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Introduction: Silence and Sound

Feb 12, 2017



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    Vedouc prce: doc. PhDr. Lubomr Spurn, Ph.D.

    Konzultant: Dr Elizabeth Eva Leach

    BRNO 2006

  • Prohlauji, e jsem pracoval samostatn, s pouitm uveden literatury.

    Martin Celhoffer

    V Brn, dne 19. ervna 2006

  • Summary

    This writing deals with one of the most problematic issues in musicology - the

    ontology of music. The themes selected are developed on the background of antinomy

    between sound and silence. This antinomy was understood as (i) audible and inaudible

    categories of music during Antiquity and Middle Ages; (ii) object (sound-concepts, sound-

    structures) and subjective condition (silence of a priori givenness of perceptual and

    cognitional faculties of a man) during the modern era. Chapters are organized chronologically

    showing the evolution of the idea of world harmony from Antiquity till the early empirics in

    the first part; and the theories of modern philosophical concern in metaphysics of music in the

    second part. The study also reflects how music was apprehended methodologically: as (i) a

    multidisciplinary field incorporating music, mathematics, astronomy, astrology; (ii) a

    philosophy of music integrating epistemology and ontology.

    Prce je vnovna snad nejdiskutabilnj otzce muzikologie - ontologii hudby.

    Vybran tematick okruhy jsou rozvjeny ve svtle zkladnho protikladu mezi zvukem a

    tichem. Tento protiklad je pojmn jako (i) otzka kategorizace hudby na slyitelnou a

    neslyitelnou v obdob antiky a stedovku; (ii) otzka vztahu objekt (zvukov koncept nebo

    struktura) a subjekt (ticho apriornch danost lidskho vnmn a umu) v obdob novovku.

    Jednotliv kapitoly jsou azeny chronologicky, v prvn sti prce vypovdaj o vvoji

    mylenky harmonie kosmu od antiky a po ran empirismus; ve druh sti o rznch

    novovkch filozofickch teorich zabvajcch se metafyzikou hudby. Prce tak odr jak

    byla ontologie hudby chpna z metodologickho hlediska: jako (i) multidisciplinrn pstup

    zahrnujc hudbu, matematiku, astronomii a astrologii; (ii) filozofie hudby zahrnujc

    epistemologii a ontologii.

  • Contents

    PREFACE.............................................................................................................................................................. 1

    INTRODUCTION: SILENCE AND SOUND .................................................................................................... 3

    PART ONE: THE COSMOLOGY OF MUSIC................................................................................................. 9

    Pythagorean tetractys ........................................................................................................... 10

    Platos World-Soul ............................................................................................................... 16

    Boethius Musica Mundana ................................................................................................. 21

    Keplers Planetary Polyphony.............................................................................................. 26

    PART TWO: POST-COSMOLOGICAL CONCEPTIONS ........................................................................... 31

    Immanuel Kant: Mind as a string instrument .................................................................... 33

    Schopenhauer: Music as representation of the Will............................................................. 40

    Phenomenology and ontology of Music............................................................................... 46

    CONCLUSION.................................................................................................................................................... 52

    BIBLIOGRAPHY ............................................................................................................................................... 56

  • Preface

    The aim of this paper is to interpret some of the basic assumptions of evaluation the

    ontological status of music. This issue never stood alone among the other disciplines, but

    became an indispensable part of our interpretations of all phenomena, whether related to

    music directly or not. When I had asked myself the question why is the ontology so

    fascinating and important for musicology, the well-known Schopenhauers statement arose in

    my mind:

    For in every mind which once gives itself up to the purely objective

    contemplation of the world, a desire has been awakened, however concealed and

    unconscious, to comprehend the true nature of things, of life, and of existence.

    (Schopenhauer, 1958, p.406)

    This concealed desire is a prime mover of our investigation. The fact that satisfactory

    answer to the questions concerning true nature of things, particularly the ontology of music,

    could not be provided does not make the attempt of this writing quixotic. The questions are

    not to be answered, simply because there is no answer to comprehend all the dimensions of

    real world in one individual theory.

    As I propose later, examining the ontology of music is not considered as independent

    and integrated theory, but might provide us with understanding of basic premises for various

    consecutive theories. A causa de cy the approach I have chosen quadrates with series of

    essays of individual subjects. The essays are organized chronologically and are arranged into

    the two parts: (1) The cosmology of music and (2) Post-cosmological conceptions. The

    cosmology of music is a gradually developing concept with common axioms. On the contrary,

    post-cosmology consists of various, occasionally contradictory theories, refusing the axiom-

    wise heritage of Antiquity and Middle ages.

    The common thread implicitly running across this writing is the idea exposed by the

    title: the antinomy between the terms silence and sound. The emphasis of this antinomy is

    given in the introduction. The reason for such a literary and not scientific title I found in

    fact, that audibility and inaudibility of music was one of the key issues in cosmology in

    accordance to ontology. On the other hand, the theories of post-cosmology deal with sound in


  • terms of sound-constructs or sound structures of musical work, and with silence in terms of

    subjective and general conditions, the question of audibility and inaudibility no more holds


    I have omitted extensive throng of philosophers and music theorists that have

    contributed to reception of the ontological issue of music. I did not intend to present a

    comprehensive enumeration of theorists, but I have pursued the tracks of some important


    Last but not least, I would like to express my thanks going to supervisors of this

    writing, Dr Elizabeth Eva Leach and doc. PhDr. Lubomr Spurn, Ph.D., who have given so

    generously of their time in granting so many advises and guidance upon my work.


  • Introduction: Silence and Sound

    It is silence here, statements like this or similar are used to describe a particular state

    of exchanging information between subject and environment. The semantic context of this

    utterance may differ: a suspicious silence, a pleasant silence, or an embarrassing silence. In

    fact, silence means relative silence as there is ultra-sound and infra-sound lying beyond the

    grasp of man ear. On the contrary, sound itself is conditional on a complicated process of

    perception. Thus silence and sound could be considered vice-versa: sound as silence and

    silence as sound. Consequently, the dichotomy of hearing is accepted as relative. While

    silence has always the same quality in terms of acoustics, sound has many forms and patterns

    sometimes evading from hearing. We built our realm of sound on the basis of contingent

    perceptual options.

    As a subject transforms environmental forces into sophisticated perception, this might

    be considered as disruption of subjects equilibrium resulting generation of information. This

    act is made through differentia, a kind of mirrored activity whereabouts conditions below

    the axis of disturbance are reflecting their nature above. Thus silence and sound refer to actual

    process rather than to particular phenomenon as an object by itself. Accounts of this, absolute

    silence and absolute sound are abstract theoretical notions.

    Silence and sound are not being further set out from acoustical point of view as might

    have been expected. The approach to these categories by means of philosophical speculation

    may uncover basic facets of our assumptions for understanding music. In addition, silence

    and sound hint at dynamical dichotomy of inherency, particularly in music. Music arises

    from a dialogue between them, from their interaction. Does music tacitly mean peculiar

    paradigm of sound otherwise can music be in specific case silent? This issue occurs when

    approaching to Pythagoras, Plato, Boethius and many other theorists.

    It is possible to imagine many things by the term music: a well-known melody, a

    popular hit, soundtrack, musical, opera, a village song or a great symphonic work. We hardly

    ever imagine something else quite different from the notion of sounding structure. Our

    understanding of music is exclusively perceptual. However, for Pythagoreans the term music


  • means mathematics. Music was for them a science par excellence, a science providing the key

    for understanding the cosmos, a universal harmony holding and uniting everything. These two

    conceptions, one based on perceptual quality and another pointing out its ontology, represent

    perceptual and meta-perceptual paradigm of understanding the term music.

    Music is unpromptedly associated with an act of hearing. Thus the purpose of music

    was described as to delight the human ear in many treatises, emphasizing the aspect of

    aesthetic perception. However, music was also considered as a science dealing with act of

    understanding and meta-sensual experience. Another conception is that of the Memoria:

    music can be stored in human memory and contrariwise restored from it. Moreover, on the

    mediaeval concept of memory is based the creative ability - inventio (Ziolkowski 2002,


    The relations of these antimonies discussed thus far are presented in table 1. The

    horizontal axis x describes the perceptual and meta-perceptual dichotomy. The vertical axis

    y divides the bilateral phenomena of sound and silence. Music is described as a set

    containing determinate areas of every category involved.

