A New Theory of Harmonic Motion and Its Application to Pre-tonal and Tonal Repertoire

ISTANBUL TECHNICAL UNIVERSITY ! GRADUATE SCHOOL OF ARTS AND SCIENCES

M.A. FINAL PROJECT

JANUARY 2015

A NEW THEORY OF HARMONIC MOTION AND ITS APPLICATION TO PRE-TONAL AND TONAL REPERTOIRE

Sami Tunca OLCAYTO

Dr. Erol Üçer Center for Advanced Studies in Music (MIAM)

Music Master’s Programme

Anabilim Dalı : Herhangi Mühendislik, Bilim Programı : Herhangi Program

Project Advisors:

Dr. Adam ROBERTSDr. Paul WHITEHEAD

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To my family, Parla and Sofia; for their love and support,

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FOREWORD

I would like to thank my colleagues in ITU MIAM, Aslı, Enis, Rina and Sibil, as they’ve never neglected me from their incredible musical tastes and mind stimulating ideas; my professors, Adam Roberts, Paul Whitehead, Jerfi Aji and Alexandros Charkiolakis, who were always available and helpful when I faced an obstacle; and finally my parents, Beste and Selçuk and my siblings, Tuna, Osman and Can, for their endless love, compassion and support, even in the darkest of times. December 2014

Sami Tunca OLCAYTO

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TABLE OF CONTENTS

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FOREWORD ............................................................................................................. ix!TABLE OF CONTENTS .......................................................................................... xi!ABBREVIATIONS ................................................................................................. xiii!LIST OF TABLES ................................................................................................... xv!LIST OF FIGURES ............................................................................................... xvii!SUMMARY ............................................................................................................. xix!ÖZET ........................................................................................................................ xxi!1. INTRODUCTION FOR A TONAL THEORY ................................................... 1!

1.1 Definition of Tonality ......................................................................................... 1!1.1.1 Components of tonality ............................................................................... 2!

1.2 Emergence of Tonality ....................................................................................... 3!1.3 Tonal Theories on Functional Harmony ............................................................ 5!

1.3.1 Rameau and harmonic motion .................................................................... 5!1.3.2 Scale-degree and function theories: harmonic identity ............................... 8!1.3.3 Neo-Riemannian Theory ........................................................................... 11!

2. PRESENTATION OF THE THEORY .............................................................. 15!2.1 The Octatonic System ...................................................................................... 15!

2.1.1 Resolution of the tritone ............................................................................ 16!2.1.2 Minor third relationship ............................................................................ 16!2.1.3 Major/minor dualism and mixture ............................................................ 17!

2.2 Three Octatonic Collections or “Functions” .................................................... 18!2.3 Root Motion and Tonal Syntax: Progressive and Retrogressive Motions ....... 20!2.4 Diatonicism and Root Motion .......................................................................... 22!2.5 Insights on Tonal Harmonic Practice ............................................................... 23!

2.5.1 Tonicization and Modulation .................................................................... 23!2.5.2 Prolongation .............................................................................................. 25!2.5.3 Chromatic chords ...................................................................................... 26!2.5.4 Major third relationship ............................................................................ 29!

2.6 Further Speculations on Progressive and Retrogressive Motions .................... 32!3. APPLICATION OF THE THEORY ................................................................. 37!

3.1 Introduction for the Analyses ........................................................................... 37!3.2 The Analyses .................................................................................................... 38!

3.2.1 Machaut, “Puis que la douce rousee” ....................................................... 38!3.2.2 Agricola, “Je n’ay dueil” ........................................................................... 39!3.2.3 Palestrina, “Agnus Dei” from Missa Papae Marcelli ................................ 40!3.2.4 Monteverdi, “Era l’anima mia” from Fifth Book of Madrigals ............... 41!3.2.5 Gesualdo, “ O tenebroso giorno” from Fifth Book of Madrigals ............. 41!3.2.6 Carissimi, recitative “Plorate colles” and final chorus “Plorate fiili Israel” from the oratorio “Jephte” ..................................................................... 42!3.2.7 Corelli, “Prelude” from Violin Sonata in E Minor Op.5 No.7 ................. 43!

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3.3 Analytical Results ............................................................................................. 44!4. CONCLUSION ..................................................................................................... 47!REFERENCES ......................................................................................................... 49!APPENDICES .......................................................................................................... 55!CURRICULUM VITAE .......................................................................................... 87!

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ABBREVIATIONS

D : Dominant function Fr +6 : French augmented sixth chord Ger +6 : German augmented sixth chord IMSLP : International Music Score Library Project It +6

: Italian augmented sixth chord M/m : Major/minor intervals T : Tonic function Tr : Tritone P : Perfect intervals S : Subdominant function RM : Root Motion + : Augmented triad Δ : Major seventh chord O 7 : Diminished seventh chord Ø 7 : Half-diminished seventh chord

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LIST OF TABLES

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Table 1.1 : Riemann's Schritte & Wechsel transformations ...................................... 11!Table 2.1 : Motion types according to root interval between chords. ....................... 21!Table 2.2 : Meeus's classification of tonal chord progressions ................................. 22!Table 2.3 : Conceptual Voice-Leadings for Root Motions ....................................... 34!Table 3.1 : Statistical results for harmonic motion ................................................... 45!

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LIST OF FIGURES

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Figure 1.1 : Example of fundamental bass progression under a chord progression ... 6!Figure 1.2 : The perfect and imperfect cadences in major .......................................... 7!Figure 1.3: An example of Riemann’s functional notation ....................................... 10!Figure 1.4: Hyer’s Table of Tonal Relations ............................................................ 12!Figure 2.1: An octatonic collection .......................................................................... 17!Figure 2.2 : The octatonic system ............................................................................. 19!Figure 2.3 : Beethoven, Symphony No.7, ii, mm.1-18 ............................................. 24!Figure 2.4 : Mozart, Piano Sonata No.7 in C Major, K309/284b, iii, mm. 1 ............ 25!Figure 2.5 : Mozart, Symphony No. 40 in G Minor, K.550, mm. 1 ......................... 26!Figure 2.6 : Wagner, “Prelude” from Tristan und Isolde, mm.1 ............................ 27!Figure 2.7 : Beethoven, Piano Sonata No.14, Op.27 No.2 (“Moonlight”), mm. 1 ... 28!Figure 2.8 : Beethoven, Piano Sonata No.21, Op.53 (“Waldstein”), mm. 257 ......... 28!Figure 2.9 : The four hexatonic systems ................................................................... 30!Figure 2.10 : The combined system .......................................................................... 30!Figure 2.11 : J.S. Bach, Prelude and Fugue in C major, BWV 846, mm. 7 ............. 31!Figure 2.12 : Beethoven, Piano Sonata No.21, Op.53 (“Waldstein”), mm. 274 ....... 31!Figure 2.13 : Summary of the table of conceptual voice-leading ............................. 35!Figure 4.1 : Tymoczko's map of tonal grammar in major ........................................ 48!

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A NEW THEORY OF HARMONIC MOTION AND ITS APPLICATION TO PRE-TONAL AND TONAL REPERTOIRE

SUMMARY

In the introduction of the paper, the components of tonality, its emergence in the Renaissance period and also important concepts from the tonal theories of last three centuries will be explained, in order to provide a background for the theory. The next chapter, which constitutes the main body of the study, is the presentation of the octatonic system. The construction of the system from components like tritone and minor third relationship, its connection with the concept of function and diatonicism will be explained in detail. After that, analytical examples from the common practice period will be presented in order to demonstrate the approach of the theory for typical tonal idioms. In the third chapter, the system will be applied to musical examples from Guillaume de Machaut, Alexander Agricola, Pierluigi da Palestrina, Carlo Gesualdo, Claudio Monteverdi, Giacomo Carissimi and Arcangelo Corelli, in order to investigate the relationship between harmonic motion and the emergence of tonality. In the final chapter, the evaluation of the theory will be made, in the context of the preliminaries from the first chapter also with the data from the analyses.

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ARMONİK HAREKETE İLİŞKİN YENİ BİR TEORİNİN SUNULMASI VE TONAL İLE TONAL ÖNCESİ REPERTUAR ÜZERİNDE UYGULANMASI

ÖZET

Giriş bölümünde tonalite kavramı tanımlanırken bileşenlerine ayrılır ve bu bileşenler ayrıca açıklanır. Sonrasında tonalitenin ortaya çıkışı, Rönesans dönemi müziği içerisinde oluşum süreci bağlamında ele alınır. Sonrasında, başta Jean-Philippe Rameau ve Hugo Riemann olmak üzere 18., 19. ve 20. yüzyıllarda müzik teorisi alanında çalışmalar yapan müzik insanlarının tonalite ve fonksiyonel armoni kavramlarına bakış açısı incelenir.

Sonraki bölüm çalışmanın ana bölümü olup, armonik harekete ilişkin bir teori olan oktatonik (sekiz-ton) sisteminin sunumundan ibarettir. Sistem, triton çözülmesi ve minör üçlü ilişkisi gibi bileşenlerden oluşturulduktan sonra armonik hareket, diyatonik ve oktatonik bağlamlar içerisinde tanımlanır ve fonksiyon kavramı, sistem ile ilişkilendirilir. Bu aşamalardan sonra sistem Barok, Klasik ve Romantik dönem müziklerinden alınan örnekler üzerinde uygulanarak; devam ettirme (prolongation), modülasyon, kromatik akorlar ve majör üçlü ilişkisi bağlamlarında açıklamalar yapılır.

Üçüncü bölümde, bir önceki bölümde detaylı bir şekilde açıklanan sistem, Guillaume de Machaut, Alexander Agricola, Pierluigi da Palestrina, Carlo Gesualdo, Claudio Monteverdi, Giacomo Carissimi ve Arcangelo Corelli gibi Geç Ortaçağ, Rönesans ve Barok dönem bestecilerinden alınan müzikal örnekler üzerinde uygulanır ve eserlerdeki armonik hareket istatistiksel olarak saptanır. Bu veriler bölüm sonunda değerlendirilip sonuç kısmına aktarılır.

Sonuç bölümünde sistem, ilk bölümde aktarılan kavramlar ve uygulama bölümünde elde edilen veriler ile bağdaştırılır. Bundan sonra teorinin doğruluk, kapsam, faydalılık, tutarlılık, pratiklik ve anlaşılabilirlik gibi ilkeler üzerinden değerlendirilmesi yapılır.

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1. INTRODUCTION FOR A TONAL THEORY

This study focuses on functional harmony, or more precisely, the harmonic motion

component of functional harmony, in order to create a new perspective on the

investigation of vertical sonorities. The employed methodology is the octatonic

system, which will be explained in a detailed way in the next chapter. The goal of

this study is not only to present the octatonic system, which provides a better

understanding of the reciprocal relationship between harmonic motion and identity;

but also to point out the increasing percentage for progressive motion while moving

from modality to tonality.

In this introductory chapter, the definition of tonality, in terms of its components will

be made, which will be followed by a short history of its emergence in Renaissance

music and a brief presentation of the concepts from different traditions of tonal

theories, which are synthesized into a theory of harmonic motion in the next chapter.

1.1 Definition of Tonality

Even though “tonality” is one of the most used terms in the musical vocabulary since

its coinage by Choron in 1810, there is no consensus on its definition among music

theorists, musicologists or musicians in general. Hyer touches on this issue and lists

eight different and equally valid definitions, but according to him, its most common

use is “to designate the arrangement of musical phenomena around a referential tonic

in European music from about 1600 to around 1910” (2008: 728). According to this

rather specific definition among others, tonality is a compositional tool or a cognitive

process that regulates both dimensions of pitch space, melody and harmony, and by

doing that, it creates a sense of directionality or goal-orientedness through different

forms of musical experience. As the temporal boundaries of the above definition

suggests, common practice tonality is also used to express the stylistic features of the

period and to avoid confusion with other meanings of the term.

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1.1.1 Components of tonality

A better understanding of tonality, in my perspective, requires a certain amount of

deconstruction. In that sense, tonality can be seen as a system of relationships

between different musical concepts. According to the definition of tonality above,

firstly, a fundamental sonority that all of the melodic and harmonic pitch events

relate to must be present: the tonic triad or its root in the most basic form. The

centricity of the fundamental sonority, whether in the form of a chord or a single

pitch class, constitutes the first component of tonality.

