney Parkinson’s Journal (1773) the editor pedantically
‘corrected’ the signature to “by I. Cook”.7
Similarly it can be noticed that U and V also share the same number as
each other.
Examples of Gematria in Bach’s Music
Smend’s theories
In 1947 Smend published four volumes of books about Bach’s Cantatas in
which he incorporated his findings from a large file of notes and
correspondence with Martin Jansen. The main part of Smend’s number
theory is to be found in the introduction to the third volume.8 Early on in his
study he draws attention to the number 14, which has since become widely
known as ‘the Bach number’.
B + A + C + H
2 + 1 + 3 + 8 = 14
Smend claims that other numbers are to be found in the structure of Bach’s
works, usually of a theological symbolism. These include 3 for the Holy
Trinity; 10 for the Commandments; 12 for the disciples and so on. As an
7 I am indebted to Garry J. Tee of the Department of Mathematics, University of Auckland, for the information in this paragraph. 8 Smend, F. Kirchen-Kantaten III (1947), pages 5 - 21
5
example of Smend’s claims about other numbers the following paragraph
gives some insight:
When Bach uses the number alphabet in his compositions, it is by no means always a matter of his own name alone, although that often occurs. Words of quite a different kind, words of a religious and theological nature, are regularly to be found also. We will do well to consider only a small number of these, in order to see our way clearly. CHRISTUS is expressed by the number-symbol 112, CREDO by 43. The ‘Credo’ of the B minor Mass, in its original form, numbers 784 (i.e. 7 x 112) bars; the sacred name of CHRISTUS is invoked 7 times therein. In the chorus ‘Credo in unum Deum’ in the same work, the word CREDO appears 43 times. The same chorus, plus the following movement ‘Patrem omnipotentem’, amounts to a total of 129 (i.e. 3 x 43) bars; i.e. Bach’s setting of the first article of faith, in the very number of its bars, states: CREDO, CREDO, CREDO.9
Ruth Tatlow has no time for Smend’s theories, condemning them chiefly
on the basis that “he gives little historical evidence to support his
statements”. She says that “most of his claims have to be taken on trust as he
does not support them with references to primary or secondary sources. The
most serious omission is an example of the natural‐order alphabet from the
seventeenth or eighteenth centuries.”10
However, an absence of historical evidence or reference to sources cannot
give reason to discount Smend’s observations: either the numbers of bars in
Bach’s scores are as Smend says they are, or they are not, and it is clear in
many cases that they are.
Further, in her appendix, Tatlow includes a list of publications that make
reference to number‐alphabets: 28 of these were published before Bach’s birth
in 1685. Although there is no evidence that Bach read any of these or knew of
them, it is known that he was a well read and educated person who died
9 quoted in Tatlow op. cit., page 9 10 Ibid., page 10
6
leaving an extensive and wide‐ranging library of books on theological and
scientific subjects.11
Tatlow also complains that Smend has not supplied an example of the
natural order alphabet from the seventeenth or eighteenth centuries. Here is
such an example:
Fig. 1.1: number alphabet by Georg Philipp Harsdoerffer, Poetischer Trichter, Nürnberg 1648
It will be seen that this number alphabet is the same as that on page 3
above, which Tatlow calls “Latin natural‐order: variant 1”. The fact that
Smend does not supply an example like this does not mean that no example
exists; neither does it invalidate his argument. It is true that there is no
evidence that Bach saw this or anything like it; but even Tatlow says:
It is extremely likely that Bach came across many different number alphabets. Techniques of gematria were well known in his day…12
11 A list of the books left in Bach’s estate can be seen in David, H. T., Mendel, A., et al. (1998) The new Bach reader : a life of Johann Sebastian Bach in letters and documents New York, W.W. Norton, page 253 12 Ibid., page 126
7
In his handbook on Bach’s Mass in B minor, John Butt sounds a useful note
of caution:
This style of interpretation [i.e. the search for gematric significance in music] is readily open to those who desire it. However it is so often the product of a sort of fundamentalism which is bound to accompany a composer of Bach’s musical genius. […] There is at once no supporting evidence for verification and, on the other hand, no possibility of refutation. There is a tendency to interpret everything in sight and even to ignore certain obvious features … if they happen to contradict the deeper spiritual message. The systems by which the information is gathered are governed entirely by whatever yields the most satisfactory results; the author selects data from bar totals, movement numbers, musical gestures and verbal instructions with no consistency which is externally imposed. Finally, one often gains the impression that Bach composed virtually for no other reason than to include ‘messages’ latent in bar totals and verbal directions; before long any consideration of musical style, its quality and structures, will cease to be of importance. Clearly it would be against the spirit of Bach’s age to eschew any reference to number and religious symbolism. But for most scholars and listeners these must surely be subordinate to a more comprehensive view of the work and its style.13
It is hard to disagree with this approach. Nonetheless the frequency with
which significant numbers appear in Bach’s music cannot be discounted. If
hard historical evidence is not to be found, one may gather circumstantial
evidence by amassing a body of instances which together point to a plausible
analysis. There may be those who would regard the apparent
correspondences between words or music and numbers as coincidences; and
indeed coincidences they may be. But if one were to keep seeing such
correspondences between words and numbers in many other places in Bach’s
music, the totality of such observations might lead a reasonable person to
conclude that Bach might have been doing this intentionally.
13 Butt, J. (1991) Bach Mass in B minor Cambridge, Cambridge University Press, page 38
8
The number 43: The B Minor Mass Credo and the Organ Prelude Wir glauben all’ an einen Gott BWV 680
As Smend points out (page 5 above), C+R+E+D+O = 3+17+5+4+14 = 43; and
no‐one can dispute the fact that Bach has set the word “credo” 43 times in the
first chorus, Credo in unum Deum, of his Symbolum Nicenum in the Mass in B
minor BWV 232. 14
Another example of the use of the number 43 to represent ‘Credo’ occurs
in the Chorale Prelude on Wir glauben all’ an einen Gott BWV 680 (the “Giant”
Fugue): the final pedal entry is comprised of 43 notes, which may or may not
be significant in a setting of the tune associated with the words “We all
believe in one true God”.
The number 3: The “St Anne” Fugue and the Organ Prelude Wachet auf, ruft uns die Stimme
The organ Fugue in E flat BWV 552 (the “St Anne” Fugue) is often
regarded as having many Trinitarian connotations. It is comprised of 3
clearly delineated sections; there are 3 separate fugue subjects; its key
signature has 3 flats; it has 3 different time signatures and yet the beat
remains proportional (suggesting, perhaps, theological ‘unity in trinity’). The
total number of bars of the three sections of the fugue is 36+45+36 = 117; 117
might be expressed as 13x3x3 – a very Trinitarian row of numbers (1333)
encapsulating unity (1) and trinity (three 3s). The total of those digits
(3+6+4+5+3+6) is 27, which might be expressed as 3x3x3; and indeed
2+7=3+3+3.15 Further, 117 might also be expressed as 27x4.3 which might lead
14 A further example of Bach’s deliberate use of numbers in the Credo (in this case, the number 84) is described in Chapter 3. 15 The foregoing information in this paragraph, together with a number of other insights referred to throughout this thesis, comes from Dr. John Bertalot's enlightening articles (Bertalot, J (2000) "Spirituality and symbolism in the music of J. S. Bach." Organists' Review 86 (3): 222-225, and 86 (4):331-336.). The text of these articles is expanded from lectures given by Dr. Bertalot in King's College, Cambridge in August, 1987; the Universities of Witwatersrand and Port Elizabeth, South Africa; Southwell Minster, England on 6 July 2000; and the Hereford Three Choirs' Festival on 18th August 2003. I am grateful to Dr. Bertalot for sharing many of his insights in private correspondence.
9
to a leap of logic equating those numbers to a cryptic statement by Bach that
“I believe” (credo=43) “in the Trinity” (27=3x3x3). Keeping Butt’s word of
caution in might, some might think that this is going too far; on the other
hand, who is to say that Bach was not saying that? The fugue comes at the
end of the Clavierübung, a collection of 27 pieces. All of this is undeniable and
the suggestion of deliberate Trinitarian connotations seems not unreasonable;
it is clear that Bach has made a point of constructing this piece around the
number 3.
The Trinitarian references in Wachet auf, ruft uns die Stimme BWV645 are
difficult to dismiss. This is a transcription for organ of the tenor aria from
Cantata 140, which was written for the last Sunday in the church year, the
twenty‐seventh Sunday after Trinity. Neatly, as was mentioned above, 27 is
3x3x3. Continuing the Trinitarian references: there are the 3 flats of the key
signature, 3 independent voices, 3 staves per system, 3 systems on each page,
3 pages in Schüblerʹs edition, and the announcement of 3 low E‐flats by the
pedal in the first bar. The total number of bars is 54: 27x2. This work is one of
six “Schübler” Chorales and more is said about both this chorale prelude and
the collection below.
The numbers 14 and 41 (the “BACH numbers”): the Mass in B Minor; the Schübler Chorales and the Chorale Vor deinen Thron
The foregoing examples are cases in which it may appear that Bach has
used gematria to generate ideas (for example, having the proportions of a
fugue relate to the number 3 as a reference to the Trinity) that are of a
primarily religious significance. Another aspect of numbers that can be seen
in his music has to do with his own name. Turning now therefore to
numerological occurrences of the number 14 in Bach’s music, there are some
examples which may seem too deliberate to be regarded as mere
coincidences. On many occasions Bach used ‘his’ number for different
10
reasons: to put his ‘signature’ on the music; to associate himself with some
aspect of the music or text at the point where the number appears; or to draw
attention to something significant about that point in the music. We have
seen how the letters of BACH give a total of 14 (page 4 above). With his
predilection for symmetry, Bach must have been pleased that J+S+B+A+C+H
gives a total of 41, which is the retrograde of 14. The total of these numbers,
14+41, is 55 and this is another number that we see frequently associated with
Bach’s name, and which will assume considerable importance in my
discussion of the length of the unfinished fugue in Chapter 3. An example of
such an appearance of the number 55 can be seen in the total number of bars
of the Gloria section of the Mass in B minor: the total is 770 bars, which can be
expressed as 14x55.
The Kyrie of the B minor Mass comprises three movements: Kyrie I;
Christe; and Kyrie II. In Kyrie I, after an opening introduction of 4 bars, there
is a big fugue 122 bars long. The Christe occupies 85 bars; and Kyrie II is 59
bars long. However, looking more closely at the final bar of the Christe, it can
be seen that Bach has deliberately left the last beat of the bar silent:
Ex. 1.1: the last bar of the Christe eleison from the Mass in B minor
It might therefore accurately be said that the Christe is 84¾ bars in length.
The bar numbers of the three movements – 122, 84.75, 55 – are
mathematically proportional in length:
122 84.75
84.75=
59
11
Each of these fractions produces a ratio in each case of 1.4. It may be
tempting to assume that this is a reference to the number 14, but it is
important to remember that Bach was most unlikely to have thought or
worked in decimals. Nevertheless, it is intriguing to note that 1.4 can be
expressed as a vulgar fraction 1⁴/₁₀: a fraction whose visual appearance on the
page may suggest the number 14 not once but twice (from reading the 1 and
the upper 4, and by adding the numbers 4 and 10 that appear in the
fraction).16 Much more important, however, is the fact that the ratio is exactly
the same in both cases, which is a convincing explanation for the fact that
Bach chose to write that rest at the end of the Christe. In both Kyries, the final
note arrives half way through the last bar, just as it does in the Christe, and
Bach has made the last note full length, emphasising the point by writing a
pause sign on each of those last notes. He has not done this at the end of the
Christe. A further instance of the number 1.4 appearing in proportional ratio
between movements can be seen in the two settings of the Chorale Aus tiefer
Noth in the Clavierübung. The first setting has 55 bars (another appearance of
the number 55) not counting repeats; the second has 75 bars. 55+75=130 (the
chorale Aus tiefer is a setting of Psalm 130). However, if the repeats are
counted the total numbers of bars are 75 and 102. Noticing the repetition of
the 75 leads to the realisation that the numbers are proportional: 75/55 =
102/75 : a ratio of 1.4. I am grateful to Dr John Bertalot for this observation. 17
There are however further Bachian numbers to consider, for example in
the Kyrie II. There is much of interest and significance in the subject of this
16 To suggest that this fraction can also be expressed as 1⅖ and that this may imply the number 55 (two 5s appearing in the fraction) is probably going too far. 17 This number 1.4 in both cases (B minor Kyries and the Aus tiefer preludes) is a rounded number: but in both cases the exact, unrounded, number is the same. Whether or not the number of the ratio, 1.4, is of any significance, the point is that the ratio is precise and that it is strictly the same in each case. This and the proportionality of the B Minor Mass Kyrie/Christe/Kyrie sequence are discussed further in Chapter 3.
12
fugue. Apart from numerical symbolism there is also visual symbolism in the
first four notes of the subject:
Ex. 1.2: the first four notes of the subject of Kyrie II
The outline of this figure is known as a chiasmus or circulatio and is a
musical figure regularly used by Bach in association with Christ (here Lord,
Kyrie) or the Cross – because one is to imagine a line drawn between the first
and fourth notes, and another line between the second and third notes. Thus
one has drawn a cross on the page, and one has also drawn the Greek letter χ
(hence the name chiasmus) which also happens to be the first letter of the
name of Christ in Greek. This is very far from being fanciful conjecture, and it
has been thoroughly documented by Dr Timothy Smith of Northern Arizona
University, who gives many examples from the Cantatas and the Passions. 18
Admittedly this may at first seem of only limited relevance in this
discussion of the number 14, until we consider the whole of the fugue subject
Ex. 1.3: subject of Kyrie II
and it can be seen that it is comprised of 14 notes. The notion that Bach has
written music in which he directly associates his own name with that of
Christ now begins to seem less far‐fetched. This point is further underlined in
the case of an entry of the subject that is singled out for special (though
subtle) attention, namely the only entry of the subject in B minor. Given that
this is the only entry of the subject in the same key as the work as a whole, it
might be argued that there could be something significant about it from the
gematric point of view.
18 Smith, T. A. (2000) "Circulatio as Tonal Morpheme in the Liturgical Music of J S Bach" Ars Lyrica (Journal of the Lyrica Society) 2000(2000)
13
Ex. 1.4: soprano entry of the subject, bar 40, Kyrie II
Totalling up the numbers associated with the letter names of this entry
gives the number 41, which as we have seen represents J S BACH. However,
this example provides another cautionary note against over‐enthusiasm in
searching for Bachian numbers, for the total of 41 can only be gained if the
note B is described as B: Bach would have designated this note H, which
leads to a different total. The naming of the notes as given in Ex. 1.4 above
also takes no account of sharps and naturals: in German notation C is
designated Cis.19
Turning to the Schübler Chorales: discussion of this set of six organ preludes
can be brief but it nevertheless shows that these works are riddled with
Bach’s numbers 14 and 41, as well as other numbers of religious significance
as described above. The information here comes from an article by Edmund
Shay.20
Bach’s personal stamp is encoded throughout the overall plan of the
published edition. There are 14 lines on the title page, 14 words in the
complete title (as there are both for the Goldberg Variations and the Canonic
Variations), 14 numbered pages in Schüblerʹs edition, 41 lines of score, and 41
words in the seven chorale titles. (There are seven because the second chorale
prelude has two titles.) The BACH musical motive (B A C B ) appears at bar
19 Further symbolism can be deduced from another aspect of the bar numbers in the Kyrie – Christe – Kyrie sequence as a whole. As a liturgical element in Lutheran worship, the Kyrie was often recited three times as a Ninefold Kyrie, and this was done in conjunction with the recitation of the Ten Commandments. Perhaps then the total number of bars, 270, may be significant in that it can be expressed as 3x3x3 x 10: the Ninefold Kyrie and the Ten Commandments. Taken by itself this might be regarded as a coincidence. 20 Shay, E. (1999) "The Schubler Chorales and the Numbers Game" The Diapason September 1999: 16
14
7 of Meine Seele erhebt den Herren, which is the precise midpoint of the entire
collection.
In the first Chorale Prelude, Wachet auf, ruft uns die Stimme, the first four
bars may contain two references to the Bach numbers.
Ex. 1.5: the opening of Wachet auf, ruft uns die Stimme
The opening pedal phrase has 7 notes, but it is repeated to give a total of
14. Similarly the number of notes in the right hand is 17+17 = 34. 34 is a
number we have not yet encountered: but 34+14 = 48 which is the product of
BxAxCxH = 48.
Lastly in this section I mention the ‘deathbed’ chorale, Vor deinen Thron
tret’ ich hiermit BWV668. This organ prelude, a revision of an earlier work,
was dictated in the last week or ten days of his life – some time between
Saturday 18 July and Tuesday 28 July 1750 – by the blind and dying Bach
most probably to his son‐in‐law and amanuensis Johann Christoph Altnikol.21
The first phrase of the cantus firmus has 14 notes. The entire cantus firmus
has 41 notes, and there are 41 imitative entries. This may lead to no other
conclusion than that Bach liked to imprint himself in his music, 22 or that he
was making a theological statement that he knew he was about to appear 21 Other possible candidates are mentioned on page 112. 22 A conclusion that seems particularly alluring when one examines the final plagal cadence at the end of this work, into which Bach has woven his own initials, JSB: the right hand has 9 notes (9=J), the left hand has 18 notes (18=S) and the pedal has 2 notes (2=B); furthermore, the last note of the cantus firmus, under which this final cadence takes places, is held for 14 beats. Bertalot, J. (2001) "Letter to the editor" Organists' Review 87(1).
15
before his Father’s throne (Vor deinen Thron); and indeed these numbers
may be mere coincidences.23 But the frequency with which the numbers 14
and 41 appear in this and other pieces cannot be discounted. When we begin
to see these numbers in significant places in The Art of Fugue, it begins to
seem more likely than not that they are deliberate.
The Bach numbers and The Art of Fugue
Herbert Kellner and the significance of the Well‐Tempered tuning system
In several articles Herbert Kellner discusses his theories about the
correspondences between numbers associated with the Wohltemperirte
(well‐tempered) tuning system developed by Werckmeister and long thought
to have been adopted by Bach, in particular for Das Wohltemperirte Clavier.24
Kellner’s papers are not light reading but his discoveries are compelling, and
even Ruth Tatlow says that his is “the most developed thesis of Bach as a
mathematician.”25 Kellner directs his attention to the authenticity of the
wording of the title “Die Kunst der Fuga”; the wording and implications of
the Nachricht or preface to the first printed edition of 1751 (which he claims –
erroneously in my view26 – reveal the secret to the number of bars required to
complete Contrapunctus 14); and a link between the temperament numbers
and the first subject of Contrapunctus 14.
Kellner’s attention to the title of the work gives us another revealing
insight into Bach’s association of his name by numbers with his work. He
23 Many other examples of numbers that are hidden to a greater or lesser degree abound in Bach’s music. The St Matthew Passion in particular is fertile ground for enthusiasts of Bach’s numbers: there are many places in which the number of bars or notes corresponds precisely to the biblical chapter and verse numbers of the text being set at that moment; and one of the more dryly humorous instances is the setting of the Last Supper, in which the words Herr, bin ichs? (Lord, is it I?) appear only eleven times – since one of the twelve disciples, Judas, remains silent in response to Christ’s statement that “one of you will betray me”. 24 Dr Bradley Lehman’s recent discoveries about Bach’s tuning system are mentioned in Chapter 3. They strongly suggest that Bach used his own tuning system, not Werckmeister’s. 25 Tatlow , page 3 26 See Chapter 3.
16
notices that in the first printed edition of 1751 the title is given as “Die Kunst
der Fuge”; and yet the manuscript version of the title page, which was
written by Bach’s son‐in‐law Altnikol, has “Die Kunst der Fuga” [sic].27
Kellner points out that Bach always used the Latin form of the word: he
never wrote “Fuge” on any of his title pages. Converting the words into
numbers, Kellner makes a striking discovery:
Die Kunst der Fuga
18 80 26 34 = 158
Johann Sebastian Bach
58 86 14 = 158
Thus Bach and his work are one and the same. Kellner concludes from this
that the title “Die Kunst der Fuga” is the correct title for the work. It does
however appear to escape his attention that 1+5+8 = 14.
The briefest possible explanation is given here of the alleged link between
the tuning system and the subject of Contrapunctus 14. Kellner explains that
the “well‐tempered” tuning system consists of 4 well‐tempered fifths (c‐g‐d‐
a‐e); 1 ‘tempering fifth’ (b – f sharp) and 7 remaining perfect fifths to
complete the tuning circle.28 This gives the relevant numbers associated with
the well‐tempered tuning system: 4, 1, 7. After referring to the ‘perfect
27 Kellner, H. A. (2000) "Johann Sebastian Bach and Die Kunst der Fuga" The Diapason March 2000: 13 28 Kellner notices that in Bach’s monogram (reproduced on the front page of this thesis) the crown has 7 jewels at the top, which may correspond to the 7 perfect fifths; and 5 jewels at the bottom, which may correspond to the 4 + 1 tempered/tempering fifths. Some other writers have claimed that there is a total of 14 (!) jewels in the crown, but that can only be claimed if the little semicircles at top left and top right of the design can be counted as jewels: they do not look like the other jewels to me, and even if they are, the trefoil at the top of the design, which may or may not be a reference to the Holy Trinity, still remains to be explained. The monogram contains other alleged numerical references: some writers claim to see 27 crosses (X) created by the intersections of the lines of the initials – JSB=27, as does SDG (‘soli Deo gloria’), often written by Bach at the end of his scores. And as has already been mentioned, 27=3x3x3 which may or may not be a reference to the Trinity.
17
numbers’ that apparently fascinated Werckmeister and Walther, he mentions
that the first perfect number is 6.29 So Kellner decides to assemble a sequence
of numbers 4‐1‐7 with its own retrograde 7‐1‐4. This results in 4‐1‐7‐7‐1‐4.
Placing the first perfect number, 6, at the centre of this construction, gives 4‐
1‐7‐6‐7‐1‐4; next he converts these numbers to the gematric equivalent letters
D‐A‐G‐F‐G‐A‐D and observes that this corresponds precisely to the first
subject of Contrapunctus 14:
Ex. 1.6: the first subject of Contrapunctus 14
Since Kellner gives no reason for his decision to assemble such a sequence
of numbers or for inserting the first perfect number into the middle of it, one
may possibly regard this as a somewhat forced conclusion; but Kellner also
notices that the Nachricht contains 417 letters, and that the second half of the
sentence, after the semicolon, gives a gematric total of 2138 corresponding
precisely to BACH: 2+1+3+8 = 14.30 Kellner also points out that the B major
prelude of Book One of the ‘48’ contains 417 keystrokes: since the b – f sharp
fifth was the crucially different fifth in the tuning system, this attention to the
numbers 4, 1, 7 may not be entirely fanciful. 31
29 Perfect numbers are numbers which are the sum of their divisors: thus 6=1+2+3 and 6 is divisible by 1, 2 or 3. The second perfect number is 28=1+2+4+7+14, being divisible in turn by 1, 2, 4, 7 and 14. 30 Kellner, H. A. (2000) "Die Kunst der Fuga: J S Bach's Prefatory Message and Implications" The Diapason (May 2000): 15 31 It all depends on the edition and the exact number of tied notes: some editions have 418 keystrokes.
18
Other occurrences of the number 14
These may be listed in a much more straightforward manner.
There are 14 Contrapuncti in The Art of Fugue.
It is very likely that Bach intended the work to be the third of his
submissions to Mizler’s society. Bach waited until he was the 14th member to
be admitted to the society,32 and to commemorate his admission he had his
portrait painted with 14 buttons on his waistcoat; in the portrait he is holding
the score of the 14th of a set of 14 enigmatic canons (the Goldberg canons).
According to another of Kellner’s articles33 there is significance in the fact
that the first two notes of the entire work are D and A: 4 and 1.
In Contrapunctus 2 there are 14 entries of the subject.
By the time the subject has reached its fully developed state, in
Contrapunctus 5 and onwards, it has through the addition of passing notes
arrived at a total of 14 notes.
Also in Contrapunctus 5, there is an unmistakable appearance of the
BACH motive (B A C B ) at bar 41 (41 = J+S+B+A+C+H). Ordinarily one
might be tempted to regard this as a coincidental grouping of notes arising
from the motivic development of the contrapuntal material surrounding the
concurrent stretto entries of the subject (alto and tenor); but the fact that
Bach’s ‘signature tune’ appears in his own bar number seems unlikely to be
an accident.34
32 The full list of members, in order of joining, can be seen at note 10 in Wolff, C. (2000) Johann Sebastian Bach : The Learned Musician Oxford, Oxford University Press, page 506. 33 Kellner, H. A. (1999) "How BACH (sic) encoded his name into Die Kunst der Fuge together with his tuning" The Diapason May 1999: 14 34 The “B-A-C-H” reference here was noticed by Jan Overduin in Overduin, J. (1998) "Bach and Die Kunst der Fuge" Ibid., (May 1998): 15; but Overduin does not draw attention to the fact that it appears
19
Ex. 1.7: Contrapunctus 5, bar 41
In Contrapunctus 6 there are 14 entries of the subject rectus and 14 entries
inversus.
In Contrapunctus 14 the second subject consists of 41 notes.
The BACH motive makes its most prominent appearance in the whole of
Bach’s output as the third subject of the 14th fugue.
Ex. 1.8: the third subject of Contrapunctus 14
Further references to the numbers 14 and 41 will become apparent in a
striking manner in Chapter 3 in connection with my discussion of the
number of bars that are required to complete Contrapunctus 14.
in bar 41. Overduin’s article claims many other appearances of the BACH motive, most of which I consider to be forced and/or coincidental.
20
The last bar
The last bar in the manuscript of Contrapunctus 14 is bar 239. To my
knowledge no other writers appear to have noticed the fact that 2+3+9 = 14.
Fig. 1.2: the last bar of Bach’s manuscript
I have shown that there are many instances in which Bach associates his
name, either through numbers or through musical notes, with something to
which he wishes attention to be drawn (or to which he is ascribing
significance in his own private way). In the light of the previous discussion,
the hypothesis from which this thesis sprang was that it may not be a
coincidence that The Art of Fugue ‘runs out’ in a bar whose number is the Bach
number. The fact that the tenor part continues by itself, together with the fact
that no rests are notated for the alto or bass (another fact that has drawn no
comment from other writers) is a clear indication that the music is to continue
– to be filled in.35 The final barline is drawn with as much confidence and
deliberation as all the other bar lines in this score: the ‘unfinished’ score was
not left without a barline. The fact that Bach has chosen this bar – a bar
referring to the number 14 as clearly as many other instances of that number
– to be the final bar he has written of this work suggests that Bach may be
telling us, in his subtle coded way, that there is something significant about this
bar. Through the use of the numbers, which refer gematrically to his own
name, he is drawing our attention to this final bar.
35 I am not suggesting that the music continues to be written on another page. See Chapter 2.
21
Dr Ruth Tatlow’s statement, quoted at the head of this chapter, that “there
is no certainty…that Bach used any eighteenth‐century number‐alphabet
forms, even assuming that he knew about them”36 is indeed true insofar as
there is “no certainty”: we cannot prove what Bach may have been thinking
or doing. Perhaps more revealing for our purposes, however, are her words
that follow soon after this statement.
Invariably using the natural-order alphabet, the puzzler allowed the inquisitive reader to discover a desirable piece of information… It would not only be difficult to hide a name in music as part of a complex mathematical puzzle, but well-nigh impossible to find, unless the puzzler left a written code indicating its presence. To my knowledge Bach left no such codes.37
(my italics)
It is my hypothesis that Bach did in fact leave a code and that code takes
the form of the encrypted presence of his name by means of gematric
conversion to numbers. The question of Bach’s use of number symbolism –
not so much for theological purposes but more to draw the observant
reader’s attention to something significant about the music – runs like a
thread throughout the rest of this thesis. In the context of the appearances of
the ‘Bach numbers’ that we have mentioned so far, in particular in The Art of
Fugue, together with those that we shall encounter during the course of the
following discussions, the fact that Bach’s ‘last’ bar carries a link to the
number 14 can be seen as a clue.
In fact the last bar is not the only thing that is strange about the conclusion
of Bach’s manuscript, and in the next Chapter I discuss a variety of aspects of
the appearance of the last page that suggest a likely interpretation: that Bach
left Contrapunctus 14 unfinished deliberately.
36 Tatlow, op. cit., page 127 37 Ibid., page 127
22
Chapter Two
‘FRAGMENT X’ RECONSIDERED This chapter considers some of the implications of the last page of Bach’s
manuscript of Contrapunctus 14, in particular the final page containing the
second half of bar 227 to bar 239, as far as is known the last bar Bach wrote of
this fugue. After examining the sources, I will consider in detail Professor
Christoph Wolff’s theory about this page, and Laurence Dreyfus’ only partial
rebuttal of it; and then I will offer my own reasons for rejecting the greater part
of Wolff’s theory, reasons that are based on a more detailed examination of the
score, and I will suggest that this is one of several factors that point to the
probability that the incomplete state of The Art of Fugue was not unintentional.
The chapter also includes a reassessment of the classification of the autograph
manuscript of Contrapunctus 14, discussing whether it is a working
composition score, a revision score or a fair copy.
The Sources
There are three principal sources: the autograph manuscript; the first printed
edition of 1751; and a second edition which appeared in 1752. This 1752 edition
was little different from the 1751 edition: the only differences between it and the
1751 edition were a new title page and a lengthy preface by Friedrich Wilhelm
Marpurg (1718‐1795), who knew Bach and his two eldest sons personally.1 The
remaining pages of the 1752 edition (i.e. the music) were printed from the same
plates as the 1751 edition.
There are however significant differences between the autograph manuscript
and the first printed edition of 1751. These are summarised by George Stauffer
1 Wolff , op. cit., page 478, note 27
23
in the introduction to a 1975 seminar report.2 These differences between the
autograph and the first printed edition have given rise to puzzlement and
speculation about Bach’s intended order for the Canons and Fugues, a matter
which Gregory Butler claims to have resolved but which is not relevant to the
present discussion.3
The Autograph: P200
The autograph manuscript is the focus of the present discussion. It is
catalogued in the Deutsche Staatsbibliothek in Berlin as Mus. ms. Bach P200 and
it consists of a binding of 20 leaves (40 sides) containing 17 pieces, including
three early drafts of the Augmentation Canon. In addition there are three
Beilagen (supplements or appendices): Beilage 1 is the final version of the
Augmentation Canon; Beilage 2 is Bach’s arrangement of the three‐part Mirror
Fugue for two harpsichords (both rectus and inversus versions); and Beilage 3 is
the autograph of the quadruple fugue, subsequently and erroneously entitled
(not by Bach) Fuga a 3 Soggetti (Fugue on 3 subjects), which we now know as the
incomplete Contrapunctus 14. All four parts of P200 – the main 20‐leaf binding
and the three supplements – are in Bach’s hand, but it seems that the three
Beilagen come from later stages of the compositional and engraving process and
were probably grouped together with the 20‐leaf binding after Bach’s death.