    Table 1.

    If we are trying to understand what the word music actually means, it would be

    unavoidable to establish an ontological dispute. In a general way, what is the nature of any

    given real particularity? The nature of something in terms of ontology refers to quality not to

    quantity and therefore is immeasurable. Therefore is not possible to define the word music as

    the succession of coded pitches in spatial and temporal organization. Anyway, how should the


  • ontology of music then be construed? As we come to a resolution that music can be

    considered as an ontological entity there is an intrinsic difference between treating this entity

    as an object or as a homologue (Wallin, 1991) between the two or more objects. Namely the

    term object associates a natural entity with spatial and temporal organization that is possible

    to perceive through the senses. On the contrary, a homologue means a correspondence which

    can be to some extent abstract as well as measurable, for instance through mathematical

    equation. Therefore, music is accepted as a multiple set of heterogeneous phenomena. In

    order to investigate the ontology the examination is focused to the correlations among them.

    Was this issue a crucial task for theorists during the antiquity and Middle Ages and

    how was it aproached? Quid est Musica? - appears in various Latin writings on music usually

    at the very beginning of treatise. The answer mostly refers to the reinterpretation of Greek

    thought that portrays music as an activity of the Muses or as a reflection of cosmic harmony

    or harmony by itself ruled by Creator - a supreme being. But the ontology of music eo ipso,

    as a discrete discipline, was never a principal and outstanding issue for such philosophers as

    Pythagoras, Plato, Plotinus or Boethius. In this sense the ontology of music had an

    interpretative quality of evaluating the aesthetic potential of music or like a theory of

    everything - integrating axiom of all phenomena.

    It is possible to provide a satisfactory answer to the question of what music is? Why is

    this question considered irrelevant in musicology and beside the point of science? The

    ultimate limit for acceptability of knowledge is given not explicitly by verification but in a

    certain extent by operative paradigm in terms of common convention. Probably the whole of

    knowledge has an abstract nature in the sense of projecting subjective notions onto reality. For

    instance, a cube with scaling of 2 x 2 x 2 units represents a three-dimensional object of

    cubage 8 units. This knowledge is mediated by empirical experience can be verified and

    simply imagined. But mathematicians are already manipulating with more than three-

    dimensional objects at the present time. That is to say, knowledge could be drawn from each

    empirical experience, and applied consequently to the field where is no direct verification

    based on classical empirics. The knowledge is not measurable by itself but through empiric

    (table 2):


  • Table 2.

    Empiricism provides a limited sum of verified data whereupon theories are construed.

    Consequently, these theories are supposed to confer a noticeable order to the particular data.

    In certain circumstances might the specific theory explain phenomena which are not possible

    to verify empirically. For example, the string field theory in physic is verified by

    mathematics and not by experiment. On the other hand, an empirically verified theory may

    fail analogous to thermodynamics. This reflection is not being an attempt to contest with

    verification in science but points out the very thin line disentangling acceptable and non-

    acceptable theories. Thus satisfactory resolution of the issue in terms of utter definition seems

    to be a quixotic investigation.

    Music is the discipline that undoubtedly has an empirical base when focusing on the

    angle of sound structure. But this positive facet might detract the understanding of its nature.

    It is not surprising that in the modern era, whenever emphasis on empiricism in terms of a

    touchstone is generally accentuated, it is currently the art most difficult to define in spite of its

    material substance. By contrast, mathematics even though dealing mostly with abstract

    numbers and theoretical dimensions eventually not possible to imagine is considered to be the

    positive science. On what matter of fact rests this paradox? It is certainly not due to music

    itself, i.e. to its acoustical structure and nature, but to the fact that music includes the

    perception and understanding of a mans subjectivity. Volker Kalisch states:

    ...if we speak about music we automatically have to deal with humans, with

    human conditions, with ourselves. And vice versa: when we start to reflect upon the

    nature of man we automatically reach a point where music intervenes. Being human

    and experiencing music are clearly inseparable. (Kalisch, 2000, p. 309)

    Therefore, music does not exclusively rest in sound and at least one facet of it is non-

    empirical. On the proviso that music includes specific human perception and understanding, it


  • will be necessary to deal with these phenomena. According to Cooke (1957, p.5), music is a

    language of sub-consciousness. Both terms are problematic alone and even more in their

    juncture. It is a literary explanation, a sophisticated metaphor and therefore it should not be

    taken au pied de la lettre. Simplistically said, a language is just a tool mediating particular

    semantic notions. Its syntax can be understood by music theory and acoustics. It is not

    necessary to emphasize how useful this would be in terms of musical analysis. But

    investigation like this cannot shed a light to the nature of music. On the other hand, inclusion

    of the assumption of subconsciousness existence seems to contribute to our ontological

    dispute. This refers on basic antimony of this discourse: sound as a conscious phenomenon,

    and silence as an unconscious phenomenon. From this point of view, Musica humana can be

    interpreted as subconscious music - inaudible consonances of human mind subordinated to

    higher quality of Musica mundana.

    However, the nature of subconsciousness - or whatever we call it - is to a particular

    extent an immeasurable part of the realm of our existence impenetrable with actual

    measurements but perhaps penetrable with imagination, inspiration and contemplation.

    Notably in the arts, imagination and inspiration have a crucial role and there are indispensable

    parts of both creative and perceptive processes. If these qualities of subconsciousness

    participate on music it might be the reason for exclusivity of phenomena such as aesthetics in

    terms of unaccountability of its nature. Does it mean that these phenomena considering being

    beyond actual methodological grasp are less important?

    As it can be seen from previous allusive explanations, an approach to the ontology of

    music can be pursued from various directions. However, the basic intention of them is to

    supply a background for interpretation of cosmology of music. A number of key notions

    (Pythagorean tetractys, Platos World-Soul, Boethius Musica Mundana, and Kepelrs

    Planetary polyphony) ranked chronologically will be presented as representation of particular

    conception of ontology. A common thread of these conceptions is the idea of cosmology and

    how this was reinterpreted in antiquity, Middle Ages, and Renaissance. The structure of study

    was deliberately set up on a number of essays and therefore it is not intended as a

    comprehensive study on ontology of music. I tried to avoid generalizations and

    simplifications although more extensive historical background would be desirable for

    advanced and more accurate research. Intention of examining ontological issues is to provide

    paradigms for understanding music, for drawing courageous parallels between the past and

    present. Aspiration of this study is not in presenting the notion of ontology as a scientific


  • method but as a cognitive-literary dialogue between mind and imagination, the reason and the

    art, and between silence and sound.



    The cosmology of music is an interdisciplinary achievement bringing together

    mathematics, astronomy (eventually astrology), and music. Although the first essay is devoted

    to Pythagorean doctrine of tetractys, we should bear in mind that the formal sources of

    Pythagoreans are running back to Babylonians (McClain, 1984). The cosmology presupposes

    a series of common premises, which were gradually shifted in pursuance of new empirical

    data. This progression is recorded in four following essays, from Pythagorean tetractys to the

    theory of Keplers harmony of the world.

    The hypotheses about the cosmology of music were set out on the basis of

    contemporary understanding of the universe. Neugebauer (1969, p.171) noted that:

    To a modern scientist, an ancient astrological treatise appears as mere

    nonsense. But we should not forget that we must evaluate such doctrines against the

    contemporary background. To Greek philosophers and astronomers the universe was a

    well defined structure of directly related bodies. The concept of predictable influence

    between these bodies is in principle not at all different from any modern mechanistic


    However, the Greek view on cosmos drew its formal conception either from

    mythology. It was generally accepted that the universe was settled by mythical beings. And

    celestial music was their expression. Forasmuch as the empirical data were missing in

    Antiquity and Middle Ages, and therefore there was no physical conception of sound, the

    antinomy of silence and sound the audibility and inaudibility of this music varied among

    authors. The theory of musical cosmology was, in fact, an attempt to discover the theory of

    everything - the universal knowledge conferring the sense to all phenomena. Sound was

    metaphorically confused with immaterial harmonious order of random phenomena.