Diatonicism is another of these elements that is frequently presented in conjunction

with the element of centricity; as the two forms of tonality, major and minor, are

defined by the quality of their tonic triad and also by the corresponding scales or

modes, which originate from the diatonic collection1. However, centricity and

diatonicism are mutually exclusive musical concepts, as there are examples of non-

centric diatonic music, like Medieval and Renaissance polyphony, in which there is a

sense of floating tonality; likewise, examples of non-diatonic centric music, as in

post-tonal works of 20th-century composers, like Bartók, Debussy and Stravinsky.

Nevertheless, the musical significance of the diatonic scale is not exclusive to the

common practice period, since at least from the times of Ancient Greeks, it has been

and still is the primary background pitch space of musical organization.

The idea of multi-layered structure in music originates from the works of dualist

theorists like Hauptmann and Riemann, but it has been fully developed into a body of

comprehensive musical concept in the theories of Heinrich Schenker. Hierarchy

among the tones of a scale and harmonic prolongation constitute the heart of this

component concept, which states that tonality exists and is perceived in several

structural levels, just as layers of different materials in visual art.

Last but not least, functional harmony is also one of these component concepts,

which constitutes the central point of this study and may be described crudely as

logical ordering of harmonies. This description creates more confusion than clarity,

as what “logical ordering” means is not clear at all. 1 The two forms of minor scale, Harmonic and Melodic minor scales are seen as derivations of Natural

minor scale, which coincides with the sixth mode of diatonic scale, Aeolian. This issue was central for

dualist harmonic theory and will be dealt with in the corresponding section.

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As Kopp suggests, functional harmony can be better understood as a concept that

unifies two components: harmonic identity (or meaning) and harmonic motion

(1995). According to this distinction, harmonic identity is the special meaning that

chords receive in relation to the tonic pitch class or chord; which is generally referred

to as three functions: tonic, dominant and subdominant. On the other hand, harmonic

motion (or action) is the transition between these meanings, or functions.

In order to help clarify this definition, an analogy can be drawn from linguistics:

syntax is the logical sequence of word types in order to create meaningful sentence.

In the context of functional harmony, word types are harmonic identities and syntax

is harmonic motion.

In short, tonality consists of separate but interrelated component concepts of

centricity, diatonicism, multi-layered structure and functional harmony. This is not to

say that tonal music consists only of these elements and other musical parameters are

trivial features. The point here is to express that tonality (not tonal music) is mostly

governed by pitch domain while other parameters - like grouping, rhythm, dynamics

and timbre - interact with it in an auxiliary manner.

1.2 Emergence of Tonality

Tonality has been thought of as a product of humanist thinking in Renaissance,

analogue to perspective in visual arts.(Korsakova-Kreyn, 2010: 15) Similarly with its

definition, there isn’t an agreement among scholars on a period for its exact origins,

ranging from 14th

to 17th

centuries (Dahlhaus, 1990: 3). In order to understand this

divergence and to provide a background for the tonal theories in the next section, it is

logical to make a brief account of changing trends in compositional practice and also

in theoretical concepts through Late Medieval and Renaissance periods.

Polyphony has a quite slow progression between 10th

and 13th

centuries, starting

from the advent of organum. The parallel 5ths and 4ths, the basic material of this

earlier form of polyphony became relatively rare in the the sacred music of Notre

Dame school and also in the secular songs of Adam de la Halle; in the period

between late 12th

century and 13th

century, which is commonly referred to as ars

antiqua. In the 14th

century, the emergence of syncopated style and the wide use of

imperfect intervals resulted with a new style of polyphony across Europe, ars nova,

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which is represented by composers like Machaut, Vitry and Landini. The beginning

of 15th century is marked by the style of Burgundian school (including composers

like Dufay and Binchois), which esteems the melodic quality of each voice, imitation

and harmonic completeness; in that sense, the works of the composers of this era

paved the way for Renaissance counterpoint. Adding to that, the four-voice texture

had become the norm around that time.

The idea of imitation and equality among voices became the focus of compositional

aesthetics in the Renaissance, through the works of masters like Ockeghem and

Josquin. In the 16th century, developments in cadential patterns and movement of

parts, handling of dissonance and the wide use of five-six voice texture resulted with

a shift of focus towards harmonic dimension. This shift from the aesthetics of High

Renaissance can be best seen in the works of Roman school, best represented by

Palestrina.

Towards the end of the 16th century, Italian madrigal style gave a new direction to

the progress of polyphony. Starting from his Third book of madrigals, Monteverdi

has been the leader of a new style that stipulates the seperation from 16th century

principles of strict counterpoint, equality among voices and treatment of dissonance,

in favor of dramatic expressiveness, with the use of freer dissonance and hierarchy in

voices. The emergence of dramatic genres laid the groundwork for the forthcoming

of instruments as accompaniment in monodic texture, which resulted with basso

continuo and figured bass practice.

The essential concept of polyphony, interval categories, have evolved through this

timespan, not only in terms of their content, but also in the implications of their use

in composition2. Franco’s intervallic classification and contrapuntal teachings were

in force until the 14th century. During the 15th and 16th centuries, Tinctoris, Gaffurius

and Zarlino elaborated the principles of counterpoint, but only in the works of the

latter the harmonic perspective has been fully emphasized. In the same period,

Glarean introduced the twelve mode system, which has been accepted by some and

rejected by others. In the 17th century, the developments in solmization and

intonation underlined the change of tonal system, from modality to tonality.

2 For reference, see “Table of Consonance/ dissonance interval- classification systems in 9th-16th centuries” in Tenney, 1988: 109.

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1.3 Tonal Theories on Functional Harmony

In this section of the introductory chapter, the speculative music theory traditions of

eighteenth and nineteenth centuries will be briefly examined under two categories,

according to their focus on either of the two aspects of functional harmony, harmonic

motion and harmonic identity. The theoretical studies in the twentieth century can be

considered either as expansions or extensions of these traditions; for instance, the

more recent works of Neo-Riemannian theorists that will also be briefly examined.

However, there are exceptions, as in the case of Schenker, who focuses entirely on

another component concept of tonality, the multi-layered structure. In that sense, his

work emphasizes large-scale contrapuntal relationships, which are hidden beneath

the surface of actual music, at the expense of the vertical aspect, the functional

harmony.

The aim here is not to make a list of all theories on tonal harmony that have been

conceived in that period and to explain them in the most detailed way, but actually is

to designate the origins of the theoretical concepts that the presented theory relies on.

With that excuse, other nonconventional and interesting ideas regarding functional

harmony, such as Kurth’s energetic-based conception of melody and harmony, and

Hindemith’s acoustically-based system of intervals will not be touched upon. For a

fully-fledged history of tonal theories, Mitchell’s dissertation (1963) should be

referred to.

1.3.1 Rameau and harmonic motion

Jean-Philippe Rameau has been widely acknowledged as the first theorist to codify

the compositional principle that will last through the common practice period or put

more simply, as the founder of tonal harmony. It is undeniable that his theoretical

works paved the way for a better understanding of harmonic tonality, even though

some of the concepts that his theory relied on were not entirely new.

Parallel to the aforementioned shift of compositional technique in the 17th century -

from interval progressions to harmony as the basis for counterpoint – chords, in place

of intervals, became the primary units of vertical dimension in theoretical works.

Starting from as early as Zarlino, triads have been acknowledged and emphasized,

but it wasn’t until the works of seventeenth-century theorists, like Harnish and

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Lippius, that 6/3 and 6/4 triads were seen as rearrangements of, and as essentially

same entities with 5/3 triads; in other words, the concept of chordal inversion made it

possible to think of harmony in terms of chords. The root, or the lowest note of a

chord in 5/3 position, has been given a central position in that perspective; as in the

work of Descartes (1618, Compendium Musicae) in which it becomes the

fundamental that generates all other intervals above it. Following these

advancements in harmonic theory, Rameau’s most important contribution to this

field is the concept of fundamental bass; in which he combines chordal theory,

diatonicism (in the form of major and minor keys), centricity of tonic triad and

resolution of chordal dissonance in order to explain the very foundation of harmonic

tonality in his conception, the harmonic motion.

In the fundamental bass progression, which consists of chordal roots in succession,

the only allowed intervals between adjacent roots are ascending and descending

perfect fifths and thirds, the component intervals of a triad. The roots of these chords

may be different from the bass as in inversions, or may not be present in the sonority

at all; in these cases, an imaginary bass pitch is considered under the actual bass. In

that way, Rameau explains any sonority either as a triad or a seventh chord.

Figure 1.1 : Example of fundamental bass progression under a chord progression

demonstrating the rule of the octave (Dahlhaus, 1990:29)

For motivational aspect in harmonic motion, Rameau relies on the ancient

contrapuntal rule of dissonance to consonance resolution; the chordal dissonance of

seventh, which may be present or implied, becomes the link between these chords. In

terms of harmonic identity, Rameau identifies three chord types, according to the

resolution of the dissonance (if there is) and the progression of the fundamental bass.

Root Progression and · 29

triads form a context that establishes a , ought, on the ,..,.... . .,-1- .... ,, ...

to be understood as a confirmation-or an attempt at a confirmation-of this principle.

Rameau does not expressly state that an imagined dissonance can be understood as a hypothetical factor and need not be deemed real or jointly heard. Yet this view can be indirectly inferred from his analysis of the following chord progression based on the regola del-l' ottava [rule of the octave]. 19

4 .,, -:::;: _fl_ =& ,,, Jllr _IT ""' -""' =i -u- :::;: -:::;: _.u!'::'. :n: <IP -u -u e-

6 6 6 5 6 e-_.=. .n

-n __o_

Basse continue

7 ::"ii>•

_I 7 7 7 7 -;;::;; -"" _fl_

-e- :::z: =& --0 Basse fondemantale

Example 3

In the second measure, Rameau interprets the first-inversion C-major chord as a fragment of a seventh chord on A so that by a resolution of dissonance and a fifth-progression of the- imagined basse fonda-mentale (A-d) he can link together the chords over e and f in the thoroughbass. According to Rameau's version of the figured bass, however, the resolution of the dissonance is irregular: the seventh (g') over the imagined bass (A) progresses upward to the octave a' instead of downward to f'. And his apparent indifference toward an illegal resolution of dissonance may serve as an indication that Rameau understood an imagined, tacitly implied dissonance to be a concep-tualized tone that did not have to be jointly heard.

Whether Rameau's theory is a theory of harmony in the 19th-century sense thus depends on the interpretation of imagined dissonances. A traditional component of Rameau's theory, the factor of linking chords by dissonances, is based on the principle of the variation of intervallic quality. And one of the basic ideas of contrapuntal theory from the 14th through the 17th century is that the variation of intervallic quality-the tendency of dissonance toward consonance, or of imperfect consonance toward perfect consonance - forms the driving force behind music's forward motion. A chain of sixths striving toward the perfection of an octave differs of course in degree, but not in principle, from Rameau's progression of seventh chords whose goal is a triad- an accord parfait.

7

The subdominante is identified with the ascending fifth motion in fundamental bass

and the added sixth as dissonance, which creates a second with the chordal fifth.

Dominante, on the other hand, receives the seventh above the root with a progression

of descending second. A chord without dissonant seventh is identified as tonique,

which doesn’t have the impulse to move like the other two categories.

Having stated that, Rameau takes the directionality of certain cadence types as the

model for harmonic motion between his chord types. The descending fifth becomes

the preferred fundamental bass motion over its ascending counterpart, which appears

in irregular cadences. On the other hand, thirds are allowed between consonant

“tonique” chords, while stepwise motions are explained with the use of interpolation

in the fundamental bass and implied dissonances.

Figure 1.2 : The perfect and imperfect cadences in major, from Rameau’s Traité de

l’harmonie (Lester, 2008: 762-763)

Looking from this perspective, Rameau’s fundamental bass seems to be a theory of

harmonic motion rather than of functions, as the harmonic identity, or chord type in

this case, is deduced from the progression of a chord to the next. In other words,

harmonic motion determines harmonic identity, not the other way around. As the

theory relies on diatonicism and centricity of tonic, chromatic harmonies that cannot

be explained as in modulatory progressions are not at all considered.

Simon Sechter (1788-1867) expanded the system of fundamental bass by employing

triads and seventh chords on all diatonic scale-degrees, in order to explain chordal

relationships. The primary root motion is still descending fifth, as exemplified in his

example of the cadence) reveals Rameau’s notion of how a sense of key is formed: thedissonances in the dominant seventh chord propel the chord toward its consonant res-olution in the triad a fifth lower; that progression defines the tonic as the point of repose.And it is the sense of motion from one chord to the next connected by a fundamentalmotion of the perfect fifth – and not merely the content of the individual harmonies –that is the essence of the cadence. This imparts a dynamism to Rameau’s theory of tonal-ity; a key is not merely a given pitch field within which harmonies and melodies move,but a harmonic focus that emerges from the dynamic of the progression.