Beilagen 1 and 3 are in what would today be called landscape format; Beilage 2 is
in portrait format.
Beilage 3
Beilage 3 is the autograph of the incomplete quadruple fugue. It has five
separate single‐sided pages and it is to the last of these that we now turn our
attention. Fig. 2.1 (page 26) shows a facsimile of page 5. A clear reproduction of
2 Bagnall, A., Baker, T., et al. (1975) "Seminar Report: Bach's "Art of Fugue": An Examination of the Sources" Current Musicology 19: 47-77 pages 47-50 3 Butler, G. (1983) "Ordering problems in J S Bach's Art of Fugue Resolved" Musical Quarterly 69(1): 44
24
the page is presented in the frontispiece to this thesis. A facsimile of all five
pages of Beilage 3 is given in the Appendix.
This last page of Beilage 3 begins with the second half of bar 227 of the
quadruple fugue and continues until the last written bar (bar 239). The music is
written in short score with two staves bracketed together for each of the two
systems. Bach has left a blank stave between the two systems. The page
contains twelve staves ruled presumably by Bach; the bottom three staves are
badly ruled, wavy and too close together. The inscription at the right is in the
hand of Carl Philipp Emanuel Bach and was obviously written some time after
Bach’s death:
“NB While working on this fugue, in which the name BACH appears in the countersubject, the author died.”4
This is the sentence which has led to so much of the romantic mythology
surrounding this work. Although Christoph Wolff says that Carl Philipp
Emanuel was “misled”,5 one may speculate with a touch of cynicism that he (C
P E Bach) may have written his nota bene deliberately in order to heighten the
mystique surrounding this ‘last’ work, given that he was trying, not very
successfully as it turned out, to sell the published editions at a profit.6 The
mention of the BACH theme appearing “in the countersubject” is also an error:
it is not a countersubject but a subject in its own right; although this error
makes little difference to anything, apart from perhaps giving an indication of
the extent to which Emanuel may have failed to understand some aspects of his
father’s work, it nevertheless appears to have drawn no comment from any
other writers on this fugue.
4 ''NB Über dieser Fuge, wo der Nahme BACH im Contrasubject angebracht worden, ist der Verfasser gestorben” 5 Wolff , op. cit., page 436 6 C P E Bach offered the plates for sale on September 14, 1756, “only about 30” of the printed editions having been sold, and it is thought that eventually they may have been melted down and sold as scrap metal. David, Mendel, et al., page 378; also Harris, R. and Norton, M. D. (1935) "The Art of Fugue" Musical Quarterly xxi(2): 166-178, page 167
26
C P E Bach’s sentence “may tempt us to think that death stayed the
composer’s pen”.7 But the evidence suggests that Bach was working on this
page much earlier than the last day or days of his life. According to Douglass
Seaton in his chapter of the aforementioned 1975 seminar report,
The handwriting of P200 tells us that the manuscript was copied during the last eight years or so of Bach's life. It is difficult to place it in any narrower time span by calligraphic evidence alone, but the writing is clear and steady, suggesting that the composer had not come to the very end of his life. Even the third and last Beilage to the autograph, the fragmentary closing fugue, still shows a steady hand. Thus the romantic idea that this work was written by the composer more or less on his deathbed does not seem to be borne out by calligraphic evidence. P200 may, in fact, have been copied considerably earlier than is normally assumed.8
Seaton also mentions evidence about the watermarks that suggests that the
paper comes from earlier rather than later:
In addition, an inspection of the paper of P200 leads to a similar conclusion. The twenty leaves of P200 are arranged in five pairs of bifolios folded together. All except the inner bifolio of the last pair (pages 35, 36, 37, 38) bear the same watermark (crowned double eagle with scepter), one which is found in various manuscripts dating from the 1730s and up to about 1742. The watermark of the paper of the other bifolio (and also of Beilage 2, the arrangement of one mirror fugue for two keyboards) is one which appears in Bach manuscripts dating from the early 1740s (arms of Eger).9
Christoph Wolff, writing after the 1975 seminar report, seems not to agree
with these earlier dates suggested by Seaton. He says (without offering any
evidence):
The composing manuscript [sc. of the quadruple fugue], which dates from 1748-49, belongs among the latest musical manuscripts from Bach’s hand …10
7 Wolff, op. cit., page 436 8 Seaton, D. (1975) "The Autograph: An Early Version of The Art of Fugue" Current Musicology 19: 54-59 (part of Bagnall, Baker, et al., op. cit.). See also Chapter 4 for a detailed discussion of the chronology of Bach’s last months. 9 Bagnall, Baker, et al. 10 Wolff, op. cit., page 436
27
Christoph Wolff’s Article: the ‘fragment x’ theory
C P E Bach’s nota bene was accepted at face value for 225 years, until
Christoph Wolff’s 1975 article threw a new and apparently compelling logical
light on the unfinished state of this fugue.11 He reminds us that this is, of course,
a quadruple fugue whose fourth subject was to have been the main Art of Fugue
subject which, in the words of Donald Tovey, “combines with the other three in
the simplest possible manner with a correctness that cannot conceivably be
accidental”.12 This discovery was first made by Gustav Nottebohm, a man also
noted for his deciphering of Beethoven’s sketch books; 13 the resulting
combination of the four subjects as given by Nottebohm and Wolff is shown in
Ex. 2.1.14
Ex. 2.1: the combination of the four subjects as given by Nottebohm and Wolff
The most important passage of Wolff’s 1975 article reads as follows:
For the composing of a polythematic fugue it is absolutely necessary to first try out the combinatorial possibilities of the various subjects. As a matter of fact, the subjects themselves have to be designed according to the rules of quadruple counterpoint. Therefore Bach had no choice but to start with the combinations of the four themes before writing the opening sections of the fugue. Consequently it is unthinkable that Bach composed the surviving fragment (Beilage 3) before he had worked out, or at least sketched, the combinatorial section of the quadruple fugue in a manuscript (hereafter designated
11 Wolff, C. (1975) "The Last Fugue: Unfinished?" Current Musicology 19(1975): 71-77 (part of Bagnall, Baker, et al., op. cit.) 12 Tovey, D. F. (1944) Essays in Musical Analysis - Chamber Music Oxford, Oxford University Press, page 87 13 Nottebohm, G. (1881) "J S Bach's letzte Fuge" Music-Welt: 234 14 Wolff "The Last Fugue: Unfinished?", page 73
28
fragment x) that originally belonged together with Beilage 3, but is now lost.15
Wolff restates his theory in his more recent book J S Bach: The Learned
Musician:
…the incomplete form in which the fugue has been transmitted does not correspond to what actually existed at the time of Bach’s death. […] His draft of such contrapuntal combinations has not survived, but a description, at least, of the composer’s intentions is included in the Obituary, which mentions the “draft” for a fugue that “was to contain four themes and to have been afterward inverted note for note in all four voices”.16 From this statement, we can reasonably surmise that The Art of Fugue at the time of Bach’s death was less incomplete than what has come down to us. But whatever the lost draft contained, it must not have been sufficiently worked out to bring the quadruple fugue to an end.17
It is a tantalizing thought that there must have been a piece of paper with the
end of the fugue – or at least a sketch thereof – written out on it. It follows as a
matter of common sense (with the benefit of hindsight, now that Wolff has
pointed it out) that the final combination of the four themes must have been
Bach’s starting point when beginning to compose this fugue.
Having suggested that the end of the fugue must have been written down
somewhere, Wolff now turns his attention to the appearance of the last page of
Beilage 3 (Fig. 2.1) and in particular to the faulty ruling of the staves in the lower
part of the page:
But the appearance of page 5 shows very clearly that Bach obviously had never planned to fill the sheet from top to bottom, in other words that he stopped writing deliberately at m. 239. The irregular and faulty ruling of the lower staff lines on page 5 did not permit the use of this part of the page for a dense fugal setting. Bach would never have started on such an untidy piece of paper had he planned to fill a larger portion of it that he did. […] Surely he used the last page only because
15 Ibid., page 73-74 16 “Seine letzte Kranckheit, hat ihn verhindert, seinem Entwurfe nach, die vorletze Fuge völlig zu Ende zu bringen, und die letzte, welche 4 Themata enthalten, und nachgehends in allen 4 Stimmen Note für Note umgekehret werden sollte, auszuarbeiten.” [His last illness prevented him, according to his draft, from bringing the next-to-last fugue to completion and working out the last one, which was to contain four themes and to have been afterward inverted note for note in all four voices.] – Wolff’s own translation. This sentence is discussed in detail in Chapter 4. 17 Wolff, C. (2000) Johann Sebastian Bach: The Learned Musician New York, Norton, page 437
29
he needed a sheet of music paper for just a few bars; since he never wasted paper, such a piece could serve his purpose.18
As supporting evidence for his last two sentences Wolff cites the appearance
of page 1 of Beilage 3: a page which demonstrates “that Bach aimed at a neat
layout and clean musical text for this fugue” (Fig. 2.2 overleaf shows page 1,
which can also be seen in the Appendix).
Wolff’s contention is that “Bach could never have planned to use page 5 for a
major text portion of the concluding fugue: he stopped at m. 239 […] because
the continuation of the piece was already written down elsewhere. […] Due to
most unfortunate and unknown circumstances that last part of the quadruple
fugue … became lost…”
Another eminent Bach scholar, Peter Williams, accepts and perpetuates
Wolff’s thinking in his recent book The life of Bach:
That still leaves a big question about the final fugue: was it finished or not? The extant manuscript’s last page was unusable, so was there once another?19
As well as Peter Williams, other writers have unquestioningly accepted and
perpetuated Wolff’s theory:
Christoph Wolff has established conclusively that this composition was, in fact, virtually finished at Bach’s death … Bach died and the quadruple fugue remained in an incomplete state even though its conclusion almost certainly existed, at least in some rough sketch.20
The permutational section proves Christoph Wolff’s hypothesis saying, that Bach completed the closing contrapunctus, at least no doubt that the combinatorial section (fragment x) was finished but then, following the composer’s death, it got lost …21
18 Wolff "The Last Fugue: Unfinished?", page 72 19 Williams, P. F. (2004) The life of Bach Cambridge, Cambridge University Press, page 158 20 Butler, G. (1983) "Ordering Problems in J S Bach's 'Art of Fugue' Resolved" The Musical Quarterly 69(1): 44, pages 55 and 58 21 Goncz, Z. (1991) "The Permutational Matrix in J S Bach's Art of Fugue: The Last Fugue Finished?" Studia Musicologica - Academiae Scientiarum Hungaricae 33(1-4): 109-119, page 118
31
Thus, in a well-known article, Christoph Wolff thought he was able to back up the hypothesis in which Bach would have finished this Contrapunctus, the end of which would have been lost.22
Christoph Wolff has plausibly argued (Current Musicology, 1975) that the fugue may actually have been finished by Bach, but that the conclusion was lost.23
Problems with Wolff’s theory
‘Fragment x’
Though one hesitates to challenge so eminent and revered a scholar as
Professor Wolff, not to mention the many experienced scholars who follow him,
I must respectfully draw attention to the fact that there are serious flaws in his
theory about the unfinished ending of Contrapunctus 14, as well as
inconsistencies and contradictions in his arguments. There are difficulties with
Wolff’s now universally accepted theory that the continuation of the fugue
existed on another piece of paper which he calls ‘fragment x’. He interprets the
words of the Obituary (given here in footnote 16, page 28) to mean that the
editors of The Art of Fugue, who included C P E Bach, had ‘fragment x’ in their
hands during the time of the engraving process but failed to realise what it was.
… C P E Bach […] must have known of or even seen fragment x, the complete draft of the combination of the themes and their inversions; […] But what happened to it?24
Wolff has no time for the editors: they were “startlingly unfamiliar with the
composer’s plans and intentions”; they made “fundamental errors” which were
“senseless” and which “prove their incompetence and carelessness”. Therefore,
he continues, “their failure to recognize the connection between the
22 Corten, W. (1988) "La dernière Fugue: piece inachevée ou ouverture sur l'infini?" Analyse musicale 11: 61-65, page 62: “Ainsi, dans un célèbre article, Chr. Wolff a cru pouvoir étayer l’hypothèse selon laquelle Bach aurait bien achevé ce Contrepoint, dont la fin aurait été perdue.” 23 Moroney, D. E. (1989) J S Bach: Die Kunst der Fuge Munich, G. Henle Verlag, page 123 24 Wolff "The Last Fugue: Unfinished?", page 75
32
fragmentary Beilage 3 and fragment x is not surprising. They considered the two
fragments as representing two different pieces…”
This seems implausible to say the least. Wolff has made several assumptions
that are open to challenge. First, the Obituary does not state, as Wolff implies,
that the editors had Bach’s “draft” in their hands: they may have known what
the draft was, but the Obituary does not say that they had it. Wolff’s translation
of the word ‘Entwurfe’ as ‘draft’ implies, in the context of discussion of a piece
of music, a draft of some or all of a composition – that is, musical notes written
down on manuscript. But the word ‘Entwurfe’ may also be translated as
‘outline’, ‘design’, ‘plan’, ‘blueprint’25 – in other words, the editors may have
had some idea of Bach’s plan or intentions for the remainder of the fugue; but
the sentence quoted from the Obituary does not in itself prove the existence of
‘fragment x’ or the editors’ alleged possession of it.26 The editors’ purported
awareness of Bach’s ‘design, plan or blueprint’ for the fugue could just as easily
have been a memory of something Bach had once said or written down in
words rather than in music. Second, if they did have ‘fragment x’ to hand, it is
simply inconceivable that a musician of C P E Bach’s stature would not have
noticed that three of the four combined subjects were the same subjects as those
in Beilage 3; further, if ‘fragment x’ were indeed “the complete draft of the
combination of the themes and their inversions”, as Wolff says (see the
quotation from him on the previous page), then how could C P E Bach or any 25 Johann Mattheson (1681-1764), who knew Bach and who corresponded with Mizler and his Society (and whose views were therefore very likely to have been known by Bach), provides evidence that the drawing up of a such an outline was a standard part of the compositional process.“DISPOSITIO is a neat ordering of all the parts and details in the melody, or in an entire musical work, almost in the manner in which one arranges or draws a building, makes a plan or sketch, a ground plan, to show where e.g. an assembly room, an apartment, a bedroom etc. should be situated. […] [The composer] should outline his complete project on a sheet, sketch it roughly and arrange it in an orderly manner before he proceeds to the elaboration. In my humble opinion this is the best way of all through which a work obtains its proper fitness, and each part thus can be measured to determine if it would demonstrate a certain relationship, similarity, and concurrence with the rest: in as much as nothing in the world is more pleasing to the hearing than that." - Mattheson, J. and Harriss, E. C. (1981) Johann Mattheson's Der vollkommene Capellmeister : a revised translation with critical commentary Ann Arbor, Mich, UMI Research Press paragraph 30, page 478 26 Other aspects of this sentence from the Obituary – its meaning, veracity and implications – are discussed in Chapter 4.
33
other editor not have noticed that one of those four themes was the main Art of
Fugue subject? This strongly suggests that the editors were not in fact in
possession of ‘fragment x’, if indeed it even existed, in which case they can
hardly be blamed for not realising that the incomplete Beilage 3 was in fact part
of The Art of Fugue (it was not until 1881 that Nottebohm discovered this!).
“Two different pieces”
A second point of Wolff’s must be challenged: his statement that “they
considered the fragments as representing two different pieces”.27 Here it
appears that Wolff has misunderstood the meaning of the phrase in the
Obituary “…bringing the next‐to‐last fugue to completion and working out the
last one…”. Various scholars, obviously including Wolff, have taken this phrase
to mean that what we now call Contrapunctus 14 was in fact to be the
penultimate Contrapunctus of the whole collection and that there was to be a
further quadruple and totally invertible fugue ‐ another mirror fugue ‐ to close
the collection. Sir Donald Tovey even went so far as to compose one on the
BACH motive, in addition to his completion of Contrapunctus 14 itself.
However, an alternative understanding, first proposed by Erich Bergel, of the
words ‘vorletzte’ and ‘letzte’ in the Obituary, is now widely accepted: the
sentence in the Obituary is understood to refer to each section of Contrapunctus
14 as a fugue in its own right – which is of course correct, since each of the
subjects is given its own exposition within a clearly marked sectional
framework. 28 Therefore the sentence refers to the third section of Contrapunctus
14 – the fugue on the BACH subject – which the Obituary correctly describes as
the ‘next‐to‐last fugue’ and as being incomplete; and to the ‘last one’, that is the
fourth and final section of the fugue which would have begun some time after
27 Wolff "The Last Fugue: Unfinished?", page 75 28 Bergel, E. Bachs letzte Fuge: die ‘Kunst der Fuge’ – ein zyklisches Werk (Bonn, 1985)
34
bar 239.29 The point made above also applies here: if the editors did have
‘fragment x’ they would have seen instantly that it was a combination of the
subjects of the extant portion of Contrapunctus 14; Wolff’s assertion that “they
considered the two fragments as representing two different pieces” comes
across as illogical.
The structure of Contrapunctus 14
This discussion of the four ‘fugues’ or sections of Contrapunctus 14 leads to a
third criticism of Wolff’s article: he has wrongly analysed the structure of
Contrapunctus 14.
It comprises three complete sections (mm. 1-115: exposition of theme I; 115-193: exposition of theme II and combination with theme I; 193-233: exposition of theme III, “B-A-C-H”) and the transition to a fourth and probably final section which begins immediately with the combination of themes I –III (mm. 233-239).30
(my italics)
It seems likely that Wolff is following Sir Donald Tovey, who has
unfortunately fallen into the same trap. Speaking of the end of section three he
writes:
Three crowded bars … end the section with a half-close. The fourth section now begins with the 3 subjects in the combination … 31
In fact the passage from bar 233 to bar 239 (i.e. the first appearance of the
combination of the first three fugue subjects) is not the beginning of the fourth
section, but rather it is still part of the third section, as is made clear by a correct
understanding of the Obituary. Gregory Butler has noticed this, though his
reasoning, based on the approximately proportional length of each section of
29 It does not automatically follow that the main Art of Fugue subject, appearing for the first time in the fourth section of Contrapunctus 14, would have had its own separate exposition before being combined with the other three subjects. Tovey says “That would be a fatal error of composition”. Tovey, D. F. (1931) A companion to 'The Art of Fugue' (Die Kunst der Fuge) of J. S. Bach Oxford University Press, London 30 Wolff "The Last Fugue: Unfinished?" , page 73 31 Tovey A companion to 'The Art of Fugue' (Die Kunst der Fuge) of J. S. Bach, page 45
35
the fugue, lacks precision.32 He correctly says that section 3 is “not quite
complete” and he theorises that the existing 46 bars of section 3 might have
extended to 52 bars, on the grounds that 52 bars is approximately two thirds of
the length of section 2, which is in turn approximately two thirds the length of
section 1. Following this line of reasoning, Butler concludes that section 4 might
be approximately two thirds the length of section 3 i.e. about 34 bars. The
question of the length of the missing section(s) of the fugue will be discussed in
Chapter 3. Suffice it to say for now that when Bach’s architecture is governed by
numerical structures, there is nothing “approximate” about them.
The appearance of page 5
Closer examination of page 5 of Beilage 3 (Fig. 2.1 above, also frontispiece)
reveals a further serious flaw in Wolff’s argument. Laurence Dreyfus correctly
points out that it is in fact only the bottom three staves of the page that are
unusable.33 Peter Williams, writing four years after Dreyfus, seems not to have
noticed this in his passage quoted on page 30 above. There is no reason why
Bach could not have continued to write up to the end of his current system and
continued on at least one more system below. If he had left one stave blank (as
he had already done after the first system on this page) he would still have had
room for another complete two‐stave system. If he had decided not to use
another blank stave, he would have had room for two complete systems; Bach
could have written at least another 20 bars of music on this page if he had
wanted to do so. This figure is based on the fact that there are 8½ bars on the
first system, and 4 bars taking up half of the second system; the second system
could have been completed with another 4 bars and there would have been
room for 8 bars on each of two possible further systems. However, page 1 of
Beilage 3 has twelve or thirteen bars per system; page 2 has between 9 and 14
32 Butler "Ordering problems in J S Bach's Art of Fugue Resolved", page 55 33 Dreyfus, L. (1996) Bach and the Patterns of Invention Cambridge, Massachusetts, Harvard University Press, page 165
36
bars per system; page 3 between 8½ and 10½; and page 4 between 9 and 11½
bars per system: so the suggestion that there is room on page 5 for a further 20
bars is a conservative one.
Dreyfus makes a compelling point when he says “The misruled paper
therefore provided no reason why Bach should have stopped copying at the
precise point where he did so.”
However, Dreyfus’ observations on the appearance of the last page go only
so far; in fact the misruling of the bottom three staves is only one of many
puzzling and inconsistent details about this page, and it is a surprise that these
details have until now drawn no comment.
Even a casual comparison of pages 1 to 4 with page 5 (Fig. 2.1, or
frontispiece) of Beilage 3 reveals immediately that the differences in these pages
are very significant – far more significant than either Wolff or Dreyfus seem to
have appreciated. The most obvious contrast between the two pieces of paper is
the general neatness and precision of one with the general untidiness of the
other. Pages 2, 3 and 4 are similar to page 1 in this respect.
The first detail to notice is that pages 1 to 4 have clearly been ruled
specifically for a keyboard short score. There is a gap after every second stave,
so that there is some space between each two‐stave system for ease of reading.
These staves have been ruled with great care and neatness, and they are all very
straight and parallel without any blotches or bumps or breaks or any other kind
of disturbance.
Reference to a variety of Bach’s other manuscripts confirms that he was in
the habit of ruling his staves and grouping them in specific ways according to
the instrumentation of the piece in hand. For example, the autograph of
Contrapunctus 1 (in P200, though it did not acquire the title Contrapunctus 1
37
until the 1751 print) shows clearly how the staves are grouped in fours, with
gaps between each system, because Bach has written this piece in open score.
Fig. 2.3: Contrapunctus 1, page 1 (autograph)
38
Similarly, the following three examples of keyboard works are very clearly
ruled for keyboard short score, just like the first pages of Beilage 3:
Fig. 2.4: Well‐Tempered Clavier Book 1, Prelude I in C (autograph)
Fig. 2.5: Well‐Tempered Clavier Book 1, Fugue in B minor (autograph)
Fig. 2.6: Fantasia in D major BWV 906 (autograph)
The following two examples from the Violin Sonatas show that when Bach is
writing for a single‐stave instrument he sets out the score with the staves evenly
39
spaced, since by definition it is not possible to group the staves into systems
with more than one stave per system.
Fig. 2.7: Fugue from Violin Sonata BWV 1001 (autograph)
Fig. 2.8: Fugue from Violin Sonata BWV 1005 (autograph)
From these examples it is clear that Bach ruled his staves specifically
according to the instrumentation of the music that was to be written on them.
This characteristic of his manuscripts is seen over and over again, whatever the
instrumentation; it is a characteristic that leaps to the eye from every page, for
example in Yoshitake Kobayashi’s collection of manuscript facsimiles in the
Neue Bach Ausgabe.34 It is also clear, therefore, that the last page of Beilage 3 was
not ruled with the intention of having a keyboard short score written on it.
Again, a detailed comparison of page 5 with pages 1 – 4 confirms this assertion.
Where page 1 has ten staves, grouped together into five systems of two staves
each, page 5 has twelve staves that are not grouped together in systems. This
presumably explains why Bach has left the third stave of the page blank: to 34 Kobayashi, Y. (1989) Notenschrift Johann Sebastian Bachs : Dokumentation ihrer Entwicklung Basel, London, Bärenreiter
40
create a gap between the systems, both for ease of reading and also, given the
high tessitura of the soprano part in bars 236 and 237, to avoid congestion on
the page. (This third stave has been misruled, with a ‘stop‐start’ mistake very
near the left margin.) Once again it is necessary to challenge Wolff’s
assumptions:
The lack of the proper spacing between the staves … was another reason for discarding the leaf as scrap paper. Bach had to skip the third staff in order to keep the systems clearly apart, while the layout of pages 1-4 was ideal for a keyboard score.35
Wolff has missed the point that pages 1 – 4 were ruled specifically for the
quadruple fugue to be written on them; he seems to assume that Bach wrote the
music on whatever paper happened to be at hand, pages 1 – 4 being suitable
and page 5 not.
The third line from the bottom of the page has a mistake in the ruling similar
to the ‘stop‐start’ error in the third stave, as if Bach has stopped ruling a short
way across the page and started again. In this case the continuation of the line is
missing in the middle line of the stave: this suggests that Bach had a special pen
with five nibs for ruling staves – a rastral – and when he ruled this stave the
middle nib was either short of ink or was not in contact with the paper. It is
puzzling that Bach did not go back and fill in the missing middle line: this is
precisely what he did on the first page of his autograph manuscript of
Contrapunctus 1. This can be seen in Fig. 2.3 above, in the tenor part of the
bottom system (bars 19 and 20), where the middle nib has failed to mark the
middle line of the stave and Bach has had to go back and fill it in afterwards,
and it is also shown here in detail:
Fig. 2.9: Contrapunctus 1 (autograph), bars 19 & 20, tenor part
35 Wolff "The Last Fugue: Unfinished?" page 76, footnote 3
41
Bach has also had to ‘repair’ the top line of the stave, seen here at the first
barline; the repair of the middle line is clearly visible in the following two bars.
This detail strongly suggests that Contrapunctus 14 (Beilage 3) was written with
the same rastral as Contrapunctus 1 – the staves at any rate. Wolff states that
the autograph of Contrapunctus 1 dates from “around 1742”36 and, as we saw
above, that the quadruple fugue dates from 1748‐49; whilst it is not impossible
that Bach could have been using the same pen for six or seven years, it is
nevertheless an interesting link between the two works which, according to
Wolff at least, were written several years apart. It also lends a degree of
credibility to the suggestion that the quadruple fugue was being worked on
well before Bach’s last days – a point that I will discuss more fully in Chapter 4.
This observation does not, however, help us to understand why Bach did not
complete the missing middle line of that stave on Beilage 3, page 5. Indeed, the
repair of missing lines in staves is seen frequently in Bach’s scores, and
particularly in many parts of the autograph of The Art of Fugue.
If, when (mis‐)ruling that page and having trouble with his pen, Bach did not
bother to go back and fill in the missing line, we may assume that he might then
have given up on that page and decided that it could not be used. Why did he
then carry on ruling two further staves? Curiously, the same stop‐start break in
the line can be observed on the stave below, and on the stave below that, which
is the bottom line of the page. What was the cause of this breaking of the line?
Perhaps an indentation or knot in the wood of the desk, or some other bump or
obstruction under the paper as it was being ruled? But if that were so, why
would Bach have carried on to make the same mistake not once but twice more?
A final detail that no other writers have considered is the bottom line of the
page and the unusual markings at the extreme corners. At first sight these
markings appear to have no connection with the musical material of the 36 Wolff Johann Sebastian Bach : The Learned Musician, page 435
42
quadruple fugue. Some possible implications of these markings are discussed in
Chapter 3.
Having shown that there are strange and possibly inexplicable
inconsistencies in the appearance of page 5, it is interesting to observe some
other characteristics of Bach’s manuscripts, in particular his habit of using up as
much as possible of each page. Page 1 of Beilage 3 (Fig. 2.2 above) is a good
example of this characteristic, as indeed are all of the examples shown in this
chapter. Figs. 2.5 and 2.6 above also show that Bach would rather fill in a small
fraction of a stave at the bottom of a page when only a few bars of music are
needed to complete the piece, rather than start on a fresh piece of paper if that
meant that only a small part of the paper would be used. An extreme example
of this is the autograph of the Cantata BWV 61, Nun komm, der Heiden Heiland,
where Bach has gone to some trouble to squash in a recitative after the end of a
chorus, even having to draw an extra 2½ bars in at the bottom of the page (see
Fig. 2.10 overleaf). Bach does this even in the autograph of The Art of Fugue
itself: on the last page of Contrapunctus 3 he has drawn in one final bar in the
bottom margin, rather than use a fresh page for just one bar of music; in this
case, the main part of the score is written in open score, but this extra bar is
notated in short score – yet another instance of economy.
A further characteristic seen in several of these scores, including the
autograph of the quadruple fugue, is Bach’s habit of writing incomplete bars
when he comes to the end of a system: he would rather write half a bar at the
end of a line and continue it on the next line than leave part of the system blank
by finishing each system at a bar line. This is difficult for modern eyes to cope
with, since current practice is not to split bars from the end of one line onto the
start of the next; but it was obviously Bach’s convention and he, and probably
all keyboard players of his day, would have been used to reading scores set out
43
in this manner. It is not overstating the matter to say that Bach was obsessive
about using up as much of each page as possible and not wasting paper.
Fig. 2.10: Cantata BWV 61
44
One might even go so far as to say that Bach was pathologically determined
to fill up his pages from the top left to the bottom right, even when it was not
possible to fill them with notes of music. In Beilage 1 (the final autograph
version of the Augmentation Canon), the bottom right of the page is filled in
with a decorative floral design, possibly because Bach wished to allow for a
page turn, since there is a rest in the right hand part which enables the player to
turn the page. (See Fig. 2.11 overleaf.) Clearly the decorative design was added
to prevent any suspicion that there was any music missing at the end of the
page ‐ a suspicion that might reasonably have arisen if the decoration were not
there and the music stopped half way along the system, leaving the rest blank.
No such decorations have been added to the end of page 5 of Beilage 3. In Beilage
1 Bach has gone to some pains – even in a handwritten manuscript of a work
destined for the printer – to indicate to the reader that the fact that the music
stops half way along the system does not mean that the music has come to an
end; in Beilage 3 he has taken no such pains, which may logically suggest the
opposite, i.e. that the music has come to an end and there is no more to come on a
following page. This is a further suggestion that the purported existence of
‘fragment x’ must be doubted.37
The use of these decorative drawings – Schmuckgraphiken – was continued in
an elaborate manner in the engraving of the 1751 and 1752 printed editions. The
second example (Fig. 2.13) shows a Schmuckgraphik incorporating Bach’s
monogram at the centre. Both examples show that Bach’s dislike of blank paper
was not confined to his autograph manuscripts.
37 In the 1751 printed edition, the layout of the page (page 49) preserves the autograph layout of the Augmentation Canon at this point, with the bottom system of the page stopping half way across the page to allow a page turn. The fact that the music continues on the next page is shown by simple directs – a minimal but nonetheless effective and intentional indication to the player or reader that the music continues on the next page. Bach’s use of directs in Beilage 3 itself, and in another supposedly ‘unfinished’ work, is discussed later in this Chapter.
45
Fig. 2.11: Beilage 1, page 2 (autograph, final version of the Augmentation Canon)
Two conclusions follow from these observations. The first is that the
appearance of page 5 is inconsistent with this characteristic of filling up the
paper as much as possible, and indeed shows an almost profligate wastefulness
in the way the music is set out on the page, particularly in comparison with the
previous four pages of the piece. The evidence shows that Bach was likely to
plan out his manuscripts so that he had enough paper for the music that he
wanted to write, and where necessary he would squash extra bars in at the end
rather than start on a fresh piece of paper. When the music came to an end
before the end of the page, Bach was anxious to fill up the empty space either
with the next movement or with a decorative design to indicate that the empty
staves were not accidental.