  • Pythagorean tetractys

    The essence of all things rests in numbers - this was a rudimental premise for

    Pythagoreans. A notion of everything penetrating immaterial numeric principle was holding

    sway among the Pythagoreans for nearly the whole millennium. Pythagoreans considered

    music as the most reputable discipline due to its possibilities of discovering the numerical

    ratios and verifying their hierarchy. However, this theorem of multilateral correlation between

    the number and music was also noted by Babylonians and Egyptians probably about two

    thousand years or even more before Pythagoras (McClain, 1984). This suggests that

    Pythagoras (c.560 - c.480 BC) was not inventor of this knowledge, but rather its importer to

    Greece, and hints on his possible scholarship in Middle East. Though Lippman (1963) holds

    the view, that as difficult as it is to specify the older constituents of Pythagorean thought, it is

    still more difficult to determine the precise achievements of Pythagoras himself. This is even

    complicated by the fact that of no authentic writing of Pythagoras has been preserved.

    Therefore, the investigation and evaluation of Pythagoras own contribution is thrown upon a

    number of secondary sources, which were written by his followers and other philosophers.

    Pythagorean thoughts were transmitted to Christian mystics thanks to Platos (c.429 -

    347 BC) writings, especially his Republic. Also Aristotle (384 - 322 BC) mentions some of

    Pythagoreans ideas in Methaphysics. Euclid (c.325 - 256 BC) presented the theory of

    numerical ratios in Elements, books 7-9 based on Pythagorean music theory. Also

    Nicomachus (c.60 - c.120 AD) discussed Pythagorean ratios in his Introductio arithmetica.

    Ptolemy (c.90 - c.168) in his Harmonics - a summary of Greek knowledge on music, points

    out Pythagoras contribution in terms of noting that music is the direct link between the

    microcosm and the macrocosm, between mans soul and the universe (Gamwell, 2004). The

    idea of the music of the spheres and codification of the regular proportional relationships

    between particular tonal pitches is also generally accredited to Pythagoras (Harap, 1938;

    Kinkeldey, 1948). Nevertheless, several myths about Pythagoras life emerged inevitably and

    became legends. Unarguably, the most famous legend is the hammer story found in

    Nicomachus Manual of Harmonics and in many other writings. Another myth considers


  • Pythagoras as the inventor of monochord, but according to Waerden (1943) and Dykast

    (2004), monochord emerged as late as in the third century BC.

    Generally, Greeks imagined the earth as a flat disc confined by the river Oceanus

    while Gods lived at the top of the mountain Olympus. The stars were assumed to be orbiting

    around the earth. Ever since the notion of their trajectories being determined by certain

    relations emerged, the particular celestial bodies were associated with particular spheres. It

    was believed that a Siren was sitting on each sphere, or respectively, each sphere was ruled by

    an individual deity. To how many spheres was the Pythagorean cosmos understood to be

    divided is disputable. Meyer-Baer (1970, p.44) suggests an interesting hypothesis on a

    possible view on cosmos:

    In the south of Italy, a number of bronze vortices, or spindles, have been

    found which consist of eight disc on an axis, their circumferences forming a kind of

    double cone. It is probable that these, like the golden tablets found in the same area,

    were used in the rites of the Pythagoreans, a sect which had several centers in southern

    Italy and Sicily. Although the details of these rites have remained a secret, it is known

    that the idea of the cosmos constructed in eight layers was a basic dogma of

    Pythagorean belief. The image of the seven or eight spheres was derived from the

    Babylonians, but the Pythagoreans added a new concept of numbers as a ruling

    principle of the cosmos and also identified them with musical intervals.

    However, no source seems to illustrate in greater detail, how is the mechanism of

    spheres in Pythagorean cosmos understood in terms of arithmos. Forasmuch, due to the lack

    of sufficient and attested data in the ancient astronomy, the organization of the universe was

    believed to follow the same rules as the organization of material in music, where the actual

    measurements confirm the conception of arithmos and harmonia. Pythagoreans assumed all

    things to be originated in numbers, and this axiom was applied to the ontology of all

    phenomena. Thus music was a key, a gateway to a higher understanding a theory of


    Pythagoreans proposed a distinction between the notions of unity - immaterial being,

    and multeity - material being. This basic antimony constitutes an ontological differentia,

    which results in the act of creation:

    ...material world, being a physical entity, must have a limit. This limit is

    inherent in the numbers 1, 2, 3, 4. They create the point, the line, the plane and the

    volume. Adding up these dimensions, 1 + 2 + 3 + 4 = 10, we exhaust the limits of


  • physical extension. There is no number following 10 that is not incapsulated in the

    tetrad. Nothing can be added that does not exist as a combination of these four

    numbers. Tetrad and decad are therefore the models of perfection. They create unity

    out of multeity, and multeity out of unity. They originated in the unlimited, absolute

    world, but when they extend into the physical world, they create a limited, yet perfect,

    unified system, a universe. (Berghaus, 1992, p.45)

    This very accurate interpretation of the tetractyl, i.e. a set of four numbers regarded as

    the source of all things, suggests the basic constituents of Pythagorean thought. The dynamic

    aspect of tetractyl and emergence of time quality is described as follows:

    Time, or the fourth dimension, enters when the idea of number is given

    physical extension. Chronological time (chronos) proceeds from eternity (aion), just

    as finite space proceeds from infinity. The metaphor for this process is creation. The

    act of creation establishes the monad in time and space. Time is a corollary product of

    the creation of the physical world, and it prevents the created universe from remaining

    static. The cosmos is dynamic, ever-changing. However, this constant motion is

    ordered. The dimension of temporality is structured just like the spatial world. The

    patterns of movement obey a scheme. All changes are phases proportionate

    propensities. (Berghaus, 1992, p.45)

    The Pythagorean conception of the genesis of universe shifted the traditional Greek

    world-view and also the role of deity to a great extent. However, the only touchstone available

    for experimental support of this doctrine was through music, especially the specification of

    numeric ratios in musical intervals. There are two basic categories of ratios: superparticular,

    defined by formula ; and reciprocal . With regard to this formulas,

    superparticular ratios are 2:1, 3:2, 4:3 etc.; reciprocal 1:2, 2:3, 3:4 etc. These ratios are

    equivocal according to string length (or cubage of air column) and frequency. Where

    superparticular ratios determine the frequency, the reciprocal determine the string length and

    vice versa:

    diapason diapente diatessaron

    string length 1:2 2:3 3:4

    frequency 2:1 3:2 4:3

    Table 3.


  • This table represents the first three harmonics. Naturally, as the ratio sequence

    continues further, frequency increases and, respectively, string length decreases. But only first

    three ratios are considered as the perfect consonants. This is mathematically proven by the

    fact that these ratios of tetractycal numbers exclusively. Moreover, tetractys (1, 2, 3, 4 /

    1+2+3+4=10) is a ground for basic numbers that constitute the ratios of consonances in

    whichever combination: 4:1= double-octave, 3:1= octave + fifth, 2:1 or 4:2= octave, 4:3=

    fourth, 3:2= fifth (addition of fifth and fourth results the octave and difference between them

    the whole tone 9:8). The fact that what is sensually perceived as consonant and plausible

    harmonious sounds can be expressed mathematically, suggests the hypothesis that everything

    has its base in numbers. For all of the above, mathematics became the most reliable method of

    codifying the rules of music and, consequently, the rules determining all phenomena.

    Providing that the whole universe originates in number, multeity of all things is

    unified by basic numerical patterns. Of course, this includes individual human beings: a body

    and a soul. It is assumed, that Pythagoras believed that listening to the music has a specific

    effect on the soul. The dynamic process in music evokes the congeneric process in the human

    soul. Censorinus (early third century AD), a neopythagorean, states that music has a similar

    effect on the soul as the stars. Similarly as music, astrology was understood to be determined

    by numbers and geometrical figures (De die natali, XI-XIII, ed. O. Jahn, Berlin, 1845. IN:

    Godwin, 1986, 19). Iambichus (c.250 - c.325 AD) points out that music contributed greatly

    to health, if it was used in an appropriate manner (Life of Pythagoras, translated by Thomas

    Taylor, London, 1818, pp. 43-8, 80-3., IN: Godwin, 1986, p.28):

    And there are certain melodies devised as remedies against the passions of the

    soul, and also against despondency and lamentation, which Pythagoras invented as

    things that afford the greatest assistance in these maladies. And again, he employed

    other melodies against rage and anger, and against every aberration of the soul. There

    is also another kind of modulation invented as a remedy against desires. He likewise

    used dancing; but employed the lyre as an instrument for this purpose.