The Perfect Cadence is one of the two basic cadential types that Rameau proposes.The other is the Irregular Cadence (cadence irregulière): a chord built on the fourthdegree of the scale moving to a tonic, in which an added sixth makes the first chord dis-sonant, propelling it toward a resolution, as shown in Plate 24.2.

Like the Perfect Cadence, the Irregular Cadence follows the normative motion of aperfect fifth in the fundamental bass, albeit ascending rather than descending. Andlike the Perfect Cadence, the Irregular Cadence reflects the mechanistic model of adissonance impelling a chord toward consonant resolution. (Rameau’s arguments forjustifying the added sixth as a dissonance comparable to the seventh in the dominante-tonique required some subtle reasoning to be discussed below.)

Rameau considers these cadences not only as the progressions that end phrases(which is the way we generally use the term “cadence” nowadays), but as the models

762 joel lester

Plate 24.1 The perfect cadence, from Jean-Philippe Rameau, Traité de l’harmonie,Book II, Chapter 5, p. 57

Cambridge Histories Online © Cambridge University Press, 2008

for directed harmonic motion in general. He saw all music as basically a series of inter-connected cadences, in which most cadential conclusions are evaded (adopting andadapting from Zarlino) because one (or both) of the chords has been altered to avoid acomplete resolution: one or both chords might be inverted, the chord of conclusionmight itself contain a seventh or an added sixth, or the third in a dominant seventhmight be minor to remove the drive of the leading tone (in which case it is no longer a“dominant-tonic,” but merely a simple “dominant” chord since it no longer has thepower to define the following chord as the tonic of a key). In all cases, though, themusic is driven onward by the motivating force of the dissonant seventh (or occasion-ally, the added sixth) involving largely fifth motion in the fundamental bass until a finalpoint of consonant repose is attained at the tonic.

With Perfect, Irregular, and evaded cadences, Rameau tried to show how the fun-damental bass proceeded primarily by fifths and thirds – the very intervals generatedfrom the fundamental string. He thereby explored the recently developed sense oftonal directionality that di◊erentiated the music of his time from that of earlier gener-ations. He was so enthralled by his ability to explain directed harmonic motion involv-ing triads and seventh chords moving by fourths and fifths as a series of real or evadedcadences that he attempted to extend these insights to all the chordal types and alltypes of harmonic connections.

“All chordal types” for Rameau meant those indicated in thorough-bass signatures.Thorough bass generally indicated chords to be played along with bass notes: conso-nant chords on the beat, and also suspensions. Simultaneities that arose from various

Rameau and eighteenth-century harmonic theory 763

Plate 24.2 The imperfect cadence, from Jean-Philippe Rameau, Traité de l’harmonie,Book II, Chapter 7, p. 65


8

“Sechterian chain.” The stepwise fundamental bass progressions are explained by the

concept of “concealed” fundamental, which is similar to Rameau’s interpolated

fundamental bass progressions. His actual contribution to fundamental bass tradition

is the conception of chromatic harmonies, which he refers to as “hybrid chords” with

notes derived from multiple keys, not as alterations of diatonic scale degree keys.

Sechter’s adaptation of Rameau’s fundamental bass theory has been widely accepted

in theoretical circles around Vienna and has been further extended by Mayrberger

and Bruckner, in order to accommodate with the chromatic harmonic practice of

Romantic period. The works of these theorists are regarded as the predecessors of

harmonic theories of Schoenberg and Schenker.

1.3.2 Scale-degree and function theories: Harmonic identity

As Bernstein notes (2008: 778) there are three main trajectories in the German-

speaking realm after the impact of Rameau’s theory of harmonic motion: the scale-

degree (Stufen) tradition that originates from theories of Vogler and Weber,

aforementioned extensions of fundamental bass theory by theorists like Sechter and

Mayrberger, and finally, the function theories of dualist tradition, which have been

fully expressed in the works of Riemann.

The fundamental difference between these three traditions lies in their conception of

tonality; scale-degree and fundamental bass traditions embrace the diatonic scale as

the theoretical basis, while function theories rely on chordal relationships. In other

words, scale-based conception of tonality defines harmonic identity and motion on

the basis of diatonicism and centricity of the tonic scale-degree; while harmonic

conception deduces diatonicism and harmonic motion according to harmonic identity

and centricity of the tonic triad 3.

The first theorist to create a systematic scale-degree theory was Georg Joseph

Vogler, who was the first to use Roman numerals as chord labels. He took Zarlino’s

senario4 as the starting point and extended it to 16th partial, resulting with the

“natural” scale and finally, he concluded that major and minor scales can be derived

3 Dahlhaus puts these different conceptions of tonality in comparison thoroughly, which are represented by Fétis and Riemann (1990:7). 4 The series of ratios that expresses all of the consonances that the ear can perceive directly: 1:2:3:4:5:6

9

out of this scale. According to his theory, any of the major, minor and diminished

triads could be formed on any degrees of these two scales; the result was an awkward

system of chords with labels in respect to their fundamental scale-degrees.

It was Gottfried Weber, a contemporary of Sechter, who gave the definitive form to

the scale-degree theories. He adopted the Roman numerals to label chords in relation

to the tonic from Vogler, but he didn’t speculate about the acoustical foundation of

harmonic relationships. In his system, major, minor and diminished triads and minor,

major, dominant and half-diminished seventh chords are labeled according to their

positions in a diatonic scale. As a result of this diatonic basis, chromatic chords were

explained with the concept of minor and major mixture; in other words, according to

this system, any non-diatonic chord is a modulation away from the tonic key.

Unlike Rameau’s theory of fundamental bass, the scale-degree theory doesn’t have

prescriptions for chord progressions, as these directly intelligible chords do not have

tendencies to progress anywhere. Apart from the acknowledgement of the

directionality of certain cadential progressions, scale-degree theory is not involved

with harmonic motion. In short, both harmonic identity and consequently, harmonic

motion are described from the perspective of diatonicism.

The idea of harmonic dualism originates from the works of Moritz Hauptmann

(1792-1868) who adapted Hegelian dialectic to musical context. In his work, logic is

the theoretical basis not only for the construction of the triads, but also for the

designation of primary triads in a key and also for the syntactical ordering of chords

in a cadential progression. In order to explain the consonance of minor triad, he

“inverted” the relationships of the root with the third and the fifth of the major triad;

for example, in a major chord, the root has a consonant major third and a perfect

fifth, while in a minor chord, this note is a major third and a perfect fifth of two other

notes (as in C-E-G and F-A�-C). (Harrison 1994: 227)

In order to explain major and minor harmony as dual counterparts, Arthur von

Oettingen (1836-1920) took this opposition of having/being to the extreme and ended

up with the idiosyncratic system of tonicity/phonicity; which consists of overtone

series with a common fundamental and next to it, the “fundamental” series with a

common overtone. Every aspect of major harmony becomes inverted in order to

explain the so-called true nature of minor harmony (ibid., 243).

10

Hugo Riemann (1849-1919) has been heavily influenced by these two theorists while

working on his own function theory. Riemann’s harmonic system consists of two

different mechanisms for harmonic identity and harmonic motion (Kopp 1995). The

funktion component of his system takes the fifth relation as its basis to designate the

primary chords of a key: subdominant, tonic and dominant functions. The functional

meaning of diatonic secondary chords and also chromatic chords are defined

according to their common tones with these primary chords. The relationship

between primary triads and such “derivative” chords are expressed via three basic

transformations, P (Variation) R (Paralel) and L (Leittonwechsel), as in Figure 1.3.

Figure 1.3: An example of Riemann’s functional notation (Bernstein 2008:798)

The purpose of these transformations on primary chords is to account for any

progression that involves major and minor triads on all twelve pitch classes based on

their functional identity; in other words, this system only takes into account the

identities of individual chords which progressions link to each other.

The component that is accounted for harmonic motion in his system is the twofold

Schritte/Wechsel, which might be considered as the simplified version of Oettingen’s

dual system of chord relationships. While Schritte progresses a major or minor chord

to a transposition of the same quality, Wechsel inverts a major chord to minor, or

vice versa, and progresses it into a transposition of the chord with the opposite

quality (Klumpenhouwer 2008: 466).

While all possible major and minor chord progressions may be expressed by these

two transformations, there is little explanation for motivational aspect of the

harmonic motion. Logical syntax of his earlier works has been sacrificed in order to

account for all chromatic major and minor chords in dualist perspective. Therefore,

as Dahlhaus points out, because of the lack of rules and norms of harmonic

progression, Riemann’s functional system is rather descriptive than logical (Kopp,

1995).

11

Table 1.1 : Riemann's Schritte & Wechsel transformations (Klumpenhouwer 2008: 471)

1.3.3 Neo-Riemannian Theory

Neo-Riemannian theorists seek to revitalize the ideas of the influential German

theorist by overcoming the limitations of dualist thinking and also with the use of

various analytical technologies in order to analyze chromatic triadic music that

juxtaposes tonal and post-tonal features. As the principles of tonal harmony are no

longer adequate solely to explain the harmonic aspect of this repertoire, six

theoretical concepts have been substituted for them in the works of Neo-Riemannian

theorists: triadic transformations, common-tone maximization, voice-leading

parsimony, "mirror" or "dual" inversion, enharmonic equivalence, and the "Table of

Tonal Relations.” (Cohn, 1998: 169) These concepts originate from the works of

nineteenth-century theorists, namely Oettingen and Riemann, who employed them in

a conception of tonality that combines diatonicism, centricity and harmonic function

with dualism. However, in order to capture the significance of harmonic function,

Neo-Riemannian theory isolates the aspects of diatonic centricity and dualism from

these concepts.

Monat Mai” from Schumann’s Dichterliebe, we can assert that the piece presents in turnthe following four tonal genera: Cs minor-major, A major, Fs minor-major, D major-minor. Moreover, the transformations given in Table 14.1 can be shown to have indi-vidual tonal value, by referring them to trajectories on one or more of thetopographies.

Secondly, the topographies form the basis from which to understand Riemann’stheory of dissonant (non-triadic) events, which derive ultimately from his conceptual-ization of tonality – that is, his four modes of tonality – along the lines presented inFigures 14.4–14.7.

Dualist tonal space and transformation in nineteenth-century musical thought 471

Table 14.1 Riemannian Transformations

I. Schritte

Transformation Interval Klang deployment Examples

11 Quintschritt P5 I to II C↑ → G↑; E↓ → A↓12 Gegenquintschritt P4 II to I G↑ → C↑; A↓ → E↓13 Ganztonschritt M2 twice I to II F↑ → G↑; B↓ → A↓14 Gegenganztonschritt m7 twice II to I G↑ → F↑; A↓ → B↓15 Terzschritt M3 I to III C↑ → E↑; E↓ → C↓16 Sextschritt M6 II to III G↑ → E↑; A↓ → C↓17 Leittonschritt M7 I to II plus I to III F↑ → E↑; B↓ → C↓18 Gegenleittonschritt m2 II to I plus III to I E↑ → F↑; C↓ → B↓19 Gegenterzschritt m3 III to II E↑ → G↑; C↓ → A↓10 Gegenterzschritt m6 III to I E↑ → C↑; C↓ → E↓11 Tritonusschritt d5/a4 twice I to II plus I to III F↑ → B↑; B↓ → F↓

II. Wechsel

Transformation Definition Examples

12 Seitenwechsel Invert a klang around I C↑ ↔ C↓13 Quintwechsel Quintschritt, then Seitenwechsel F↑ ↔ C↓14 Sextwechsel Sextschritt then Seitenwechsel G↑ ↔ E↓15 Leittonwechsel Leittonscritt then Seitenwechsel C↑ ↔ B↓16 Ganztonwechsel Ganztonschritt, then Seitenwechsel G↑ ↔ A↓17 Terzwechsel Terzschritt, then Seitenwechsel C↑ ↔ E↓18 Tritonuswechsel Tritonusschritt, then Seitenwechsel F↑ ↔ B↓19 Gegenterzwechsel Gegenterzschritt, then Seitenwechsel C↓ ↔ E↑20 Gegenganztonwechsel Gegenganztonschritt, then Seitenwechsel C↓ ↔ D↑21 Gegensextwechsel Gegensextschritt, then Seitenwechsel E↑ ↔ G↓22 Gegenquintwechsel Gegenquintschritt, then Seitenwechsel G↑ ↔ C↓23 Gegenleittonwechsel Gegenleittonschritt, then Seitenwechsel C↓ ↔ B↑


12

According to Cohn, the origins of Neo-Riemannian theory lies in Lewin’s

transformational system, which consists of two classes of transformation: contextual

inversion that maps a major or minor triad to its related triad, and shift

transformation, which shifts a triad left or rightward on a series of generalized

intervals (ibid., 170). Hyer takes Lewin’s system as the basis and derives from it the

triadic transformations of PAR (P) , REL (R), LT (L) and DOM (D). The nineteenth

century idea of geometric representation of harmonic relationships reappears in the

form of “Table of Tonal Relations” in Hyer’s work (Figure 1.4), in which triadic

transformations can be seen as vectors between triadic triangles (ibid., 172).