46
Fig. 2.12: 1751 edition, page 12, end of Contrapunctus 4 and Schmuckgraphik
Fig. 2.13: 1751 edition, end of Contrapunctus 8 and Schmuckgraphik incorporating “JSB” monogram
The second conclusion is that when Bach began to write up a fair copy of a
composition he planned it out in advance, ruling out the staves in a custom
47
format for the piece in hand (be it a single line score for a violin, a keyboard
short score, open score or a full setting for choir and orchestra) and being
careful to ensure that he had exactly enough paper – neither too little nor too
much – to fit the length of the piece. The question then arises whether the
manuscript of the quadruple fugue is a fair copy or not. The question obviously
has a direct bearing on any consideration of the unfinished ending: if it is not a
working composition manuscript, the unfinished ending becomes more
difficult to explain.
What category of score is Beilage 3?
Robert Marshall’s 1972 study of Bach’s manuscripts provides a most helpful
basis for this discussion.38 Although it concentrates on the manuscripts of the
vocal works, it begins with some observations which apply to Bach’s
autographs generally. As Marshall notes in his Preface, the problem of
classification arises from the fact that
Bach was a remarkably ‘clean’ worker, and that, while his compositions rarely seem to have sprung forth fully grown from the head of their creator, the manuscripts contain few traces of the genesis of the compositions notated in them.
He goes on to quote a passage from Spitta’s biography:
Bach’s scores do not give the impression that he made many sketches or experimented with the basic ideas in advance, as did, for example, Beethoven. The appearance of the scores suggests that they were written after the work concerned was thought out internally by the composer thoroughly and extensively, but not to the extent that the composer did not conceive any further during the act of writing.39
Since Marshall’s introductory summary cannot be improved by
paraphrasing it, I give the following direct quotation from the beginning of his
first Chapter.
38 Marshall, R. L. (1972) The Compositional Process of J. S. Bach: A Study of the Autograph Scores of the Vocal Works Princeton, Princeton University Press 39 Spitta, J. A. P. et al., (1884) Johann Sebastian Bach, his work and influence on the music of Germany, tr. by C. Bell and J.A. Fuller-Maitland Lond. (volume 2, page 180); quoted in Marshall op. cit., (Preface)
48
The German Bach literature since Spitta and the BG has traditionally distinguished two types of autograph score: The Reinschrift or fair copy, and the Konzeptschrift or composing score. Although one understands intuitively what is meant by these terms, it is virtually impossible to create any infallible, unambiguous criteria which would serve to relegate every manuscript either to one group or the other.
In the clearest and most extreme cases, one can indeed recognize Reinschriften with considerable certainty by their meticulous calligraphic script and by the virtual absence of any corrections that cannot be interpreted as slips of the pen or momentary lapses of attention. On the other hand, a typical composing score reveals, through a relatively hasty handwriting, an often large number of formative corrections, and the occasional presence of sketches and drafts, that Bach most probably wrote these manuscripts while conceiving the music contained in them.
The appearance of an autograph score, however, can be affected by such unknowable biographical factors as the degree of “inspiration” and patience at the command of the composer at the moment he penned a particular manuscript or composition. When Bach's invention was flowing most freely, he was capable of composing an almost flawless score; and, if he wished, he could set it down in a careful calligraphic hand, so that an observer can hardly determine whether the manuscript is a fair copy or a composing score. Conversely, Bach at times copied from a pre-existent source in an inattentive, hasty manner that produced a score that was neither calligraphic nor free of corrections.
The two ideal types should therefore be considered the extremes of a veritable continuum of gradations formed by the 170 extant autograph scores which represent every degree of calligraphy and every variation in the density and significance of corrections. For the entire composing and/or music-writing process was subject to the vagaries of inspiration and the moods of the composer.
Marshall goes on to mention the four categories of classification proposed by
Georg von Dadelsen40, and he presents them in the following schematic form:
l. HANDWRITING CRITERIA
a) Calligraphic: careful horizontal and vertical placement of all symbols; upright stems; evenly and rationally spaced note-heads and symbols, often varying in size according to temporal value (i.e., whole notes are larger than the heads of half-notes, quarter-notes larger than eighth-notes, etc.)
40 Dadelsen, G. V. (1958) "Beiträge zur Chronologie der Werke Johann Sebastian Bachs" Tübinger Bach-Studien 4-5
49
b) Non-calligraphic (Gebrauchsschrift): tilted stems; irrationally, unevenly spaced note-heads and symbols; frequently ambiguous placement of symbols (especially crucial in regard to pitch)
2. COMPOSITIONAL CRITERIA
a) Fair copy (Reinschrift): very few or no formative corrections; corrections usually limited to rectifying copying errors or slips of the pen; no sketches or drafts
b) Composing score (Konzeptschrift): relatively numerous corrections of a formative nature, plus the presence of sketches and drafts.41
Marshall then introduces a further, new category of score, one that is
particularly relevant to the present discussion.
In addition to the variety of fair copies and composing scores, one clearly “intermediate” manuscript type appears often enough to merit a specific designation here. Many manuscripts of the early cantatas, and again those of the 1730's and '40's, reveal a calligraphic penmanship and contain comparatively few corrections. Those corrections present are for the most part minor – concerned perhaps with text underlaying – or “grammatical” in nature. Furthermore, they are usually concentrated in particular lines the voice parts, say – or in particular sections of movements, e.g., either the A or B section of a da-capo aria. These features suggest that in his early period (particularly the pre-Cöthen years) Bach worked a great deal from preliminary drafts which may have been subsequently destroyed, and they confirm our knowledge that the composer made extensive use of earlier material in his compositions of the 1730's and '40's. The autographs of all these works can perhaps best be described as “revision copies”, reflecting a stage in the compositional process more advanced than that observed in the composing scores of the Leipzig church cantatas of the 1720's, but hardly to be classified together with the famous calligraphic Reinschriften of, say, the Brandenburg Concertos or the Well-Tempered Clavier. In the Weimar and Mühlhausen cantatas one often has the impression that the invention of themes and overall design of many movements had been worked out in advance and that only the final touches were then added in the revision copy. In at least two instances the preliminary drafts or scores were apparently so complete that Bach had a copyist write out large portions of the final manuscripts. We may assume that Bach's slower pace in Weimar – the church cantatas were mostly written at four-week intervals between 1714 and 1716 – in comparison with the weekly production of his first three years in Leipzig, encouraged a relatively leisurely routine and consequently a more conscientious penmanship. By the same token his slower routine in Leipzig after 1727 and his increasing use of pre-existent material in the cantatas of this
41 Marshall, op. cit.
50
period, occasionally enabled him to employ the services of a copyist to write out large portions of a score. […]
While the introduction of the term “revision copy” will often explain the appearance of a particular score and the type of corrections it contains better than the terms “fair copy” and “composing score”, there remain nonetheless a number of elegant, carefully penned autographs, particularly from the pre-Cöthen period but also from later years, which elude any attempt at such classification. A particular manuscript may be a highly “inspired” composing score, a revision copy, a rather careless fair copy, or a fair copy that had been subjected to yet further revision. In the case of such problematic manuscripts there can rarely be any certainty that one of these designations any more than the others accurately reflects the biographical circumstances surrounding the production of the scores.
To these three broad categories of manuscript – the composing score, the
revision score and the fair copy – must be added another, namely a copy
prepared for use by an engraver, called an Abklatschvorlage. The word
Stichvorlage seems to be synonymous with Abklatschvorlage. It seems that Beilage
3 is one of those manuscripts that defies classification, since it can on the one
hand be described as an “elegant, carefully penned autograph” that matches
von Dadelsen’s “Handwriting Criterion 1a” above (careful placement of all
symbols, evenly spaced, etc), but on the other hand it does contain a number of
corrections.
The differences of opinion between Douglass Seaton, Davitt Moroney and
Christoph Wolff – three writers who have published studies of the manuscript –
illustrate the extent of the confusion around this point. Seaton’s article, referred
to at the beginning of this Chapter, speaks of the main part of P200 (not the
Beilagen42) and, after discounting the possibility that the manuscript is a revision
copy, “one stage of the Art of Fugue in a succession of versions leading to the
original print as the completed form”, on the grounds that “such copies were
not normally retained when a work was further revised, and manuscript copies
42 Seaton makes the extraordinary statement that “the three Beilagen need not come into the picture here … for they do not form an integral part of the manuscript.” (Seaton, page 55)
51
of works which were later printed were preserved only on rare and special
occasions”, he goes on to say:
Clearly this is not a composition score. For some of the fugues it seems to be a fair copy; there are hardly any corrections in the first pieces. For later fugues the manuscript must have served as a revision copy, since there are sections which are quite heavily corrected. The whole manuscript is in a clean calligraphic hand and is easily read, though the writing becomes a bit more squeezed in the later sections. We can confirm that one use of P200 was as a source for the printer’s copy; this is shown by the presence of the copyist’s casting-off marks and notes such as the crosses on page 18 […]. The original intention was not, apparently, that P200 should serve as a revision or rough copy, but as a fair copy. […] We may hypothesize that P200, when it was made, was conceived as the final fair copy of a completed work.43
Though this paragraph focuses on the ‘main part’ of the autograph
manuscript, Seaton does mention Beilage 3 itself:
The writing [sc. of P200] is clear and steady, suggesting that the composer had not come to the very end of his life. Even the third and last Beilage to the autograph, the fragmentary closing fugue, still shows a steady hand.44
Wolff agrees with Seaton that P200 is an “autograph fair copy”.45 But he says
that the manuscript of the unfinished quadruple fugue is a composition score:
That Beilage 3 represents the composition manuscript of the fragmentary last fugue is shown by some characteristic corrections and the overall graphic appearance.46
He does not say what these corrections are (I discuss them below), and he
does not specify what he means by “the overall graphic appearance”. In fact the
overall appearance of the manuscript – i.e. pages 1‐4 – is one of extreme care
and neatness; there are very few “characteristic corrections”, no more or no less,
on average, than the main part of P200 over which Seaton and Wolff are in
agreement that it is a fair copy. Why then does Wolff say that Beilage 3 is a
composition score? He says himself that “pages 1 – 4 of Beilage 3 demonstrate
43 Bagnall, Baker, et al., pages 55-56 44 Ibid. 45 Wolff Johann Sebastian Bach : The Learned Musician, page 435 46 Wolff "The Last Fugue: Unfinished?", page 73
52
that Bach aimed at a neat layout and clean musical text for this fugue”.47
Further, he says that Beilage 3 is written on the same “type of paper that was
used for the engraver’s copy (Abklatschvorlage)”; that “the paper must have been
left over from a supply of ruled paper for the engraver’s copy”; and that
“Beilage 3 was written at a time when the preparation for the printing was
already in progress, even more precisely, while or after the printer’s copies of
the canons were made”.48 These statements seem to contradict the notion that
the manuscript is a composition score.
Wolff unfortunately even contradicts himself on this specific point. He is
credited as the author of the chapter on Bach’s life for the Leipzig period
1739‐50 in Grove Music Online (the internet version of the New Grove Dictionary
of Music and Musicians), and in discussing The Art of Fugue he writes:
Bach had been unable to complete the fair copy of the last movement, a quadruple fugue, and so the fugal cycle ends with an unfinished movement.49 (my italics)
In various statements in his 1975 article, Wolff seems to imply that Beilage 3,
together with ‘fragment x’, were intended as the copy for the engraver. In a
footnote, speaking of Beilage 3, he writes “it had to be easily legible for the
copyist, who eventually had to transcribe it into open score for the printer’s
copy”.50 Wolff is suggesting that, “because the continuation of the piece was
written down elsewhere, namely in fragment x” there was no need for Bach to
continue writing on page 5, and the printer would have pieced the two pages
together.
… it seems that the surviving source material offers no better solution to the problem than the following; Beilage 3 (mm. 1-239 of the last fugue) was originally to be supplemented by another manuscript with
47 Ibid., page 72 48 Ibid. 49 Wolff, C. Bach, §III: (7) Johann Sebastian Bach 9. Leipzig, 1739–50. <http://www.grovemusic.com.ezproxy.auckland.ac.nz>, (web site) Grove Music Online ed. L. Macy (Accessed 6 December 2005) 50 Wolff "The Last Fugue: Unfinished?", page 76, footnote 2
53
the remainder of the piece, comprising the combination of the four subjects which had to be composed first.51
As evidence for this theory, Wolff draws the parallel between Beilage 3 and
the early and late versions of Contrapunctus 10, which in the 1751 printed
edition appears with 22 new bars added at the beginning. These 22 bars do not
appear in the autograph manuscript and are obviously a later addition by Bach,
but the autograph of the new bars has not survived. Plausibly, Wolff suggests
that
The autograph of these 22 bars has not survived, but it must have been a “fragment” – similar to Beilage 3 – which had to be pieced together, by a copyist, with the original version of the double fugue in the autograph (P200, pp. 14-16) to produce the printer’s copy.52
Is it likely that, if this scenario is correct (i.e. that a copyist was expected to
piece together ‘fragment x’ with the first five pages of Beilage 3) Bach would
have expected the copyist to work from a working composition manuscript,
rather than a revised score or a fair copy? Wolff has contradicted himself on
several levels and seems to have vitiated his own argument.
A further detail of the content of page 5 also weakens Wolff’s theory on this
point. If it were true that the continuation of the fugue beyond bar 239 was to be
found on another sheet of paper, and the printer was to be expected to piece the
two manuscripts together, what would the printer or copyist be expected to
make of bar 239 itself as Bach has left it?
51 Ibid. 52 Ibid., page 74
54
Fig. 2.14: Contrapunctus 14 (autograph), last two bars
It can be seen that only the tenor part has been completed and that the
soprano, alto and bass parts have not. Can they be assumed to be silent, and if
so why has Bach not written any rests? If they are not to be silent, why has Bach
not written their notes? If the music continues on another page, how would the
printer know what to put in this bar to complete it in accordance with Bach’s
intentions? The fact that Bach has not completed this bar is in itself strong
evidence that the music does not continue on another page.
If indeed Bach had begun by writing the conclusion of the fugue on another
piece of paper, with the intention that it would later be joined on to the
beginning of the piece that was to be composed afterwards, how would Bach
have known to begin his conclusion at precisely bar 240? Further, if the
continuation did not begin at precisely bar 240, but was merely a sketch, outline
or draft of the combinatorial possibilities of the subjects, why did Bach not
continue on page 5 with the bars that follow bar 239? As I showed earlier, there
is room on that page for at least another 20 bars of music, more if necessary, and
Bach was not averse to squashing his music into a page when necessary.
Furthermore, if the ‘piecing‐together’ theory is correct, would it not be
reasonable to assume that Bach would have written a note to the printer to
explain how to accomplish the join? In several places Bach’s proofing comments
and instructions to the printer can be seen clearly, and they show that he kept a
meticulous eye on the engraving process. The appendix to Wolff’s 1975 article
55
mentions a pencil marking in Bach’s hand, revealed only by infra‐red
photography, which is “an instruction to the scribe of the printer’s copy to
double the note values of the first triple fugue, as shown in the original
edition.”53 The first double fugue has a similar instruction in ink; there are
crosses on pages 17 and 18 (see Fig. 2.11); a note “Corrig[i]rt” on page 19; an x
on page 29; and a note on the proof sheets written after Bach’s death by his son
Johann Christoph Friedrich says
Canon p[er] Augmentationem contrariu motu.
NB: the late Papa had the following heading engraved on the plate: “Canon per Augment: in Contrapuncto all octava”, but he had crossed it out on the proof plate and put it in the above-noted form.54
Even in Beilage 3 itself, on the reverse of page 4, there is a list of 25 corrections
referring to pages 21‐28, 31, and 33‐35; each correction has been systematically
crossed out as it has been attended to.55
In addition to the absence of any written instruction to the printer, there is a
further factor which strongly argues against the likelihood that the music
continued on another page, and that is Bach’s use of ‘directs’ in his manuscript.
Directs can be seen clearly at the right‐hand end of every system of the
autograph manuscript of Contrapunctus 14, warning the reader of the notes
that are to come at the beginning of the next system. Even at the top right of the
last page of Beilage 3 Bach has written directs which refer to the next (and last)
system on the page. If, as Wolff suggests, the music were continued on
‘fragment x’ and the printer were expected to know this, why then has Bach
written no directs (or any other indication) at the end of bar 239?56
53 Ibid., page 77 54 quoted in Wolff Johann Sebastian Bach : The Learned Musician, page 436. The original German is “N.B. Der seel. Papa hat auf die Platte diesen Titul stechen lassen, Canon per Augment: in Contrapuncto all ottave, er hat as aber wieder ausgestrichen auf der Probe Platte und gesetzet wie forn stehet.” 55 This list of corrections may be in the hand of C P E Bach. 56 See also footnote 70 (page 68) for a further instance of an apparently unfinished work which does include directs at the end of the last surviving page.
56
Bach was careful and detailed in his instructions to the printers. How
plausible therefore is the suggestion that the purported join between page 5 and
‘fragment x’ would have been left in such an ambiguous and unexplained state?
The suggestion that Bach would have allowed ‘fragment x’ to be lost also seems
highly improbable.
In the Critical Commentary to his edition of The Art of Fugue,57 Davitt
Moroney suggests (without reference to Wolff’s suggestion that Beilage 3 was
intended as the copy for the engraver) that “there must have been an
intermediate Ms in open score, prepared for the engraver”.58 Moroney draws
attention to an error in the engraving process, in which the soprano and alto
parts in bars 95‐6 were switched around in the print: “this error is due to the 2‐
stave original Ms having these 2 crossing parts on the same stave. Such a
thematic error proves that Bach did not supervise the preparation of the open
score Ms for the engraver, and confirms use of the Ms as main source.”59 This
error is perpetuated in Christoph Wolff’s own edition of the work, where on
page 69 a dotted line clearly directs the soprano A to the next note D, which
actually belongs to the alto even though it is on the top of the texture.60 There is
no excuse for this error since the alto line at that moment is a statement of the
fugue subject.
Clearly there is considerable disagreement over the classification of the
autograph of Contrapunctus 14. In summary of the foregoing: Seaton says that
the main part of P200 is “the final fair copy of a completed work” but that it was
also “a source for the printer’s copy”; he does not attempt to classify Beilage 3
but says that it “shows a steady hand”. Wolff states in one article that it is “the
composition manuscript” and in another article that it is “the fair copy”; he also 57 Moroney, op. cit. 58 Ibid., page 114 59 Ibid., page 124 60 Wolff, C. (1986) (ed.) Die Kunst der Fugue BVW 1080 (Band II: Spätere Fassung des Originaldrucks) Frankfurt, New York, London, C F Peters
57
implies that it was the printer’s copy. Moroney says that it cannot have been the
printer’s copy but that there must have been another, intermediate copy, now
lost, for the printer.
Examination of the manuscript
In January 2006 I travelled to the Staatsbibliothek in Berlin to examine the
autograph manuscripts of the whole of The Art of Fugue at first hand. They are
kept in two boxes: the first box contains all of P200, the main part of the work as
described at the beginning of this Chapter. This box is labelled:
Mus. ms. P200 BWV 1080
Die Kunst der Fuge
The second box contains only the three Beilagen, including the autograph of
Contrapunctus 14 (Beilage 3). In the context of the present discussion, the
Staatsbibliothek’s labelling of this second box, presented here exactly as it is
written, is striking:
Mus. ms. P200/1 BWV 1080
Die Kunst der Fuge (Abklatschvorlage für den
Druck).
Considering the disagreement among scholars over the classification of this
particular manuscript, it is interesting to note that it is specifically labelled as
being “The Engraver’s Copy for the Printed Edition” (Abklatschvorlage für den
Druck).
Examination of the manuscript shows clearly that Douglass Seaton is correct
to describe the main part of P200 as either a fair copy or, later in the manuscript,
as a revision copy. It is true that there are not many corrections or alterations in
the early part of the manuscript, and such as there are more likely to be minor
copying errors; for example, in the score of what is now Contrapunctus 5 (called
58
“IV.” [sic] in the autograph), at bar 39‐40 the barline bulges to the right,
suggesting that Bach must have written the soprano and alto parts first –
presumably copying from another manuscript – and forgot to leave room for
the bass part which is in quavers and therefore takes more room. In “vii.”
(Contrapunctus 6) there are three instances of corrections written out in letter
names because the original notes and their corrections have been made illegible
by the initial attempt to change them. In “VI.” – the early, shorter version of
what became Contrapunctus 10 – some sharp signs have been squashed in
above or below the noteheads, suggesting that they were added after the notes
were written. In these and similar minor alterations to the score, the ink is
clearly the same as the notes themselves and the alterations were obviously
made at the same time that the score was written.
Later in this part of the manuscript, as Seaton mentions, there are a few
changes that are more significant. Perhaps the most heavily corrected
movement is “viii” (Contrapunctus 7), with a few examples of short semiquaver
figurations revised, but more substantial reworkings of bars 39 and 40 (whose
original text has been rather vehemently crossed out) and bars 51 and 52. In
both of these cases Bach has written an extra temporary stave below the system,
with the bar lines extended down to it from the bottom of the system.61 The
colour of the ink of these alterations is noticeably lighter than the rest of the
manuscript, making it clear that these changes were made at a different, later
time. Lighter ink can also be seen in other places, for example in “x.”
(Contrapunctus 8) where Bach has written an instruction underneath the tenor
part of bar 135 to say that it should be an octave lower.
It is beyond doubt that the main part of P200 matches von Dadelsen’s
description of a fair copy (Reinschrift): “very few or no normative corrections;
61 These extra staves were written not with a rastral like the rest of the score, but each of the 5 lines of the stave was drawn freehand, separately and, it has to be said, rather untidily.
59
corrections usually limited to rectifying copying errors or slips of the pen; no
sketches or drafts.” It is equally beyond doubt that it does not match the
description of a composing score (Konzeptschrift): “relatively numerous
corrections of a formative nature, plus the presence of sketches and drafts”.
Since Christoph Wolff has stated (in one of his articles, though not in
another) that the autograph of Contrapunctus 14 is a composing score, it should
be possible to make an objective assessment by comparing it to a) von
Dadelsen’s terminology and b) the rest of P200. The first task is to consider the
handwriting criteria. As can be seen in Fig. 2.2 above, or in the Appendix, and
as was noted by Seaton, the handwriting is strong and clear. The notes are
written confidently and cleanly, accurately lined up both vertically and
horizontally. The stems and barlines are perfectly upright. The ruling of the
page is neat and even, set out in short score with gaps between the systems. The
noteheads vary in size according to their temporal value: semibreves are large
and fat, minims a little less so, and minims whose stems are downward
consistently exhibit the characteristic of Bach’s later years in which the stem
descends from the middle of the notehead rather than from the side. In general
the use of the ink is bold and generous. This is clearly a calligraphic score.
Although Robert Marshall allows for the possibility that certain of Bach’s
composing scores may have a calligraphic appearance, the overall graphic
appearance of Beilage 3 is so clean, strong and confident that it looks very like a
careful and neat copy, not, as Wolff says, a score in which Bach is composing as
he goes along.
The characteristics described in the preceding paragraph are particularly
prominent on page 1 of the manuscript and in the whole of page two, with the
exception of the one significant alteration in the whole manuscript, towards the
bottom of page two (bars 112‐114): I discuss this alteration in detail below and
in Chapter 3. There is no other minor correction before bar 130 (in the second
60
bar of page 3). All subsequent corrections are minor. A comparison of page 4
with page 1 shows that by page 4 the handwriting has become marginally
thinner, a little more ‘spidery’. The fact that there are no corrections in the early
part of the manuscript, together with the fact that the handwriting seems to
become marginally more ‘tired’ by page 4, suggests to me that this manuscript
was written out at one sitting and is very clearly not a composition score but
some form of copy from a composition score. Any composer or copyist,
particularly one in his mid‐ to late‐60s, would be getting writer’s cramp after a
stint of several hours of copying music.
Moroney’s Critical Commentary lists all the variants or corrections in the
manuscript of the quadruple fugue. For the sake of completeness, and to show
that all but one of them are minor, I reproduce Moroney’s list here.62
110 B: this bar was originally a whole note D; altered when the following bars were recomposed
112-14: these 3 bars were originally only 2; this version was crossed out and the three replacement bars written in tablature letters at the bottom of the page.63
114 S: this bar was originally half note D, half note rest (with half notes in the T and B as well); the 4 Soprano notes A, A, G, F were added as an afterthought. The mark which has traditionally been interpreted as a tie between the 2 As is in fact the half note rest (sitting on the 5th line) from the early version. The 2 As are not tied in the print either. Comparable rests in the T and B have been “ballooned” into half notes and, in the case of the Bass, tied to the previous note.
114 T 3: could be read as a B or an A in Ms
130 A: was originally 4 eighth notes, tied G, A, G, F, 2 quarter notes E, D, corrected by hand
144 T 5: was originally half note, altered by hand to quarter note, quarter-note rest
62 Moroney, op. cit., page 124 63 It should be noted that Moroney merely lists this change: he offers no suggestion for the reason for it. To my knowledge no other writers apart from Moroney have mentioned the existence of this particular alteration in the manuscript, and none has explained it. In Chapter 3 I will show that the reason for the change was not a musical one (i.e. a change made by Bach because he was dissatisfied with the notes he had originally written) but a structural one, having to do with the length of the fugue as a whole.
61
165 T 1: was originally a half note, altered by hand to quarter note, quarter-note rest
190-93 seems originally to have been as in Example 2, but the text is hard to read due to erasures [Moroney gives in his Example 2 a slight variant of the bass part: it can be seen on page 139 of his commentary.]
209-208: barlines only every 4 half notes; hence the system of half bars used here
222 B 1: was originally half note, altered to quarter note, quarter-note rest
239: Ms ends here (6 bars after the ending in the print).
It can be seen that the great majority of these ‘corrections’ occur later in the
piece rather than earlier; and also that the great majority of them are very minor
alterations to note values. Another similar correction not listed by Moroney or
in any other critical commentary has been observed in a recent article by Ewald
Demeyere, who rightly draws attention to the first soprano note of bar 101,
which appears originally to have been written as a minim and subsequently
filled in to make a crotchet.64 Fig 2.15 shows an example of a similar alteration:
bar 165, beat 1 which does indeed appear to show, as Moroney says, that the
tenor note (the soprano is silent in this passage) was originally written as a
minim, and Bach has filled it in to make it a crotchet.
Fig. 2.15: Contrapunctus 14 (autograph), bar 165
This kind of ‘mistake’ which is subsequently ‘corrected’ is entirely consistent
with the idea that this manuscript is a copy: it is the kind of mistake that would
be very easy to make when copying – to draw a minim instead of a crotchet –
especially if the writer were beginning to tire a little after writing for several
64 Demeyere, E. (2006) "The Art of Fugue: Expanding the Limits!" <http://www.bach-cantatas.com/Articles/AOF-Demeyere.htm> (website): accessed 12 May 2006
62
hours. The only possible alteration of any musical significance listed by
Moroney is the purported change to bars 190‐193 and even here it is not
possible to say that this was a change of heart by the composer who
subsequently revisited that place and decided to ‘improve’ it: it could just as
easily have been a correction that was needed because of some slip of the pen
while copying.
It is also relevant to note that in the case of this particular manuscript of this
fugue, all the changes are seen to be exactly in the same ink as the rest of the
piece, unlike the changes in the main part of P200 which, as was noted above,
are in lighter ink and were clearly written on a different occasion; the changes
in Beilage 3 were clearly made at the same time and in the same ink as the music
itself was written down. Even the strong crossing‐out of bars 112‐114 and their
replacements notated in tablature in the bottom margin (page 2) are in the same
ink.
It is now possible to consider the second of von Dadelsen’s categories: the
compositional criteria (the handwriting criteria were considered above). He
describes a Konzeptschrift or composing score as being characterised by
“relatively numerous corrections of a formative nature, plus the presence of
sketches and drafts”: clearly this does not apply to the autograph of the
quadruple fugue. In fact this autograph matches much more closely the
description of a fair copy or Reinschrift: “very few or no formative corrections;
corrections usually limited to rectifying copying errors or slips of the pen; no
sketches or drafts.”
Combining these observations of the appearance of the manuscript – both the
handwriting and compositional criteria – it seems indisputable that Beilage 3 is a
calligraphic fair copy. It is probably not possible to state definitively that it is a
final fair copy, and – despite the Staatsbibliothek’s unequivocal description of it
63
as an Abklatschvorlage – Moroney’s idea that there must have been a version in
open score for the engraver is persuasive. It could be possible to make a case
that this score is some form of revision copy. But – with all possible respect to
Professor Wolff – I have to disagree with his claim that this is a composing
score. As I have mentioned, even Wolff himself describes it as a fair copy in his
Grove article.
The point is crucial to a correct understanding of the unfinished bar 239. The
universally held belief that the piece is unfinished because of Bach’s illness and
death can only have any validity if the manuscript is an unfinished composing
score. If – as the evidence seems overwhelmingly to suggest – the manuscript is
some form of copy, it seems almost impossible to imagine that Bach (or any
other composer) would have set about making a copy of a work that was not
yet finished in its original composing score. The work must have been finished
if the composer had made a copy of it. To this statement it might be objected
that if the manuscript is indeed a copy, the use of such an untidy and scrappy
piece of paper for the last page is a contradiction, since by definition a copy is
supposed to be neat. It would seem illogical, even capricious, to use such a
piece of paper as part of a copy. This fact can be added to the now long list of
strange inconsistencies about the appearance of this manuscript, a list to which
I now continue to add.
More on the appearance and ruling of page 5
Earlier I concluded that page 5 could not have been ruled with the intention
of having short score keyboard music written on it. In the light of the foregoing
observations it is also possible to conclude that – since it is clear that Bach ruled
up his manuscript paper with the layout of systems specific to the type of
instrumentation being used, and since the autograph of Contrapunctus 14 is
obviously some form of copy of the composition and not a working
composition score – page 5 was not ruled with the intention of having
64
Contrapunctus 14 written on it. The careful ruling of pages 1‐4 makes this clear:
page 5 does not resemble pages 1‐4 in any of the essential characteristics – the
neatness of the ruling; the number of staves per page; the grouping of the
systems; or the number of bars on each system.
A further detail that supports this statement is the size of the paper. Pages 1‐
4 of Beilage 3 are 21 x 34 cm (‘landscape’ format), and indeed the pages of the
main part of P200 are 33 x 21 cm (‘portrait’ format). But page 5 of Beilage 3 is 20
x 35.5 cm – a different size of paper from pages 1‐4.65 This strongly suggests that
page 5 was not intended to ‘belong with’ pages 1‐4; at the very least, it is one of
several factors that draw attention to the fact that almost everything about page
5 is different from its predecessors.