    Therefore, the instrumental and vocal imitation of the music of the spheres was

    considered as the early music therapy (particularly in terms of mental health), through which

    the harmonia between body and soul and between soul and cosmos was achieved. Censorinus

    (Godwin, 1986) mentions that Pythagoras himself might have been used to play his lyre

    before sleep and upon waking up to imbue his soul with its divine quality.


  • Another disputable issue emerges in connection with audibility of music of the

    spheres. Various sources refer to the myth of Pythagoras exceptional ability to hear celestial

    harmony. However, there are two quite different suggestions, and both can be found in

    Simplicius (fl. first half of 6th c.) commentary on Aristotles On the Heavens (trans. by Th.

    Taylor in The Theoretic Arithmetic of the Pythagoreans, London, 1816, IN: Godwin, 1986,


    Pythagoras, who is reported to have heard this harmony, should have his

    terrestrial body exempt from him, and his luminous and celestial vehicle, and the

    senses which it contains, purified, either through a good allotment, or through probity

    of life, or through a perfection arising from sacred operations, such a one will perceive

    things invisible to others, and will hear things inaudible by others.

    Though Simplicius is quite skeptical about this possibly influenced by Platonic view

    on disputable materiality of spheres. He assumed that if any sound emerges from superlunary

    spheres (i.e. those of immaterial celestial bodies), it should be neither percussive nor

    destructive but it activates the powers and energies of sublunary sounds (i.e. material), and

    also affects the senses in terms of harmonization. Accordingly, there are the two types of

    sound in the Pythagorean theory: material and immaterial. If then, sound is not passive there

    [i.e. in superlunary spheres] it is evident that neither will the sound which is there be passive.

    However, Simplicius suggested yet another view: he mentions that Pythagoras might have

    said that he hears the harmony of the spheres as understanding the harmonic proportions in

    numbers, of the heavenly bodies, and that which is audible in them.

    The proportions which constitute human beings are homologous to proportions of the

    universe, nature and music. The Pythagorean theory, leastwise as comes to us, incorporates an

    ontological issue. From this point of view it was not only descriptive analysis of music, but

    searching for primordial principle that extends into the material world and becomes

    multeity (Berghaus, 1992, p.45). The musical sound is considered as an audible

    representation of this principle. On the other hand, ontological immaterial music rests in

    number in the state of rest and becomes audible through the dynamics of tetractys - sound

    emerges from silence, music of the spheres becomes audible and recognizable on the basis of


    Pythagoras, in fact, established the numerical basis of acoustics. The generalizations of

    intervallic proportionality were also applied to dance, sculpture and architecture. The visual

    arts follow proportionality defined by the golden section, tetragonal figure designed on the


  • basis of Pythagorean numeric ratios. Pythagorean conception of cosmic harmony influenced

    Neo-Platonists, astronomers, astrologers, humanist scholars, poets and musical theorists. His

    thoughts, as driven on by his followers, traveled in two directions: those who followed his

    mathematical conception and those who followed his ethos theory (Harap, 1938).


  • Platos World-Soul

    Prior to number becoming a matter, the idea appeared. Platonian idea provides a

    sophisticated plan for numeric underpinning of the material world. It is believed that all of

    Platos (c429 - 347 BC) authentic writings are preserved. The most remarkable sources of his

    thoughts on music are Timaneus, Laws and Republic. However, Plato mentions the term

    music (mousike) in many places in his dialogues and the meaning of this term may vary from

    often condemned sensual pleasure to speculative music as a supreme wisdom (Fubini, 1990).

    The immense range of Neo-platonic sources also accounts for the further development of his

    ideas during the Middle Ages and Renaissance.

    Generally, Plato was continuator of two Greek conceptions of music: (i) the

    Pythagorean numeric ontologism and (ii) the Damonian doctrines of musical ethos. Platos

    approach to music was marked by his tendencies of searching the ideal social structure. Thus,

    he implemented the ideas about music into wider context of education and politics. He was a

    traditionalist, censorious towards the individualism in modern music, which might break

    integrity of the society. On the contrary, he viewed the traditional music as acieraging the

    society, having a collective character and therefore being an important element of the social

    sustainability. Plato criticized decadent style of music, apparently in practice in Greek society,

    which he perceived as corruptive, threatening the social welfare of society. These ideas about

    the social role of music are explored on the background of musical ontology, which is the

    synthesis of Pythagoreism and Platos doctrine of ideas, particularly the doctrine of the world

    soul as a universal plan of all things described in Timaneus.

    In his Laws, Plato provides the first epistemological antimony between rationalism and

    sensualism, when criticizing astronomers for theirs methods of verifying the arguments

    through sensual experience. However, his rationalism is not well marked but is rather based

    on intuitivism and mysticism. The most remarkable of all Platonian thoughts is undoubtedly

    the doctrine of the ideas. Ideas are non-material archetypes, primordial prototypes, invariable,

    perennial units that are qualitative superior to all phenomena and subjects of visual and

    sensible world, which is only the imperfect imitation of them. Whereupon the reality is

    divided into two spheres: (i) world of ideas, which are the primordial existence - an


  • ontological basis of all phenomena and (ii) world of spatial-temporal phenomena, which can

    be recognized through the senses. The Platonian ontology can be summarized in four basic


    All things have an autonomous existence.

    The ontologisation of the notion: the subject of every notion has to be the idea - an

    inalterable being.

    The hypostasis of the notion: notions exist outside the world.

    The knowledge has rational nature and not sensual in terms of recalling the divine

    substance settled in the human soul.

    The first preserved formulations of the theory of cosmology can be found in Platos

    Timaneus (the word music is not mentioned, but Plato used the term harmony, Kinkeldey,

    1948), and in his Republic. The Platos view on Cosmos was based on the same assumptions,

    but differs from that of the Pythagorean view in some details. The main difference is that

    Plato defined the model for physical universe - a world-soul, as mediation between numerical

    base of all things and physical universe. Therefore numbers became a matter through world-

    soul, which is constructed in accordance to a numeric plan. The physical lay-out of the

    universe is very similar to that of the Pythagorean one. The whole universe has a shape of

    globe as the most perfect form in terms of symmetry. Cosmos consists of eight circular orbits

    - spheres: the first sphere consist of fixed stars running in opposite direction to the others, the

    second is occupied by Saturn, follows by Jupiter, Mars, Mercury, Venus, Sun, and the eight

    sphere is occupied by Moon.

    And of each of its circles there was seated a Siren on the upper side, carried

    round, and uttering a single sound on one pitch. But the whole of them, being eight,

    composed a single harmony. (Republic, X, 614b-619b, translated by Thomas Taylor in

    The Works of Plato, London, 1804, vol. I, pp. 466-75., IN: Godwin, 1986).

    Therefore the individual spheres are not producing basis for musical cosmic scale,

    but are moving in harmonious proportions. Godwin (1986, p.3) suggests that:

    If we allow that these spheres may represent not only the physical orbits of

    the planets, but also the powers of those planetary archetypes - the Mercurial, the

    Solar, etc. - within the human soul itself, then the journey [journey of souls described

    in the Myth of Er] takes on new meaning as an inner pilgrimage through the layers of


  • the psyche to the divine center within, which is also the all-encompassing

    circumference of the Spirit.

    This may hint at possible contemporary conceptions of astrology, reflecting very close

    relations between macrocosm and microcosm and the universal validity of the basic creative

    principles displayed through the visible cosmos.

    The universe, human beings and also music are built upon the world-soul, in

    accordance to its harmonious proportions. The structure of the world-soul consists of the two

    basic numerical ratios: 1:2, 1:3 and of their squares, which constitute the basic numerical

    consequence: 1, 2, 3, 4, 8, 9, 16, 27, 32 and so on. This sequence can be divided into two lines

    according to double or triple squares ordering from 1 to 729 (Dykast, 2005):

    2 4 8 16 32 64 128 256 512 1

    3 - 9 27 - 81 243 - 729

    Table 4.