Figure 1.4: Hyer’s Table of Tonal Relations (Cohn, 1998: 172)

Having a parallel trajectory, Kopp (2002) focuses on chromatic mediant relationships

while Cohn (1996, 1997, 2012) relies on parsimonious voice-leading in hexatonic

systems (Figure 2.9). Other recent works from theorists like Lerdahl (2001), Rings

(Tonality and Transformation, 2011) and Tymoczko (2011) take conceptual distance

in pitch-space between sonorities as the theoretical basis and make use of complex

mathematical formulas or highly sophisticated geometrical models. Due to these

indirect analytical interfaces, the aspect of perceptibility in the music is often

questionable in transformational theory. Also, the combination of common-tone and

grouping principles creates a false impression of equivalence and interchangeability

between distantly related chords. Adding to that, there is often very limited

explanation for the preference of certain progressions, and exclusion of some others,

in the common-practice tonality.

Harrison is another theorist who follows Riemann’s path, but his distinctive approach

is essentially different from others. Contrasting to transformational theory, his work

is based on a restatement of dualism, which has been expressed as the major-minor

F# - C#- G# -D# -A#

B F C. D A" / D L R

Db Ab Eb Bb F -C

Fb- Cb- Gb Db Ab

Figure 2

cate Hyer's four transformations as they act on a C-minor triad. Each of the three contextual inversions inverts a triangle around one of its edges, mapping it into an edge-adjacent triangle. P, for Parallel, inverts around a horizontal (perfect fifth) edge, mapping C minor to C major; R, for Rel- ative, inverts around a secondary diagonal (major third) edge, mapping C minor to Eb major; and L, for Leading-tone-exchange, inverts around a main diagonal (minor third) edge, mapping C minor to Ab major. The fourth transformation, D (for dominant), transposes a triangle to the vertex-adjacent triangle to its left, mapping C minor to F minor. (Note that the D transformation is redundant, since it is produced by a composition of L followed by R.) Transformational directions on the Tonnetz are invariant, although the direction of the arrow is reversed when contextual inversions are applied to major triads. The Tonnetz thus provides a canonical geometry for modelling triadic transformations.

The objects and relations of Hyer's Tonnetz, unlike those of most nineteenth-century antecedents, are conceived as equally tempered, a cir- cumstance that Hyer acknowledges by substituting enharmonically neu- tral integers for the enharmonically biased pitch-class names favored by Lewin (and used in Figure 2). Under equal temperament, the horizontal axis of Pythagorean fifths becomes the circle of tempered fifths, and the diagonal axes of justly tuned thirds become the circles of tempered major and minor thirds respectively. The table becomes circularized in each of its dimensions, and the entire graph becomes a hypertorus. Such a conception greatly enriches the group structure of the transformations, which Hyer explores in detail.

Hyer's appropriation of the Tonnetz intensifies the relationship between triadic transformational theory and nineteenth-century harmonic theory. Versions of the Tonnetz appeared in German harmonic treatises

172

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13

opposition in “sonorious materials”: keys, scales or chords (Harrison, 1994: 15).

Another fundamental in his theory is the idea of functions, namely tonic, one lower

and one upper dominant, subdominant and dominant. In his theory, the primary triads

on first, fourth and fifth degree create the link between dualism and functions; in

which the major and minor diatonic scale degrees are assigned with individual

functional roles according to their appearance in these primary triads. Secondary

triads and other sonorities that consist of different combinations of diatonic tones and

their “projections” are labeled according to this perspective and in harmonic

successions, functional discharge occurs in each voice individually.

The inevitable result of this approach is the functional mixture, when the tones of a

sonority carry multiple meanings from different functions 5. In such situations, the

relative power of each function depends on the amount of tones assigned to each

function, as well as other parameters that articulate them, like register, timbre and

dynamics. In other words, the answer to how such a sonority “functions” in relation

to the tonic is context dependent and thus, open to interpretation. In connection to

this, the need for the referential point, the tonic, in order to assign harmonic identity

is, in my point of view, another important limitation. Harrison presents two strategies

in that sense: position-finding and position-asserting; but in passages where tonal

disunity is prominent, there is a great possibility for obscurity because of multiple

interpretations that are equally likely, according to different key centers.

Nevertheless, the accuracy of his theory by means of its resolution in chromatic pitch

space is remarkable.

While Riemann used the transformations (P, R, L) in order to designate the

derivation of sonorities from primary triads, neo-Riemannian theory makes use of

them as operations in harmonic successions that transform or progress one chord into

the next. In that sense, components of functional harmony that used to be examined

seperately (as with transformation and Wechsel/Schritte systems in Riemann) have

been fused together. However, the motivational aspect and syntactical rules of

functional harmony, or in other words, the underlying principles of tonality have not

been explained in a satisfactory way. As a result of this, analytical works of neo-

Riemannian theorists appear to be descriptive rather than prescriptive. 5 Dominant seventh chord is an example for this: In C major, G-B-D are base, agent and associate of Dominant function, while F is the base of Subdominant.

14

15

2. PRESENTATION OF THE THEORY

Before going through the technical details of the theory, it is crucial to explain

beforehand, in most basic sense, what the theory consists of, what its assumptions are

and how it aims to contribute to the existing body of tonal knowledge. The theory

makes use of the octatonic system, which provides a model for harmonic motion in

triadic and occasionally tetradic musical contexts. It explains progressive and

retrogressive motions in terms of transitions from one octatonic collection to another,

and by doing that, it bypasses the necessities of diatonicism and centrality of tonic.

The motions between chords are identified with respect to the interval between

adjacent roots; therefore, the basis of the theory is the concept of root motion, which

has its origins in Rameau’s fundamental bass theory. This subject has been explored

further by Schoenberg (1954), Sadai (1980), Meeus (2000) and Tymoczko (2003,

2010); but in all cases, the pitch space is almost entirely diatonic. The contribution of

the presented theory to this lineage is the underlining of the octatonic system, which

proves to be a useful vehicle while venturing into the chromatic pitch space.

2.1 The Octatonic System

The octatonic system consists of three octatonic collections. An octatonic collection

has eight pitch classes with alternating semitones and whole tones between its

constituents; for example the octatonic collection that starts from C is: C - C♯ - D♯ - E

- F♯ - G - A - B�. Because of its symmetrical nature, there are only three different

octatonic collections (CI, CII and CIII in Forte’s terminology, starting from C♯, D

and D♯). After the third reiteration, the fourth maps onto the first collection, when we

take into account enharmonic equivalence.

The octatonic system, which constitutes the basis of the theory, should be seen as an

a posteriori analytical apparatus rather than an a priori phenomenological law that is

evident in nature. Therefore, in order to justify the theoretical significance of the

system, its construction from basic tonal elements will follow.

16

2.1.1 Resolution of the tritone

Rameau in his Treatise (1722) shows the resolution of a dominant seventh to a major

or minor tonic as the most basic progression of tonality, dominante-tonique. The

dissonances of the dominant seventh chord propel to the consonance with the root a

fifth below; in his terminology, the minor dissonance is the chordal seventh (4),

while the major dissonance being the leading-tone (7). In another perspective, these

dissonances, which are labeled in reference to the bass note, may be seen as pitch

classes that are a tritone apart.

This interval is called “minor fifth” in Fétis’s terminology, as there appears to be

nothing “diminished” with an interval that can be found within the diatonic scale.

Therefore the “appellative” consonance between tonic-fa and dominant-mi degrees6

has a central role in his explanation of the fundamental law of tonality: the resolution

of this consonance (!) to a third or a sixth in the tonic chord.

Another important aspect of the tritone is its symmetrical nature; there are only six

pairs of tritone related pitch classes, the seventh maps onto the first pair. This quality

lays the ground for the “tritone substitution” concept; the interchangeability of tritone

related dominant seventh chords, which are assumed to be cognitively similar

because of their common tones7. A reharmonization technique that is frequently used

in jazz, this concept explains the augmented sixth chord as the tritone pair of

dominant seventh chord on second degree: the dominant seventh of the dominant, or

simply II7 8. Consequently, the common pitch classes resolve in the same way in

tritone related chords.

2.1.2 Minor third relationship

Another highly symmetrical structure that has only three different forms, the

diminished seventh chord, is frequently used in tonal harmony, as it can be found on

the seventh degrees of harmonic or melodic forms of minor mode and is also used 6 In the ancient solmization system of Guido (11th century), the pitches that have a pitch a semitone below and above are sung as “fa” and “mi”, respectively , in relation to the “root” of the hexachord. 7 For example: G7 and D�7 chords share tritone apart pitch classes: F and B (C�). The third of the first chord becomes the seventh of the second chord and vice versa. 8 For example: in C minor context, augmented sixth chords are: A�– C – F ♯ (It+6) and, D (Fr+6) or E� (Ger+6). If the German augmented sixth is taken as the representative out of the three and reinterpreted enharmonically, A�7 is the result, which is tritone related to D 7 (II7).

17

extensively for tonicization purposes. The diminished seventh chord can be thought

of as three adjacent minor thirds and also as two tritone pairs9.

This property forms a kinship that is similar to the aforementioned tritone pairing.

The minor third related chords also have two pitch classes in common10. The circular

ordering of minor third related dominant seventh chords, which also contains tritone

pairs, would give us the one of the three octatonic collections of the system.

Figure 2.1: An octatonic collection

While minor third related dominant chords are not commonly used in successions or

as substitutions in tonal harmony, the theoretical significance of this structure will be

explicit when the last component is integrated into the system.

2.1.3 Major/minor dualism and mixture

For centuries, the consonance of a minor triad has been a central topic in speculative

music theory. Starting with Enlightenment philosophers, Descartes and Rameau,

followed by dualists like Hauptmann, Oettingen and Riemann, there has been a great

effort to justify this situation using the natural properties of sound, overtones and

resonance. Despite this long-lasting and ongoing struggle, considering its vast use in

Western music tradition, the validity of a minor triad is a cultural fact that is

impossible to dismiss.

From this perspective, major and minor triads can be seen as dual counterparts that

are equal in validity. However, the hybridization of minor tonality in Western

9 For example: B°7 as B / F and D / A� tritone pairs. 10 For example: G7 and B�7 chords share pitch classes D and F.

18

tradition - with the use of major dominant degree and three different scales -

problematizes the dualistic view of major and minor tonalities. The solution for this

problem is to distinguish a minor triad (or chord) from a minor key (or tonality).

A minor key consists of primary triads that are mostly minor, but as in the case of

melodic minor - in which only the tonic triad is minor - major and minor primary

triads are interchangeable. This property was not exclusive for the triadic quality of

primary triads; what we know as modal borrowing (or “mixture” as in Schenker)

made it possible to use major scale-degree chords in minor keys and vice versa. This

technique, which has been utilized thoroughly in Romantic music, led to the total

dissolution of parallel major and minor keys into one chromatic key.

Going back to the octatonic system, the ideas of equality, interchangeability and

mixture can be integrated, in order to accommodate these features of chromatic

harmony. The pitch class content of an octatonic collection allows us to substitute

minor thirds in place of major thirds above the roots of the dominant seventh chords,

which results with another occasionally used chord, the minor seventh. In addition,

minor third related half-diminished chords and French augmented sixth chords11 can

also be derived from such a collection, in the presence of pitch classes that are a

tritone apart from the roots. In short, an octatonic collection can be thought to consist

of four minor third related tetrads that are either dominant, minor or half-diminished

seventh chords. For the sake of simplicity, only the root of the chords that belong to

the octatonic collection are shown in the next figure (Figure 2.2).