Is it plausible that Bach ruled up four pages of manuscript, thinking that that
would be a sufficient number of pages to accommodate the completed
Contrapunctus 14? Since we know that it is a copy (fair or revision), not a
working composition manuscript, Bach obviously knew exactly how long the
piece was, so this seems unlikely. If we assume for a moment that Bach had in
fact underestimated the amount of paper needed, why when he got to the end
of page four would he have taken up such a scrappy piece of manuscript that
does not match the preceding pages? Surely, if this were indeed a fair copy
destined for the printers, such an untidy page would not have been acceptable.
Is it plausible that Bach literally did not have another piece of paper in the
house on which to rule new staves, so that he was forced to hunt about to find a
stray scrap of paper, one that he had perhaps discarded on a previous occasion
because it was misruled?
65 The dimensions of Beilage 1 (final version of Augmentation Canon) are c.21 x 33.5 cm (‘landscape’). The dimensions of Beilage 2 (arrangement of three-voice mirror fugues for two harpsichords) are c.35.5 x 22 cm (‘portrait’).
65
Mention of the printing or engraving process invites consideration of yet
another inconsistency: is it likely that Bach would have embarked on the
process of having a work of his engraved and published, even going to the
extent of having the plates drawn up and carefully proof‐read by himself, if the
work in question were not finished? The answer surely must be No. What
composer would send in an unfinished work to the printers? Is Professor Wolff
really suggesting that Bach would have fed his last great fugue, the climax of a
monumental work, to the printers in a piecemeal manner, on different bits of
paper (at least 3 different lots of paper, if the ‘fragment x’ theory is to be
believed)? If the claim is that Bach started the printing process even though
Contrapunctus 14 was not finished (a most unlikely scenario at best), a person
making that claim might suggest that Bach said to himself “It will be easy to
finish that fugue; there is not much left to do; I can easily get it finished by the
time the engravers will need the fair copy; they may as well start now on the
engraving of the plates for the early pieces in the collection, and by the time
they come to engrave Contrapunctus 14, I will have it finished and ready for
them.” But if this is indeed what was going through Bach’s head, the obvious
question is: why then did he not finish it, if it was such an easy thing to do in
the time available?
Each of these detailed observations taken individually might be able to be
dismissed as insignificant, but cumulatively and taken together they point
inexorably to the fact that Christoph Wolff’s ‘fragment x’ theory does not stand
up to scrutiny and is illogical. The more plausible conclusion is that ‘fragment
x’ never existed, and that Bach never intended Contrapunctus 14 to be
continued (on paper) beyond bar 239. He began the engraving process because
Contrapunctus 14 is finished as it stands: Bach intends the piece to be left exactly
as it is. His decision to use such an odd piece of paper for his ending, together
with all the other inconsistencies that I have mentioned, is a clue to the
66
observant reader, just like the appearance of the number 14 in the last bar that
was discussed in Chapter 1, that there is more to that last page than meets the
eye.
If one bases one’s conclusions on a logical assessment of what can be seen in
the appearance of the manuscript and what can be noticed in the music itself or
its structure, one may well come to a conclusion that Bach left the fugue
unfinished deliberately. That conclusion in itself, however, does not answer an
intriguing question: Why? As we have seen above, Laurence Dreyfus pointed
out the fact that only the bottom three staves of page 5 of the manuscript were
unusable. I agree with him in so far as he says that Bach could have continued
to write on that page if he had wished to do so. However, the explanations that
Dreyfus offers for the fact that Bach did not do so seem to me to enter the
realms of the esoteric, if not the bizarre. First, in his note 31 (page 256), Dreyfus
refers to an article by Matthew Dirst and Andreas Weigand (“Baroque
Forecasting: On Completing J S Bach’s Last Fugue”, Santa Fe Institute for Studies
in the Sciences of Complexity, 1993, pages 151‐177) in which Dirst suggests that
the short scoring for the final fugue in the autograph “was due to Bach’s
attempt to fill in pages already allotted to the engraver”.66 This may have an
element of truth in it, but Dreyfus goes on to say:
If in composing the fugue, Bach saw that its length would never fit on the two available pages [allotted by the engraver], he may well have given up in despair and never returned to complete the project before his illness.
66 A number of writers have been exercised by the fact that the autograph of Contrapunctus 14 is written in short score, whereas the other four voice fugues are in open score. Gregory Butler’s article seems to corroborate Dirst’s theory in that he shows that the engraver had allowed six pages and a completed Contrapunctus 14 (if Butler’s claim that the completed length would have been 273 bars is correct) “would have fit nicely on six pages”. (Butler "Ordering Problems in J S Bach's 'Art of Fugue' Resolved", page 55). The true explanation for Bach’s use of short score, however, may well be more prosaic: my suggestion is that it was simply to save paper, since such a very long piece would have used up a great deal of paper – 10 or 12 pieces – if written in open score. Further, I have already drawn attention to Davitt Moroney’s proposal that there must have been a subsequent copy for the engraver which would have been written in open score.
67
This seems to take an extraordinary leap into speculation about Bach’s state
of mind. Given that Bach had begun work on The Art of Fugue several years
before, in the early 1740s, it seems most unlikely that he would ‘spoil the ship
for a ha’p’orth of tar’, giving up so easily for the want of a small number of bars
– especially since, as Wolff originally suggested, he must have had a very clear
idea of how the fugue was to be concluded. In all Bach’s output there is no
other example of his having left a work unfinished for this kind of reason; on
the contrary, he was well used to finishing all his projects thoroughly and
efficiently. One has only to think of the hundreds of church cantatas, all written
to a strict weekly deadline, to see that Dreyfus’ explanation does not seem
plausible. It is clear that The Art of Fugue was a monumental project that must
have been dear to Bach’s heart: a treatise on counterpoint that in many ways
summed up his life’s work, stubbornly written over several years at a time
when such counterpoint was very much out of fashion. To suggest that Bach
would give up so easily, just because he might have thought that he was
running out of space on a piece of paper, is untenable.
In the main part of his text, Dreyfus continues his attempt to psychoanalyse
Bach, suggesting that at the end the composer was overcome by a sense of
modesty:
…it is certainly worth considering whether the composer had second thoughts about issuing a work in which his boldly embroidered family emblem was proclaimed, no matter how appropriate and well-deserved this proclamation. So even if he could easily have finished filling in the details of the final fugue, of which no-one can have the slightest doubt, one can argue that Bach chose through his inaction to leave it unfinished at his death.67
Not only does Dreyfus here contradict his earlier explanation quoted above,
by saying that it would have been easy for Bach to have filled in the details, but
the unfounded ascription of self‐deprecation to Bach seems out of place in a
67 Dreyfus, op. cit., page 167
68
work that sets out to be one of musical analysis. When Dreyfus mentions Bach’s
“boldly embroidered family emblem” he is of course referring to the BACH
theme, the third fugue subject of Contrapunctus 14; but as shown in Fig. 2.13
above, Bach suffered no pangs of modesty when he had his own initials
elaborately included in the design of his printed edition. As if his argument
were not fanciful enough, Dreyfus continues:
Perhaps, then, Emanuel Bach’s mythic inscription – suggesting that Bach died after composing the passage where his name appeared in the countersubject – contains within it a kernel of truth: that is, that as a composer of fugues, Bach had in fact died at this very moment.68
Peter Williams offers a different but equally speculative theory:
…is it possible that, as with another late fugue (in C minor, for organ), the composer lost interest in it? Despite the awesomely ingenious way it harmonises its three themes so far, their relentless exploitation can be found progressively less enthralling.69
Williams does not specify which late organ fugue in C minor he means. It
cannot be either BWV 574 (Fugue on a theme by Legrenzi) or BWV 575 since
both of these are complete and are in any case thought to date from Bach’s pre‐
Weimar and Weimar periods respectively (1708–1717) and are therefore not
‘late’ works. He probably means the fragment of the fugue BWV 562, but there
is no evidence whatsoever that the reason this particular fugue is incomplete is
that Bach “lost interest in it”; indeed, Williams’ own three‐volume study of
Bach’s organ music suggests that it was in fact finished but the last page or
68 Ibid., page 168. Perhaps Dreyfus is being whimsical here? His idea is reminiscent of comments made by William Malloch, who wrote the following flight of fancy in the programme notes to a recording of his own orchestration of The Art of Fugue: “Bach leaves us at the crossroads. He knows that his life goes on elsewhere; he knows that the music will remain alive and well without his help, and he invites us to continue practicing the art of counterpoint. But surely if one is to take up Bach's invitation and apply it to this unfinished fugue, it would not be appropriate to round off the fugue in the conventional sense, since it is a work designed to remain open in the end. For myself, I have added an 'unfinish' at the point the manuscript leaves off. It is planned as a natural outgrowth from the point the whole orchestra unites, in mid-range, in praise of Bach's name. After Bach's notes come to an end, the orchestral sections go their separate ways, fuguing away forever, past the horizon line; Bach ascends to Heaven, and we hear his sweet admonition to us to begin anew." I am grateful to Dr Bradley Lehman for bringing this passage to my attention. 69 Williams, op. cit., page 158
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pages have been lost.70 This fugue can hardly be presented as compelling
corroborative evidence that Bach may have “lost interest” in completing
Contrapunctus 14. The suggestion seems even more unlikely when considered
from another angle: he had been working on this collection of fugues for several
years and up to bar 239 of Contrapunctus 14 he had written a total of 1716 bars
of music, not including the canons. If for the sake of argument we assume (with
Gregory Butler) that a mere 6 + 34 = 42 bars of music are needed to complete the
work as a whole,71 bringing the total number of bars to 1756, then Williams is
suggesting that Bach got bored after completing 97.7% of a major work and
abandoned it within 2.3% of the finish. Lastly, if Williams does not find the
“exploitation” of the material “enthralling”, that is a subjective opinion, not
evidence that Bach had lost interest.
It seems remarkable that these Bach scholars should offer up these reasons
for the incompleteness of Contrapunctus 14: that the composer gave up in
despair because he did not have enough paper; that he was overcome by
modesty; that he metaphorically ‘died’ at bar 239; or that he got bored. In the
light of these speculations, the suggestion that he left the piece unfinished
deliberately seems far more credible.
70 “In P 490 [sc. the manuscript of BWV 562] the fugue takes the last of the four sides of the MS, the final bar followed by directs to the next page, showing that the fugue was either continued/completed and lost, or planned but not executed. Other incomplete works of Bach do not have a full texture up to the point at which the work stopped (e.g. the fragment ‘O Traurigkeit, O Herzeleid’ or the incomplete movement of The Art of Fugue), which suggests that BWV 562ii was once longer; moreover, at least one other organ work in an autograph score (BWV 548) breaks off during the fugue, which was then written out to the end by a copyist.” Williams, P. (1980) The organ music of J.S. Bach Cambridge, Cambridge University Press, Volume 1, pages 202-203 71 See Chapter 3 for a more precise number.
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Conclusions
I agree with Professor Wolff only insofar as the notion that something
relating to the conclusion to Contrapunctus 14 may have been written down on
a piece of paper – ‘fragment x’ – and that that piece of paper has been
unaccountably lost. However I believe that ‘fragment x’, if it existed, was more
likely to have been only a sketch or outline of the final combination(s) as part of
the initial stages of the compositional process, rather than a fully worked
completion of the rest of the fugue. There is no evidence that the editors knew
of or saw ‘fragment x’ even if it did exist; indeed a logical analysis suggests that
they could not have seen it, since if they had they would certainly have known
what it was and would have related it to the extant portion of Contrapunctus
14. There is no evidence for the suggestion that ‘fragment x’, if it existed, was
designed to be a continuation of the extant page 5 of Beilage 3. Wolff’s theory
about the misruled paper does not stand up to scrutiny. It is clear that Bach
could have continued to write on that page if he had chosen to do so. The fact that
he did not continue to write on that page may therefore lead to a plausible step
of reasoning that suggests that he deliberately chose not to do so. The
appearance of the last page is in many respects completely different from the
previous four pages, and from the vast majority of Bach’s other scores. Other
scholars have offered far‐fetched theories to explain why the fugue is
unfinished. In contrast, if one assumes that Bach left it unfinished deliberately,
far fewer speculative leaps have to be made to explain the fact that it is
unfinished. This hypothesis is able to explain all the anomalies around the
autograph and the engraving history without any intervening steps in the
argument having to be made. The question remains: why would he leave it
unfinished deliberately?
71
I am convinced that one explanation seems to be the most likely. The Art of
Fugue as a whole is an exercise in progression from simple to complex, starting
with the relatively straightforward Contrapuncti 1 – 4, introducing inversion of
the subject in Contrapuncti 3 and 4. Then there is a group of three fugues (Nos.
5 ‐7) that employ techniques of stretto, augmentation and diminution. The next
fugues, Contrapuncti 8 – 11, are essays in double and triple invertible
counterpoint. The astonishing technical achievement of the Mirror Fugues, Nos.
12 and 13, takes compositional virtuosity to a level unsurpassed before or since.
The four Canons – however they are meant to fit into the overall order (a
question that has baffled many scholars) – also show in themselves a
progression from simple to complex, culminating in the Augmentation Canon.
Contrapunctus 14 is designed to be the grand finale to the whole collection. It is
in itself an enigma, since its main point, the introduction of the main Art of
Fugue subject in quadruple combination with the other three subjects, is
withheld by Bach (though as we shall see in Chapter 4, a ghostly outline of it
hovers over Bach’s last written bars).
Whatever Bach’s motivations may have been in creating this composition,
one of them may well have been to write a didactic work; even if that was not
his motivation, it is nevertheless the result of what he has written: The Art of
Fugue is in effect a textbook on counterpoint and fugue. Certainly C P E Bach
and Marpurg had this in mind when they wrote to promote the sale of the
published editions. Emanuel’s advertisement includes these sentences:
… one knows how fruitless instruction is without illustration, and experience shows what unequally greater advantage one draws from practical elaborations rather than from meagre theoretical direction. The present work, which we announce to the public, is throughout practical and indeed accomplishes what many skilful men have suggested in their writing over the years. […] What special understanding, however, one can attain of the art of composition by studying scores, both with respect to harmony and melody, those who
72
have been fortunate in proving themselves will demonstrate with their examples.72
Similarly, Marpurg wrote in his preface to the 1752 edition:
… this work has given the occasion to examine more closely the nature of the fugue, and to compare with it the rules which have in the meanwhile been drawn up governing its construction. […] Since the rules of fugue have generally been treated together with the other principles of musical composition, many music lovers who do not have at hand the great and lengthy works on composition may thus find satisfaction. […] It is to be hoped that the present work may inspire some emulation…73
There is no doubt that The Art of Fugue is a very complete and systematic
treatise on fugue, even if Bach did not set out to write it with that purpose in
mind. Bach takes the reader, student or performer on a progressive and
systematically organised journey from simple to complex, all built upon the one
subject. When we come to the unfinished ending of Contrapunctus 14, the
hypothesis that I want to explore is that Bach is saying to the reader: “I have
shown you thus far what can be done with this material. Now it is for you to go
and work out how to finish it off yourself – and I have left you some hidden
clues to let you know that that is my intention, and to give you hints of what
can be done in the conclusion. That is why I have not written out the ending
myself.”
Perhaps it was not an accident of words that led Marpurg to say in his first
paragraph “in this work are contained the most hidden beauties possible to the
art of music” (my italics).74
As a kind of sidelight on the claim that it may well have been Bach’s wish
and intention that Contrapunctus 14 should be completed by the student or
performer, I conclude this Chapter with a quotation from Michael Ferguson,
72 David, Mendel, et al., pages 256-258 73 Ibid., pages 375-377 74 Ibid., page 375
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some of whose work we shall consider in detail in Chapter 4. Mr Ferguson is an
American organist who has devoted many years of study and scholarly
research into The Art of Fugue: this has included the composition and
publication of a completion for Contrapunctus 14, and the release of a
commercial recording of the work played on the organ. This recording includes
an essay on the work in the CD liner notes; the following quotation, extracted
from three places in that essay, expresses eloquently the idea of “Art of Fugue
as Game”.75
Bach also loved an improvisatory game called ‘leading astray’, where two organists seated at different organs in the same room, would improvise a fugue together. One would begin, then break off; the other would try to take it up without missing a beat, and add a twist to the counterpoint before handing it off to the other, who would have to add still more complexity, and so on – until one would lose the musical ‘thread’, and have to drop out.
[…] As a game of ‘leading astray’, the Art of Fugue is a most extreme example. … The progression of complexity involved is hardly a matter of a few steps along a single contour of development. A better analogy would be the differentiating ‘complexification’ of cells that results in a living creature. There, every new ‘division’ of ‘the idea’ has a fresh purpose, tending more and more toward consciousness and light. In the Art of Fugue, the complexification tends toward ever greater power of expression, stronger climactic moments, and the rarely achieved possibility of sustained intensity of emotion. Beginning with an unusually stark theme, Bach makes every development of it seem necessary, even inescapable, with the compelling emotional logic of a dream. Each movement prepares for the next, and all of them reveal an orientation to one and the same focal point. Bach’s delight in sending his notes on the chanciest adventures, and then gathering them home again with his unerring instinct for the final cadence, has here enormous scope for play.
[…] Any composer who feels drawn to this unique musical game can also feel the encouragement that I felt, in knowing that Bach would both have wanted a response, and have been pleased if it went well. For does not that awesome moment in Contrapunctus 14, when Bach’s notes break off, also resemble another kind of moment that his young son Carl Philipp Emanuel often shared with him? When they heard an organist playing a fugue, Bach would whisper to his son how it ought to go, according to the potential of the theme. He would also,
75 Ferguson, M. (2003) Art of Fugue (CD recording, liner notes), record label Birolius 1001/2, Minnesota, USA
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joyfully, as Emanuel remembered, nudge his son when the organist did as Bach had hoped.76
It is true that everything in this passage is speculative, and that it comes from
a CD booklet rather than from an academic article or textbook; it is equally true
that we shall never know what was in Bach’s mind, and that C P E Bach’s letter
to Forkel (footnote 76, below) does not refer specifically to The Art of Fugue.
Nevertheless, the comparison between what is described in Emanuel’s letter
and the unfinished ending of Contrapunctus 14 presents an attractive similarity.
In the context of the inconsistencies and questions that surround the manuscript
of this fugue and that have been surveyed in this Chapter, the suggestion that
Bach may be inviting the student or performer to fulfil his expectations for a
conclusion may seem less fanciful or whimsical than the hypotheses that some
other writers have offered to explain the work’s incomplete state.
76 The reference is to C P E Bach’s letter to Forkel of December 1744: “When he [JSB] listened to a rich and many-voiced fugue, he could soon say, after the first entries of the subjects, what contrapuntal devices it would be possible to apply, and on such occasions, when I was standing next to him, and he had voiced his surmises to me, he would joyfully nudge me when his expectations were fulfilled.” Quoted in David, Mendel, et al., page 397. There is a similar reference in the Obituary: “He needed only to have heard any theme to be aware – it seemed in the same instant – of almost every intricacy that artistry could produce in the treatment of it.” Ibid., page 305
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PART TWO: AFTER BAR 239
Chapter Three
THE BACH NUMBERS, PROPORTIONAL RATIOS AND THE MARKINGS ON THE LAST PAGE: THE LENGTH OF THE QUADRUPLE FUGUE
CONCLUSIVELY ESTABLISHED
Introduction
Though there are several puzzling questions about The Art of Fugue, one fact
that cannot be disputed is that Contrapunctus 14 as it stands is incomplete. This
statement is true irrespective of whether Bach left the piece unfinished
deliberately, or whether a continuation of the piece did or did not exist on
another page. However, there is no agreement among composers and
musicologists about how long the fugue should be if an attempt to complete it
is to be made. The most widely accepted theory to date comes from Gregory
Butler’s important article in which he concludes that “approximately” 40 bars
are needed to complete the fugue.1 I believe that it should be possible to arrive
at a more definite answer, one that does not need the qualification of the word
“approximately”. In this chapter I will show that it is in fact possible to establish
conclusively that Bach’s intention was for a precise number of bars to complete
the fugue, and that he has left us a number of clues to show us the way to that
information. The exact number can be corroborated on a number of levels, and
in particular I will show, by means of an analysis of markings on the last page
of Bach’s manuscript, that he may have supplied the information in coded form.
As far as I know I am the first to comment on these markings, and certainly the
1 Butler "Ordering Problems in J S Bach's 'Art of Fugue' Resolved", page 55
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first to propose a theory about them. The number of ‘missing’ bars can be
derived from a correct understanding of the architectural proportions of the
sections of the fugue: previous writers have not achieved complete accuracy in
their analysis of this aspect of the music. An explanation of a correction made
by Bach on page 2 of his manuscript also corroborates this new theory. I will
show that my analysis results in some remarkable numerical correspondences,
lending weight to the claim that it is correct; and I will show that Bach had done
similar things elsewhere.
Bach’s use of Gematria or Numerology: brief overview
Bach’s habit of using numbers in various ways in some of his music, and in
particular its relevance to The Art of Fugue, is discussed in detail in Chapter 1.
For the purposes of the present chapter, however, all that is needed is a brief
reminder of the most important of the ‘Bach numbers’. First is the number 14,
resulting from the numbers associated with the letters of Bach’s name:
B + A + C + H = 2 + 1 + 3 + 8 = 14
We saw earlier that the number 14 appears frequently in Bach’s music and in
The Art of Fugue in particular, not least in the 14th fugue.
The second important number is 41, which results from the letters
J + S + B + A + C + H = 41
A noticeable feature of the number 41 is that it is the retrograde of 14.
On page 10 it was noted that another number frequently seen in Bach’s
music is 55, which is the result of 14 + 41 = 55. This number will be seen to
assume a crucial importance in the present discussion.
77
Varying lengths of completions of Contrapunctus 14
A surprisingly large number of composers have written completions to
Contrapunctus 14, some more successful than others. This is not the place for a
discussion of the merits of them or a description of their contents, and I confine
myself to mention of the lengths – the number of bars that the composers have
added after Bach’s final bar 239. Walter Kolneder’s 1977 book about The Art of
Fugue lists twenty completions.2 A more recent article by Jan Overduin surveys
nine published completions, mentioning the lengths of some but not all of
them.3 The following is a selective survey of the lengths of some completions.
The longest is by Ferrucio Busoni in his Fantasia Contrappuntistica (1912).4 He
adds 672 new bars, though this is not really a completion but rather a new work
that incorporates Bach’s surviving 239 bars without bringing in the main Art of
Fugue subject. Overduin describes it as “a spiritual commentary on
Contrapunctus 14”.
The eminent Swiss organist Lionel Rogg has written and recorded a
completion that sounds most convincing and very Bachian (though some
aspects of it are open to challenge).5 It is 46 bars long. It is undated but it must
have been written before M. Rogg’s recording of it in 1969.
Perhaps the most famous completion is that of Sir Donald Francis Tovey
(1931).6 At 78 bars (not 80 as Overduin states) it is one of the longer attempts,
along with that by the American organist Michael Ferguson, whose completion
2 Kolneder, W. (1977) Die Kunst der Fuge : Mythen des 20. Jahrhunderts Wilhelmshaven, Heinrichshofen 3 Overduin, J. (2001) "Nine Published Completions for Keyboard of BWV 1080, 19, from 'The Art of Fugue' by J. S. Bach" The American Organist 35(1): 78-82 4 Busoni, F. (1912) Fantasia Contrappuntistica New York, Dover Publications 5 Rogg, L. (?1968) Art de la Fugue: conclusion hypothétique. The completion is unpublished, but I am most grateful to M. Rogg for sending me a copy of his manuscript. There are some slight differences between the manuscript and M. Rogg’s recording. 6 Tovey, D. F. E. (1931) Die Kunst der Fuge by J S Bach London, Oxford University Press
78
takes 98 bars.7 Longer still is the “reconstruction” by Zoltán Göncz: 111 bars.8
The blind organist Helmut Walcha published a completion 71 bars long,9 and a
completion 96 bars long by Dr Adel Heinrich was published in the journal of
the Riemenschneider Bach Institute.10
Davitt Moroney published his own completion in conjunction with his
edition of The Art of Fugue for Henle in 1989. In his Critical Commentary to his
edition, Moroney refers to Butler’s article, which he has obviously taken as a
guide when judging the length of his completion: “… the missing portion
cannot have been intended to contain more than about 40 bars, at most. (I have
added 30.)”11 This is a good example of the way statements by academics and
musicologists can take on a life of their own, developing accretions and
assumptions over the years as they are adopted and paraphrased by one writer
after another. In fact Butler’s article does not contain the words “at most”: it
stops at “approximately 40”, which is not precisely the same as Moroney’s “not
more than about 40 at most”.
In a somewhat esoteric article of May 2000, Herbert Kellner states that the
ideal length of the fugue should be 238 bars – that is, an extra 37 bars.12 His
reasoning is based on (a) a numerological analysis of the words and letters of
the Nachricht (preface) to the 1751 printed edition (there are 37 words in the
second half of the Nachricht, the total of whose letters is 2138 [=BACH]), and (b)
on an idea that the fugue is constructed around a midpoint at bar 138/139.
Nothing is to be gained here by analyzing Kellner’s arguments in detail; but I
7 Ferguson, M. (1990) Contrapunctus XIV, A Completion of J S Bach's Unfinished Quadruple Fugue from The Art of Fugue, newly edited for organ according to Bach's autograph manuscript and the 1751 original edition Saint Paul, Minnesota, Holbrook & Associates 8 Göncz, Z. (1996 & 1997) "Reconstruction of the Final Contrapunctus of The Art of Fugue" International Journal of Musicology 5 & 6: (vol 5): 25-93; (vol 6):103-119 9 Walcha, H. E. (1967) Bach: Die Kunst der Fuge Frankfurt, Peters 10 Heinrich, A. V. (1986) "Completion of the 'Incomplete' Quadruple Fugue (Die Kunst der Fuge)" Bach (Quarterly Journal of the Riemenschneider Bach Institute) XVII(4): 30-42 11 Moroney, op. cit., page 123 12 Kellner, H. A. (2000) "Die Kunst der Fuga: J.S. Bach's Prefatory Message and Implications" The Diapason 91(5): 15-17
79
will suggest a more convincing alternative to his conclusion that 37 bars are
required to complete the fugue.
Though many of these completions contain a great deal of inventive, stylish
and justifiable material, most of them are too long. Even without mentioning
any musicological or technical considerations, it is clear from the intensity and
drama of Bach’s final pages that the piece must be drawing near to its
conclusion by the time we arrive at his final written bar. There can be no doubt
that a big final climax is not far away and that the heat generated by the wild
and dramatic chromaticism of Bach’s last pages cannot be sustained for much
longer. As Tovey himself puts it, “we are certainly entering upon the final
section of the work, and nothing must stop its flow.”13 The longer completions,
like Göncz’s and Tovey’s, lose this dramatic impetus and seem to settle into
lengthy note‐spinning in their determination to cram in as much contrapuntal
wizardry as possible. Unfortunately Tovey seems to contradict himself a few
paragraphs later when he says “… it requires a peroration of at least 80 bars,
and this gives plenty of room for all the variety we can obtain from the now
incessant whirl of the quadruple counterpoint.”14
It can be taken as read that the third section of the fugue needs to be
completed, that the fourth needs to be begun with the introduction of the main
Art of Fugue subject, and that this section would at some stage present the four
subjects in combination, in accordance with Nottebohm’s discovery; the
Obituary can be interpreted to mean that the four themes would also be
presented in inversion during this fourth and final section of the fugue.15
Presentation of these subjects means by definition that a certain number of bars
will be needed to accommodate them – the second subject, the longest, is seven
bars long – and there will have to be some episodes and perhaps a coda at the 13 Tovey A companion to 'The Art of Fugue' (Die Kunst der Fuge) of J. S. Bach, page 45 14 Ibid., page 46 15 This point is discussed in detail in Chapter 4.
80
end. There is a lot of contrapuntal material that can be included, and this has
led some of the completions to run away with themselves and go on for too
long. But the need to include these essential elements at the end of the fugue
must be held in balance with the need to preserve the dramatic tension inherent
in Bach’s last bars and to bring the music to a satisfying climax without losing
momentum. A range of approximately 30 – 50 bars would seem to be about the
right length for this. Anything above 50 bars is almost certainly going to be too
long from the point of view of climax. This is a very vague and broad range,
and it ought to be possible to narrow it down to something more exact. Gregory
Butler’s researches come a step closer to achieving that.
Gregory Butler’s article
Mention has been made of Butler’s article and, although I disagree with his
conclusion about the length of the fugue, I have followed his method which is
based on a proportional analysis of the lengths of the sections of the piece.
Through a fascinating description of his detective work in examining evidence
that the original page numbers had been erased from the plates before printing,
to be replaced by new numbers, Butler argues compellingly that he has been
able to reconstruct the originally‐intended pagination scheme for the 1751
printed edition and thereby to establish the correct order of the fugues and
canons – a question that has dogged the work largely because of the differences
in the order and content of the pieces between the autograph manuscript and
the 1751 print.
Both the early (autograph) and later versions of Contrapunctus 10 (as we
now know it) were printed in the 1751 edition. The inclusion of the early,
shorter version was obviously a mistake by the editors and it should not have
been included in the print, since the later version with its 22 new bars at the
beginning obviously represents Bach’s final intentions for that fugue. Butler
writes:
81
Since Contrap: a 4 [i.e. Contrapunctus 10, early version], presently occupying the three pages immediately preceding the augmentation canon, pages 45 to 47 inclusive, presents the version of the second double fugue found in P200, and thus duplicates material which appears earlier in the print, Bach could never have intended it to be included in the collection. Therefore, in his original ordering scheme, something else must have been intended to fill the gap of six pages, from page 45 to page 50 inclusive.16
Butler explains that the editors, unsure after Bach’s death of his intentions for
the collection and, in particular, unsure of whether the “Fuga a 3 soggetti”, as
they called it, was part of The Art of Fugue, had to decide what material to
include and what order it should be in.
The arrangement for two keyboards of the second mirror fugue, the “unfinished” Fuga a tre soggetti and the compensatory chorale, Wenn wir in höchsten Nöten sein, are all in the hand of the same scribe and must also have been added after Bach’s death, since he would never have included an arrangement of material presented earlier in the collection, an incomplete work, or an unhomogeneous chorale. Of these four posthumous pieces only one could conceivably have been destined for inclusion in the collection had Bach lived to see it through the press. And it is this work which would, I submit, have occupied the six pages between the mirror fugues and the canons – the incomplete quadruple fugue. […] Since Plates 51 to 59 had already been paginated, Schübler [the engraver] was left with a gap of six pages in the original scheme. Whether he asked Bach for material to fill out the collection or whether manuscript material on Bach’s work table after his death was sent to the printer by Bach’s family is not clear. At any rate, Schübler decided to fill in the gap of six pages in the original ordering with material at hand. Since the quadruple fugue was in an incomplete state, it could not be inserted in its originally intended position, and for aesthetic reasons it was decided that it should appear at the end of the collection.17
Having demonstrated convincingly that six pages had been allotted to
Contrapunctus 14, Butler takes his argument further to make a prediction about
how long the fugue would have been if it were completed.