    The number one represents the fundamental number from which all other numbers are

    generated. The ascending sequence of these numbers (1, 2, 3, 4, -, 8, 9, 16, 32, -, 64, 81, 128,

    243, and 256) constitutes the numeric plan of word-soul expressed in numeric ratios:


    The individual ratios of this sequence determine the musical intervals:

    1 = unison 64:81 = major third

    1:2 = octave 81:128 = minor sixth

    2:3 = fifth 128:243 = major seventh

    3:4 = fourth 243:256 = half-tone.

    4:8 = octave

    8:9 = whole tone

    9:16 = minor seventh

    16:27 = major sixth

    27:32 = minor third

    32:64 = octave


  • The last two numbers are not included in the world-soul, because their ratio 512:729

    defines diminished fifth which is the delimitative point for acceptance. The first four numbers

    of this sequence are, in fact, the numbers of Pythagorean tetractys. In general, the closer the

    ratio is to its source (number 1) the more perfect is the ratio and the more plausible is the

    musical interval based on it.

    The fact that ratios of world-soul can be applied to determine the musical intervals

    indicates the ontological status of music and its inextricable link to mathematics. In

    accordance to Platonian doctrine, audible music is just imperfect imitation of music of the

    spheres, music that rests in the silence of world-soul. On the contrary, inaudible music is

    expression of harmony and symmetry and can be considered as a general quality of

    phenomena, and as a desirable state of human soul. Thus, musician is not only the one who

    plays or sings, but also the one who has reached the perfect harmony within his soul (Fubini,

    1990, p.39) and who has acquired the capability of higher understanding closer to vision of

    the subject-matter. Moreover, playing the instrument does not necessarily mean performing,

    but has the symbolic meaning:

    The lyre ultimately became a popular symbol of the harmony of the cosmos. It

    appears in the hands of Muses, or Eros, or Orpheus... The lyre becomes the means for

    harmonizing the cosmos, and the figure that governs the cosmos rules it by playing the

    seven strings of the lyre, harmonizing the seven spheres. (Meyer-Baer, 1970, p.68)

    Nevertheless, Plato understood music as being in a close juncture with the other arts:

    poetry, dance and theatre. Music is audible and non-audible (as sound and silence)

    representation of world-souls numeric proportions. Each art displays the same proportionality

    in different way. However, Plato proposed ontological contrast between material and non-

    material, visible and non-visible, audible and non-audible world through antimony of idea and

    matter on the same numerical base. Platonian ontological hierarchy can be designed as


    Number: ratios, mathematical formulation - analogous to probable patterns of

    oscillation of the string, which is the most fundamental known particle in physics.

    Idea: primordial prototypes - analogous to genetic patterns.

    Matter: audible and visible things - analogous to spatial-temporal phenomena.

    Plato believed that music has a magic influence on human soul. Every human soul has

    been wandering in supernal world of ideas and numeric entities before its birth (this can be


  • understood in terms of modern science as genetic being). When came into material and non-

    eternal world, the soul is recollecting the past through experience and gradually penetrates

    through the layers of the soul analogous to the spheres of the universe. This might be the

    aspect of inner journey in the Myth of Er (as suggested Godwin above). The harmony within

    music echoes the harmony within the soul, and at the same time the harmony of the whole

    universe (Fubini, 1990). The motion in music is corresponding to the motion within the

    human soul and therefore music has not primarily the function of entertainment but serve as a

    tool for refinement of the human affects and for the ethical improvement. Thus music rounded

    out the divine plan in humans. Platonian tradition came to Middle Ages as the antinomy of

    idea and matter, soul and body, and became transformed by Christian conception of world-

    view as being synonymous with the terms of good and evil. Therefore Neo-Platonist

    philosophers doubted the sensual perception of music and appreciated music as a

    mathematical discipline.


  • Boethius Musica Mundana

    The influence of Boethius conception of musica on the following generations of Latin

    writers on music is undisputable. He is regarded by modern scholars as perhaps the most

    profound writer and translator of Greek tradition into Latin culture, needless to say, besides

    the other philosophers of the early Christian Age, as were Iambichus of Chalcis, Augustine,

    Martianus Capella and Cassiodorus. At first, one question comes to foreground: to which

    extend is Boethius translator and compiler of Greek knowledge, or contributor of new ideas

    and theories. According to Karpati (1987, pp.6-7) there was no comprehensive work in Latin

    which could contain high-level theoretical matter and from which Greek music - theory could

    be thoroughly learnt at the time of Boethius. Contemporary Roman writings were merely

    focused on grammar and rhetoric, in not comparable range to mathematical field, wherein

    music was counted. This might be the reason, why Boethius devoted himself to the

    mathematical disciplines. He put the four mathematical sciences under the umbrella term

    quadrivium (quadruvio vestigatur, chapter: Proemium, in quo divisio mathematicae, De

    institutione arithmetica, book I) for the first time, and which includes arithmetic, geometry,

    astrology and music. He regarded arithmetic as the superior discipline to the others:

    Quae igitur ex hisce prima discenda est nisi ea, quae principium matrisque

    quodammodo ad ceteras obtinet portionem? Haec est autem arithmetica ... ut animal

    prius est homine (De institutione arithmetica, book I).

    One might expect that the first of his writings was De institutione arithmetica, which

    is written after Nichomachus work on arithmetic reflecting Pythagorean view of ontology. It

    is not known whether he wrote also about geometry and astrology, but treatise on music De

    institutione musica is preserved, even incomplete (five books preserved from supposed

    original seven books), and become the most quoted treatise on music during the Middle Ages.

    De institutione musica is probably written after Nicomachuss lost work on music

    Introduction to music and the fifth book is a compilation of Ptolemys Harmonics. His treatise

    is a synthetic work reflecting Neo-Pythagorean and Neo-Platonic tradition. He understood

    music as a pure theoretical discipline - philosophical speculation. Counting music as a

    quadrivium discipline he contributed to music theory in terms of mediaeval systematical


  • knowledge and his Latin typology of music based on Greek cosmology become the general

    theoretical framework of mediaeval concept of music theory. This background information is

    important for proper understanding of different methodology of music.

    The concurrence of various disciplines, particularly those of quadrivium, in one

    common universal knowledge, simply imply that his thoughts about music should be scanned

    not only in De institutione musica, as it is usually realized, but as Chamberlain (1970)

    correctly suggests, also in other sources, notably in De institutione arithmetica and in

    Consolatio philosophiae. According to exploring the ontology of music, we can found

    fragmentary information in De institutione musica as Boethius himself noted that he will

    provide the further explanations of musica mundana and musica humana, but he did not

    mention where. Boethius, analogous to Pythagoras and Plato, understand numbers as

    primordial principle. According to his Christianity, of course, numbers become the primary

    tool of creation:

    ...Conditor deus primam suae habuit ratiocinationis exemplar et ad hanc

    cuncta constituit, quaecunque fabricante ratione per numeros adsignati ordinis

    invenere concordiam (De institutione arithmetica).

    The fact that numbers are construed as primordial beings is also important for

    ontology of music as the musical intervals are derived from numeric ratios. In chapter De

    substantia numeri he states that omnia quaecunque a primaeva rerum natura constructa sunt,

    numerorum videntur ratione formata. Therefore Boethian conception of music is based on

    mathematical underpinning of nature and influenced by methodology of quadrivium, which

    reflects the analogy of epistemological organon to the four senses, four seasons, four

    elements, and to the four creatures below the throne of God. It means that there are four ways

    how to approach to the reality according to hierarchy of quadrivium:

    Arithmetic: ontology - superior to all phenomena

    Geometry: pure forms - spatial phenomena

    Astrology: cosmos - spatial-temporal phenomena

    Music: perceivable quality of spatial-temporal phenomena

    Boethius, probably under the influence of Aristotle, emphasizes the sensual dimension

    of music and considers the ability of perception through the senses as a natural attribute of all


  • living creatures. However, he assumes that perception itself is linked to the mind rather than

    to the object and poses ontological issue:

    It is obvious that we use our senses in perceiving sensible objects. But what is

    the exact nature of these senses in connection with which we carry out our actions?