2.2 Three Octatonic Collections or “Functions”

Three octatonic collections can be arranged circularly, in order to construct a

constellation of minor third and tritone related seventh chords. It is also possible to

perceive the system on a triadic level, so that the chordal seventh is absent from the

sonority; this means that major, minor and diminished triads are also subject to our

octatonic categorization.

11 As dominant chords with �5 alteration, for example: C – E – G�– B�

19

Figure 2.2 : The octatonic system

As it is possible to observe from the figure, all pitch classes are exclusive to one

collection, which means that each of the octatonic chords that are constructed on all

of the twelve chromatic pitches belongs to a single collection 12.

The grouping of these chords into three categories is only one part of the theory. The

system promises significant insights when we turn our attention to harmonic activity.

As an example; the paradigmatic I-IV-V7-I harmonic progression in C major can be

seen as a counter-clockwise circuit around the octatonic system; from the collection

on top to the bottom left, then right and finally to the initial one. The direction of the

circuit doesn’t change if we substitute ii in place of IV as the subdominant, or

transpose to the parallel minor, the dominant or any other closely or distantly related

key 13.

12 Major, minor triads and dominant, minor and half-diminished seventh chords, which are available

in an octatonic collection. The special condition with diminished triads and seventh chords will be

explained later. 13 The pitch classes on the figure symbolize the roots of all the aforementioned octatonic chords.

20

At this point, I propose that the three conventional functions of Tonic, Dominant and

Subdominant, whether in the sense of abstract categories as in Riemann or of chord

types as in Rameau, map onto these three discreet octatonic collections, according to

the tonic pitch class. Similar to the mechanistic approach of latter theorist, the

transitions between collections are motivated by dissonances, which may be present

(actual dissonance) or implied (contextual dissonance).

2.3 Root Motion and Tonal Syntax: Progressive and Retrogressive Motions14

If the notion of root interval is combined with the octatonic system, there are only

three options where one chord leads to another:

1. To stay in the same collection; RM: m3 up, m3 down or tritone up/down.

2. To progress into “next” collection; RM: P4 up, M3 down, M2 up, m2 down.

3. To retrogress into “previous” collection; RM: P4 down, M3 up, M2 down,

m2 up.

The reason for labels as next and previous is the accumulation of the “progressive”

motion in common practice tonal music. Progressive root motions are the ones that

are grouped under second category with the addition of root descent of a minor third

from the first category. Retrogressive motion, which is the dual opposite of

progressive motion, consists of root motions under third category and the root ascent

of a minor third. The motion between tritone related chords is a special case in which

the motion is neither progressive, nor retrogressive or both at the same time. In that

sense, according to the system, tritone root motion creates a harmonic stasis in which

the functional meaning of the chord remains unchanged.

The symmetrical structure of the octatonic collection creates difficulty in assigning

progressive and retrogressive motions to ascending and descending minor third

relationships within the system; however, the descending minor third is seen more

frequently in harmonic practice. Tymoczko uses the term “pre-subdominant” for the

use of vi in progressions like I-vi-ii-V-I and I-vi-IV-V-I (2003). Progressions like

14 Carter’s (2005) terminology of “Progressive / Retrogressive” in place of Meeus’s (2000) “Dominant / Subdominant” has been adopted for the two opposite directions of harmonic motion, in favor of terminological clarity rather than qualitative description. In other words, the use of this terminology in this work is a technical consideration and is free of any chronological or cultural implications.

21

ii-IV in major and i-III in minor are far less frequent than grammatical I-vi; therefore

the descending minor third is labeled as progressive motion, while its ascending

counterpart as retrogressive.

In line with the aforementioned root motion theories, the system assigns preferred

harmonic direction to progressive motion in tonal music. This doesn’t mean to say

that tonal music consists solely of progressive motion; there is always a certain

amount of retrogressive motion, while a significant portion of it consists of motion

from tonic to dominant directly, without a mediating subdominant. This situation has

been explained by giving tonic chord a primary status that enables it to progress

anywhere within the diatonic boundaries. Another important factor next to direct T-D

motion is the repetition of progressive T-S or S-D progressions, in which

retrogressive motion occurs between the first and second statements.

Irregular retrogressive motion that doesn’t fit into these two main sources can also be

observed in the common practice repertoire; but these exceptions do not affect the

primacy and frequency of progressive motion in tonal syntax, especially in less

contrapuntal, more harmonic forms of Baroque and Classical periods. The issue of

symmetrical harmonic motion in pre-Baroque music will be explored throughly in

the next chapter.

In short, harmonic identity and motion (progressive/retrogressive) are explained via

the membership of a chord to one of the three octatonic collections and the transition

from one collection to another, respectively. Minor third and tritone relationships are

exceptions to this scheme as they retain the octatonic collection or function. Table

2.1 shows the grouping of root intervals according to progressive and retrogressive

harmonic motions.

Table 2.1 : Motion types according to root interval between chords.

Motion Type Root Interval

Progressive Motion m2! M2" m3! M3! P4"

Retrogressive Motion m2" M2! m3" M3" P4!

Harmonic Stasis Tritone " or !

22

2.4 Diatonicism and Root Motion

Meeus, in his article (2000) groups tonal chord progressions into two categories;

primary and substitute; in line with the ideas of Rameau and Schoenberg, he gives

primacy to fifth relation and defines progressive and retrogressive motions (as

Dominant and Subdominant progressions, in his terminology, respectively)

according to root intervals between chords.

Table 2.2 : Meeus's classification of tonal chord progressions (2000).

This table of root intervals in respect to dominant and subdominant progressions is

similar to the one for the octatonic system; however there is an important difference

of specific and generic intervals in the categorizations. The generic intervals of

traditional root motion theories rely on the diatonic pitch space and create a false

impression of equivalence between absolute intervals that share the same ordinal

number (for example; minor and major thirds); contrastingly, the octatonic system

presents specific intervals in chromatic pitch space. While perfect fifth (or fourth),

minor/major thirds and major second coincide in two categorizations, a discrepancy

occurs with minor second and tritone.

In diatonic context, it is possible to exclude these two problematic root intervals, for

example, by avoiding the use of iii and root position vii° chords in a major key 15. In

one perspective, their infrequent use can be connected to their potential to obstruct

the flow of progressive accumulation in the common practice tonality. In addition,

since the through-bass treatises of 17th century, vii° chords have been seen as

rootless dominant chords instead of independent harmonies, because of their lack of

a perfect fifth. In this theory, I also adopt this point of view for diminished triads that

resolve up a semitone, which explains the resolution as a root ascent of perfect

15 Tymoczko, in order to create a comprehensive grammar for diatonic major harmony, excludes it from his list of harmonies (2003). Fétis relies on counterpoint when he forbids its use; he points out that E and B, both mi-degrees, would create a parallel fifth in their “obligatory” resolutions to F and C (1994).

12/1/2014 MTO 6.1: Meeus, Toward a Post-Schoenbergian Grammar of Tonal and Pre-tonal Harmonic Progressions

http://www.mtosmt.org/issues/mto.00.6.1/mto.00.6.1.meeus.html 3/6

Sadaiï¿½s categories to two, each of which including one "principal" and two "substitute"progressions. This view actually returns to Rameauï¿½s conception of the dominant and subdominantfunctions, so that the categories may be renamed as "dominant" and "subdominant":

CATEGORY MAIN PROGRESSION SUBSTITUTES

Dominant A fifth down A third down or a second up

Subdominant A fifth up A third up or a second down

[8] Let us return to the paradigmatic tonal phrase I -->> IV -> V -->> I. The first and the thirdprogressions, I -->> IV and V -->> I, are ordinary dominant progressions. The second, IV -> V,now appears as a substitution for a dominant progression. The characteristics of this phrase are thefollowing:

1. it is formed of dominant progressions exclusively; 2. it includes one substituted progression.

These I take to be normal features of any good tonal progression. I -> II -->> V -->> I differs fromthe previous one only in the position of the substituted progression. More extended phrases evidencethe same features, usually with a larger number of ordinary dominant progressions, but always with atleast one substituted progression:

I -->> IV -> V -->> I I -> II -->> V -->> I I -> VI -->> II -->> V -->> I I -->> IV -> V -> VI -->> II -->> V -->> I etc.

(Arrows pointing to the right indicate dominant progressions; subdominant progressions would beindicated by arrows pointing to the left).

[9] The reason why the phrases include at least one substituted progression is that this is the necessarycondition for the phrase to return to its starting point, the tonic. A continued series of unsubstituteddominant progressions, following a cycle of fifths, indeed, could only inexorably lead away from theoriginal tonic. A well formed tonal phrase can therefore be defined, from the point of view ofharmonic progressions, as a series of dominant progressions of which at least one is substituted.

[10] A graphic device may help illustrate this. Example 1 shows both a I-IV-V-I and a I-II-V-Iprogressions. The bass roots are arranged on horizontal lines following the cycle of fifths, andconnected either by thick lines (main progressions) or by dotted lines (substituted progressions, with avertical dotted line linking the implied root to the real one). Dominant progressions appear as linesdescending from left to right. The overall Z shape of these figures, which I take to be characteristic ofa well formed tonal phrase, differs only in the position of the dotted lines.

[11] Example 2 (MIDI) shows the same graphic presentation applied to a real example, Bachï¿½schoral Gottlob es geht nunmehr zu Ende (BWV 321, Bpf 192). One will note the two subdominantprogressions (appearing as ascending thick lines) in the second system, characteristically followed bycompensating dominant progressions, in what Schoenberg calls "a mere interchange" or, in Sadaiï¿½sterminology, forming a b a patterns, V-II-V in mm. 9-10, I-V-I in mm. 15-16. This choral counts 25progressions in all, of which 23 (92%) are dominant progressions according to my definition; 7 ofthese (about 30%) are substitutions. Although it may be difficult to obtain statistics for the tonalcorpus as a whole, I consider these figures to be quite typical of the common tonal practice.

[12] Subdominant progressions in some cases may be more frequent, but they are normally not found

23

fourth, not a minor second. Similarly, diminished seventh chords are interpreted as

dominant minor ninth chords.

Nevertheless, as we enter the realm of chromatic harmony in Romantic music,

diatonic root motion theory loses its explanatory power in reference to the generic

intervals. The octatonic system aims to provide the remedy with the use of specific

intervals for root motion.

2.5 Insights on Tonal Harmonic Practice

A theory that doesn’t communicate with the music, I believe, is doomed to fail.

Therefore, the last section of this chapter consists of analytical excerpts from the

common practice period that examplify how the theory approaches some of the basic

tonal concepts. It is noticeable that the majority of harmonic activity consists of

progressive motion, which is in line with a previous statement about tonal harmonic

motion.

Another point to mention in these short examples is that the analytical focus is

mainly on the surface details, or foreground elaborations in Schenkerian perspective.

The reason for that is to grasp the harmonic intricacy in the local level, especially

with the use of chromatic chords; however, the system is available for use on

different levels of structural depth.

2.5.1 Tonicization and Modulation

Tonicization, or the use of secondary dominants, is a temporary departure from the

stability of the tonal center and its diatonic pitch space, while modulation can be

described as a transition from one key to another. According to Schenker, however,

the difference between tonicization and modulation is a matter of scale rather than of

type. Taking this view into account, the excerpt from the second movement of

Beethoven’s Seventh Symphony is analyzed in the home key on top of two local

keys (Figure 2.3).

24

Figure 2.3 : Beethoven, Symphony No.7, ii, mm.1-18 (Piano arr. by Franz Liszt)

(IMSLP)

While functions coincide for relative keys of A minor and C, this is not the case for

fifth related A minor and E minor: C – B progression in measures 10 - 12 has been

analyzed as T – S and S – D, respectively. Ignoring the involvement of different

functions, both motions are progressive according to the system.

In order to create stronger tonal coherence between two keys, Reger suggests

modulation through the use of a pivot chord that functions as subdominant in the new

key (2007). The purpose of this instruction is to create chains of S – D – T motions in

different key interpretations, superimposed on top of each other. Adding to that and

recalling the circular arrangement of functions in the system, tonic can be seen as

dominant of subdominant (as in I7 – IV) or as subdominant of dominant (as in vi – II7

– V). What this shows is that, the crucial part of functional thinking is the direction

of motion between different functions, rather than the individual meanings of them,

or how the collections are labeled, which are dependent on the local tonic.

25

2.5.2 Prolongation

According to Caplin, “A harmonic prolongation is created when a single harmonic

entity is perceived in the listener’s imagination to be sustained through time, despite

the presence of an intervening chord (or chords) of different harmonic meaning”

(1989: 25). An important element of tonal harmony, prolongation makes it possible

to think of functions in different structural levels.