As it appears in the print in its incomplete state, this work occupies five pages. It seems clear that the finished version would have fit nicely on six pages.18
16 Butler "Ordering Problems in J S Bach's 'Art of Fugue' Resolved", page 54 17 Ibid., pages 55 and 58. The unfinished fugue was printed on pages 61 – 65 inclusive. 18 Ibid., page 55
82
He also mentions that the average number of bars on each of those five pages
is 46.19 Davitt Moroney states that “none of the five engraved pages contains
more than 47 bars”.20 The following table shows the exact layout and
dispositions of bars on each of the five pages of the print. Each page has three
systems, each of which contains a varying number of bars. It can be seen that
Bach’s habit – alluded to in Chapter 2 – of writing half bars at the ends (and
therefore also the beginnings) of some systems is continued here in the print.
Table 1: layout of bars and systems on each page of the unfinished fugue (printed edition)
Page number Bars per system (top/middle/bottom) Total bars per page
17 + ½
½ + 14 61
14 + ½
46½
½ + 14 + ½
½ + 15 62
14
44½
16 + ½
½ + 15 63
15 + ½
47½
½ + 15 + ½
½ + 15 64
15
46½
17
16 65
14 + ½
47½
Average number of bars per page: 46½
A refinement of Butler’s and Moroney’s statements is therefore possible: the
average number of bars per page is 46½, and no page contains more than 47½
19 Ibid. 20 Moroney, op. cit., page 123
83
bars. Fig. 3.1 below shows the first page of the fugue in the print (page 61), to
illustrate the layout of the pages.
Fig. 3.1: page 61 of the printed edition: first page of the unfinished fugue
To verify that the work would have fit onto six pages of the printed edition,
Butler discusses the proportions of the sections of the fugue, and his work here
brings him close to the correct answer.
If we examine the relative proportions of the three extant sections of this fugue, we notice a consistent diminution in the lengths of successive sections. Moreover, section 2 (78 measures) is almost exactly two-thirds the length of section 1 (115 measures), and section 3, not quite complete, occupies forty-six measures and conceivably in its complete state would have occupied approximately two-thirds the length of section 2 (52 measures). Adhering to the same proportions, section 4 may well have occupied approximately two-thirds the length of section 3, that is, approximately thirty-four measures.21
Butler’s conclusion therefore is that “approximately” another 6+34=40 bars
are needed to complete the fugue. The editors of the print decided to leave out
the last six bars of Bach’s manuscript, preferring to end the piece on the
Phrygian cadence of bars 232‐233, and so if those six bars (bars 233‐239) were to
be reinstated:
21 Butler "Ordering Problems in J S Bach's 'Art of Fugue' Resolved"
84
This would leave approximately forty-six measures for the concluding sixth page which is exactly the average number of measures per page for the first five pages as they appear presently in the print.22
Refinement of Butler’s Theory
Butler’s conclusion is immensely plausible and convincing, but it seems a
pity that it is so qualified by the repeated use of the word “approximately”. If it
is indeed true that the proportions of this fugue are governed by ratios of length
between the sections, as Butler suggests, then it seems more likely that such
ratios would be exact rather than approximate.
I am sure that Butler is right to observe that the sections of the fugue appear
to be in proportion to one another, and to extrapolate his observation to predict
the length of the unfinished sections. In Chapter 1 two other examples of Bach’s
music being constructed according to proportional ratios of the bar numbers
were noted.23 To recap briefly: in the Mass in B minor, the three movements of
the Kyrie (Kyrie I – Christe – Kyrie II) sit in a diminishing proportional
relationship to one another, a ratio of 1.4 existing between the relative lengths of
each movement 122 : 84.75 : 59. It was noted also that the last bar of the Christe
contains only three beats of music, or 0.75 of a bar, giving the precise length of
the piece at 84.75 bars, rather than 85 bars. Similarly it was observed that the
two Clavierübung settings of the chorale Aus der Tiefe also exist in a proportional
relationship of 1.4 when their bar numbers are accurately totalled.
Strictly speaking, the ratio involved in the Kyrie sequence is 1.43 rather than
1.40. It is certainly true that 1.43 can be rounded to 1.4; it is also true that the
exact ratio applies in both cases, i.e. 122/84.75 = 1.43 and 84.75/59 = 1.43.
Likewise, in the case of the Aus der Tiefe settings, the exact ratio is 1.36, a
number that can similarly be rounded to 1.4; but the exact ratio of 1.36 applies
in both cases, so the relative lengths of each piece must be deliberate. For the 22 Ibid. 23 See pages 11 – 12.
85
113 114 115
&c.
purposes of the present discussion it is the fact that the movements are in
precise proportional relationship to one another that is important, more
important than the appearance of the number 1.4 which may tempt a
suggestible observer into seeing 1.4 as a reference to the ‘Bach number’ 14. The
idea is alluring but, as I mentioned in Chapter 1, Bach was not likely to have
worked or thought in decimals.
Can we therefore find a precise proportional relationship between the
sections of Contrapunctus 14? I believe that we can if we look more carefully at
the structure of the existing portions of the fugue, keeping in mind what has
already been noted (in the context of the B minor Christe) about the importance
of observing precise lengths of bars, even down to fractions of bars.
The “approximate” nature of Butler’s prediction derives from the fact that he
has not been sufficiently precise in his analysis of the lengths of sections 1 and 2
of the fugue. In the passage quoted on page 83 above he states that section one
is “115 measures” long and section two is “78 measures”. However,
examination of the score reveals a subtly different total for each section.
Ex. 3.1: Contrapunctus 14, bars 113 ‐ 115
The crucial point is that there is an overlap between the end of section one
and the start of section two. Section one actually finishes on the first beat of bar
115: it is therefore accurate say that section one is not 115 bars long, but is in fact
114¼ (114.25) bars long.
Section two of the fugue begins with the entry of the second subject, seen
here in the alto starting on the second beat of bar 114. Before we can make an
86
192 193 194
&c.
accurate statement about the length of section two we must look at the end of
the section, shown here in Ex. 3.2:
Ex. 3.2: Contrapunctus 14, bars 192‐194
Section two ends half way through bar 193. The true length of section two
then is: ¾ of bar 114 + bars 115 to 192 + ½ of bar 193: a precise total of 79¼
(79.25) bars. Butler’s statement that section two is 78 bars long is not sufficiently
accurate.
I have thus established that the precise lengths of the first two sections of the
fugue are: section one at 114.25 bars; and section two at 79.25 bars.
These two sections are therefore in a proportional ratio of 1.4 – just like the
Kyrie sequence in the Mass in B minor and the two Aus der Tiefe chorale settings.
Once again one may be tempted to see the appearance of the Bach number, 14,
in this the 14th fugue of the collection. (I have noted elsewhere that the second
fugue subject consists of 41 notes, the gematric equivalent of J S BACH.24)
The ratio of 1.4 shown here is, however, a rounded figure, like the
Kyrie/Christe/Kyrie and Aus der Tiefe sequences. The exact ratio of these two
sections is 1.44. If Bach’s intention were that the fugue was to be constructed on
proportional ratios then it would be reasonable to expect the ratio to continue
between sections two and three, in the same way that it has operated between
sections one and two.
Taking the length of section two at 79.25 bars as the starting point, it can
therefore be assumed that 79.25 represents 1.44 times the length of a completed
24 See page 19.
87
section three.25 Performing this calculation results in a length for section three of
precisely 55.00 bars.
At the beginning of this chapter, and in Chapter 1, I drew attention to the
significance of the number 55 in Bach’s music, since it is the sum of 14 (BACH)
and 41 (J S BACH). Given that this third section of the fugue is built on the
BACH theme (B A C B ); that it is part of the 14th fugue in the collection; and
that it stands in a proportional relationship of 1.4 both to the section preceding
it (which is built on a subject of 41 notes) and (as I am about to show) to the
section following it, the discovery of the precise number 55 (14 + 41) as the
projected length of section three locks into place with (to borrow a phrase from
Tovey) “a correctness that cannot conceivably be accidental”.
Before moving on to a consideration of the length of section four and of the
entire fugue, there is one more point to make in support of my claim that
section three needs to be 55 bars long. Bach has left us 46 bars of section three
before the score breaks off at bar 239. The final seven bars contain the first
appearance of all three fugue subjects given in combination. It is reasonable to
assume that this one statement of the triple combination is not the only one
there should be at the end of the third section, before the fourth section of the
fugue begins with the appearance of the main Art of Fugue subject. For example,
in the two other fugues with three subjects, Contrapuncti 8 and 11, Bach does
not present only one combination of the first subjects before bringing in the
final subject. In Contrapunctus 8, the second section of the fugue (bars 39‐94)
presents the first two subjects in different combinations seven times before the
final subject appears. In Contrapunctus 11 also, the first two subjects are
presented in combination seven times between bars 89 and 146, which is where
25 Butler correctly notices that neither the first appearance of the triple combination (bar 233) nor Bach’s last written bar (239) represents the end of section three and the start of section four. In fact section three is not yet finished by bar 239 – a point not understood by Christoph Wolff or Donald Tovey. See Chapter 2 for a fuller discussion of this point.
88
all three subjects are presented simultaneously for the first time. One can
predict with confidence that at least one more permutation of the triple
combination would be presented at the end of section three of Contrapunctus
14 . Gregory Butler predicts that a further 6 bars are needed to complete section
three (see the quotation on page 83 above), bringing the total length of section
three to 52 bars. But the second fugue subject is seven bars long and that
therefore is the amount of space needed to present a triple combination: in
Butler’s scheme a second presentation of the triple combination would not be
possible and his section three would have to be completed with episodic
material.26 But according to my scheme – a total of 55 bars, i.e. 9 additional bars
– there is room for a brief episode (bar 239 and perhaps 240) to lead into a
second triple combination before section four begins with the appearance of the
main Art of Fugue subject.
Turning now to calculating the length of the final section of Contrapunctus
14: since the first three sections are in a proportional relationship of 1.44 to one
another, it follows that section four should follow the same pattern. If section
three’s 55 bars represent 1.44 times the length of section four, then section four
must be exactly 38 bars long.
We therefore have an additional 9 bars to section three (Bach’s 46 bars + 9 =
55) and 38 bars for section 4. 38 + 9 = 47. The exact number of bars needed to
complete Contrapunctus 14 is 47.
It will be noticed immediately that 47 bars would fit precisely onto the sixth
page allotted by the engraver.
In the light of my hypothesis that Bach left the work unfinished deliberately,
a sceptical observer might at this point ask why the engraver would have
allotted a page to something that Bach had no intention of supplying. No 26 That would not be unprecedented. Both the end of section one and the end of section two are episodes.
89
definitive answer can be offered to this point, except to say that all the evidence
suggests that there was considerable confusion about what Bach’s intentions
were, especially after his death when the engraving process was being
completed. It seems clear that Bach cannot have communicated his intentions
for Contrapunctus 14 to the engraver. Butler’s article (see the quotation on page
81 above) speculates that the engraver might have asked Bach for material to fill
out the collection or that manuscripts on Bach’s work table might have been
sent to the engraver by the family. Butler also says in the same passage that
Bach “…would never have included…an incomplete work”: this assumption
can now be read in a new light, i.e. that Bach intended Contrapunctus 14 to be
included in the collection, but, despite its truncated appearance, he did not
regard it as ‘incomplete’ because it is complete as it stands, at least as far as
Bach himself intended to take it. One possibility could even be that Bach might
have expected the sixth page to be left blank, as a further indication that the
music is to be completed by the student or performer, and this might be the
reason that the engraver allotted six pages to the engraving of a manuscript that
contained only enough music to fill five pages: but this is speculation, and there
is no evidence to support or refute it.
My claim that the total number of ‘missing’ bars is exactly 47 is backed up by
the fact that each of the four sections now stands in a proportional relationship
of 1.4 to its neighbours – quite appropriate in the 14th fugue, a fugue which
incorporates Bach’s own ‘signature tune’, the BACH subject. Further, the ratios
result in an exact number of 55 (14 + 41) for the BACH fugue itself (section
three). Lastly, the projected total fits precisely – not “approximately” – into the
space shown by Butler to have been originally made available for it by the
engravers.
90
The proportions of the fugue can be represented in diagrammatic form, as in
Fig. 3.2 below.
Fig. 3.2: the structure of a completed Contrapunctus 14
Since section two ends and section three begins half way through bar 192, the
total length of the fugue will be, strictly speaking, 192.5 + 55 + 38 = 285.5. If the
strictest possible interpretation of these totals is to be observed, then the last bar
will contain a minim (with a pause) and a minim rest, just like the last bar of
Contrapunctus 8 and Contrapunctus 10; it is possible that, if the last bar were to
be filled with a full semibreve and a pause (as might be expected at the
conclusion of a large cycle of pieces), the overall architectural proportions of the
fugue will not be significantly disturbed.27
As was mentioned earlier, Herbert Kellner’s theory was based on an alleged
midpoint, in which bar he claimed to see significance. Taking the correct total as
285.5 bars, as in the diagram and paragraph above, the midpoint bar is 143 – or,
to express it more exactly, 142.75. Two observations can be made about this bar.
First, the notes that are sounded ¾ of the way through bar 143 (i.e. on the fourth
beat: bar 142 + ¾) are F, A, C and D. The gematric conversion of these letters is
6, 1, 3 and 4, giving a total of 14 – the Bach number appearing at the precise
midpoint of the fugue according to my claim of the correct total number of bars.
27
91
Second, the alto part contains a hint of the BACH motif (B A C B ) –
admittedly one that is broken up by two other notes ‐ as can be seen in Ex. 3.3.
One might even go so far as to say that the midpoint of the fugue is surrounded
by the BACH motif:28
Ex. 3.3: bars 143‐4, showing a hint of the BACH motif, and the notes D, A, C and F (=14) sounding in the fourth beat
These correspondences seem to me to be convincing, but if further evidence
is needed I now present my explanation of the alteration Bach made to the
second page of his autograph manuscript.
Explanation of a “correction” in the manuscript
As mentioned in Chapter 2, in January 2006 I travelled to the Deutsche
Staatsbibliothek to examine at first hand Bach’s autograph manuscript. As part
of the discussion in Chapter 2 I stated that all but one of the various alterations
and corrections made by Bach in the autograph of Contrapunctus 14 are minor
– the majority of them are notes that were initially written as minims and then
28 Mention was made of numerical correspondences and symbolism in the Schübler Chorales in Chapter 1. The Schübler Chorales is another set where Bach can be seen to have signed his name with the BACH motif at the exact midpoint of the collection, namely bar 16-17 of the “Magnificat” Chorale Meine Seele erhebt den Herren BWV 648, in which the BACH motif can be seen in the left hand. Noticed and fully explained by Randolph Currie in Currie, R. N. (1973) "Cyclic unity in Bach's Sechs Chorale: a new look at the Chüblers [sic], Part I" Bach (Quarterly Journal of the Riemenschneider Bach Institute) 4(1): 26-38, quoted by John Bertalot in Bertalot, J. (2000) "Spirituality and symbolism in the music of J S Bach I" Organists' Review 86(3): 222-225 As a further link between the Schübler Chorales and The Art of Fugue, it may be worth mentioning that The Art of Fugue was engraved by Johann Heinrich Schübler, the younger brother of Johann Georg Schübler who was the publisher of the Schübler Chorales. This fact is mentioned by Davitt Moroney in his Critical Commentary as being the finding of Wolfgang Wiemer (“Die wiederhergestellte Ordnung in Johann Sebastian Bachs Kunst der Fuge: Untersuchungen am Originaldruck” Wiesbaden, Breitkopf & Härtel, 1977). The Specificatio of Bach’s estate includes a ‘liability’ of 2 thalers and 16 groschen to “Mr Schübler”, which is presumably a payment for the engraving of the last few plates that were engraved in the second half of 1750 after Bach’s death. The Specification of the Estate can be seen in translation on page 255 of David, Mendel, et al. Since Bach included his BACH motif at the midpoint of one work published by Schübler, it is not fanciful to show that he intended to include it in another.
143
B A
C H
&c.
92
‘filled in’ to be crotchets, or other similar alterations to note values; I mentioned
that there is only one major compositional change and that is towards the
bottom of the second page, where Bach has crossed out two bars and notated
three new ones in tablature in the bottom margin of the page. This alteration
can be seen on the fourth system of the second page of the facsimile given in
Appendix 1 and for convenience I give a detail of it here in Fig. 3.3:
Fig. 3.3: Contrapunctus 14 (autograph, page 2), showing the original bars 111 and 112 crossed out and three replacement bars notated in the bottom margin
In the presence of the original manuscript one can see clearly what are the
notes that Bach originally wrote before deciding to cross them out: I have
transcribed them directly from the original and they are given first in Ex. 3.4
overleaf, with the right hand part in Bach’s original soprano clef, and then in
Ex. 3.5 the same passage is given with the right hand part transposed into the
treble clef for ease of reading:
93
Ex. 3.4: Contrapunctus 14 (autograph, page 2), showing the original music of bars 111 and 112 before Bach crossed them out
Ex. 3.5: The same passage using treble and bass clefs
These two bars (111 and 112 in Figs. 3.4 and 3.5 above) were replaced by the
three new bars notated in the margin, giving the final reading that was used in
the 1751 print and all subsequent editions, as seen in Fig. 3.6 in which bars 111‐
113 are a transcription of the tablature notation in the margin:
Ex. 3.6: the three new bars (111, 112 and 113 in this example) that replaced bars 111 and 112 in the previous examples; these three bars are a transcription of the tablature notation in the margin.
It seems a curious and puzzling change for Bach to have made, and it does
not seem easy to find a musical reason for the rejection of the original bars 111
and 112. If Bach had not made the change, but had left the score with those two
bars unaltered, would scholars and analysts today point to them and identify
them as a weak moment? Perhaps it might be argued that the new bar 111
110 111 112 113 114
&c.
110 111 112 113 114
&c.
110 111 112 113 114 115
94
strengthens the approach to the following cadence (bars 113‐114) by the move
to the subdominant that was not present in the original; and it might be argued
that the new bar 113 is rhythmically very slightly more interesting than the old
bar 112. But these are tiny points of argument and I am convinced that if the
change had not been made, nobody would ever have criticized the old bars 111
and 112 as being weak.
My analysis of this alteration – something that nobody else has done before –
is based on two crucial observations about this passage. First, the change has
been made not to improve the notes, but to replace two bars with three bars.
Second, the location of the change is significant: it is right at the point of the end
of the first section of the fugue and the start of the second section. The ink and
the handwriting at this place in the manuscript give no indication that these
changes were made subsequent to the writing out of the music itself: it seems
clear that the alteration was made at the same time as the rest of the manuscript
was written. (I have shown in Chapter 2 that the more significant alterations
Bach made to the main part of P200 are in a lighter ink and appear to have been
made at a later time than the writing of the manuscript itself.)
The replacement of two bars with three new ones, at the exact point of the
transition between the end of section one and the start of section two, strongly
corroborates my theory that the precise length of each section of the fugue is of
vital proportional importance. Since there is no powerful musical reason for the
change (the new music is scarcely an improvement on the old), the only
remaining possible reason for the change must be a structural one. Bach has
lengthened the first section of the fugue by one bar because he realized that it
needed to be longer by one bar in order for his scheme of strictly diminishing
proportional ratios to work properly.
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A further possibility is that – since the notation of the original music (the
original bars 111 and 112) bears no sign of hesitation or mistake of any kind and
gives every appearance of having been written as strongly, neatly and
confidently as all the music before and after it – Bach has done this deliberately:
not as an afterthought, but as a clue to the alert reader that the length of each
section of the music provides the solution to the question of how much music is
necessary to bring the unfinished fugue to completion.
A further possible corroboration of the number 47 as the number of bars
needed to complete the fugue, and of the claim that Bach has left coded
information as a clue to that number, lies in the unusual markings at the very
end of the manuscript.
Markings on the last page of the autograph: a possible numerological interpretation
One aspect of the last page of Bach’s manuscript that appears to have
escaped the attention of all those who claim to have made a close study of this
work is the strange markings at the bottom left and bottom right corners of the
page. So much attention has been paid to the ruling of the staves, the number of
bars on preceding pages, the supposed existence of ‘fragment x’ and the many
other anomalies that I have mentioned that it seems extraordinary that nobody
has noticed these odd additions to the page. These ‘doodles’, for want of a
better word, can be seen at the bottom of the page:
96
Fig. 3.4: Beilage 3, page 5, showing ‘doodles’ at the bottom corners of the page (see also the Frontispiece to this thesis for a clearer reproduction)
These markings are shown in closer detail here:
Fig. 3.5: bottom left Fig. 3.6: bottom right
Clearly the marking at the left is some kind of indefinable ‘squiggle’, while
that at the right is clearly a succession of notes, of which the first four are
semiquavers. To the immediate left of the notes is what appears to be a rest, or
it may be a blot: it remains ambiguous even on the closest possible examination
of the original. These markings are represented more clearly here:
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Ex. 3.7: graphic representation of the markings
Do these markings have any meaning, and if so what might it be? Bach was
not in the habit of littering his scores with meaningless doodles. The outline of
the notes bears no relation to any thematic material in Contrapunctus 14;
neither can anything be divined from the rhythmic value of the notes or of the
rest. What is to be made of the squiggle at the left? It is not easy to say what it
is: but it can be said with confidence what it is not. It is not a clef – and that is
perhaps the clue to the interpretation of the notes. If we do not know the clef,
we cannot name the notes. The left‐hand marking occupies the position that a
clef would occupy, and yet it is not a clef; therefore the letter names of the notes
written further along the same stave cannot be identified. Perhaps this is
precisely the point that Bach is making by writing these markings at the very
end of his score of Contrapunctus 14. The notes have meaning, but Bach has
disguised it. I believe that these markings are a coded clue to a number – the
number of bars required to complete the fugue.
Gematria applied to the markings
Since the notes at the bottom right‐hand corner of the page bear no thematic
or rhythmic relation to the music of Contrapunctus 14, it is clear that they must
provide some other function. Bach has written a squiggle where he could have
written a clef, so the letter names of the notes are not immediately obvious. By
replacing the left‐hand squiggle with different clefs, we can assign letter names
to the notes at the right.
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If the clef is taken as a bass clef, then the notes are A C E G C. If we use a
soprano clef the notes are D F A C F. If we use a treble clef then the result is F
A C E A; and so on. If Bach is giving us a numerical clue, then we can assign
the corresponding gematric numbers to each letter name and come up with a
total.
D F A C F Soprano clef
4 6 1 3 6 = 20
F A C E A Treble clef
6 1 3 5 1 = 16
G B D F B Alto clef
7 2 4 6 2 = 21
E G B D G Tenor clef
5 7 2 4 7 = 43
A C E G C Bass clef
1 3 5 7 3 = 19
This appears to be of little help. None of these numbers suggest anything
significant or helpful. If, however, we take note of the fact that, for some reason,
Bach has included a sharp sign before one of the notes, then remembering that
in German notation a sharp is described by the suffix “‐is”, a different result is
yielded. Thus G becomes G I S; C becomes C I S; E becomes E I S.
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F A C E IS A Treble clef
6 1 3 5 9 18 1 = 43
G B D F IS B Alto clef
7 2 4 6 9 18 2 = 48
E G B D IS G Tenor clef
5 7 2 4 9 18 7 = 52
A C E G IS C Bass clef
1 3 5 7 9 18 3 = 55
The interesting number appears when we apply this to the soprano clef –
perhaps the most obvious clef to use, since Bach’s score of Contrapunctus 14 is
written using that clef.
D F A C I S F Soprano clef
4 6 1 3 9 18 6 = 47
Thus a gematric application of the ‘doodled’ notes using a soprano clef
results in the number 47 – the same number that I have arrived at by means of
the calculation of strict proportional ratios, and that was allowed for by the
printer.
At this point it would be prudent to be reminded of the note of caution
sounded by John Butt in the passage quoted on page 7: “there is a tendency to
interpret everything in sight … the author selects data…with no consistency
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which is externally imposed.” It is true that, as I have shown, if indeed a
numerological interpretation of the markings is intended, there are several
possible interpretations. There is in fact no irrefutable proof that the markings
are even in Bach’s own hand (though my own observation of the original
manuscript suggests that every aspect of the ink – its colour and the weight and
pressure of the strokes – is identical to the rest of the writing on that page).
Nevertheless, the facts that these markings occur at the very end of the
manuscript; that they appear to have been written at the same time and in the
same ink as the rest of the music on that page; and that an interpretation of
them using the same (soprano) clef as the clef in which the music is written
yields precisely the same number as not only the number yielded by the
extrapolation of the proportions of the four sections of the fugue, but also the
exact number of bars allocated by the engraver, may cumulatively suggest that
my interpretation of the markings is correct.
Corroborative examples
The Mass in B minor
The claim that Bach has left us a coded or cryptic clue to a significant number
might seem stronger if it can be shown that he has done something similar
elsewhere. There is such an example, in the manuscript of the Mass in B minor.
In the chorus Patrem omnipotentem, part of the Credo, can be seen a little number
84 alongside the conclusion of the chorus, immediately after the final bar on the
bottom stave. This is shown in Fig. 3.7 overleaf. At the beginning of this chorus
Bach has the choir reiterate the words Credo in unum Deum, which he has
already set in the preceding chorus and which strictly speaking do not belong
in this one. The result of this is that in this chorus there are now 14 words (that
number again!) and 84 letters in this movement. Furthermore, there are 84 bars of
music in this chorus. Among other correspondences and gematric claims about
this number, Friedrich Smend points out that 41 + 43 = 84 [(J+S+B+A+C+H) +
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(C+R+E+D+O) = 84] – thereby further explaining the reintroduction of the
words from the previous chorus: perhaps another example of Bach associating
himself through numbers with the sentiments expressed by the words.29 Again,
29 Smend’s observation is quoted in Tatlow, op. cit. The suggestion is that by means of these numbers Bach is embedding in his music – a setting of part of the Creed – a statement that “I, J S Bach, believe”. Smend does not appear to notice the fact that there are now 14 words in the chorus.
Fig. 3.7: The manuscript of the “Patrem omnipotentem” chorus from the Mass in B Minor, showing the number 84 written at the end of the score
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it is necessary to be cautious. It has not been proved that this number 84 is in
Bach’s own hand: it may even have been written by a later copyist, counting the
bars of the chorus to plan his own copying (though there are no other similar
numbers written in the manuscript of the Mass in B minor). It does however
seem to be the case that, whether it was written by Bach himself or not, this
number 84 refers to more than just the number of bars in the chorus, and it is
not impossible that it is a clue to the observant reader that the number is
significant in the chorus. One who notices the number written at the end may
go searching in the music for examples of its appearance, and in this case they
are there to be found. The parallel between this example and the cryptic notes at
the end of the last page of Contrapunctus 14 is clear: it is just that in the latter
case Bach has added a further level of ‘coding’ to his clue, rather than simply
writing the number 47.
The title page of The Well‐Tempered Clavier
In a recent article, Dr Bradley Lehman of Virginia, USA chronicles his
discovery of the code to Bach’s tuning system.30 The code is hidden in an
apparently irrelevant looping line at the top of the title page of The Well‐
Tempered Clavier, a line which for over two centuries had either not been noticed
by anyone or had been regarded as nothing more than a flourishing artistic
decorative line. The title page is shown in Fig. 3.8 and a detail of the line can be
seen in Fig. 3.9.
30 Lehman, B. (2005) "Bach's extraordinary temperament: Our Rosetta Stone" Early Music 33(1): 3-23
103
Fig. 3.8: The title page of the Well‐Tempered Clavier
Fig. 3.9: detail of the loops at the top of the page31
The clue to the existence of a hidden code is the letter C that can be seen in
the last complete loop at the right of the drawing. Dr Lehman discovered that
by turning the drawing upside down, the sequence of loops and the number of
revolutions in each loop can be seen to correspond to the details of Bach’s
specific method for tuning keyboards. 31 Images from Lehman, B. (2005) <www.larips.com> (web site), reproduced by kind permission of Dr Lehman
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Fig. 3.10: the drawing turned upside down, showing correspondence with Bach’s tuning system32
Dr Lehman’s discovery and his method of decoding the diagram are
described in detail in his fascinating article, as is the tuning system that results
from interpreting the drawing; these details are very technical and are
principally concerned with the minutiae of baroque keyboard tuning, which is
not our subject here. For present purposes it is enough to give this as a very
powerful example of Bach’s habit of hiding something of tremendous
significance in a marking on the page that may easily escape notice and which
may appear at first glance to be both innocuous and irrelevant. The preceding
sentence could just as easily have been written about the markings at the
bottom of the last page of Contrapunctus 14.
The Musical Offering: “Quaerendo invenietis” and the Houdemann Canon BWV 1074
Though it is known today from its dedication as the Musical Offering BWV
1079, Bach’s collection of pieces (based, like The Art of Fugue, entirely on one
subject) was originally titled by him Regis Jussu Cantio Et Reliqua Canonica Arte
Resoluta. This is an acrostic, spelling the word RICERCAR, an old‐fashioned
name for a fugue, and each of the fugues in the Musical Offering is called
Ricercar. This is one example of a double meaning being disguised, but an even
more interesting one comes in the form of the ninth of the ten canons. This
extraordinary fragment of music is designed to be read by one player reading in
the alto clef with the page the right way up, followed at some distance of time
by another player reading from the bass clef with the page upside down. But
32 Ibid., reproduced by kind permission of Dr Lehman
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Bach has not specified the time interval, and the only clue is the words
“quaerendo invenietis” written over the top.33
Fig. 3.11: Musical Offering, Canon No 9 “quaerendo invenietis” Graphic © 1996, Timothy A. Smith, used by permission34
In his website, Dr Timothy Smith of Northern Arizona University gives one
possible realisation of this cryptic canon (Ex. 3.12)35:
Fig. 3.12: one possible realisation of Canon No. 9 Graphic © 1996, Timothy A. Smith, used by permission36
Here again, then, is an example of Bach giving only the barest possible clue
to the realisation of his music, leaving it to the reader to work out what he
intends from his coded, cryptic information. Surely the injunction “quaerendo
33 The information in this paragraph, together with the images in Ex 3.11 and 3.12, comes from Smith, T. (1996) The Canons and Fugues of J S Bach <http://jan.ucc.nau.edu/~tas3/bachindex.html>, accessed 10 December 2005 34 Graphic from <http://jan.ucc.nau.edu/~tas3/musoffcan9real.gif> 35 Smith The Canons and Fugues of J S Bach <http://jan.ucc.nau.edu/~tas3/bachindex.html> 36 Graphic from <http://jan.ucc.nau.edu/~tas3/realcan9mo.html>
106
invenietis” – seek and ye shall find – is no less appropriate to the conundrum of
the unfinished quadruple fugue than it is here.