    And what is the actual property of these objects sensed? The answers to these

    questions are not so obvious; and they cannot become clear to anyone unless the

    contemplation of these things is guided by a comprehensive investigation of reality.

    (De institutione musica, Introduction, IN: Godwin, 1986, p.44)

    As Woodcock mentions (1943, p.32), The battle between reason and the senses still

    rages. Boethius presumes that through the mathematics it is possible to gain the

    understanding of the essence of all sensually perceived phenomena:

    ...if someone sees a triangle or square, he can easily identify it by sight. But

    what is the essence of a triangle or a square? This he must learn from a mathematician

    (Godwin, 1986, p.44).

    But when hearing, perceived sounds are not only recognized in terms of their

    structure, but also in terms of aesthetic judgment made by subject. This may be plausible if

    the are in the form of sweet and well-ordered modes, or implausible if the sounds heard are

    unordered and incoherent. Therefore, as Boethius states, the other disciplines (quadrivium)

    are related to the investigation of truth and music to speculation and morality: musica vero

    non modo speculationi verum etiam moralitati coniuncta sit. The basis for such an aesthetic

    judgment is considered in fact that amica est enim similitudo, dissimilitudo odiosa atque

    contraria. This can be understood as the principle of similarity between the structures

    within the subject and perceived object who are supposed to have, consequently, generic

    numerical origin. With accordance to the principle of similarity, as it was explaned by Neo-

    Platonists Plotinus, Boethius clearly adopt Platos ethos theory:

    For there is no greater path whereby instruction comes to the mind than

    through the ear. Therefore when rhythms and modes enter the mind by this path, there

    can be no doubt that they affect and remold the mind into their own character.

    (Godwin, 1986, p.45)

    The ethos theory is unavoidable consequence of hierarchy of phenomena and of

    generic numerical underpinning of the reality. Music is presupposed being everywhere, in

    both macrocosmic and microcosmic dimensions. Thus Pythagorean and Platonic thoughts

    about music and cosmology are prerequisites for Boethian classification of music. This,


  • perhaps the most quoted classification of music in mediaeval literature, can be found not only

    in De institutione musica, but as Chamberlain (1970) suggests, the other relevant definitions

    can be found in Consolatio. Either few further explanations can by drawn from De institutione

    arithmetica. Boethius describes in De institutione musica three types of music (Tres esse

    musicas): musica mundana, musica humana and musica instrumentalis. This typology is not

    quantitative, but either qualitative in the terms of dimensional ontology. Thus the music

    mundana and humana have different ontological status as instrumentalis. Musica

    instrumentalis is in its definition closest to the modern term of music and therefore is not

    further explained in this essay. On the contrary, the two other types of music are disputable in

    terms of ontology. Musica mundana includes three distinctive forms:

    The first type [musica mundana], that is the music of the universe, is best

    observed in those things which one perceives in heaven itself, or in the structure of the

    elements, or in the diversity of the seasons. (Godwin, 1986, p.46)

    Either according to Consolatio (Chamberlain, 1970), musica mundana consists of

    three forms or categories:

    1. Motions of celestial bodies - not audible for human ear.

    2. The binding of the elements - inaudible harmony.

    3. The alteration of the seasons - inaudible.

    In De institutione arithmetica Boethius explanes the generic numerical base for

    elements, seasons and celestial movements: Hinc enim [numerorum ratione formata] quattuor

    elementorum multitudo mutuata est, hinc temporum vices, hinc motus astrorum caelique

    conversio. However, Boethius does not distinguish the category of sound in such ontological

    hierarchy as the types of the musicas themselves. This regards the first type of musica


    How could it possibly be that such a swift heavenly machine should move

    silently in its course? And although we ourselves hear no sound - and indeed there are

    many causes for this phenomenon - it is nevertheless impossible that such a fast

    motion should produce absolutely no sound, especially since the orbits of the stars are

    joined by such a harmony that nothing so perfectly structured, so perfectly united, can

    be imagined. (Godwin, 1986, p.46)

    Chamberlain (1970, p.82) assumes that the philosophic tone of this music, like that of

    the world, would seem to be largely physical and mathematical. This confirms that Boethius


  • emphasizes the perceptual aspect of hearing and construed it as a part of world music. Thus,

    as Woodcock (1943) suggests, evaluation of consonance depends on both ear and science.

    In De institutione musica he only poses questions with regard to musica instrumentalis

    but does not provide a satisfactory answer:

    For what unities the incorporeal existence of the reason with the body except a

    certain harmony (coaptatio) and, as it were, a careful tuning of low and high pitches in

    such a way that they produce one consonance? What unites the parts of mans soul,

    which, according to Aristotle, is composed of a rational and irrational part? In what

    way are the elements of mans body related to each other or what holds together the

    various parts of his body in an established order? (Godwin, 1986, p.47)

    However, with reference to Consolatio, musica humana, similarly as mundana, has

    three forms, too (Chamberlain, 1970, p.82):

    1. The fit proportioning or blending (coaptatio and temperatio) of the

    incorporeal life of reason with the body, like one musical consonance of high

    and low notes.

    2. The second is the joining of parts within the soul itself, of rational and irrational


    3. The third is both the thorough mixing of the elements and the fixed proportioning

    of members in the body alone.

    This type of music is inaudible, and therefore rests in silence. Boethian conception of

    musica humana is far from simplification that of the harmony between the body and soul. It

    indicates the structural convexity within the living system on the basis of highly sophisticated

    mediation which Boethius assumes being conveyed by mathematical proportions. The four

    basic elements of genetic information and their combinations are the basic constituents of

    such living system - a microcosm itself, what seems being analogous to Pythagorean tetractys

    as the source of macrocosm. Boethius was much more Pythagorean as it is thought according

    to his treatise De institutione arithmetica. His classification of music should not be

    understood literary, but through the background information of arithmetic, numerology, Neo-

    Pythagorean and Neo-Platonic mysticism.


  • Keplers Planetary Polyphony

    Johannes Kepler (1571-1630), German astronomer and philosopher, was the last

    adherent of Greek tradition of Music of the spheres. His prolific and courageous attempt was

    to link together the idea of harmonious organization of the cosmos and the latest astronomical

    observations based on empirical data. However, Keplers first great work Mysterium

    Cosmographicum (1596) reflects deviation from Ptolemaic cosmological theory in favor of

    the heliocentric theory of planetary motion invented by Nicolaus Copernicus. The

    contemporary astronomers, such as Galileo Galilei, and Tycho de Brahe, took advantage of

    modern telescope equipments due to the improvement of optics. In 1600, Kepler became the

    assistant to Brahe in well-found astronomical observatory in Prague. After the death of Brahe

    in 1601, he assumed his position as court astronomer and mathematician to Rudolf II, and he

    had opportunity to use the most modern telescope in that period.

    Kepler sketched a comprehensive hypothesis of geometrical organization of planetary

    motions and distances among their orbits. He knew only five planets: Saturn, Jupiter, Mars,

    Venus, and Mercury. If there are included also Sun, Earth, and Moon, it is similar planetary,

    respective spherical, system as in Greek cosmology. However, the role of deities has shifted

    under the influence of new theories and observations. The spherical motion focused around

    the Earth became the orbital motion of particular planet around the Sun and was not

    autonomous henceforward. The Sun became not only the center of this new view of cosmos,

    but he considered it as the basic motive power which decreases proportionally with the

    distance of particular planet from the Sun. Later on, Isaac Newton established his theory of

    gravitational force on Kepler's theories and observations.

    During his service on the court of Emperor Rudolf II he wrote one of his major works

    Astronomia Nova (1609). This work contains formulations of two of so-called Keplers three

    laws of planetary motion, which are best known nowadays, although Kepler himself regards

    his explanations of worlds harmony more important than his formulations of the three laws

    of planetary motions (Brackenridge & Rossi, 1979). The first law is that he discovered the

    planetary orbits have to be elliptic and not circular as he had presupposed before. The Sun is

    stationary focus of these elliptic orbits. The second law states that an imaginary line from the


  • Sun to a planet terminates equal areas of an ellipse during equal intervals of time, accordingly

    an orbital speed of a planet is not constant and depends on distance from the Sun: the closer a

    planet comes to the Sun, the more rapidly it moves. The formulation of the third law was

    introduced in Harmonices Mundi (1619), it states that the ratio of a cube with face equal to a

    planets distance from the Sun and the square of the planets orbital period is a constant valid

    for all planets.