As an example of a typical tonal idiom, in measures 3 - 6 of the previous example

(Figure 2.3), tonic harmony is prolonged in the presentation phrase of the main

theme. Together with the intervening dominant between two tonic chord, a T-D-T

progression is present in local level. While the progression from tonic to dominant is

retrogressive, the second part of the progression is progressive. On this perspective,

these opposite directions can be thought to cancel each other and to create a

harmonic stasis, which we define as prolongation.

Subdominant and dominant functions can also be prolonged just as the tonic; for

example, in measure 8 of the previous example, a second inversion tonic chord is

labeled as a dominant six-four chord, which resolves to its root position in the next

measure. This technique has been used extensively in the common practice period in

order to achieve better voice-leading and to prolong dominant functions.

The intervening chord doesn’t have to be from the dominant side of the prolonged

function. For instance, tonic prolongation can also be created with the use of

subdominant function, as in the excerpt from the third movement of Mozart’s Piano

Sonata No.7 in C Major. Again the second inversion triad makes it possible to

interpret the subdominant harmony as a neighbor chord that is used in favor of

prolongational purposes. The resulting progression of T – S – T creates a similar

harmonic stasis, in which the only difference is the position of the progressive and

retrogressive motions.

Figure 2.4 : Mozart, Piano Sonata No.7 in C Major, K309/284b, iii, mm. 1-5 (IMSLP)

26

2.5.3 Chromatic chords

As the system covers diatonic and chromatic pitch spaces, secondary dominant,

mode mixture and other chromatic chords can be interpreted in functional terms. This

approach may be seen as an exact opposite of Riemannian transformations, as the

focus is on the progression or action of these chromatic chords rather than their

derivation from primary chords, their identities or meanings. Having stated that, I

would like to proceed with the analyses of well known passages from the repertoire,

which have instances of chromatic chords with varying degrees of complexity.

In the continuation phrase of the well-known theme of Mozart’s Symphony No.40, a

half-diminished seventh chord on sixth degree comes after thirteen measures of tonic

prolongation, which leads to the German augmented sixth and ultimately to the

cadential progression that has several measures of dominant prolongation. This

somewhat pivotal chord is actually diatonic in melodic minor, but its functional

meaning comes out when we identify it as an octatonic relative of the tonic chord and

the igniter of the T-S-D-T cadential progression.

Figure 2.5 : Mozart, Symphony No. 40 in G Minor, K.550, mm. 1-22 (Piano arr. by August

Horn) (IMSLP)

27

The famous “Tristan chord” from the prelude of Wagner’s opera is a sonority that is

very similar to the chord in the previous example, a half-diminished seventh chord

that resolves downward by half step 16. The functional property of the chord doesn’t

change if the chord is interpreted as a French augmented sixth chord (F-A-B-D♯)

with G♯ as an appoggiatura, as both G♯ and A belong to the subdominant collection.

In a similar logic, as both A♯ and B belong to the dominant, the chord on the

downbeat of the third measure can be interpreted as an altered dominant chord or a

regular dominant seventh chord with a passing dissonance.

Figure 2.6 : Wagner, “Prelude” from Tristan und Isolde, mm.1-3 (Reduction from

Wikimedia: http://commons.wikimedia.org/wiki/File:Tristan_chord.png)

A less controversial and complex chromatic chord, the Neapolitan, is the only legacy

of once popular Phrygian mode that survived through the common practice period.

Due to its common positioning right before the dominant and its common tones with

the primary subdominant chord, the Neapolitan has been, I suggest, misinterpreted as

a subdominant chord. As the octatonic system shows, the Neapolitan chord belongs

to the dominant function as it is the tritone pair of the primary dominant chord.

The Neapolitan chord that proceeds directly to the tonic - a progression that would

create the anathema of good voice-leading: parallel fifths - was unavailable for

composers of the common practice period. However, as voice-leading gradually lost

its governing position in favor of harmonic extravaganza in Romantic period, major

triads and dominant seventh on lowered second degree that resolved directly to tonic,

similar to augmented sixth chords that resolve to dominant, started to appear 17.

16 This similarity has been brought to my attention by Smith’s 1986 article: “The Functional Extravaganza of Chromatic Chords”. 17 An example for such progression is in the second measure of Reger’s “Straf mich nicht in deinem Zorn”, Op. 40, No. 2, mm.1-2 as Harrison points out (1994:117).

28

An evidence that supports this idea is that the Neapolitan chord almost always

appears after a subdominant chord in progressions. In that perspective, N6 – V

progression is actually a harmonic stasis rather than a progressive motion, which

obscured its dominant function in the first place. A typical use of the Neapolitan

chord is shown in the excerpt from the theme of Beethoven’s “Moonlight” Sonata.

Figure 2.7 : Beethoven, Piano Sonata No.14, Op.27 No.2 (“Moonlight”), mm. 1-5 (IMSLP)

Other chromatic chords such as the diminished triad in the next example from

Beethoven’s “Waldstein” Sonata (Figure 2.8) would avoid an easy interpretation and

an appropriate Roman numeral label below it, as it resolves atypically to the

dominant chord that shares the fifth degree (G) as a common note. In a reductive

way, it could be explained as a prolongational (or non-functional) harmony for

dominant, however, the octatonic perspective shows that the diminished triad on the

fifth degree belongs to the same collection with the dominant harmony. If we look at

the pitch content of the chord, B�or A♯ may be interpreted as the raised ninth and D�as the raised eleventh (or flattened fifth) of an altered dominant chord.

Figure 2.8 : Beethoven, Piano Sonata No.21, Op.53 (“Waldstein”), mm. 257-260 (IMSLP)

29

This example shows that unusual harmonies as such may be interpreted as altered

dominant chords18 that originate from the pitch classes of an octatonic collection,

thus from one of the three functions.

2.5.4 Major third relationship

Augmented triads, major seventh chords, ninth chords and other extended tertian

sonorities are not found in the octatonic system. However, these sonorities can be

analyzed from the perspective of major third relationships in combination with the

octatonic system.

Similar to minor third relationships of the chords within an octatonic collection,

major third related major and minor triads form another symmetrical collection: the

hexatonic. This collection can also be considered to consist of two minor second

related augmented triads 19. Due to its symmetrical nature, there are only four

hexatonic collections; the fifth iteration maps onto the first collection.

Parallel to the way the octatonic system has been presented previously, the hexatonic

system may be constructed out of the four hexatonic collections. Cohn, in his articles

(1996, 1998) and his recent book “Audacious Euphony” (2011), relies on this system

in his voice-leading analyses of chromatic harmonies (Figure 2.9). However, there is

a fundamental difference between these two systems: a circuit around one of the

hexatonic cycles is a function changing progression as opposed to a function

preserving one in the octatonic cycle 20.

Taking this into account, octatonic and hexatonic systems can be collided in order to

achieve a combined system of chords, which is organized as functions and transitory

areas between them. Figure 2.10 represents the combined system graphically:

continuous lines represent the three familiar octatonic collections, while the dashed

lines between the members of different octatonic collections show these transitory

areas. The telescopic circles represent the hexatonic collections, which may be

18 Alterations and extensions like�9, ♯ 9, �5/ ♯ 11 and 13, and also different combinations of them. 19 Pitch constituents of C major, E major and A� major chords are: B – C – E� – E – G – A�, constituting the hexatonic scale. This scale that alternates semitones with minor thirds in each step could also be acquired with C, E and A�minor triads or with B – C , E�– E or G – A�augmented triad pairs. 20 For instance, according to octatonic system, F – A – C♯ major or minor chords that belong to one hexatonic collection represent S – T – D functions respectively.

30

expanded outwards infinitely, or can be curved to create a three-dimensional torus.

For the sake of simplicity, however, the combined system is presented in its most

basic form.

Figure 2.9 : The four hexatonic systems21 (Cohn 1996:17)

Figure 2.10 : The combined system 21 Cohn’s use of the term “system” is different from mine; he regards the cyclic ordering of major third related chords that belong to the same hexatonic collection as a hexatonic system, while I reserve the term for the entire constellation of four (or three in octatonic) collections, or cycles.

31

According to this combined system, non-octatonic tertian sonorities are interpreted as

combination of triads from different octatonic collections. For instance, as in the next

example from Bach’s C Major Prelude, a C major seventh chord has been analyzed

as a transition between, or as a functional mixture of tonic and dominant, in reference

to its triadic constituents, C major and E minor.

Figure 2.11 : J.S. Bach, Prelude and Fugue in C major, BWV 846, mm. 7-9 (IMSLP)

Another non-octatonic chord, the augmented triad is one of the least commonly used

chords of tonal vocabulary because of its ambiguous character; however, there are

some rare instances of its occurrence, as in the next excerpt from the coda of the first

movement of Beethoven’s “Waldstein” Sonata (Figure 2.12). Due to its symmetrical

nature, an augmented triad can be thought as a chord with three possible roots and

therefore, three possible functions. Keeping that in mind, I propose that the

functional meaning of an augmented triad should be understood as a transition

between the preceeding and succeeding chords.

Figure 2.12 : Beethoven, Piano Sonata No.21, Op.53 (“Waldstein”), mm. 274-275 (IMSLP)

32

Other extended tertian sonorities can also be dissected into their constituent triads

and analyzed accordingly. For instance, a dominant major ninth chord on G in C

major context has a functional mixture of S and D 22. The relative force of these

individual functions is affected by parameters other than pitch, like vertical ordering

and timbre, and will not be explored further in this study.

Tonal ambiguity will increase gradually if tertian superimposition is continued, and

when the point of functional saturation is reached, all seven pitches of a diatonic

collection 23 will sound together. In such a situation, it is not possible to think of

motion between different functions anymore, as all three of them would be present in

a single musical moment 24. The extensive use of such chords in the works of post-

Romantic and modernist composers in the late nineteenth and early twentieth

centuries, led to the dissolution of functional harmony and therefore, tonality.

2.6 Further Speculations on Progressive and Retrogressive Motions

In the previous sections of this chapter, the octatonic system has been presented as

well as its perspective upon the typical idioms of tonal harmony. In this last section,

the aim is to investigate why progressive motion, not retrogressive, is the preferred

harmonic motion in tonal harmony and also to see if the theory can provide an

explanation for this within its own limits. In order to do that, the voice-leading aspect

of the two categories of harmonic motion has been examined, using the theoretical

device of conceptual voice-leading.

Conceptual voice-leading basically consists of a harmonic succession in which an

octatonic seventh chord is connected to a perfect triad with the most parsimonious

voice-leading possible. As stated before (see 2.2), the seventh of the second chord

doesn’t have to be present in the actual music; but it has been included in the

procedure, in order to explain the motivational aspect of harmonic successions as

such, a strategy that is adopted from Rameau. Accordingly, the second chord has

22 While G major presents Dominant function, B half-diminished seventh and D minor present Subdominant function. 23 Quasi-diatonic collections like Harmonic minor, Harmonic major and Melodic minor, which can be achieved with different combinations of minor and major primary triads, are also included. 24 For example, a sonority like G-B-D-F-A�-C-E (G 11�13) has G major, F minor and C major chords; the primary triads of each functions.

33

been determined as a triad that represents the point of repose, in which there is no

motivation to move elsewhere.

The procedure for the conceptual voice-leading of such a harmonic succession is as

follows:

1. The first chord is a seventh chord which consists of root, major or minor third,

diminished or perfect fifth and minor seventh.

2. The second chord is a perfect triad with a major or minor third.

3. All pitches of the seventh chord are present.

4. Every pitch of the seventh chord moves to the closest pitch in the triad, taking

into account the chromatic alterations; if there is any common tone with the triad,

it is sustained.

5. All pitches of the triad are present, one of them is doubled according to step 4.

The combination of every conceptual voice-leading with C major/minor as the

second chord and their groupings according to harmonic motions result with the table

of conceptual voice-leadings (Table 2.3).

The fourth step of the procedure is crucial in constructing the voice leading table and

interpreting its outcomes. According to this guideline, the pitches are allowed to

move only by major or minor seconds, taking into account the chromatic alterations

of the thirds of both chords and fifth of the first 25. The motions of twelve pitch

classes can be summarized as in Figure 2.13; the encircled pitch classes move to the

same pitch class (or pair of pitch classes, E� and E). One important point to mention

is the dual roles of D and F, which is a result of their equal distance to two of the

constituents of the second chord (major seconds to C – E (�) and E (�) – G,

respectively). The double emploi is the reason for the alternatives in descending

major third and perfect fourth successions. The fifth step, on the other hand, prevents

the occurrence of such alternatives in descending major second and ascending perfect

fourth root motions; in which there is a possibilty of doubled roots and fifths.