The “Houdemann” Canon BWV 1074, possibly the most famous of Bach’s
compositions in his lifetime, is a similar example. Once again, only the most
minimal clues, here in the form of clefs and key signatures written upside down
and on different parts of the stave, are given to help the reader work out the
realisation of the puzzle canon. Some of the possible solutions to this canon are
explained in an accessible style in the recent book “Evening in the Palace of
Reason” by James Gaines.37
Fig. 3.13: puzzle canon BWV 1074
Conclusion
The correct number of bars needed to complete Contrapunctus 14 is now
definitively established as 47. This number has been arrived at by following
Gregory Butler’s lead but by analysing the proportional relationships of the
fugue more precisely than he did. This analysis has resulted in a number of
correspondences that are unlikely to be coincidental, including the facts that the
‘Bach number’ 14 appears as the ratio of proportion in all four sections of the
37 Gaines, J. R. (2005) Evening in the Palace of Reason : Bach meets Frederick the Great in the age of Enlightenment London, Fourth Estate
107
fugue; that the application of the precise ratios results in a total length of exactly
55 (14 + 41) bars for the third, ‘BACH’, section; that this length allows room for
a second presentation of the triple combination at the end of the third section;
that the overall length of the fugue fits precisely into the space allotted for it by
the engraver; and that it fits with an artistic understanding of a correct sense of
the placement of climax.
As corroborative evidence for this claim, I have said that the only significant
alteration made by Bach to his autograph manuscript of this fugue was made
specifically to lengthen the first section of the fugue by one bar, showing that
the exact length of the sections of the fugue is important; further, Bach’s
markings at the end of Beilage 3 may be a clue to the fact that there is a number
of significance to be discovered. In this case, however, Bach has disguised the
clue: rather than simply writing a number 47 at the bottom of the page he has
written a few notes of no apparent relevance to the music, and one of them has
a sharp sign attached to it; these notes have no clef, only an indeterminate
squiggle. A gematric totalling of the names of these notes under the same clef
that the music is written in, allowing also for the sharp sign, produces the
number 47 – the exact number that is consistent with the diminishing
proportional ratio of each section of the fugue and which fits precisely onto the
engraver’s page. It may be an indication that Bach is telling us in his own
oblique way that 47 bars are needed to bring the fugue to a proper conclusion.
Lastly, I have drawn further parallels to places where Bach has similarly
disguised his meaning in a coded and cryptic way, strengthening the evidence
in favour of my claim that not only are the markings at the end of the score a
coded clue to the number 47, but also that Bach has used similar coded clues
elsewhere.
I
II
III
Chapter Four
QUESTIONS RAISED BY THE OBITUARY, AND NEW SOLUTIONS TO THE PROBLEM OF THE INVERTED
COMBINATIONS Having established the length of the missing final section of Contrapunctus
14 as 47 bars, it is now possible to reconsider what the content of those 47 bars
was intended to be.
Contrapunctus 14 is a quadruple fugue – a fugue with four subjects. Only the
first three subjects are present in the extant portion of the fugue. During the
fugue as Bach has left it these four subjects are presented both singly and in
combination with one another. The first three subjects of Contrapunctus 14 are
given here:
Ex. 4.1: The three subjects of Contrapunctus 14
The first subject is given its own exposition on a grand scale for the first 114¼
bars. The order of voices in which the subject enters is BTAS. During this first
section of the fugue it appears sixteen times in its rectus form; and six or seven
times in its inversus form, depending on whether the soprano in bars 98‐102
counts as a disguised statement of the inversus subject. This subject also
appears in stretto with itself in a variety of permutations, including stretti that
include simultaneous use of the rectus and inversus forms.
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233
The second subject is given its own exposition starting on the second beat of
bar 114; this second section of the fugue runs through until half way through
bar 193. The order of voices in which the subject enters in the exposition is
ASBT. Subject II appears eight times in its rectus form and none in its inversus
form. There are no stretti but there are four combinations with the first subject,
the last of which (bars180‐186) is a three‐part combination with subject II in the
bass and a stretto of two rectus statements of subject I above it.
In the third section of the fugue, starting half way through bar 192, subject III
is given its own exposition, with ten statements of the rectus form and two
statements of the inversus form. The order of voices in which the subject enters
in the exposition is TASB. There are several instances of subject III being given
in stretto with itself.
Only at bar 233 do all three subjects combine together for the first and only
time, in the arrangement ()‐II‐III‐I [i.e. ()=free material in the S; subject II in the
A; subject III in the T; subject I in the B. This convention for describing the
layering of the subjects will be followed in the rest of this chapter.]
Ex. 4.2 The triple combination at bar 233
It was not until 1881 that Gustave Nottebohm discovered that the main Art of
Fugue subject is able to be combined with the first three subjects, and that it
must therefore have been the fourth subject of this quadruple fugue.151
151 Nottebohm, op. cit.
110
229
(A)
(S)
Ex. 4.3: The quadruple combination as given by Nottebohm
In a 1998 article Professor Walter Corten persuasively claims to see the ghost
of the main Art of Fugue subject in the soprano part of Bach’s last bars:152
Ex. 4.4: bars 229‐238, soprano voice, following the outline of the main Art of Fugue subject
Since this outline of the main Art of Fugue subject appears above the triple
combination that is given explicitly, it could plausibly be interpreted as a hint
from the composer that the main subject does in fact fit in combination with the
other three subjects.153
Nottebohm’s discovery ties in with two other sources of information about
Bach’s intentions for the remainder of the fugue: The Nekrolog or Obituary, and
J N Forkel’s 1802 biography of the composer.
152 Corten 153 As always in these apparent correspondences, one should approach them with a healthy dose of scepticism and caution. The curious discovery that the subject of the D minor organ fugue BWV 565 happens to lock into place precisely with the main Art of Fugue subject does not necessarily mean that BWV 565 is to be combined with the existing three subjects of Contrapunctus 14!
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The Obituary appeared in the last ever issue (Vol. IV, Part I, 1754) of Lorenz
Mizler’s musical journal Lorenz Mizlers Musicalische Bibliothek, under the title
“Monument to Three Late Members of the Musical Society” — the last member
being Bach. The author is uncredited but C P E Bach later told Forkel that he
had written it with the help of Sebastian’s pupil J F Agricola. Though not
published until 1754, the Obituary was written in the latter half of 1750, within
months of Bach’s death.154 The New Bach Reader calls it “the richest and most
trustworthy early source of information on Bach’s career.”155
Describing The Art of Fugue, the Obituary has this to say:
This is the last work of the author, which contains all sorts of counterpoints and canons, on a single principal subject. His last illness prevented him from completing his project of bringing the next-to-last fugue to completion and working out the last one, which was to contain four themes and to have been afterward inverted note for note in all four voices. This work saw the light of day only after the death of the late author.156
When Forkel comes to mention this in his Biography of Bach, he says almost
the same thing.
The last fugue but one has three themes; in the third, the composer reveals his name by B A C H {=B ACB }. This fugue was, however, interrupted by the disorder in the author’s eyes, and as the operation did not succeed, was not finished. It is said to have been his intention to take in the last fugue four themes, to invert them in all the four parts, and thus to close his great work.157
Forkel also gives a little more information about Bach’s illness:
The indefatigable diligence with which, particularly in his younger years, he had frequently passed days and nights, without intermission, in the study of his art, had weakened his sight. This weakness continually increased in his latter years till at length it brought on a very painful disorder in the eyes. By the advice of some friends who placed great confidence in the ability of an oculist who had arrived at
154 Even though the Obituary was not published until 1754, Mizler is known to have received a copy of it in Warsaw in March 1751, so it must have been written by then. David, Mendel, et al., page 297, note 28 155 Ibid., page 297 156 Ibid., page 304. The original German text (not including the first sentence of the above quotation) can be seen in Chapter 2, footnote 16 (page 28 above.) 157 Ibid., page 466
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Leipzig from England, he ventured to submit to an operation, which twice failed. Not only was his sight now wholly lost, but his constitution, which had been hitherto so vigorous, was quite undermined by the use of perhaps noxious medicines in connection with the operation. He continued to decline for full half a year till he expired on the evening of the 30th July, 1750, in the 66th year of his age. On the morning of the tenth day before his death, he was all at once able to see again and to bear the light. But a few hours afterwards he was seized with an apoplectic fit; this was followed by a high fever, which his enfeebled frame, notwithstanding all possible medical aid, was unable to resist.158
Possibly Forkel is expanding on another, more contemporary source, namely
Marpurg’s preface to the second edition of The Art of Fugue. As was described in
Chapter 2 above, the work was first published in 1751 with a short preface. It
was reissued in 1752, using the same plates for the music but now with a longer
preface by Marpurg which contains these sentences:
Nothing could be more regrettable than that, through his eye disease, and his death shortly thereafter, he was prevented from finishing and publishing the work himself. His illness surprised him in the midst of the working out of the last fugue, in which with the introduction of the third subject, he identifies himself by name.159
Another contemporary source of information about The Art of Fugue is the
newspaper advertisement placed in May 1751 by C P E Bach to promote sales of
the new printed edition of the work. There is a brief reference to the composer’s
eyesight and the ‘deathbed’ Chorale, but it says nothing about the unfinished
fugue or Bach’s supposed intentions for the completion.160
It is worth looking closely at the wording of the Obituary and of Forkel’s
Biography in an attempt to determine how accurate it is and whether it can be
relied upon in respect of the question of the combinations of the four subjects.
158 Ibid., page 430 159 Ibid., page 376 160 The full text of the advertisement can be seen in Ibid., page 256.
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“This is the last work of the author…”
This statement is not strictly true. It is well known that the Chorale Prelude
Vor deinen Thron tret’ ich hiermit BWV 668a (a revision of Wenn wir in höchsten
Nöten sein BWV 668) was Bach’s last composition, almost certainly dictated on
his deathbed some time in the last ten days of his life, between 18 and 28 July
1750. Most texts tell us that J C Altnikol, Bach’s son‐in‐law, was the amanuensis,
but Wolff says that Johann Nathanael Bammler and Bach’s boarding pupil
Johann Gottfried Müthel are also possible candidates.161
Yoshitake Kobayashi, the principal scholar of Bach’s manuscripts and
handwriting, also lists the late version of the Sonatas for violin and harpsichord
BWV 1014‐1019 as having been written after the unsuccessful cataract
operations and he dates them specifically as coming from after March 1750,
whereas he lists the unfinished fugue (Contrapunctus 14) as having been
composed after August 1748 and before October 1749. During this period he
also lists the completion of the Mass in B minor (from the Credo to the end); the
flute sonata BWV 1031; and copies in Bach’s hand of a motet by J C F Bach and
Handel’s Brockes Passion.162 According to Schweitzer, Bach also worked during
his illness on the revision of the Eighteen Chorales for organ and made a “new
and improved fair copy” of the Canonic Variations.163
Having shown that Contrapunctus 14 was not strictly speaking Bach’s final
work, it is nonetheless fair to point out that the authors of the Obituary should
not be blamed for making the assumption that it was. Presented with an
unfinished score that clearly dated from the composer’s final period – the only
unfinished score of that last period – it was not unreasonable of them to come to 161 Wolff Johann Sebastian Bach : The Learned Musician, page 508, note 81 162 Herz, G. (1990) "Yoshitake Kobayashi's Article, 'On the Chronology of the Last Phase of Bach's Work - Compositions and Performances: 1736 to 1750' - An Analysis with Translated Portions of the Original Text" Bach (Quarterly Journal of the Riemenschneider Bach Institute) XXI(1): 3-26. Based on Kobayashi, Y. (1988) "Zur Chronologie der Spätwerke Johann Sebastian Bachs. Kompositions- und Aufführungstätigkeit von 1736 bis 1750" Bach-Jahrbuch 74: 7-72 163 Schweitzer, A. (1923) J.S. Bach London, Adam & Charles Black, Vol. 1, page 223
114
the conclusion that it was the last piece he had been working on and that the
reason it was unfinished was because he became too ill to complete it. The
inaccuracy of the statement that it was “the last work of the author” does not
necessarily invalidate the veracity of the Obituary or of the other statements
that it makes about The Art of Fugue. However, the next phrase that the authors
used also needs to be examined carefully.
“His last illness prevented him from completing…”
This comment in the Obituary can be taken in conjunction with the
statements by Forkel and Marpurg quoted above about Bach’s declining
eyesight and his associated health problems. In Christoph Wolff’s book a
chapter entitled “The End” chronicles as much as is known about Bach’s
condition in the last year of his life.164 On 4 April 1749 Bach was well enough to
perform his St John Passion, but by May 6 he issued a receipt in which he put
only his signature on the document, the main text of it having been written by
one of his sons; this may indicate that he would have struggled to write the
whole thing out himself. In June a letter from the Prime Minister of Saxony, no
less, predicted Bach’s imminent demise and attempted to put forward his (the
Prime Minister’s) own personal musician for appointment as Kantor of the
Thomaskirche; though the exact nature of Bach’s illness at this point is
unknown, it must have been serious to provoke such actions from the Prime
Minister and from the town council, who immediately (and somewhat
insultingly to Bach) arranged auditions for the Prime Minister’s candidate even
while Bach was still very much alive and still in office. But Bach clearly
recovered from whatever this illness was, for by 25 August he was performing
his Cantata BWV 29 for the annual council election service, a Cantata whose
showy obbligato organ part in the Sinfonia and in one of the arias was in all
probability played by Bach himself. The handwriting of letters signed by Bach
164 Wolff Johann Sebastian Bach : The Learned Musician, pages 442-451
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in December seems to indicate that writing was giving him trouble, leading
Wolff to conclude that Contrapunctus 14 must have been written “no later than
the first weeks of 1750” – perhaps narrowing down the timeframe set for this by
Kobayashi. Wolff goes further to say that “some limited and unstressful work
may have continued until about Easter 1750”. (Easter fell on 29 March in 1750.)
The English oculist Sir John Taylor, who was later to operate on Handel’s eyes,
performed the first operation on Bach’s eyes between 28 March and 1 April
1750; a second operation took place between 5 April and 8 April. It seems that
Bach went downhill from here and was “almost continually ill” not for “full
half a year” as stated in Forkel’s Biography [n.b. not the Obituary as Wolff
erroneously has it165] but for a quarter of a year. Nevertheless, he was obviously
well enough to accept a new pupil, Johann Gottfried Müthel (mentioned above
in connection with the dictation of the ‘deathbed’ Chorale), who moved in to
Bach’s house on 4 May. Finally, we know from Forkel that Bach had a
temporary recovery of his sight ten days before he died and that he was able to
dictate the ‘deathbed’ Chorale, even though it was a revision of an existing
piece under a new title, not a new composition, and therefore perhaps not a
very onerous or taxing project.166
From the foregoing we can conclude that Bach may have been unable to
work between May/June 1749 and August 1749. After recovering, he was
working, perhaps with some difficulty, between August 1749 and the first
weeks of 1750, perhaps until the end of March 1750. From April 1750 he was in
serious decline but was nonetheless well enough to take in a new boarding
pupil at the beginning of May, and there is no evidence that he was completely
incapacitated, though Forkel states that he was completely blind after the
165 Ibid., page 449 166 Having said that it may not have been very onerous or taxing, it is important to point out that Vor deinen Thron is very elaborate in terms of its numerological content, with several appearances of the numbers 14 and 41, some which are discussed in Chapter 1; and it is a most tightly constructed contrapuntal complex. So even though it is only a short piece, Bach’s mental faculties must have been in very good order, even if his physical abilities were weak at this time.
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second operation. Finally, we know that in his last days he was well enough to
dictate music because he obviously felt that it was important to him to get this
particular piece down on paper before he died.
In Chapter 2 above it was noted that that Douglass Seaton suggests an early
date for the manuscript of Contrapunctus 14; Wolff disagrees with him, but
even he states that “the composing manuscript … dates from 1748‐1749” (my
italics).167 And I have already mentioned that Kobayashi places it before October
1749, and after August 1748.
The statement that Bach’s illness and death are the reason that the fugue is
unfinished does not stand up to scrutiny. It is surprising, to say the least, that so
many scholars have accepted the statement at face value without looking at the
dates involved. According to the information presented here, Contrapunctus 14
could have been written between August 1748 and June 1749. It is worth
stressing that he could, therefore, have been working on the fugue as early as
one year and eleven months – almost two years – before his death. Perhaps it is
worth reminding ourselves at this point of Seaton’s statement, and my own
direct observation of the original, that the manuscript is in a strong calligraphic
hand – contrasting with the shaky handwriting in the documents described by
Wolff that come from 1749 and 1750.168 If he had not worked on it during this
period (August 1748 – June 1749), or if he had begun it but not completed it, he
had a further opportunity between August 1749 and the beginning of 1750,
perhaps as late as Easter (29 March). Further, if he were well enough to take on
a new pupil in May 1750 – after the two gruesome eye operations – and to teach
that pupil, even if he was blind, surely he would have been well enough to
arrange for Contrapunctus 14 to have been completed, perhaps by dictation, if
he had wished to do so. Indeed, the Obituary explicitly states that it was his 167 Wolff Johann Sebastian Bach : The Learned Musician, page 436 168 See Chapter 3 for a full discussion of the handwriting of the manuscript, including the debate about whether it is a composing manuscript or not.
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“last” illness that prevented him from completing the work, which suggests that
the work was extant but not complete before his “last” illness. This suggests in
turn that the period August 1749 – early 1750 may be more likely than the
earlier possible period of August 1748 – June 1749, since we know that Bach’s
previous illness was between June and August 1749. It seems clear that
Contrapunctus 14 must have been brought to its extant state (239 bars) by the
time of the eye operations or shortly thereafter at the very latest: around
March/April/May 1750. How plausible is it therefore that Bach was unable to
finish the ending of this great cycle of pieces, when I have shown that the
chronology strongly suggests that he would have been capable of finishing a
short passage of music? (I have shown in Chapter 3 that 47 bars are all that
remain.) The ‘deathbed’ Chorale is 45 bars long, and Bach was, as we know,
capable of dictating that much in his last few days, some time between 18 and
28 July.
It is clear that Bach would have had the time to finish the fugue if he had
chosen to do so, and that his illness and death can no longer be offered as
reasons for its incompletess. This lends further weight to my hypothesis that
Bach chose to leave the work as it is.
“the next‐to‐last fugue”
This has been discussed in more detail on page 33 above. The phrase “next‐
to‐last fugue” led some writers, including Sir Donald Tovey, to conclude that
Contrapunctus 14 was the penultimate fugue in the collection, and that it was to
be followed by another fugue with four subjects that would be invertible.
Because the exact words of the Obituary are “the last [fugue] which was … to
have been afterward inverted note for note in all four voices”, Tovey assumed
that this further fugue was to be invertible in its entirety (not just its subjects)
after the manner of the two Mirror Fugues (Contrapuncti 12 and 13). Tovey
composed such a fugue, using the BACH motif as one of the subjects. But, as
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was discussed in detail in Chapter 2 (page 33), this was a misunderstanding of
the meaning of the words in the Obituary; a correct understanding was to treat
each section of Contrapunctus 14 as if it were a fugue in its own right.
The passage in Forkel’s Biography is obviously derived from the Obituary,
since it too speaks of “the last fugue but one” being unfinished, and then goes
on to mention “the last fugue” which was to have four “themes” that would be
inverted in all the parts. In this case the wording specifically says that it was the
themes that were to have been inverted, not the entire fugue as a whole as the
Obituary seemed to imply.
But Forkel had clearly failed to take note of Marpurg’s 1752 Preface to the
second printed edition of the work, quoted above, in which Marpurg
specifically mentions “the last fugue, in which, with the introduction of the third
subject, he identifies himself by name” (my italics), making it clear that
Contrapunctus 14 was in fact the last fugue of the collection and not the
penultimate one.
Once again, we see a slight confusion in the understanding of C P E Bach,
Agricola and Forkel in these two passages (i.e. the Obituary and the Biography).
This was not strictly speaking Bach’s last work; neither can they be correct
when they say that illness was what prevented him from finishing it; and now
we see that they did not realize properly that Contrapunctus 14 was indeed the
last fugue of the collection. Marpurg knew Bach well and they corresponded
with each other; since Marpurg was invited to contribute the preface to the
second printed edition, it seems more likely than not that he had a clearer
understanding of Bach’s intentions, perhaps communicated to him directly
either in conversation or in correspondence.
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“the last one was to contain four themes…”
It has already been mentioned and is by now well known that the main Art of
Fugue subject is the fourth theme or subject of the quadruple fugue, and that it
would have made its appearance at some point in the remaining portion of the
composition. I showed in Chapter 3 that 9 bars are needed to complete section 3
of the fugue and that the fourth section would begin with the appearance of this
fourth subject immediately thereafter i.e. at bar 249. Tovey stated (certainly
correctly) that the fourth subject would not be given its own full exposition,
basing this opinion on the dramatic pacing of the music. Since I have shown
that 38 bars remain for the fourth section of the fugue, a complete exposition of
the fourth subject before it combines with other subjects would take 5 x 4 = 20
bars, not including any codettas. That would leave only 18 bars for other
contrapuntal combinations, each of which needs a minimum of 7 bars. If there
were to be only one inversus and one rectus quadruple combination, that
would leave only 4 bars remaining to be filled by episodes and a (possible)
coda: obviously not enough. It is clear, therefore, that Tovey is correct on this
point and that when the fourth subject enters it will begin to combine with
other subjects (not necessarily quadruply at first) almost immediately.
At the beginning of this Chapter, I mentioned the order of voices in which
each subject enters at its exposition.
Fugue/section 1 B T A S
Fugue/section 2 A S B T
Fugue/section 3 T A S B
Fugue/section 4 ?
It can be seen that Bach has systematically worked his way through the
available permutations, leaving only one permutation left if each entry is to be
in a voice that has not been used before: S B T A. This may suggest that the
fourth subject will make its first appearance in the soprano and that its next
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three appearances will be, in order, in the bass, tenor and alto. This may apply
even if the fourth subject is not given its own independent exposition before
being combined with the other subjects. This is the ‘permutational matrix’
identified by Zoltán Göncz in his two articles on what he calls the
“reconstruction” of the unfinished fugue.169 On the other hand, it may be worth
noticing that these orders of entry of voices correspond to three of the first four
Contrapuncti.
Contrapunctus 1 A S B T
Contrapunctus 2 B T A S
Contrapunctus 3 T A S B
Contrapunctus 4 S A T B
If there is a parallel between the first four fugues of the collection and the last
four (accepting that Contrapunctus 14 is made up of four ‘fugues’), then it may
be equally plausible to suggest that the order of voices in which the fourth
subject will enter will be S A T B, rather than S B T A. For reference, I give the
order of voices in which the subjects enter in their expositions in all the fugues.
Contrapunctus 1 A S B T
Contrapunctus 2 B T A S
Contrapunctus 3 T A S B
Contrapunctus 4 S A T B
Contrapunctus 5 A B S T
Contrapunctus 6 B S A T
Contrapunctus 7 T S A B
Contrapunctus 8 T B S
Contrapunctus 9 A S B T
Contrapunctus 10 A T B S
Contrapunctus 11 A S B T
Contrapunctus 12a S A T B
Contrapunctus 12b B T A S
Contrapunctus 13a S A B
169 Göncz, Z. (1991) "The Permutational Matrix in J S Bach's Art of Fugue: The Last Fugue Finished?" Studia Musicologica Academiae Scientiarum Hungaricae 33(1-4): 109-119; Göncz "Reconstruction of the Final Contrapunctus of The Art of Fugue" op. cit.
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Contrapunctus 13b A B S
Contrapunctus 14i B T A S (= Contrapunctus 2)
Contrapunctus 14ii A S B T (= Contrapunctus 1)
Contrapunctus 14iii T A S B (= Contrapunctus 3)
Contrapunctus 14iv S B T A or S A T B (= Contrapunctus 4)
From this chart it can be seen that Bach has taken care to include as much
variety as possible in this particular matter throughout the course of his
collection. It may or may not be significant that the order of entries in
Contrapunctus 14, the last four fugues, is noticeably paralleled by the order of
entries in the first four fugues. I mention this here only to show that Göncz’s
claims on this matter are not the only possibility.
The Rectus Combination of the Four Subjects
“… was to contain four themes”
Nowhere does the Obituary, Marpurg’s Preface or Forkel’s Biography state
that the four subjects or themes are to be combined together. Nevertheless it
seems incontrovertible that that must have been Bach’s intention. Ex. 4.3 from
above is reproduced here for convenience:
Ex. 4.5: The quadruple combination as given by Nottebohm
One of the clues pointing to the validity of Nottebohm’s theory is the last
four notes of subject 4, seen here in bars 4 and 5 of the soprano part: they keep
the rhythmic movement going at exactly the place where it is needed, the other
three parts having come to rest at the second half of the bar.
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306
From now on I shall refer to the four subjects in Roman numerals, as in Ex.
4.1 above: so the fourth subject, the main Art of Fugue subject, is designated IV.
A problem with the combination given in Ex. 4.5 – and quoted exactly by
Wolff in his 1975 paper170 – is the consecutive fifths between S and A in bar 3.
Admittedly they are consecutives in which one of the fifths is perfect and the
other diminished, a form of consecutives which seems not to have bothered
Bach and which is often said to be acceptable.171 But these consecutives
particularly bothered Tovey when he set about making his completion of the
fugue. To avoid them, he made a change to the rhythm of IV:
Ex. 4.6: Tovey’s third and last rectus combination
Tovey chooses to adopt this form of IV, with its altered rhythm, for all of its
appearances in his completion of the fugue, even when it is not strictly
necessary to avoid consecutives. He also uses this altered rhythm when he uses
inversus forms of IV. This rhythm of IV was also adopted in a completion by
David Schulenberg (1992)172 and by Michael Ferguson in his completion of the
fugue: indeed Ferguson’s one and only rectus combination is identical to this
one of Tovey’s.173
A strict teacher of harmony and counterpoint might argue that this little
change to the rhythm does not remove the consecutives. Lovelock, for example,
170 Wolff "The Last Fugue: Unfinished?" 171 Some further examples of these kinds of consecutives are given later in this Chapter. 172 Listed in Overduin, page 80. See note 29 for a further commentary on this. 173 bar 311 of Ferguson’s completion
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55
58
says dogmatically “Suspensions do not remove consecutives”174 – and this is a
case of a suspension that resolves upwards, of which more will be said below.175
Other objections might be brought to bear on this means of avoiding the
consecutives: if indeed Bach’s intention was to bring back the Art of Fugue
subject as the grand finale to his collection, a masterly contrapuntal peroration
at the end of the work, would he not have been more likely to design the first
three subjects in such a way that they could be combined with IV without the
need to tinker with it? To be fair, Bach himself – a higher authority than
Lovelock – has used this rhythm once before, in two successive instances in
Contrapunctus 3, starting at bar 55:
Ex. 4.7: Contrapunctus 3, bars 55 ‐ 61
In the first instance, in bar 57, it is clear that the reason for the change in the
rhythm in the alto is to avoid the consecutives that would otherwise have
resulted between A and B, even though one of the fifths would have been
perfect and one diminished. In the second instance, starting in the S at bar 58,
the changed rhythm is not necessary to avoid consecutives, but has been
preserved since this entry of the subject is an answer to the previous one. In
both cases, an alternative justification for the change to the rhythm could be
that Bach wished to continue the quaver movement at these points, which
174 Lovelock, W. (n.d.) First year harmony : complete Brundall, Norwich, W. Elkin Music Services, page 65 175 In order to avoid accusing Tovey of not knowing how to resolve a suspension, one might alternatively analyse the first quaver of the third beat of bar three of Ex. 4.6 as a diminished seventh chord whose components resolve correctly in both directions.
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would have ceased if the rhythm of the subject had been unchanged. However,
the avoidance of consecutives seems the more likely explanation. If Bach
wished to avoid consecutives – even perfect/diminished ones – at this point in
Contrapunctus 3, it is reasonable to assume that he may equally have wished to
avoid them in a quadruple combination such as Ex. 4.5. Tovey, when discussing
his completion and mentioning his change to the rhythm of IV, makes no
reference to this point in Contrapunctus 3, and it is not clear that he had noticed
it.
Bach’s habit of foreshadowing future musical events – a hint of something in
one piece being brought to fruition in a later one – has been noticed in a number
of places.176 Several foreshadowings in The Art of Fugue can be found, and could
well be made the subject of further study; a case could be made for this example
as a foreshadowing of the rhythmic change necessary to avoid the consecutives
in the quadruple rectus combination of Contrapunctus 14.
Three questions now arise in connection with the combination presented by
Tovey and followed by Ferguson and others. First, is the rhythmic change
necessary? Second, do the consecutives matter? Third, can one or more rectus
quadruple combinations be found that do not contain consecutives (or other
problems) and that do not require a change to the rhythm of IV or any of the
other three subjects? The answer to the first question can be found by
answering the second and third questions, so I digress now to a brief
consideration of Bach’s own use of consecutives.
Consecutives in Bach’s Music
Perhaps surprisingly, especially to students of harmony who have been
brought up to believe that consecutive fifths and octaves are a heinous crime,
Bach himself was not above writing them from time to time. Here are two 176 For example, see Shatzkin, M. (1984) "Thematic Anticipations in Bach" Bach (Quarterly Journal of the Riemenschneider Bach Institute) XV(2): 25-34
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examples from the chorale harmonizations that include consecutive perfect
fifths.
Ex. 4.8: Chorale “Gott sei uns gnädig und barmherzig”, showing consecutive fifths between T and S in the last bar
Ex. 4.9: “Einʹ feste Burg ist unser Gott”, showing consecutive 5ths between S and B in the last bar
In her book that exhaustively catalogues such curiosities, Augusta Rubin lists
26 similar examples from the chorale harmonizations alone.177 These two are
selected from that list by way of illustration. In Ex. 4.8 it may be that Bach
decided that a more interesting and high tenor part (bar 2, 4th beat to bar 3, 1st
beat) that is harmonizing a very static melody was preferable to a duller tenor
part that might have avoided the consecutives. The consecutives in Ex. 4.9 are
harder to explain since the antepenultimate bass note could easily have been D
rather than F : possibly the alto’s octave syncopated leap was the striking effect
that Bach considered most important. The other voices had to be arranged
around that, and Bach may have considered the consecutives a small price to
pay for such strong inner parts.
Consecutive fifths can be seen in bars 64 and 167 of the Ricercar a 3 from the
Musical Offering. They are consecutives in which one is augmented and the
other perfect. Ex. 4.10 gives bar 64 as an example.
177 Rubin, A. (1976) J. S. Bach: the modern composer Boston, Crescendo Pub. Co., pages 287-293
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15
18
Ex. 4.10: Ricercar a 3 from the Musical Offering, bar 64
Two examples of such consecutive fifths can be found in The Art of Fugue,
specifically in Contrapunctus 6.