    According to his discoveries, Kepler believed that the universal harmony could be

    found in planetary structure. He devoted himself fully to research the geometrical laws

    accountable for distribution, organization, and motions of celestial bodies. In The

    Cosmographic Mystery he posed the question why there are only five planets in the solar

    system. He found the reason in the fact that there are only five regular three-dimensional

    solids with identical faces all joined with identical angle. These solids are: (1) Cubus, (2)

    Tetrahedron, (3) Dodecaherdon, (4) Icosihedron, and (5) Octohedron. In the fifth book of

    Harmonices Mundi, De harmonia perfectissima motuum coelestium, chapter De quinque

    solidis regularibus he drew a parallel between the intervals among the planetary spheres and

    these regular solids in following order: Saturn - cube - Jupiter; Jupiter - tetrahedron - Mars;

    Mars - dodecahedron - the Earth; the Earth - icosihedron - Venus; and Venus - octahedron -

    Mercury. Further investigation proved that data fits only approximately and not exactly.

    Nevertheless, Kepler believed that God the Creator is the Geometer and speaks in the

    universal language of geometry (Brackenridge & Rossi, 1979, p.113).

    The Keplerian geometrical forms, which are in basic correspondence to musical ratios,

    are derived from astrology. It comes to this that the layout of two particular planets or stars

    forms a geometrical or a non-geometrical angle by connecting the imaginary vectors of these

    bodies with the Earth. (Brackenridge & Rossi, 1979) A harmonic angle is the angle derived

    from one of eight harmonic ratios of motion (0o, 60o, 72o, 90o, 120o, 135o, 144o, and 180o;



  • Table 5. (according to Brackenridge & Rossi, 1979, p.114)

    Implicitly, these harmonic angles are indubitable due to inevitability of their causal

    existence, but their interpretations are axiomatic. The circle is divided by inscription of

    regular polygons into identical fragments, which constitute the ratios and angles, as it is

    presented in the schemes above. These angles and ratios implied from them are the basic

    terms of astrological interpretation of planetary constellation:

    Constellation conjunction opposition trine quartile sextile quintile biquintile sesquaquadrate

    Angle 0o 180o 120o 90o 60o 72o 144o 135o

    Interval unison octave fifth fourth minor





    sixth minor sixth

    Ratio 1:1 2:1 3:2 4:3 6:5 5:4 5:3 8:5

    Table 6.

    Kepler wrote: Perhaps God himself has invariably expressed these proportions

    [regular polygons inscribed into circle] in bodies and motions (Harmonices Mundi Lber III,


  • Source: Godwin, 1986, p.149). He accepted all the harmonic ratios derived from harmonic

    angles as musical consonances, in accordance to music theory of early 17th century.

    Kepler at first had believed that planets orbits are circular, but latter the observations

    data did not fit to this assumption and hint on the new theory that these orbits have to be

    elliptic (the first law of planetary motion) and therefore imperfect. This imperfection must

    have some reason and Kepler proposed a bit curious hypothesis: each planet does not

    represent particular tone (because the distance from the Sun is not constant) but melodic

    pattern in dependence of the range between the aphelius (farthest point from the Sun) and

    perihelius (closest point from the Sun) distances. This hypothesis of music of the spheres was

    new at all and based on empirical data. He adjudicated the musical scale the fundamental

    sequence of ascending and descending notes for each planet in correspondence with their

    elliptical orbits, and within the range equal to the deviation of perihelion and aphelion


    Table 7.

    The individual notes of these melodic patterns are selected points of permanently

    changing pitches (in terms of planetary glissando) within its range. The closer is the planet

    to the sun the higher is the pitch. The lowest note of each pattern represents aphelius point,

    the highest note perihelium. Therefore each pattern repeats simultaneously with one orbital

    period which is different for each planet (according to Kepler observations presented in

    Harmonices Mundi): Mercurius 87 days, Venus 224, the Earth 365, Mars 686, Jupiter 4332,

    and Saturnus 38769 days. The musical range is very small in the case of Earths orbital

    course - only the semitone mi fa. According to Keplers interpretation, the semitone sounds


  • sad, and symbolizes the decline and corruption of earthly sphere with its imperfection (misery,

    famine, Gamwell, 2002). Forasmuch as each planet has its own melodic subject, the music of

    the spheres, of Musica mundana, has to be polyphonic. Walker (1967, p.232) noted that:

    For Kepler just intonation and polyphony had finally prevailed because they

    were natural, that is, they corresponded to the archetypes in the mind of God, on

    which the created world was modeled, and which are also in the mind of man, the

    image of God.

    Keplers intention was primarily not focused to prove celestial music, but to find and

    prove harmonic proportions of planetary system in terms of their spatial and temporal

    organization. His speculations about music were based on astronomical research and

    astrologic interpretation of the universe. Is the Keplerian analogy among the regular solids,

    harmonious angles and musical consonances a metaphor or a real correspondence between the

    macrocosms and unifying principle based on universal harmony? Majority of Keplers

    observations and conclusions are valid till nowadays, but his cosmic polyphony is symbolic,

    because he did not believe that celestial harmonies are audible: The source of the human

    response to music is to be sought in the soul and intellect, not in physical matter (Harmonices

    Mundi Lber III, Source: Godwin, 1986, p.149). The important part of Keplers theories about

    music is his interpretation of a sound:

    How a plucked string will transfer its sound to another unplucked one if they

    are consonant with one another, but leave it immobile if they are dissonant. Since this

    cannot be the work of any mind, because a sound thus caused has neither mind nor

    intellect, it follows that we must say that it is caused by a simultaneity of motions... If

    the speed of one string can serve to move the other one tuned to it, though it be

    untouched as far as the eye can see, will not the same speeds of both strings serve for

    the pleasant stimulation of the ear, because the latter is in some way moved uniformly

    by the two strings, and two impulses from the two sounds or vibrations meet at the

    same moment? (Harmonices Mundi Lber III, Source: Godwin, 1986, pp.150, 151)

    The medium of transferring particular vibration Kepler took as species immateriata -

    the etheric emanation from object. Movement of e.g. a string causes this phenomenon. If a

    string of particular length and tension is potentially influenced by species immateriata, then it

    would be perhaps possible to transfer the powerful cosmic vibrations into human mind. The

    idea of simultaneity of motions was later treated by Athanasius Kircher in his great work

    Musurgia Universalis (1650). He described an imaginary instrument enneachord of nature

    on which every string, if plucked, resounds through all levels of the beings.



    As soon as philosophy became an autonomous discipline, by the result of establishing

    the positive natural sciences based on empirics during the 18th and 19th centuries, the

    metaphysic dimension of music was apprehended in separation from theory and practice of

    music, and from other sciences such as astronomy, mathematics and physics. Jean-Philippe

    Rameau noted in Preface to his remarkable Treatise on Harmony (1722):

    However much progress music may have made until our time, it appears that

    the more sensitive the ear has become to the marvelous effects of this art, the less

    inquisitive the mind has been about its true principles. One might sad that reason has

    lost its rights, while experience has acquired a certain authority. The surviving

    writings of the Ancients show us clearly that reason alone enabled them to discover

    most of the properties of music. Although experience still obliges us to accept the

    greater part of their rules, we neglect today all the advantages to be derived from the

    use of reason in favor of purely practical experience... Conclusions drawn from

    experience are often false, or at least leave us with doubts that only reason can dispel

    (Rameau, 1722/engl. ed. 1971, p.xxxiii).

    Contrary to the cosmology of music, where all disciplines were integrated in their

    methodology, the disciplines of modern science forked into arts and exact science. The

    methodology used in post-cosmological conceptions of musical ontology stems from

    philosophy: (i) epistemology, concerned with role of subject in cognition, (ii) ontology

    focused on subjective perception in terms of a priori givenness, and on the objectivity of

    music. Therefore the ontological issues lost their simplicity based on common premises, and

    became sophisticated disputes trying to comprehend the complexity of all phenomena and

    provide an acceptable paradigm for them.