25 For example, F# to E�is an augmented second, which is prohibited; therefore F(#) pair in descending major second root motion goes to G.

34

Table 2.3 : Conceptual Voice-Leadings for Root Motions

!!

E! G! B! !!D!! ! ! !!

E ! G! C! !!

E ! G! B! !!D !! ! ! !!

E! ! G! C! C!!!

C! ! E! G! !!B!! ! ! !!

C! E! ! G! C!!!

B ! D! F! !!A!! ! ! !!

C! C! E! ! G!

G! B! D! !!F!! ! ! !!

G! C! C! E !!!

G ! B! D! !!F !! ! ! !!

G! C! E! ! G!

A! C! E! !!!G!! ! ! !!

G! C! E! ! G!!!

F! A! C! !!E!! ! ! !!

! G! C!! E

B! D! F! !!A!! ! ! !!

C! E ! G! G!!!

D! F! A! !!C!! ! ! !!

E! ! G! G!! C!!!

G!!!!or!!E!!

!!!!!!!!!!!

P4"!

M3!! m3!!

M2"! m2!!

M3"! m3"!

P4!!

M2!! m2"!

F! ! A! C! E!! ! ! !!

G! ! C! E !!!Tr!

C!!!!or!!!G!!!!!!!!!!!

Progressive

Retrogressive

Harmonic Stasis

C!!!!or!!E!!

!!!!!!!!!!!

35

Figure 2.13 : Summary of the table of conceptual voice-leading

The superiority of root over other constituents of a chord is an axiom that proves to

be useful for interpreting the table of conceptual voice-leadings. In successions of

progressive motion (with the exception of an alternative in descending major third)

the root of the triad is doubled, which is a result of the contrary motion between B(�)

and D(�) pairs. On the retrogressive side, the third or the fifth of the triad is doubled,

which is a result of contrary motions of F (♯) and A(�) or D( ♯ ) and F(�) pairs,

respectively. In addition, the absence of the root of the first in the second chords of

the progressive successions creates an essential harmonic contrast between them. In

root motion of a tritone, the two alternative thirds of the first chord, A and B(�) point

towards different pitch classes, the fifth and the root, respectively; which explains the

diffused, static character of not-progressive, nor-retrogressive “harmonic stasis”.

Another important point is that, in progressive successions, with the exception of

descending minor second26, all of the “dissonant” minor sevenths resolve

downwards. However, out of six successions of retrogressive motion and harmonic

stasis combined, only in one does the seventh resolve “properly.” In accordance with

the Rameauian perception of dissonances as sevenths above supposed basses, the

treatment of dissonances (both actual and contextual ones) may be seen as the reason

for the prominence of progressive motion in Palestrina style counterpoint.

In conclusion, the conceptual voice-leading shows that, the principles of counterpoint

in Late Renaissance, as codified by Zarlino (1968) (such as the preference of

complete sonorities as opposed to “hollow” sounding fifths, of four-voice texture

which requires doubling of a pitch in the triad, of contrary motion between voices

and also the treatment of dissonance) may be the reason for coming into prominence

of progressive motion rather than its retrogressive counterpart.

26 The augmented sixth spelling in this succession is an adjustment for the irregular resolution of the dissonant seventh.

36

37

3. APPLICATION OF THE THEORY

3.1 Introduction for the Analyses

In the previous chapter, the theory which defines progressive and retrogressive

harmonic motions in relation to the octatonic system, has been presented.

Furthermore, the analytical examples towards the end demonstrated that the typical

idioms of common practice harmony correspond with the model that is presented by

the system. In this chapter, examples from the pre-tonal and tonal polyphonic

repertoire will be analyzed, in order to examine the relationship between the

harmonic motion and tonality, or more specifically, the individual contribution of

harmonic motion to the emergence of tonality. The examples, which range from a

Late Medieval motet to a Renaissance Mass movement and a Baroque sonata

prelude, reflect the multiplicity of genres and textures within the polyphonic setting

of the specified period. The statistical results for the harmonic motion will be

evaluated at the end of this chapter.

The method that is employed here, in fact, proves to be quite useful in many aspects.

The analyst is not bound to the diatonic scale as the norm and to interpret the

chromaticism of leading-tones and musica ficta alterations as deviations from it. In

addition, the relativistic aspect of the system makes it possible to consider modal

harmonies directly, without relating them to a tonal center.

The employed analytical method, which focuses on chords and root motion between

them, may seem to be an anachronistic approach that hardly reflects the intention and

the technique of the composer. At this point, I argue that, this criticism can be

pointed towards any analytical endeavour; as the whole process of analysis involves

a certain amount of deconstruction, in order to trace the fundamental concepts and

rules that govern the music at hand. This is not to say that the compositional

technique is ignored altogether; as in some passages, such modal features will be

emphasized. The aim here is, to position the analytical scope from the perspective of

a listener, who is assumed to be well-versed with functional harmony.

38

In the analyses, the progressive and retrogressive motions are defined according to

the root interval betweeen successive harmonies; but due to the linear compositional

technique of pre-tonal music, the designation of individual harmonies turns out to be

a troublesome task, in the absence of clear phrase structure and harmonic rhythm.

The musica ficta alterations over the staff create another difficulty, as their

interpretation varies not only in different editions of the same piece, but also in

different performances of the same score. With these aspects taken into account, the

harmonies are determined in reference to their prominence, metric placement and

their intervallic content is interpreted in the octatonic perspective, which allows for

alternation of the third, fifth and in some cases, the root. On the scores that can be

found in the Appendix, prolongational harmonies, such as 6/4 triads, are either

reduced or shown in parantheses, while the progressive and retrogressive motions are

labeled with counter-clockwise ( ) and clockwise ( ) symbols, respectively,

indicating the direction of rotation in the octatonic system. In the rare occurrences of

harmonic stasis, or the root motion of a tritone, the ceasura symbol takes place of the

rotational symbols. In the chord labels, parantheses are used for the alternation of

major-minor chord qualities and interpretations of diminished chords.

3.2 The Analyses

3.2.1 Machaut, “Puis que la douce rousee”

The motet that dates back to the second half of the 14th century has a three-voice

texture and its isorhytmic structure provides the compositional unity. The initial and

final pitches of F in the limited ambitus of the tenor give the clue for authentic

Lydian mode. The unmistakebly modal features of linear and harmonic dimensions

show themselves as early as the first phrase, even though the first regular parallel

cadence 27 occurs on G at the end of the second phrase. Other occurrences of parallel

cadence between F# -6 and G are in measures 25, 66, 77 and between E-6 and F in

measures 51, 99, 107 and 145, as well as the final cadence of the piece.

27 The parallel cadence (also known as the double-leading-tone cadence) involves a progression from a 6/3 sonority to an 8/5 sonority. Typically, the tenor descends a step while the top two parts ascend in a parallel way , thereby giving the name to the cadence. Fuller underlines the structural importance of this so called directed progression, which according to her, has a syntactical succession of tendency and resolution that is motivated by the imperfect to perfect interval progression (1992).

A

B - C G C E - C A -

D - G - D - F

E - C G# ° A - D -

38

35

32

��6 A - � � � �66

�

�6 6

� � �

�(E7)

6� � � � �64 A

B - C G C E - C A -

D - G - D - F

E - C G# ° A - D -

38

35

32

��6 A - � � � �66

�

�6 6

� � �

�(E7)

6� � � � �64

39

The progression at the end of the first phrase (m.8-9), which sounds like a deceptive

cadence to our modern hearing, should be seen as a variation of the regular parallel

cadence. Instead of going to F# in order to create the tendency sonority, the triplum

leaps up to dissonant G. In the next measure, C# in the motetus resolves to D, while

the triplum and the pre-existing tenor line descend to F and B�, respectively.

The establishement of multiple tonal centers, which is a characteristic feature of

Machaut’s music according to Fuller (1998) and Bain (2008), is achieved by these

“stepwise” cadences. Contrastingly, root motion by an ascending fourth is infrequent

in general and in fact, there isn’t a single occurrence of C major, or the dominant in

the home “key” of F. The use of a stepwise, cantus firmus tenor line as the

compositional and harmonic foundation28 is the primary reason for this condition.

As this example shows, musica ficta alterations, non-triadic sonorities as well as

irregular dissonances create complication in such a chordal analysis. Nevertheless,

the symmetry between progressive and retrogressive motion can still be observed.

3.2.2 Agricola, “Je n’ay dueil”

A contemporary of Josquin des Prez, Alexander Agricola (1445 - 1506) is one of the

most renowned composers of his time, whose style is representative of the High

Renaissance polyphony. The piece dates back to the second half of the fifteenth

century and has a four-voice polyphonic setting, while its extensive use of imitation

is reminiscent of Ockeghem.

The entry of the countertenor that starts with the fifth degree of Dorian (A) is

imitated at the octave by soprano and tenor, while the bass imitation is at the fourth

(or the lower fifth). The emphasis on A-B� semitone throughout the piece gives a

Phrygian character to the piece, but in cadences where A major leads to D minor

(such as the final cadence of the A section and of the piece 29 in m. 28), the strong

impression of D Aeolian obscures the boundary between these modes.

28 While Zarlino points out the tenor as the compositional (or contrapuntal) foundation, he reserves the harmonic foundation for the lowest voice of the texture, which defines the quality of the intervals above it: either the tenor or contratenor bassus (or the bass) (Lester, 2008:754). 29 In this cadence, the use of unprepared dominant seventh that resolves “correctly” is a striking feature, which exemplifies its use at least a hundred-years earlier than Monteverdi’s “Cruda Amarilli”.

40

The modal cadence in m. 6-7 is similar to the variant parallel cadence in the previous

piece in terms of root motion, but here the outer voices carry the parallel major thirds

between root position A and B� major chords, therefore, sounding more like a

deceptive cadence in functional hearing. Other examples of this cadence are in m.34

and also, between E and F major chords in m. 20-21.

In terms of overall harmonic motion, the symmetry between progressive and

retrogressive motions is intact, as there are almost equal occurrences of both

directions, when the form of the piece (ABBCAA) is taken into account.

3.2.3 Palestrina, “Agnus Dei” from Missa Papae Marcelli

The next analytical example is a movement from the famous Pope Marcellus mass

(1562), which represents the contrapuntal style of the 16th century master. The

hallmark of Renaissance polyphony, imitation, is still used as the means for integrity.

The entry of the contralto, which is followed by soprano and bass entries, hints at

Mixolydian mode; the ending, however, is on C major which points at Ionian mode.

In other words, the piece either starts on the dominant that resolves to tonic in the

second harmony (m.3) or the regular ending of G in the contralto line is accompanied

with C and E in place of B and D.

Putting aside the issue of modality in a polyphonic setting as such, the prominence of

descending fifth root interval and progressive harmonic motion in general is

significant. For instance, the passage between measures 25 and 32 can be seen as a

chain of progressive motion, while the repeated progression at the end of the piece

creates an accumulation of progressive motion (m. 42 – 46 and m. 46 – 50), which

enhances the pull towards the ultimate C major. The final succession - a cadence

imparfait in Rameau’s terminology - is a retrogressive motion that counterbalances

the consistency of the previous bars with its position, which attaches a modal ending

to the piece. In general, descending fifth cadence takes the place of modal parallel

cadence, which frequently occurred in previous pieces and not even once here.

Contrasting with the previous examples, the asymmetry that leans towards the

progressive side is a distinctive feature to which Palestrina’s contrapuntal style and

treatment of dissonance contributed significantly.

41

3.2.4 Monteverdi, “Era l’anima mia” from Fifth Book of Madrigals

The pivotal figure in between the periods of Renaissance and Baroque, Monteverdi is

commonly regarded as the creator of modern music, or of tonal harmony. The piece

is from his fifth book of madrigals, which includes an introduction that serves as the

declaration text of Seconda prattica (1605). The piece has a five voice texture that is

typical for madrigal setting, while there is also a key signature that implies D minor.

The tonal organization of harmony in this first section is remarkable; the quasi-

homorhytmic texture has a notably tonal progression in D minor: i – iv6 – V!!!!!!! – i .