Ex. 4.11: Contrapunctus 6, bar 15
Ex. 4.12: Contrapunctus 6, bar 18
These are more directly related to the matter under consideration, since they
come from The Art of Fugue and since they are examples of consecutive fifths
where one fifth is diminished and the other perfect, like the consecutives in the
quadruple combination shown above in Ex. 4.3 and Ex. 4.5. Since it is clear that
in The Art of Fugue as a whole Bach has set himself the strictest possible
boundaries, allowing himself no contrapuntal latitude whatsoever (for example,
by always insisting on stating the main Art of Fugue subject in its entirety
whenever it appears, even when it would have been convenient to make
alterations to it in some cases), it might therefore be arguable that since he has
allowed himself these kinds of consecutives in one part of the collection, they
might equally be allowable in another. If Tovey and others had followed this
line of thinking, they could have allowed the consecutives (Ex. 4.3) to stand and
they need not have gone to the lengths of altering the rhythm of IV in an
64
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attempt to avoid them. On the other hand, Bach’s alteration to the rhythm of the
subject in Ex. 4.7 may suggest that he preferred to avoid even these kinds of
consecutives when it was possible to do so.
Cases could therefore be made to defend both Nottebohm’s combination (Ex.
4.3 and 4.5, with its consecutives) and Tovey’s version (with the rhythm of IV
altered to avoid the consecutives, as Bach had done in Contrapunctus 3).
However, I want to say that, since it is certain beyond any reasonable doubt
that it was Bach’s intention to combine these four subjects, he would surely
have designed them in such a way that they were capable of being combined
without breaking any rules of musical grammar, harmony or counterpoint and
without the need for alterations to be made in either the notes or the rhythm of
any of the subjects. The task is to search for one or more combinations that fit
these requirements.
To illustrate this point it is interesting to observe Bach’s procedure in the two
other fugues of the collection that have three subjects: Contrapuncti 8 and 11. In
each case, the subjects are first presented individually and/or in a variety of
double combinations; the triple combination of all three themes is reserved for
the end of the fugue where it is presented in a number of different
permutations. The following tables show the layering of voices that Bach uses
in his various permutations of the combinations in these fugues. Two things can
be noticed: first, he is taking care to introduce as much variety as possible;
second, and more importantly, the last combination of each fugue is layered
from top to bottom in the same order as the fugue subjects were each presented
for the first time. Thus in Contrapunctus 8, subject I was first, II second and III
third so at the end they are presented with I on top, II in the middle and III at
the bottom. The same can be seen at the end of Contrapunctus 11.
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Contrapunctus 8 S A B
Bar 147 I II III
Bar 152 III I II
Bar 158 II III I
Bar 169 III II I
Bar 181 I II III
Contrapunctus 11 S A T B
Bar 145 II III I
Bar 174 I III II
Bar 179 I II III
Does it therefore follow that in the final quadruple combination of
Contrapunctus 14 the subjects will be layered from top to bottom in the same
order as they are presented in the fugue i.e. S‐I A‐II T‐III B‐IV (I‐II‐III‐IV)? The
idea seems alluring, but it presents problems.
Ex. 4.13: The subjects layered I‐II‐III‐IV
Here there are no difficulties with consecutives, but S and A are congested
and keep running into one another. Removing this problem by putting the S up
an octave only creates another problem, since that renders the stretch too wide
for the music to be playable by two hands on one keyboard, a requirement
shown by Gustav Leonhardt to be within the strict parameters that Bach has set
for this work.178 A more serious problem, though, is the weak and un‐Bachian
chord on a strong beat (in bar 5) that results from having IV in the bass. In
178 Leonhardt, G. (1952) The Art of Fugue, Bach's last harpsichord work; an argument The Hague, M. Nijhoff
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previous fugues, when IV (i.e. the main Art of Fugue subject, as we should
properly call it in earlier Contrapuncti) is in the bass, Bach arranges the
harmony in the vast majority of cases so that the last note of the subject is the
bass of a root position chord, usually with a suspended third. A few examples
of the last note of this subject being the bass of a chord can be found (eg
Contrapunctus 3, bar 55; Contrapunctus 4, bar 43; Contrapunctus 5, bar 27) but
this is usually as part of a cadential formula and, significantly, is not part of a
contrapuntal combination with other subjects that continue after the end of IV,
as in the above example.
Examination of all Bach’s triple combinations in Contrapuncti 8 and 11 (Ex.
4.14 and 4.15, overleaf) shows that in almost all cases (all but one), and in both
final cases, the subjects are presented unadulterated and in exactly the form in
which they were presented at their first appearances in each fugue.
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145
145
(the same passage transposed into D minor for reference)
174
179
Ex. 4.15: The three triple combinations from the end of Contrapunctus 11
Here it can be seen (Ex. 4.14) that only in the first two combinations of
Contrapunctus 8 has Bach altered the notes (i.e. the accidentals) of some of the
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subjects; in all other cases, including those in Ex. 4.15, the subjects are presented
in exactly their original forms without any kind of alteration. To this may be
added the triple combination given by Bach at the end of the existing portion of
Contrapunctus 14 (bar 233) as shown in Ex. 4.2 above.
To return to the search for an effective rectus quadruple combination we
should look again at Sir Donald Tovey’s work, to see if we can improve on what
he has done by obviating the need for his rhythmic change to subject IV.
Tovey’s main bugbear was the consecutive fifths created by this layering of the
subjects:
Ex. 4.16: The consecutives created by layering IV above II
As we saw earlier, Tovey’s solution (pace Lovelock) was to alter the rhythm
of IV by syncopating it, delaying the second note of bar 3 by a quaver and
thereby avoiding the consecutives. As mentioned earlier, Bach himself had
done almost exactly the same thing in Contrapunctus 3 (Ex. 4.7 above). But this
is at the expense of considerably distorting the outline of IV.179
Ex. 4.17: Tovey’s solution to the problem of consecutives
179 In his article op.cit., Jan Overduin describes a completion by David Schulenberg (1992) in which Schulenberg also follows Tovey’s lead by altering the rhythm of IV. “Schulenberg defends the change by pointing out that Bach himself altered the rhythm of theme I in order to combine it more smoothly with II … this is a spurious argument, however, since making a small change only once (to a different theme!) is quite a different matter from the permanent alteration of the main theme of the work.” In this I agree with Overduin. Schulenberg’s reference to a change in the rhythm of I is a reference to Bach’s bar 158 in the tenor. Tovey says of this passage: “When I is combined with other themes it is singularly difficult to manage in any part other than the bass, and Bach has to double-dot its second note before he can find a proper bass for it.” (Tovey A companion to 'The Art of Fugue' (Die Kunst der Fuge) of J. S. Bach, page 43-44.) These words of Tovey will prove to be prophetic: the following discussion will show that only two rectus quadruple combinations work convincingly, and both have subject I in the bass.
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It seems to have escaped Tovey’s attention that by layering the subjects in the
reverse order, with II above IV, the consecutive fifths vanish, to be replaced by
fourths.
Ex. 4.18: The subjects layered with II above IV: the consecutives vanish
It may reasonably be presumed that Bach would not have written, and
would not have intended us to write, consecutive fifths if there was no good
reason to do so. Here is a way of layering the subjects in a way that creates no
consecutives and that obviates the need for Tovey’s change to the rhythm of IV.
It therefore follows that in any quadruple rectus combination, II must be above
IV. It follows further therefore that IV can never be in the S (in a combination
that includes II, whether quadruple or not). We saw earlier that IV can never be
in the bass (in a quadruple combination) because it results in a weak chord.
There is a further detail that Tovey and his followers have missed. With one
exception180, every person who has attempted to complete Contrapunctus 14
has to my knowledge taken at face value the combination given by Nottebohm
(Ex. 4.3 above) and has used the original form of the main Art of Fugue subject as
IV. However, by Contrapunctus 5 Bach has introduced his 14‐note version of
that subject and that is the version he uses almost exclusively from then on (the
only exceptions being Contrapunctus 9; the first Mirror Fugue, Contrapunctus
12; and the Canon alla Decima, which revert to the original form). As was
shown earlier in Chapters 1 and 3, Contrapunctus 14 contains many
numerological references to the numbers 14 and 41, as well as including Bach’s
name as a musical motif and at a theoretical midpoint which itself carries an
allusion to the number 14. It was noted that it is the 14th fugue; that the second 180 Göncz "Reconstruction of the Final Contrapunctus of The Art of Fugue"
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subject has 41 notes; that Bach’s final bar gives a total of 14; and I have shown
that the third section is convincingly established as having a length of precisely
55 (14+41) bars. Surely then it makes complete sense for Bach to reintroduce his
main subject in its form containing 14 notes. If we look again at Ex. 4.18 above
we notice a slightly ugly ‘running‐in’ in the second beat of bar 2 (though to be
fair this would be considerably mitigated if the two themes were placed an
octave further apart); the fourth beat of bar three also has a nasty dissonance.
Both of these small problems disappear when we introduce the 14‐note version
of IV:
Ex. 4.19: How the 14‐note form of IV fits in the counterpoint
In fact II looks like it was specifically composed to sit with this 14‐note form
of IV. (It will be noticed that the introduction of this form of IV creates
consecutive fifths at the end of the first bar: this point will be attended to later
in this Chapter.)
We are now in a position to survey the possible permutations of the
combinations and by a process of elimination to arrive at one or more that fit
the requirements of obeying all the laws of counterpoint, harmony and
grammar whilst not altering the notes or rhythms of the subjects, being playable
by two hands on one keyboard and – less definably – being musically
convincing.
Ex. 4.13 above is a logical starting‐point since it presents the subjects in I‐II‐
III‐IV order. Setting aside for a moment the difficulties presented by the weak
chord in this arrangement, it enables us to see a remarkable property of the first
subject, which is that it is also capable of working in double counterpoint at the
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twelfth with the other subjects, when it is in any part other than the bass.
Compare Ex. 4.13 with Ex. 4.20:
Ex. 4.20: The subjects layered I12‐II‐III‐IV
The problem of the weak remains, but it is remarkable that Bach has
designed subject I so that it works with the others at the twelfth as well as at the
octave. In Ex. 4.13 the second note of subject I, in the S, presents an awkward
clash with the first note of III in the T, and this is removed in this version. In
order to try to get around the problem of the weak chord, this permutation
may be tried: II‐I‐IV‐III:
Ex. 4.21: The subjects layered II‐I‐IV‐III
Now further problems are encountered: the weak chord of Ex. 35 has been
replaced by an equally weak in the second half of bar 4. The S and A are
terribly congested and run into one another very crudely, especially in bars 5
and 6. This problem is removed if subject I is put at the twelfth, but the
difficulty of the chord is not cured by that:
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271
Ex. 4.22: The subjects layered II‐I12‐IV‐III
I have systematically worked out of all the permutations in order, starting
with I‐II‐III‐IV, then I‐II‐IV‐III, I‐III‐II‐IV and so on through all the available
possibilities. I do not propose to present every single one of them here – that
would be tedious – but for present purposes it is enough to say that time and
again we encounter difficulties, whether they be forbidden consecutives, weak
s, arrangements that cannot be played, or congested parts.
Since Tovey presents three rectus quadruple combinations in his working, it
will be instructive to consider each of them. His first one is at bar 271 of his
completion.
Ex. 4.23: Tovey’s first rectus combination
Tovey has set this passage of his completion starting in the key of G minor;
to make it easier to compare it with previous examples, I give the passage here
transposed into D minor.
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271
Ex. 4.24: Tovey’s first rectus combination (I12‐III‐II‐IV) transposed to D minor
Here Tovey has drawn on the ability of subject I to be put in counterpoint at
the twelfth, in the S. In bar 3 of the bass part his syncopated alteration to the
rhythm of IV can be seen, even though in this case no consecutives would have
resulted if he had used the unadulterated form of the subject. At the beginning
of bar 5 he has the unstylistic chord, though he gets off it as soon as possible
with rapid quaver movement. Most surprising of all, however, is the presence
of consecutive fifths between A and T in bar 6 – precisely the thing that Tovey
has been anxious to avoid, so anxious that he is prepared to alter the rhythm of
IV even when he does not need to. In his G minor version of this combination
(Ex. 4.23) he may have considered that the second interval, B ‐F , coming from a
perfect fifth C‐G, is somehow less severe a transgression, since the ear may hear
it as a minor sixth rather than an augmented fifth; but the fact remains that it is
a fifth, and he has written consecutive fifths when he has shown himself
elsewhere to be anxious to avoid them. For all these reasons, this first rectus
combination of Tovey’s can be discounted.
Tovey’s third and final rectus combination, IV‐II‐III‐I, comes at the end of his
completion, followed by a coda, at bar 306.
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306
Ex. 4.25: Tovey’s third and final rectus combination: IV‐II‐III‐I
This has several points in its favour: IV, the main Art of Fugue subject, is
presented prominently at the top of the texture, as seems appropriate as a final
statement at the end of the work. The use of I in the bass gives a solid
foundation to the whole. But he has still found it necessary to use his
syncopated rhythm for IV in bar 3. Apart from this change, this combination is
to all intents and purposes identical to Nottebohm’s combination as given in Ex.
4.5 above. The layering of this combination is exactly the same as that used by
Michael Ferguson at the end of his completion. (Every other quadruple
combination offered by other composers – that I am aware of – uses Ex.4.5,
complete with its consecutives.) However, since I am convinced that the 14‐note
form of IV is the correct one, this combination will not be possible: introducing
the 14‐note form here would, as can be seen at a glance, create consecutive fifths
between S and A in the last beat of bar 1.
There is a quadruple combination that meets the requirements that I have
described. In fact there are two – one of them is used by Tovey and it will be
discussed shortly. First though I present this version of the rectus quadruple
combination, which so far as I know has never been used or mentioned before
by anyone else.
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Ex. 4.26: II‐IV‐III‐I
Here is a combination which avoids consecutives, preserves Bach’s fugue
subjects in their unadulterated forms, uses the 14‐note form of IV, has a strong
bass line, avoids congestion of the voices, is playable by two hands on one
keyboard, and is musically convincing. Though the dissonance in bar 3 between
T and B may seem extreme, it is identical to that used by Bach in bar 235 (see
Ex. 4.2 above).181 This is obviously only the skeleton of the contrapuntal
combination: the second half of bar 5, for example, clearly needs to be filled in
with rhythmic movement, and this can be supplied by the alto; likewise the
music needs to continue to be propelled forward, and that is why in the
following conjectural filling‐in of the alto I have put a B on the first beat of bar
7, where the note A might have been more predictable. The alto part in bar 5 is
derived directly from the S in Bach’s bar 237.
181 One is reminded of a quotation from Johann Philipp Kirnberger (1774), who wrote of Bach’s contrapuntal daring: “At times, even four tones in succession may be dissonant. When the movement is fast and the melody light and easily grasped this may pass; only in slow tempo it would be intolerable. But even in fast tempo it must not occur too often, particularly if other voices were to rush along in similar manner; for it makes the music highly confused. It is better in this matter to take as a model Capellmeister Graun, the most euphonious and thoughtful writer of beautiful vocal melody, than Handel or J S Bach. The latter was the most daring in this respect, and therefore his things require a quite special style of performance, exactly suited to his manner of writing, for otherwise many of his things can hardly be listened to. Anyone who does not have a complete knowledge of harmony must not make bold to play his difficult things; but if one finds the right style of performance for them, even his most learned fugues sound beautiful.” Quoted in David, Mendel, et al., page 367-368
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&c.
&c.
Ex. 4.27: II‐IV‐III‐I with conjectural continuation
It may be objected that the consecutive fourths in the last beat of bar 1 sound
bald; but it should be remembered that here too either the T or B, or possibly
both, are available to fill in. A downward quaver scale in the second half of bar
1, in either voice (A‐G‐F‐E) arriving on D at the beginning of bar 2, would serve
perfectly. I propose that this quadruple rectus combination would appear
somewhere early in the fourth section of a completed Contrapunctus 14, not at
the end. It might therefore not necessarily be set in the key of D minor, as it is in
the foregoing examples; indeed it might be thought that in D minor II is sitting
a little high. (I have placed it in the upper octave so as to avoid the congestion
with the alto that would result if it were in the lower octave.) The combination
works well when transposed down, as here for example into the key of A
minor.
Ex. 4.28: II‐IV‐III‐I with conjectural continuation, transposed into A minor
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290
I have mentioned Tovey’s first and third combinations, and now I turn to his
second one – a good one – in an attempt to make a couple of small
improvements in it. He presents it at bar 290 of his completion.
Ex. 4.29: Tovey’s second rectus combination II‐III‐IV‐I
It is a layering of II‐III‐IV‐I in the key of A minor. It is similar to Ex. 4.26 but
has the inner parts interchanged, resulting in a more even vertical distribution
of the voices. However, Tovey has once again used his syncopated rhythm of IV
when it is not necessary to do so: because II is above IV in this case, there is no
danger of consecutives. A detail is that the last note of subject I in the bass has
been altered. The other shortcoming of this combination is that it does not use
the 14‐note form of IV. It is possible to incorporate the 14‐note form of IV and
reinstate the correct rhythm; if the music is transposed down into D minor there
is this splendidly strong and dark result:
Ex. 4.30: II‐III‐IV‐I using the 14‐note form of IV
Of course the same comments as were made above about the need to fill in
free material in the A and T apply here also. Since this and Ex. 4.26 are the only
rectus quadruple combinations that ‘work’ according to the strict parameters
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that I have set for them (which I believe to be in accordance with Bach’s own
strict parameters), I propose this combination, a variant of Tovey’s second, as
the final contrapuntal complex at the end of a completed Contrapunctus 14. The
succession of very strong root position chords at the beginning of every bar but
one, in the tonic key of D minor, makes for a very strong and confident
peroration. It would be followed by a coda, perhaps over a tonic pedal point,
that might include some prominent final reference to IV as a grand conclusion
to the entire work.
The Inversus Combination of the Four Subjects
“…and to have been afterward inverted note for note in all four voices”
The problem of combining the four subjects of Contrapunctus 14 in their
inverted forms is a considerably more intractable one than that of combining
the rectus forms, so much so that the majority of composers who have
attempted completions (including Moroney, Rogg, Walcha and Göncz) have not
included an inverted quadruple combination in their workings at all.182
However, the solution is there to be found for those who are prepared to look
hard enough; it is testament to Bach’s incredible skill that he has been able to
design his subjects so that they can lock together not only in their rectus forms,
with one of them working in double counterpoint, but in the ways they also
combine in their inversus forms, which I will consider now. The two scholars
who have offered the most thoughtful and convincing contribution to this
question of the inverted combination are the American organist Michael
Ferguson, and Sir Donald Tovey, both of whom have published their own
completions to Contrapunctus 14. Both completions have many merits, and
both are open to criticism: for example, both are too long; and Ferguson’s is
182 Rogg, for example, in his unpublished completion, presents a clever bit of counterpoint in which the inverted form of IV (IVinv ) combines with the rectus forms of II and III; but he does not attempt a simultaneous combination with subject I as well. Similarly, Göncz states that only IVinv should be used in combination with the rectus forms of I, II and III.
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designed to be played on the organ with pedals and in places is not able to be
played by two hands on one keyboard.183 My purpose here, however, is not to
offer a detailed critique of their workings but to focus on the way they have
dealt with the inverted combination. I will propose that the most convincing
result can be achieved by taking elements of both Tovey’s and Ferguson’s
thinking and combining them together, to produce a new solution to the
combination that is presented here for the first time.
Before looking in detail at the inversion of the subjects, it is worth examining
the exact wording of the Obituary: “…and to have been afterward inverted note
for note in all four voices”. This statement follows the statement that the last
fugue would have contained four themes, so it can be presumed that by
“afterward” the Obituarists mean ‘after the introduction of the fourth theme’.
This tells us that at the beginning of section four of the fugue, subject IV will be
introduced almost certainly in immediate combination with one or two other
subjects (but not necessarily immediately in a quadruple combination, as we saw
earlier). It is at some point after this introduction of IV (and presumably after
the first presentation of a quadruple rectus combination) that the subjects will
be inverted in all four voices. The significance of the phrase “note for note” will
become apparent during the course of the following discussion. 183 Ferguson defends his decision to write his completion for organ with pedals in Ferguson, M. (2001) "Letter to the Editor" The American Organist 35(4): 18-24 “We have known for a long time that Mattheson, upon the initial publication of The Art of Fugue, hailed it specifically as an organ work. And the newspaper advertisement for the original edition, recently rediscovered, states that it was specifically arranged by its composer for performance on the organ. There can, therefore, be no credible scholarly argument for rejecting, out of hand, completions specifically for organ.” The text of C P E Bach’s advertisement can be seen in David, Mendel, et al., page 257: the actual wording is “… arranged for use at the harpsichord {Clavier} or organ.” This does not seem to specify the use of organ with pedals, in fact rather the contrary. And Ferguson’s position flies in the face of the evidence presented by Gustav Leonhardt in his book which suggests compellingly that the music was intended for two hands on one keyboard. However, this particular debate is not pertinent to the present subject, except inasmuch as the restriction that the music be playable by two hands on one keyboard, once accepted, will have a bearing on attempts to achieve an inversus quadruple combination: it is possible that one might be able to assemble an inversus quadruple combination that ‘works’ in a technical sense but which cannot be played by two hands on one keyboard. Such a combination would therefore be invalid if Leonhardt’s position is correct, as I believe it is. Beyond that, Ferguson’s and Leonhardt’s opposing opinions may be left to one side without further comment, except to say that it is of course obvious that music written for harpsichord may well in many cases translate to the organ, sometimes with and sometimes without pedals; whether a performer chooses to play the music on the organ is a matter of individual taste.
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It may be worth mentioning that neither the Obituary nor the other sources
(Forkel and Marpurg) say that the subjects are to be combined! They merely say
that the subjects are to be inverted. Nevertheless, it is a reasonable assumption
that an inverted quadruple combination will follow a rectus quadruple
combination in this last section of Contrapunctus 14. Tovey was in no doubt; “I
am absolutely certain that Bach intended to invert all 4 themes in the quadruple
counterpoint”.184 Since The Art of Fugue as a whole is, among other things, an
exercise in steadily increasing contrapuntal complexity, this would seem to be a
natural progression from the extraordinary achievement of the Mirror Fugues
(Contrapuncti 12 and 13), in which two entire fugues (each having just the one
subject) are invertible; to write a fugue with four subjects which combine both
rectus and inversus would be a logical way – perhaps the only way – to trump
the dazzling compositional virtuosity of the Mirror Fugues. Further, if it can be
shown that the four subjects can be combined in inversion, then it is reasonable
to assume that they were intended for that.
Inverting the subjects
Bach has demonstrated his method of inverting fugue subjects early in the
collection: it happens for the first time in Contrapunctus 3. It is a
straightforward process in which the tonic is replaced by the dominant; in the
main Art of Fugue subject (IV in Contrapunctus 14) the second note is a fifth up,
so in the inversion the second note is a fifth down, and so on. From
Contrapunctus 5 onwards Bach uses both rectus and inversus forms of the
subject. Contrapunctus 11 is a fugue on the inverted forms of the subjects of
Contrapunctus 8. The scale of inversion can be expressed thus:
184 Tovey A companion to 'The Art of Fugue' (Die Kunst der Fuge) of J. S. Bach, page 45-46
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I
Ex. 4.31: the standard scale of inversion
By following this scale of inversion the inverted form of subject I (designated
hereafter Iinv) can be established:
Ex. 4.32: subject I and Iinv
Simlarly, II is straightforward to invert.
Ex. 4.33: subject II and IIinv
In the case of III, Tovey makes a statement that this subject operates on a
different scale of inversion from the others.185 The scale he gives is
Ex. 4.34: inversion scale for III as given by Tovey
Presumably Tovey has been led to give this scale because of the position in
which IIIinv is seen for the first time in the portion of the fugue that Bach has left
us, at bar 213 in the alto.
185 Ibid., page 44. He restates this position without explaining it in Tovey, D. F. and Foss, H. J. (1944) Essays in musical analysis : Chamber music London, Oxford University Press : H. Milford, page 88.
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222
Ex. 4.35: First appearance of IIIinv as given by Bach in the alto at bar 213
But this is merely an instance of IIIinv presented in a different position
because of the context at the time. Its second (and last) appearance in Bach’s
writing, in the bass at bar 222, is in a different position again, but this has not
led Tovey to say that a second (or rather third) scale of inversion is operating.
Ex. 4.36: Second and final appearance of IIIinv as given by Bach in the bass at bar 222
When Tovey goes on to give his examples, and when he uses IIIinv in his own
working, he does not use it in either of these positions or according to the scale
he gave (Ex. 4.34) but in the standard inversion resulting from the inversion
scale shown in Ex. 4.31 and used for all the other subjects. Respectfully, I
disagree with his statement about the scale of inversion used for III. In fact there
is no reason not to use the standard inversion scale, giving the inversion of III
as:
Ex. 4.37: subject III and IIIinv
A further point to note about Ex. 4.35 and 4.36 (Bach’s own uses of IIIinv) is
that they are not strict inversions of the whole subject: the second half of the
original subject trails off into longer notes. But in each case this subject is being
presented on its own and not in combination with any other subjects. We are
told in the Obituary that Bach’s intention was “afterwards” to invert the
subjects “note for note”: the fact that he has not inverted this subject note for
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note in an earlier part of the composition does not necessarily mean that he
would have done exactly the same thing later on.
Subject IV inverts straightforwardly:
Ex. 4.38: subject IV and IVinv
More on subject II: Bach’s sometimes irregular resolution of suspensions
Before moving on to consider how these inverted subjects might combine,
there is more to be said about II. At the opening of this Chapter it was stated
that subjects I and III are presented by Bach in their inverted forms as well as
their rectus forms. But II does not appear at all in an inversus form during the
course of the extant portion of Contrapunctus 14. Since II contains suspensions,
which by definition resolve downwards, it follows that IIinv contains
suspensions that appear to resolve upwards. This may present a problem.
Tovey partially addresses this point by giving as an example a short extract of
II:
Of these notes he says that they
…show every sign of being constructed for inverse-contrary counterpoint. Melodically they are artificial, though construable enough, and their evident purpose is to make accented discords that resolve both ways, downwards within one crotchet in the original position, downwards at the minim when inverted.186
A reference to Ex. 4.33 above will show that this is a convincing argument,
and that in IIinv the minim D can be persuasively argued to resolve
ornamentally onto the next minim C by way of those intervening quavers and
186 Tovey A companion to 'The Art of Fugue' (Die Kunst der Fuge) of J. S. Bach, page 46
148
semiquavers. Unfortunately, however, Tovey has chosen to stop his little
illustrative extract (above) at the minim B and has not shown the continuation
from it: a further look at Ex. 4.33 will show that the minim C in IIinv (the result
of inverting the minim B in II) cannot by any stretch of the imagination be said
to resolve downwards at the minim or any other interval of time. The only
conclusion therefore is that this particular note resolves upwards. Since that is
irregular, it is necessary to see if Bach has set any precedents elsewhere for this
kind of resolution of a suspension: it would be unfortunate if a proposed
inverted combination were to be invalidated by this kind of technicality.
The ‘bible’ of musical grammar in Bach’s day, unsurpassed as a textbook for
the teaching of strict counterpoint, was Gradus ad Parnassum by Johann Joseph
Fux (1660‐1741). It contains an interesting passage:
Joseph.― Before I begin the exercises may I ask, if you don’t mind, whether the retardation or ligature into the dissonance is also to be used in ascending?
Aloys.― You raise a problem which is harder to untangle than the Gordian knot. I shall deal with it later because, being still at the threshold of the art, you would not now wholly understand it. […] For the time being, as your teacher I advise you to resolve all dissonances down to the next consonance.187
This seems to suggest that there are some circumstances in which
suspensions may resolve upwards, although it comes from a chapter dedicated
specifically to Fourth Species Counterpoint after the manner of Palestrina.
Further, it is known that although Bach himself owned a copy of Fux, he did
not like to use it for teaching: a letter from C P E Bach to Forkel of January 13
1775 says “In composition he started his pupils right in with what was practical,
and omitted all the dry species of counterpoint that are given in Fux and
187 Fux, J. J. and Mann, A. (1944) Steps to Parnassus : the study of counterpoint London, Dent, page 59-60
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others.”188 It could be argued, therefore, that not much can be deduced from the
fact of Bach’s ownership of Fux’s book.
In Dr Ebenezer Prout’s exhaustive Harmony: Its Theory and Practice we read:
Some theorists call these [sc. suspensions resolving upwards] “Retardations”; but there seems no reason for giving a different name to them. By far the most common upward suspension is 7-8 – the suspension of the root of the chord by the note below it.189
This kind of upwardly‐resolving suspension – which Prout calls “by far the
most common” – is precisely what can be seen in bar 6 of Ex. 4.33 above.
In her book J S Bach: The Modern Composer, Augusta Rubin states:
Inasmuch as the Retardation creates an accented ascending discord, that is to say a suspension effect that resolves upward, it is alien to the spirit and tradition of contrapuntal movement. Thus, as may be expected, it is very rarely encountered…190
Note that Rubin says “rarely encountered” – not “never encountered”.
Other modern theorists have little or nothing to say on the subject of
irregular resolution of suspensions. Aldwell and Schachter, for example, say
only “other upward‐resolving suspensions are less common” – again it should
be noted that the existence of such suspensions is not denied.191 Other theorists
such as Forte192 and Lester193 are silent on this subject. Even a book totally
devoted to the theory of suspensions has nothing to say on the matter.194
Nevertheless it is clear that Bach is prepared to overlook the strict rules from
time to time. Having said that, it is also important to take a long view of the
188 David, Mendel, et al., page 399 189 Prout, E. (1903) Harmony : its theory and practice London, Augener, 16th edition, page 152, paragraph 357 190 Rubin, op. cit. 191 Aldwell, E. and Schachter, C. (1989) Harmony and voice leading San Diego, Harcourt Brace Jovanovich 192 Forte, A. (1962) Tonal harmony in concept and practice New York, Holt Rinehart and Winston 193 Lester, J. (1982) Harmony in tonal music New York, Knopf : Distributed by Random House 194 Komar, A. J. (1971) Theory of suspensions; a study of metrical and pitch relations in tonal music Princeton, N.J., Princeton University Press
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resolution of dissonances and not to look too narrowly at a particular example.
This is a trap into which Prout has fallen, for example, by quoting bar 9 of the
Prelude No 1 from the Well‐Tempered Clavier, Book Two:
Ex. 4.39: WTC, Book 2, Prelude 1, bars 9 ‐ 10
The tied G (here marked *) is described as a retardation by Prout, and
indeed at first sight that is exactly what it appears to be. But Prout has missed
the point that by ‘resolving’ upward onto the A, the G has ‘resolved’ to another
dissonance (the A being dissonant with the bass E), which ultimately resolves
correctly on the second beat of the next bar; functionally, it is merely a
rhythmically‐displaced dissonant as a neighbour chord to V, and this is
probably how more modern theorists like Forte or Lester would read it.
Similarly Ex. 4.40, an example given by Rubin, might be interpreted as a
retardation (a suspension resolving upwards). The suspension of the tenor on
the third beat certainly produces a striking clash with the alto: but in reality it is
only an ornamental resolution of the tenor C (second beat), resolving correctly
downwards to the B at the end of the bar.
In Ex. 4.41, another example given by Prout, the E (bass, second beat of bar 2)
appears to resolve upward to the F (second half of second beat of bar 2) by way
of an ornamental G. One might also alternatively interpret the resolution as
taking place a whole minim after the suspension, i.e. on the third beat. It is
Ex. 4.40: Verlieh’ uns Frieden gnädiglich (Riemenschneider No. 91)
9
*
&c.
151
certainly true that in both of these places a complete consonance is arrived at,
which may suggest that the resolution of the suspension has taken place;
however an alternative analysis of this passage is that the bass F resolves
correctly downwards to D on the fourth beat of the bar (with a voice‐exchange
with the soprano as shown), or on the first beat of the next bar.
Ex. 4.42 is an interesting case: one would naturally expect the suspended F to
fall to E. Although the ear may hear a ‘correct’ resolution that ultimately arrives
on E on the third beat (especially on an instrument like the harpsichord where
the differentiation of the voice‐leading is not easy to make clear), it is important
to notice that aural resolution of the suspension is actually supplied by a
different voice from the voice which has been suspended.
These examples therefore can be seen to be at best ambiguous. But a
fascinating new article by Ewald Demeyere of Antwerp (April 2006) focuses
specifically on Bach’s “Expanding the Limits” in The Art of Fugue in his
treatment of dissonances, ’mistakes’ against voice leading, chromaticism,
technical aspects of fugal writing, sequences and linear intervallic patterns,
rhythmic motion and rhythm, and metre, all as compared to the expectations
raised by a strict understanding of contrapuntal ‘rules’ as expressed by Fux and
Ex. 4.41: WTC, Book 2, Prelude 11
Ex. 4.42: WTC, Book 2, Prelude 11 bars 15‐16
12
*
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similar treatises.195 The author says that “Bach, in his Art of Fugue, pushed
every textbook rule and concept to the limit.” It is “quickly evident that Bach
was not too concerned with existing rules (certainly not when breaking them!)”.
The whole of M. Demeyere’s article is learned, thorough and compelling, but
for the purposes of the present discussion I have selected only one small part of
it to illustrate a most irregular – not to say unclassifiable – resolution of a
suspension.
Bach’s relaxed attitude towards rules does not limit itself to (hidden) successions of fifths and octaves (as well as to the ottava battuta), but also shows itself in, at times, highly irregular resolutions of dissonants (seventh and fourth) and his chord choice. The most striking liberty that Bach allows himself is a dissonant ligatura (seventh) in the soprano, which does not resolve stepwise and downwards, but which makes an ‘incorrect’ ascending leap of a fourth. I have found two examples of this in the Art of Fugue: Contrapunctus 10, bar 21 and Contrapunctus 14 bar 102.196
The example from Contrapunctus 14 bar 102 is explained as acquiring its
resolution three bars later; but the suspension in Contrapunctus 10, bar 21
defies analysis by any conventional textbook rule. It is given here in Ex. 4.43:
In fact this suspended A (soprano, bar 21 beat 1) does not resolve at all: it
simply stops! The next soprano note, D, is not the resolution of the suspended A
but rather is a harmony note in the ‘wrong’ register, perhaps resulting from a
register shift of the tenor and bass from the first beat of bar 21 to the second
beat. Exactly this kind of register shift will appear in my solution to the inverted
quadruple combination at the end of this chapter.
195 Demeyere, op. cit. 196 Ibid., in the section “Mistakes against voice leading” (My Ex. 4.43 is from Demeyere’s Ex. 16.)
Ex. 4.43: Contrapunctus 10, bars 20‐22
20
153
Demeyere also notes the soprano A in bar 7 of Contrapunctus 1 as resolving
“incorrectly” – it should fall to a G – a passage that is repeated in the tenor and
bass in bar 15.
These last examples in particular show clearly how Bach was prepared to
push expectations around suspensions to their limits: a point that will become
relevant in the subsequent discussion of the inverted quadruple combination.
Combining the four inverted subjects
My first attempt to combine the four rectus subjects (Ex. 4.13) was in the
order I‐II‐III‐IV, so the order Iinv‐IIinv‐IIIinv‐IVinv seems a logical place to start
when attempting an inversus quadruple combination. Perhaps not surprisingly,
difficulties are immediately encountered.
Ex. 4.44: Iinv‐IIinv‐IIIinv‐IVinv
There are consecutive fifths in bar 3 between alto and bass, and again in bar 6
between alto and tenor. From this we may deduce that IIinv may never be above
IVinv, because the same problem will recur each time. Similarly, because of the
consecutives in bar 6, IIinv may never be above IIIinv. It would be tedious to
present every possible permutation of the combinations; suffice it to say that
nearly all of them present problems of one kind or another, almost always in
the form of consecutives and/or congested parts. For the purposes of this
discussion we may bypass the lengthy process of elimination by trial and error
and move directly to one straightforward combination that on the surface
154
appears to be successful without breaching any of the strict parameters:
IVinv‐IIIinv‐Iinv‐IIinv.
Ex. 4.45: IVinv‐IIIinv‐Iinv‐IIinv
This appears to breach no rules, and it is (just) playable by two hands on one
keyboard. As before, this shows only the combination of the subjects: certainly
the soprano at the end , as well as possibly the alto and tenor at the beginning,
would need to be filled in with free material. However, although it ‘works’ in
the strict sense of musical grammar, it is not musically convincing or
satisfactory. Subject IIinv seems ill at ease as a bass, and in particular the second
note of IIinv in bar 3 is awkward. Also, the major seventh between alto and tenor
in bar 3 seems harsh and crude; where the conjunction of these two subjects in
the rectus version, producing a minor second, is tolerable, this interval is not.
The tenor D might in this case also be analysed as a suspension against the alto
C , but if that it what it is then it resolves upwards; although I have shown that
such a procedure is not unprecedented in Bach, in this particular case it is still
unsatisfactory. A further comment is that the texture is muddy with the bass
hovering low in its range: if a decision to use this combination were taken,
however, during the course of a fugue it would not necessarily appear in this
key. If it were transposed up into G minor, for example, the muddiness would
disappear.
155
263
As a second example of this kind of combination, we may cite the first of the
two inversus quadruple combinations used by Tovey, this one at bar 263 of his
completion.
Ex. 4.46: Tovey’s first inverted combination
It is clear that in his attempt to make this combination work, Tovey has got
himself into some difficulty. The first thing to notice is the recurrence of his
altered rhythm for IVinv, in the alto in bar 3, even though in this case no
consecutives would have resulted from using the unaltered version. IIinv as a
bass is just as awkward as it was in Ex. 4.45, and in bar 5 it can be seen that he
has made what should be a C into a C , presumably because his alto line,
coinciding with the tenor B , has nowhere else to go but to the C (it cannot go
to an A because that would create consecutive octaves with the tenor, and it
cannot go to a C because that would be an augmented second; so his only
option was to alter the bass). In bar 6 he has also made the fourth bass note an
F instead of F . Lastly, he has altered the rhythm of the first note of III in the
tenor (bar 3) to avoid the harsh clash between the tenor C and the soprano D.
So in this short passage there are four examples of Tovey choosing to ignore the
Obituary’s information that the subjects are to be inverted “note for note”.
Michael Ferguson comments on straightforward inverted combinations of
the subjects:
An inversion of all four themes seems required by statements made by those who had knowledge of Bach’s intentions. And yet, if inverted as is typically done (in the minor mode), this has an unlovely effect that
156
no-one has ever believed Bach would approve for use in his greatest fugue.197
Zoltán Göncz has a similar opinion that he expresses in his two articles cited
earlier:
A lot of scholars understand the words of Mizler’s obituary literally and include the inversion of all four subjects later on. These combinations make a strange impression and lack the necessary force of conviction. Besides, the various completions frequently contain inadequate elements of style, forced stretto, inconsistencies of form.198
But their success is doubtful. These combinations sound unfamiliar and they do not carry enough conviction either.199
Though phrases like “an unlovely effect”, “a strange impression” and
“sound unfamiliar” are not the words of technical musical analysis, it is
nevertheless hard to disagree with them. In order to achieve a successful
combination, it is going to be necessary to look beyond straightforward
layerings of the subjects and to deal with the counterpoint in a more
sophisticated manner.
Tovey’s second inverted combination provides us with the first clue of the
way forward: he presents it at bar 280 of his completion in the order IIIinv‐ IIinv‐
IVinv‐ Iinv. Ex. 4.47 shows the skeleton of that layering.
Ex. 4.47: IIIinv‐ IIinv‐ IVinv‐ Iinv
197 Ferguson "Letters to the Editor", op. cit. 198 Göncz "Reconstruction of the Final Contrapunctus of The Art of Fugue", page 56-57 199 Göncz "The Permutational Matrix in J S Bach's Art of Fugue: The Last Fugue Finished?", page 110
157
Leaving aside for a moment the fact that this combination contains
consecutives in bar 3, we can use this example to demonstrate a further
remarkable contrapuntal property of subject I. Earlier I showed that in its rectus
form it is capable of being combined with the other subjects not only at the
octave but also in double counterpoint at the twelfth, when it is in any part
except the bass. We now discover that the bass of Ex. 4.47 can be transposed
down a fifth while the other voices retain their current positions. In its inversus
form, subject I works not only at the octave but also appears to work in double
counterpoint at the twelfth.
Ex. 4.48: IIIinv‐ IIinv‐ IVinv‐ Iinv‐12
Bach’s skill in designing such contrapuntal wizardry is awe‐inspiring. At a
stroke, the dissonance that we have mentioned before, in the previous example
between soprano and bass in bar 3, has vanished. (Admittedly the consecutives
are still present, but this combination is only being used as a temporary
example.) A more serious problem has now presented itself, however, and that
is the presence of the soprano B in bar 4 sounding against the bass B . In his
final working of this combination, Tovey has no qualms about changing that
soprano B into B , to make it fit with the bass.
158
280
Ex. 4.49: Tovey’s second inversus combination
This is a great improvement on Tovey’s first combination, but he has had to
alter the notes of III and he has once again used his syncopated rhythm for IV –
necessary in this case to avoid consecutives. Once again, we are not able to say
that this is a “note for note” inversion of the subjects.
Before leaving this particular area of discussion we should consider a variant
of this layering: in Ex. 4.47 we had IIIinv‐ IIinv‐ IVinv‐ Iinv, and in Ex. 4.48 we saw
how this (almost) works with Iinv in counterpoint at the twelfth. Ex. 4.49 showed
how Tovey adapted this layering in his working. Ex. 4.50 shows the
combination IVinv‐IIIinv‐ IIinv‐ Iinv.
Ex. 4.50: IVinv‐IIIinv‐ IIinv‐ Iinv
This arrangement has the considerable merit of containing no consecutives.
But the first beat of bar 3, with subject I in the bass, results in a weak and
unacceptable chord. Also, tenor and bass are very congested. However, the
discovery that subject Iinv can be transposed removes these difficulties, as shown
in Ex. 4.51.
159
Ex. 4.51: IVinv‐IIIinv‐ IIinv‐ Iin‐12
The problems of the weak chord and the congestion have disappeared,
only to be replaced by the same problem that was encountered in Ex. 4.48: the
false relation now between alto and bass in bar 4, that can only be ‘fixed’ by
altering the third note of III. The significance of Ex. 4.51 will become apparent at
the end of this discussion: it is the one layering that holds the key to a
successful inverted quadruple combination. But for now, we are still in a
position where we cannot say that we have inverted the subjects note for note.
Perhaps therefore it is time to revisit the wording of the Obituary.
Why would the authors of the Obituary – one of whom was Bach’s eldest son
– have used the phrase “inverted note for note in all four voices” if they did not
mean exactly that? That Bach was involved in such a complex and elaborate
composition over a period of several years must surely have been a topic of
domestic conversation in the Bach household in Leipzig. No work of this
contrapuntal complexity had been attempted before, and it is unthinkable that
Bach would not have discussed his plans for this piece with his sons. The
Obituary was written within months of Bach’s death while such matters would
have been fresh in the authors’ minds. They would hardly have imagined or
invented such a statement if Bach himself had not told them about it. The
balance of probabilities must surely lie in favour of this part of the Obituary
being correct. Therefore, it must surely be possible to find a way to invert the
subjects note for note. The question then becomes, what exactly does “note for
160
note” mean? Michael Ferguson provides the answer, and the second clue on the
way to a totally successful combination, with a new interpretation of the
meaning of the words “note for note”. The following passage revisits and then
continues Ferguson’s earlier quotation.
An inversion of all four themes seems required by statements made by those who had knowledge of Bach’s intentions. And yet, if inverted as is typically done (in the minor mode), this has an unlovely effect that no-one has ever believed Bach would approve for use in his greatest fugue. However, if strictly inverted – semitone for semitone, something no-one before me had attempted – the mode changes to the major, and acquires that darkly brooding quality that only Bach could achieve while writing in a major key. For this discovery, which at a stroke decisively confirms Nottebohm’s hypotheses about the inclusion of the missing main theme, as well as the inclusion of this incomplete fugue as integral to The Art of Fugue, I cheerfully claim sole credit.200
Ex. 4.52 sets out the subjects inverted semitone for semitone as Ferguson
suggests, rendering each one in the major mode.
Ex. 4.52: The four subjects inverted semitone for semitone
200 Ferguson "Letter to the Editor", op. cit.
161
286
Ferguson’s inverted quadruple combination can be seen at bar 286 of his
completion. He has chosen to set it in the key of B major – a very reasonable
choice, since it is one of the keys that features already in Bach’s existing portion
of the fugue.201
Ex. 4.53: Michael Ferguson’s inverted quadruple combination
There can be no doubt that this sounds very much more convincing than any
of the inverted combinations we have seen hitherto. There are a couple of
details to mention about subject II: the last (tenor) note of bar 1 is a G where a
strict semitonal inversion from the original C (before transposition into B
major) would have resulted in a B (i.e. G after transposition). This minimal
alteration is necessary to avoid introducing any sense of the minor mode that a
G would bring into a B major tonality. In bar 3, the second note of the tenor
part should, according to a strict inversion, be an octave higher; presumably
Ferguson has put this note in the lower octave to avoid having it strike against
the soprano, which might disrupt the sense of line of subject Iinv. With great
respect to Mr Ferguson, I still have to say that I am troubled by the Toveian
syncopation of IV (necessary here in this layering to avoid consecutives), and by
201 For obvious reasons much of the fugue is set in D minor, with frequent passages in A minor and the subdominant key, G minor. There are also brief excursions into B major and F major. So Bach’s key structure for Contrapunctus 14 can be seen to be relatively simple: the usual tonic, dominant and subdominant relationships, with the use of major keys whose roots are a third above and below the tonic note of D minor. To put it another way, the key centres are ordered with typically Bachian symmetry around the tonic: minor keys a fifth above and below the tonic; and major keys a third above and below the tonic. Since Ferguson’s combination is in the major mode, it makes sense for him to have chosen one of these two major keys for it.
162
the alteration to the rhythm of the first note of III to avoid the ugly clash with
the soprano that would have resulted if it had been a minim.
I am certain that Michael Ferguson is on the right track with his discovery of
the interpretation of “note for note” as “semitone for semitone”. But I want to
achieve a combination that meets all the other strict parameters that I have set
out in the discussion of the rectus combination. IV should be in its original
rhythm and I believe that it should be presented in its 14‐note form, so as to be
consistent with the rectus combinations. I do not want to have to alter the
rhythm or pitch of any of the subjects, so that it will be possible to say truly that
the subjects have been inverted “note for note”. It must be playable by two
hands on one keyboard, and must avoid congestion, confusing or muddled
voice leading, chords and muddiness of texture. Finally, I am sure that the
observation made earlier about subject Iinv’s ability to work in counterpoint at
the twelfth is important: if Bach has designed his counterpoint with that
property inherent in it, it is reasonable to suppose that he intends it to be used.
The first task therefore is to find a way of layering the subjects in such a way
that there are no consecutives, so that there will be no need to adjust the rhythm
of IV. Such a layering was encountered in Ex. 4.50. We may therefore take
Ferguson’s subjects as he has inverted them, semitone for semitone, leaving out
his free material, and layer them in the same way as Ex. 4.50: IVinv‐IIIinv‐IIinv‐Iinv.
At the same time, IV can be restored to its original rhythm.
Ex. 4.54: Ferguson’s inversion layered in the order IVinv‐IIIinv‐IIinv‐Iinv
163
With the subjects layered IVinv‐IIIinv‐IIinv‐Iinv, a very neat arrangement and
perhaps an obvious one (since it is the inversion of I‐II‐III‐IV, the order in which
the rectus subjects were presented!), the main Art of Fugue subject is
appropriately displayed with prominence at the top of the texture. We may
now bring in subject I’s capacity to work in counterpoint at the twelfth: in order
to keep it within the reach of the fingers it will have to go up a fourth, rather
than down a fifth. At the same time, we can restore IV to its 14‐note form.
Ex. 4.55: The inversion layered in the order IVinv‐IIIinv‐IIinv‐Iinv‐12, with the 14‐note form of IV
With magnificent and thrilling precision, like an intricate piece of clockwork,
the false relation that was formerly encountered (as in Ex. 4.51) between the
third note of III and the fourth note of I, has disappeared. This in itself is a very
strong indication that this is the correct combination. These subjects now lock
together with, to use Tovey’s memorable phrase once again, “a correctness that
cannot conceivably be accidental”. The false relation is not the only problem to
have vanished, for the second note of subject I (alto, bar 3) was formerly a B ,
causing an ugly clash with the second note of III (bass, bar 3), which is why
Ferguson had altered the rhythm from a minim to a crotchet, just as Tovey had
done in his first inverted combination, as seen in Ex. 4.56. With the bass in this
new position there is now no reason not to reinstate the first note of III to its
original value. Likewise, there is now no reason that the second tenor note of
bar 3 cannot now revert to its true position one octave higher, where it should
be according to a strict inversion, since it will not now be in any danger of
colliding disruptively with another voice. Having made these changes, I can
164
now present what I believe to be the definitive solution to the problem of the
inverted quadruple combination. The subjects have indeed been inverted note
for note in all four voices, in strict accordance with the wording of the Obituary.
Tovey’s use of subject Iinv at the twelfth in the bass, and Ferguson’s switch to the
major mode were the pointers to this solution; while I acknowledge these
important origins of my combination, it is tempting to echo Ferguson’s words:
“for this discovery I cheerfully claim sole credit”:
Ex. 4.56: The solution to the inversus quadruple combination, as proposed by Indra Hughes
This solution presents an interesting analytical question around the
treatment of the bass E in bar 3, specifically the third beat, and then the bass F
on the fourth beat. This inverted form of I (Iinv) as presented in this combination
with the other voices creates what appears to be a suspension of the bass E
from the second beat of the bar into the third beat, where the chord changes
above it and the bass note remains constant. If this is indeed a suspension then
it is one that resolves upwards rather than ‘correctly’ downwards. Earlier in this
Chapter I discussed Bach’s irregular treatment of resolutions; Ex. 4.43 in
particular is a striking example of a suspension whose resolution defies
classification, making this bass note seem like a relatively tame infraction of the
‘rule’ that says that suspensions should almost always resolve downwards by
step.
A second, admittedly rather forced, analysis of this note could identify this
bass E as a pedal: in Joel Lester’s harmony textbook a pedal is defined as “a
165
3
E:
V4
2
V7
I
&c.
note, often in the bass, sustained through one or more chords. Pedals do not
resolve during the chord to which they are nonharmonic.”202 The bass line in
bar 3 above matches this description precisely, even though it is only sustained
through two chords, thus providing a possible way to sidestep the analysis of
that note as an irregularly‐treated suspension.
A third and much more secure analysis is simply to consider the harmonic
framework and the voice‐leading in bar 3 of the extract. Taking the local tonic
as B , the first two beats of bar 3 are obviously a subdominant chord (chord IV).
On the third beat, the chord changes to a dominant seventh in third inversion
(V ), the bass remaining constant to provide the seventh of the chord. On the
fourth beat of the bar, as the bass moves up to F, all that has happened is that
chord V is re‐stated in a different position (i.e. root position, V53), the seventh
of the chord (E ) being transferred through a voice‐exchange from the bass to
the soprano, where it resolves ‘correctly’ in a conventional manner. Bach has
used exactly this procedure elsewhere, namely in the Sinfonia (Three‐part
Invention) No 12 in A BWV 778: in bar 4 of this Sinfonia the first beat is V
(with the seventh of the chord obviously in the bass) and the second beat is V53,
the seventh of the chord having been exchanged to the alto. This passage is also
interesting because, considering the direction of the bass line from the end of
bar 3, Bach has created a passing chord which one would normally expect to
move downwards; here, though it moves upwards.
Ex. 4.57: Three‐part Invention No 12, bars 3 and 4
202 Lester, op. cit., volume 1, page 79
166
This procedure is similar to the unusual register‐shift that was encountered
and discussed in Ex. 4.43 above (page 151); though that is not a strict voice
exchange, the similarity of the way the notes ‘shuttle’ within a chord in both
examples is striking.
It is a relatively straightforward matter to supply conjectural free material to
‘fill in the gaps’ around the contrapuntal framework of Ex. 4.56. As we saw with
the rectus combination, the consecutive fourths in the first bar seem bare and
need to be dressed. The first beat of bar 2 cannot stand as it is, with the effect of
a 6‐4 chord; the alto will need to supply a D (any doubts about the doubling of
the third in a first inversion chord can be dispelled by reference to any number
of examples, such as bar 45 of either version of Contrapunctus 12). The second
half of bar 5 needs to be filled in by rhythmic movement which obviously must
come from the soprano. Here the introduction of an E gives the chord a minor
seventh flavour, helping to prevent too foursquare a feeling to the harmony,
and the consecutive dissonances in the last full bar are in accordance with
Mattheson’s opinions mentioned earlier. I have chosen to send the soprano to
an A on the last note: a straight triad seems too plain and final, even though in
context the music would quickly move on. The slide to a dominant seventh is a
bold move, one I believe not out of keeping with the dramatic heat (or “searing
passion”, as Ferguson puts it in the Preface to his completion203) of this
extraordinary piece of music. Here then in Ex. 4.58 is my solution to the
inverted quadruple combination with a suggested, conjectural filling‐in of the
non‐thematic material.
203 Ferguson Contrapunctus XIV, A Completion of J S Bach's Unfinished Quadruple Fugue from The Art of Fugue, newly edited for organ according to Bach's autograph manuscript and the 1751 original edition
167
Ex. 4.58: The combination complete with suggested non‐thematic material
An outline of a possible completion
This thesis attempts to throw some new light on the end of The Art of Fugue. I
have suggested that there is evidence that Bach left the work unfinished
deliberately as an invitation to posterity to complete the work, and that he may
have left some information about some of the content of any completion,
including the number of ‘missing’ bars. In this Chapter I have suggested that
solutions to both the rectus and inversus quadruple combinations can be found
that are consistent with the information provided by the historical sources,
including the Obituary, Forkel’s Biography and the Prefaces. Putting these
theories together, I believe that it is now possible to make convincing
predictions about what some of the essential ‘building blocks’ of a completion
might be; but since I make no claim to be a composer myself I do not include a
full attempt at a completion of my own, confining myself rather to a
consideration of how we might use the information and musical material that
Bach himself has left us. The following description and outline score provide a
sketch of the possibilities that arise from the discussions in the foregoing pages.
It should be stressed, however, that much of the detail of what follows is not
intended to be set in stone: questions such as the key in which a combination
might be set, or the exact bar number in which a subject or combination
appears, are of necessity subjective opinions and might well be adjusted
168
depending on the free episodic material that a composer attempting a
completion might construct around them. The conclusions I have drawn in this
thesis suggest an outline of a completion that might look like this:
Bach’s last written bar is bar 239. Evidence suggests that a further 47 bars should be added. Bars 239-241: episode Bars 242-248: a second permutation of the triple combination of
subjects I, II and III. Bach’s first such combination (bars 233-239) was in the layering II-III-I so this second combination might be in a different layering. This combination would conclude the third section of Contrapunctus 14. See pages 87-88.
Bar 249: start of section 4 Bars 249-252: statement of IV, most likely in D minor in the
soprano, perhaps in immediate combination and/or stretto with one or more other subjects. See pages 120-121.
Bars 254-258: answer (IV likely in the bass or alto - see page 121).
Bars 259-265: First rectus quadruple combination (II-IV-III-I), perhaps in A minor. See Ex. 4.28, page 140.
Bars 265-266: episode modulating to B flat Bars 267-273: Inversus quadruple combination (IV-III-II-I),
perhaps in B flat major. See Ex. 4.58, page 166. Bar 274: episode modulating to D minor Bars 275-281: Second rectus quadruple combination
(II-III-IV-I) in D minor. See Ex. 4.30, page 141. Bars 282-286: Final statement of IV, likely in the soprano,
perhaps over a tonic pedal; Tierce de Picardie.
The following score (overleaf) shows an outline of the skeleton of these
possible musical events, without including any of the free episodic material.
239 240 241 242 243 244
Bach's last bar
A second triple combination: I-III-II
245 246 247 248 249 250
Start of section 4:statement of IV
251 252 253 254 255 256
Possible combination with III
Answer
257 258 259 260 261 262
Quadruple rectuscombination
169
263 264 265 266 267 268
Quadruple inversuscombination
269 270 271 272 273 274
275 276 277 278 279
Quadruple rectuscombination
280 281 282 283 284 285 286
Final statement ofIV over tonic pedal
170
171
CONCLUSIONS
The apparently unfinished ending of Contrapunctus 14 presents the
performer or listener with an unsettling and frustrating experience, an
impression that is even more graphically heightened in performances or
recordings that break off at bar 239 without any attempt to complete the music
or round it off in any way. The emotive effect of this moment is all the greater
given the widely held belief that these were the last notes that Bach wrote and
that they somehow capture and depict the moment of the composer’s death.
The knowledge that Bach was a highly methodical and logical composer, one
who was used to working to tight deadlines, and not a person who was lazy or
disorganized, makes the suggestion that the reason the music is unfinished is
because he died, or was too ill to complete it, seem very plausible: surely Bach
would have completed this monumental work if he had been able. It is hardly
surprising that the literature is in universal agreement on this position, a
position stated in Bach’s own Obituary and subsequently perpetuated in
successive books, journal articles and recording sleeve notes. To suggest an
alternative hypothesis seems almost heretical.
Nevertheless, this thesis springs from an attempt to revisit the available
evidence, based on my conviction that the conventional statements and
interpretations of the evidence are flawed or misleading. The starting point was
the observation of the fact that Bach’s last written bar is bar 239, a number that
gives the total 14, a number of undisputed significance to the composer and one
which appears so many times during the course of The Art of Fugue that its
appearance in this last bar seemed more likely than not to be deliberate. Since
Bach has used numbers in other places in his music subtly to draw attention to
something that may not immediately be apparent to the casual observer, it
seemed possible that he was doing the same here in this last bar.
172
The very striking differences of appearance between the last page of the
manuscript and the first four pages also seem to demand reassessment. The fact
that eminent scholars are not in agreement about what category of manuscript
the autograph is, together with the many other anomalies and inconsistencies
that I have detailed in Chapter 2, cumulatively suggest that previously held
assumptions about the unfinished ending of the music cannot be justified. In
particular, Professor Christoph Wolff’s well known ‘fragment x’ theory, highly
plausible though it first appears to be, does not seem to withstand logical
analysis. In the light of my presentation of the known chronology of Bach’s last
months, which shows that Bach had a considerable amount of time and health
in which the music could have been completed, the suggestion that Bach chose
not to continue the music on that page of manuscript seems more plausible than
the suggestion that he died over it or that a subsequent page was lost. My
hypothesis is first, that Bach has left the end of the work open deliberately, with
an invitation to posterity to work out the completion, rather in the manner of
the improvisatory tests described by his son; and second, that he has encoded –
perhaps capriciously and certainly enigmatically – a number of clues in his
manuscript to indicate to posterity that that was his intention.
In the second half of the thesis I propose that further hints or clues are
embedded in the manuscript, clues that supply information about how one
might go about completing the music in some accordance with Bach’s
intentions. The question of how much music is needed to complete the work is
one of the most important things that a person attempting a completion needs
to have answered; in Chapter 3 my claim is that Bach has given us the exact
number of bars needed (47) and that that number is strikingly corroborated on a
number of levels, not only by the precise architectural proportions of the music
but also by markings at the end of the score, on which I believe I am the first to
comment.
173
In the final Chapter I turn to the question of some of the musical content of a
proposed completion, suggesting that the statements in the Obituary and other
sources, contrary to the opinions of some writers, are able to be reconciled with
the existing counterpoint. In particular, the words “inverted note for note” are
able to be interpreted literally and, building on the work of Michael Ferguson
and Donald Tovey, my proposal is that Bach has designed his counterpoint so
cleverly that only one inverted quadruple combination is possible within the
very strict parameters that can be seen to apply throughout the whole work.
An attempt at a full completion is outside the scope of this thesis, but I have
given an outline of the main contrapuntal events as they might appear in a
completion. I share Michael Ferguson’s view that Bach would have actively
wanted people to attempt their own completions, and that he would have
relished the knowledge that students and performers were working out their
own endings. No one completion can be said to be completely ‘right’ or
‘wrong’, and in particular the free episodic material is necessarily open to the
individual and subjective ideas of each person. But I believe that certain
fundamental ‘building blocks’ of the musical content of the ‘missing’ bars –
specifically the rectus and inversus quadruple combinations – will be an
essential component of any completion; I believe that with my proposals about
these combinations, together with the evidence I have presented about the
intended length of the music, we may be a step closer to a fuller understanding
of Bach’s intentions.
Indra Hughes Auckland, New Zealand
20 June 2006
174
APPENDIX: FACSIMILE OF MUS. MS. P200, BEILAGE 3
(the autograph manuscript of
Contrapunctus 14)
The original manuscript is held at the Staatsbibliothek in Berlin. The
Staatsbibliothek does not permit the taking of photographs of manuscripts, but
I am grateful to the library staff for allowing me to take copies of the microfilms
of the manuscript. The facsimile on the following pages is taken from those
microfilm copies: the quality of the reproduction is not high, but it is high
enough to enable the reader to see the detail of Bach’s handwriting and the
changes that he made to his score.
180
BIBLIOGRAPHY
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Harcourt Brace Jovanovich Bagnall, A., Baker, T., et al. (1975) ʺSeminar Report: Bachʹs ʺArt of Fugueʺ: An
Examination of the Sourcesʺ Current Musicology 19: 47‐77 Bergel, E. (1985) Bachs letzte Fuge: die ʹKunst der Fugeʹ ‐ ein zyklisches Werk Bonn Bertalot, J. (2000) ʺSpirituality and symbolism in the music of J S Bach Iʺ
Organistsʹ Review 86(3): 222‐225 Bertalot, J. (2000) ʺSpirituality and Symbolism in the music of J S Bach IIʺ
Organistsʹ Review 86(4): 331‐336 Bertalot, J. (2001) ʺLetter to the editorʺ Organistsʹ Review 87(1) Bitsch, M. (1976) ʺMÉLANGES: Deux canons énigmatiques de J. S. Bachʺ Revue
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