The tonic chord at the end of the progression (m.9) functions as a pivot chord for the

repetition of the same progression in the subdominant key of G minor. The inner part

of the piece has a livelier contrapuntal writing and correspondingly, faster harmonic

rhythm. The frequent V-I cadences in different keys like C major/minor (m. 19, 60),

G minor (m. 21, 28, 40), F major (m. 25, 34) and in B�major (m. 36) signify an

exploration of different key areas in the form of modulation, or at least of

tonicization. Also, the frequent use of V!!!!!!! progression contributes into the sense of

centricity in these local key areas.

The peculiar final cadence of the piece inverts the bass motion of the usual V-I

cadence, while the chromatic connection is between B� and A. In the absence of the

ascending “leading-tone” (C#), the penultimate harmony can only be interpreted as a

rootless dominant seventh on natural seventh degree in second inversion; a rather

awkward label for diatonic Roman numeral analysis. In the perspective of

octatonicism and functions however, the harmony can be interpreted simply as a

dominant.

The accumulation of progressive motion in outer sections heralds the arrival of tonal

harmony, but the persistance on some modal features, especially in the middle

section, decreases the steepness of the asymmetry.

3.2.5 Gesualdo, “ O tenebroso giorno” from Fifth Book of Madrigals

The first two chords of the piece (1611) strike the listener with chromaticism right

away, which is a technique favored by the composer. Aeolian mode is the prime

suspect in terms of modality, even though major thirds have been used in initial and

final sonorities. The dominant in the second measure that is prepared by a suspension

42

doesn’t reach its destination directly, due to the syncopated imitiations in the alto and

countertenor; the “tonic” solidifies only after the entry of the bass in the second half

of the fourth measure. Tonal ambiguity persists until a somewhat proper V-I cadence

in m.12, which forms the transition to the fast-moving, imitative second section of

the piece. One extraordinary harmonic moment is in m.9, in which the “outside” C#

minor and F# major chords, which create an expectation of B minor, lead to G major

in first inversion deceptively, with a proper descending fifth in the bass.

Harmonically less extravagant middle section comes to a close in around m.32, in

which the introduction of slower rhythms implies a change of texture. The

accumulation of progressive motion that starts from m.34 creates a pull towards the

final cadence between m.38 -39. The subdominant 6/4 chord in penultimate measure

serves to prolong tonic harmony with the use of pedal bass; therefore, there isn’t a

real harmonic motion.

To sum up, the chordal outer sections contrast with the imitative middle section that

is reminiscent of the older style. As a result, the disposition of harmonic motion is

closer to symmetrical, demonstrating the essential relationship between texture and

harmonic motion.

3.2.6 Carissimi, recitative “Plorate colles” and final chorus “Plorate fiili Israel”

from the oratorio “Jephte”

After Monteverdi’s impact, dramatic musical forms of opera and oratorio rose to

prominence in the Early Baroque. One of the first masterpieces of the latter form is

written by the most important representative of the Roman school, Carissimi, and is

titled as Historia di Jephte (1648). In order to show how the parameter of texture

affects the degree of asymmetry in harmonic motion, two successive movements of

this spectacular work will be analyzed and the statistical results will be presented

separately.

The first of these movements is in the recitative style, in which the solo vocal melody

is accompanied by the basso continuo line 30. In the first section, the A- B� semitone

is emphasized in the melody, which gives it a Phrygian character. The flow of

30 As there are no figures under the bass line in this edition, the analysis has been carried out by “realizing” the accompaniment according to the implied harmony in the voice part.

43

progressive motion is interrupted in the second phrase by F major, which retains the

Phrygian quality in the local key center of E, which gets reinterpreted as the

dominant of the original key. The second section starts with a phrase in G minor,

which is followed by a cadential phrase that accumulates progressive motion until

the arrival of A� major. This chord functions as Neapolitan sixth in this local key

and there is also the typical harmonic stasis between two dominants (see 2.6.3). The

third and fourth sections of this recitative movement remain in the “sharp” pitch

space of A minor and closely-related G major, as the Phrygian semitone disappears.

Remarkably, both sections consist entirely of progressive motion.

In the choral movement, the repeated first phrase is harmonized over a descending

bass tetrachord. The first inversion minor dominant in the resulting progression

serves as a “passing” harmony between tonic and first inversion subdominant chords,

nevertheless these successions are labeled as retrogressive. The next phrase, which

follows the half cadence, evokes a modal sense of harmony with the root motion of

an ascending fifth. After this phrase, the majority of the harmonic motion is

progressive, while almost all of the retrogressive motions occur between the

repetitions of cadential progressions.

The use of tonal harmony that serves for dramatic purposes, in conjunction with

modes and older style of polyphony that are preserved in Church music, reflects the

mixture of different aesthetical attitudes in mid-17th century. The third and fourth

sections of the recitative movement are especially important, as the disappearance of

the modal character directly results in an accummulation of progressive motion.

3.2.7 Corelli, “Prelude” from Violin Sonata in E Minor Op.5 No.7

Commonly regarded as the first tonal composer, Corelli is most famous for his

instrumental writing, especially for his sonatas for violin and basso continuo setting.

In the prelude of this piece (1700), rounded binary form that consists of two repeated

parts has been employed. Typically, the first part modulates to the dominant and ends

with a perfect authentic cadence. The second part, on the other hand, starts with a

descending fifth sequence that modulates to the subdominant and ends with a proper

cadence in the home key.

As the notion of tonality is complete, it is possible to perform a Roman numeral or

functional analysis; but for the sake of consistency with the previous examples, the

44

harmonic motion will be examined in terms of two directions. In the first part, the

only occurrence of retrogressive motion is in measure 18, when the local tonic of G

goes directly to its dominant. The sequence that initiates the second part possesses

the rest of the retrogressive motions; thus, the overwhelming majority and

accumulation of progressive motion in the piece exemplifies the regular tonal

harmonic motion.

3.3 Analytical Results

The statistical results show that harmonic motion gets increasingly asymmetrical, as

we move chronologically towards the common practice period (Table 3.1).

Obviously, this progression is not linear throughout the timeline, as the Gesualdo and

Carissimi examples indicate; however the tendency to accumulate progressive

motion becomes apparent at least after the 16th century. Exemplified here with

Palestrina’s work, the recognition of harmonic organisation in this period is

remarkable and deserves a full study by itself.

A fundamental association must be made between harmonic motion and tonal

system. Undoubtedly modal pieces of Machaut and Agricola have perfect symmetry

in terms of harmonic motion, while tonally-perceptable modes of Mixolydian –

Ionian mixture and Aeolian in Palestrina, Gesualdo and Monteverdi examples may

be a factor for the asymmetry in various degrees. The recitative movement of the

Carissimi example puts forth an important example in that sense; the second half of

the movement, in which the Phrygian quality has been abandoned, consists entirely

of progressive motion, while the same accompanimental texture has produced

retrogressive motion in the first part. The accummulation of progressive motion in

Corelli exemplifies harmonic motion in major/minor tonality.

Nevertheless, as the intersectional contrasts in Gesualdo and Carissimi examples

indicate, an important correlation should be also made between polyphonic texture

and symmetry of harmonic motion. In homophonic and monodic sections of these

pieces, the accumulation of progressive motion is more prominent than contrapuntal

sections that utilize imitative polyphony.

45

Table 3.1: Statistical results for harmonic motion

Machaut (before 1377)

Agricola (before 1506)

Palestrina (1562)

Monteverdi (1605)

Gesualdo (1611)

Carissimi A (1648)

Carissimi B -

Corelli (1700)

Retrogressive 42 118 20 23 33 19 19 6

Progressive 39 119 43 46 49 73 36 47

0%#

10%#

20%#

30%#

40%#

50%#

60%#

70%#

80%#

90%#

100%#

46

47

4. CONCLUSION

The reconciliation of the two harmonic perspectives of earlier traditions that are

presented in the first chapter (namely Rameauian harmonic motion and Riemannian

harmonic identity) results with the octatonic system, which reveals the reciprocal

relationship between these two factors of harmonic function. The construction of the

system makes use of the principles of resolution of tritone and common-tone

relationship between sonorities, ideas that have their origins in these two respective

lines of thought. In that sense, the study aims to put forward a different perspective

upon functional harmony and to expand its range defined by traditional theories.

As the short analytical examples in second chapter indicate; octatonicism proves to

be a useful scope while examining functional harmony in place of diatonicism, which

handicaps all existing theoretical views, in conjuction with the centricity of tonic. A

perspective that will be explored thoroughly in future studies asserts that, the

“background” scale concept might be differentiated for two dimensions of pitch

events; in other words, melodic dimension can be thought to be governed by diatonic

space, while its harmonic counterpart, by octatonic space.

As Kopp points out, an ideal theory of harmony explains how chord progressions are

determined and goal-oriented in tonal music (1995). The presented theory asserts

that, the accummulation of progressive motion is a distinctive feature of functional

harmony in tonal music. However, this doesn’t mean that tonal harmony consists

solely of progressive motion; successions such as I-V in which tonic goes to

dominant without an intermediary subdominant, repetitions of cadential progressions

(I-ii-V - ii-V-I), sequences and other irregular successions that are based upon bass

functionality (rather than of root functionality) produce retrogressive motion.

Nevertheless, progressive motion and its accummulation prevail as the norm in tonal

harmony until the extravaganzas of the Romantic period. Despite the speculations in

the last section of the second chapter, the question of why progressive motion is

preferred for accummulation rather than retrogressive remains to be open until

further studies on the subject.

48

Tymoczko accurately points out that tonal musical syntax depends largely on local

harmonic relationships; therefore, challenges the idea of recursive organisation,

which is manifested by Schenker and prolonged by Lerdahl and Jackendoff with the

adaptation of linguistic techniques (2010: 19). His statistical approach upon

harmonic grammar in tonal music results in a model (Figure 4.1), which represents

the majority of the progressions that are found in the common-practice repertoire,

even though its shortcomings due to its foundation upon diatonicism and centricity.

The root motion perspective of the octatonic system is fully compatible with this

model, when aforementioned exceptions are taken into account.

Figure 4.1 : Tymoczko's map of tonal grammar in major (2010: 9)

The evaluation of the presented theory in six criteria (Brown, 2008: 15) shows that

the system provides accurate and consistent insights on harmonic motion, while the

scope of harmonic analysis is expanded to cover sonorities of pre-tonal and post-

tonal periods. The simplicity of the system shows itself even in the limited, two-

dimensional space on a paper, in which the relationships between all chords built on

twelve pitch classes can be analyzed in the most elementary way: by referring to the

octatonic collection or to the root interval. The system communicates with the

aforementioned, existing tonal theories by means of shared concepts (such as

dissonance resolution, voice-leading efficiency, common-tone relationship and root

motion) and provides new insights on harmony of modal / tonal dichotomy,

therefore, is coherent and fruitful.

The system, however, is not limited to be used in the presence of conventional

chordal structures. As the octatonic collection can be presented in various pitch

combinations in both dimensions, there is also the possibility of using the system in

conjunction with other methodologies, such as Schenkerian analysis and set-theory,

in order to analyze Romantic and post-tonal music (Newton’s 2013 article on

Schoenberg’s pre-serial music is examplary in that sense). For the sake of scope and

depth of the study, this goal is left to be fulfilled in future projects.

49

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APPENDICES

APPENDIX A: Analytical scores 31 1) Machaut: De bon espoir 2) Agricola: Je n’ay dueil 3) Palestrina: Agnus Dei 4) Monteverdi: Era l’anima mia 5) Gesualdo: O tenebroso giorno 6) Carissimi A: Plorate colles 7) Carissimi B: Plorate Israel 8) Corelli: Prelude! APPENDIX A1

31 All scores are from IMSLP, except Machaut motet, which is from “Geschichte der Mensural Notation von 1250-1460” by Johannes Wolf, p.36-41 (Accessed via openlibrary.org).

56

Machaut:

57

58

59

60

61

APPENDIX A2 Agricola:

62

63

64

65

66

APPENDIX A3

Palestrina:

67

68

69

70

71

APPENDIX A4

Monteverdi:

72

73

74

75

76

APPENDIX A5

Gesualdo:

77

78

79

80

APPENDIX A6

Carissimi A:

81

82

83

APPENDIX A7 Carissimi B

84

85

86

APPENDIX A8 Corelli:

87

CURRICULUM VITAE

Name Surname: Sami Tunca Olcayto

Place and Date of Birth: Istanbul, 20.08.1989

Address: Gayrettepe Mahallesi, Göktürk sokak 4/33, Istanbul

E-Mail: [email protected]

B.Sc.: Istanbul University Forestry Faculty Department Of Forest Industry Engineering - July 2011

A New Theory of Harmonic Motion and Its Application to Pre-tonal and Tonal Repertoire

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