    Metaphysic of music is apprehended from various points of view, represented by three

    subjects selected: (i) Immanuel Kants integration of rationalism and empirics, his rejection of

    metaphysics, (ii) Arthur Schopenhauers modern cosmology understanding music as a

    direct copy of everything-penetrating will, and (iii) phenomenology of music providing


  • several solutions of the ontological issue. Music was apprehended as a perceptual

    phenomenon, i.e. subjective perception of objective sound concepts. The metaphysic

    dimension of this perception was sought no more in cosmology but in human subject.

    However, the antimony of silence and sound became less and less recognizable and a new

    antinomy appeared: objective sound and perceived sound. The inaudible music of universe

    was lost.


  • Immanuel Kant: Mind as a string instrument

    One of the most influential philosophical theories concerning metaphysics and

    transcendental aesthetics was set up by Immanuel Kant, notably in his writings The Critique

    of Pure Reason (1787) and The Critique of Judgment (1790). Although Kant is not generally

    regarded as philosopher of music, his contributions in the field of epistemology (rational and

    empirical), relations of cognition and perception, explanations the categories of object and

    subject are indisputable. The consequences of his proposed ideas on philosophy and aesthetics

    of music became essential concepts for following generations of scientists and philosophers.

    His works, especially the three of Critiques, are a bridge between rationalist and empiricist


    Kant stated what was formerly indicated by Descartes, that objective knowledge can

    be apprehended exclusively as a consequence of the cognitive acts of a subject. Rational

    human subject is situated in the centre of cognition. Although he understood the aesthetics as

    a science concerning the perceptional quality of objects, rational order of the world could

    never derive from accumulation and organization of sensual perceptions. While in traditional

    cosmological theories the human subject is a part of whole universe, a mirror of it,

    microcosmical representation of macrocosm, and therefore the same rules are applied for

    them, Kant drew a methodological difference between them, and not a parallelism.

    Consequently, ...he situates beauty not in the object but in a psychological experience of the

    subject: the determining ground of the beautiful can be no other than subjective (Reed,

    1980, p.566).

    If the intelligibility of objects in the world can be brought about only via

    synthesizing acts of the mind, the mind can no longer be just the imitator of pre-

    existing objects... Kant rejects claims about nature's inherent structure, arguing that we

    ourselves give the law to nature as it appears to us, so that we cannot know nature as

    it is in itself (Bowie, 2006).

    This is the most fundamental contradiction to cosmology of music. The correlation

    between the human being and the universe has categorically been changed. Kant made a

    separation of supersensible realm from phenomenal world. Therefore, his conception of


  • transcendental aesthetics (Critique of pure reason) does not presuppose any existing forms of

    beauty, whether natural or artificial (Reed, 1980, p.568). The gateway to understanding lies

    in exploration of cognitional facility of a man. Thus ontology of music is concerning with the

    human subjectivity, especially in its pure form of cognition.

    He made a distinction of the knowledge into (i) categorial (a priori): pure without

    addition of empirics, independent from experience and impressions; (ii) empirical: originated

    in experience. He noted out that even if any knowledge does not anticipate experience in

    terms of temporality, knowledge itself is not entirely originated in experience. This

    contradiction is resolved by the fact that the quality of our cognitive faculty contributes by

    itself to knowledge. The mathematics serves as an example that a priori recognition is

    possible without the dependence of experience, even though mathematical judgments

    themselves are synthetic. Therefore reason is faculty providing a priori principles of


    The transcendental knowledge, about to be a priori possible, is concerned with no

    objects or phenomena, but with the process of our recognition of objects. The objects are

    given to us through the medium of the senses which provide us with perception. A pure form

    of this perception is possible to deduce as follows: if we eliminate from the representation of a

    solid (i) what reason thinks about it, and (ii) sensations like color, impenetrableness etc.; the

    remanentia of the representation are shape and spatial dimension (Kant: Critique of pure

    reason). Similarly, the time is not regarded as empirical notion derived from experience,

    because we could not perceive the actual time and the consecutive time if they are not based

    on a priori notion of time. Therefore space and time are essential notions, substances of all

    perceived sensory data. These substances are not autonomous as well as attributes of

    phenomena, but are the form, an inner sense of a mans perception. As a result of

    implication this theory into ontology, the real world was split into (i) phenomena -

    appearances, experience, and (ii) nomena - things themselves. All of the nomena are

    incognosible, thereupon the existence of primordial musical structures are rather uncertified.

    On this account, the existence of music, or musical phenomena, is conditional to human mind.

    However, Kant was rather skeptic about music as a mind becoming phenomenon. In

    his Critique of Judgment, 51 Of the division of the beautiful arts he proposed the

    classification of the fine arts into three categories: the arts of speech - rhetoric and poetry;

    the formative arts - sculpture, architecture, and painting; and the art of the play of

    sensations - Tonkunst. The first two categories present no analytical problems, but in the


  • third (play of sensation: tone and color) Kant finds that there is no certain utterance whether

    colors and sounds consist of disordered bundles of sensations, or possess determinate forms:

    If we think of the velocity of the vibrations of light [color] or in the second

    case of the air [tone], which probably far surpasses all our faculty of judging

    immediately in perception the time interval between them, we must believe that it is

    only the effect of these vibrations upon the elastic parts of our body that is felt, but that

    the time interval between them is not remarked or brought into judgment; and thus that

    only pleasantness, and not beauty of composition, is bound up with colors and tones.

    (Kant, 51, p.169)

    This statement is like to be a phenomenological analysis of composition, the same

    issue employed medieval writers on music, but in rather different conception in grammar of

    music (add reference!) The issue was related, similarly to Kant, to the question whether

    music consists of some kind of dividable commposita or not, et inde whether music is art

    conveying particular meaning or not. Parret (1998, p.260) on this account suggests:

    It is clear that music requires no activity of thought, since it has no semantic

    component: a musical sequence has no propositional content such as is found in

    language (including the most beautiful language possible, namely poetry). Music

    does not make us think; rather, it causes us to reflect and to dream more than any other


    According to this task, Reed (1980, pp.569-570) mentioned the problem of the nature

    of sound in connection with the first and second editions of the Critique: In the first he

    [Kant] says ich zweifle gar nicht that sound vibrations possess a form, but in the second he

    says ich zweifle gar sehr. If sound vibrations have a form, than it would be possible make a

    judgment and music would be considered as fine art. Kant offered a compromise: music

    possess to a particular extend both form and sensation. However, Kant placed music at the

    lowest place among the other arts because it merely plays with sensations and therefore is

    not constituted in pure rational acts (see phenomenology of music):

    The formative arts produce lasting impressions (Eindrcke), while music

    produces only transitory impressions. And this oscillating degree of presence

    appears to be related to the difficulties Kant has had in situating music in his analytical

    categories. However, both comments are governed by a structure that continually

    discounts the worth of music with respect to that of the bildenden Knste, which thus

    represent a kind of mean of presence (even though we are to take no interest in their

    existence) between a glut and a dearth. (Reed, 1980, p.575)


  • It seems to be paradoxical that Kant understood music as sound and perception, and

    the fact that music have from this point of view transitory, not lasting existence, led him to

    exclude music from rational speculation and therefore he abases music as an formative art.

    His theory lacks some kind of higher paradigm of music (e.g. cosmology) in terms of

    rational rule that would be able to confer judgments about its quality. However, music is still

    a cultural phenomenon in that it proposes to human beings new orders of perception.

    While music is abased as a play of sensation not conveying rational form, the

    cognitional faculty of a man itself can be understood in musical terms. Kant often used in

    his Critique of Judgment such terms as harmony, especially harmony of cognitional

    faculties in accordance to the ability of aesthetic judgment. Both Reed (1980) and Parret

    (1998) proposed a speculative (at least) interpretations of Kants text:

    When one takes account of the materiality of Kants text, it is evident that for

    one of the most complicated elements of his terminology, Gemth, musical imagery is

    omnipresent: the mind (Gemth) is presented to us as itself a musical instrument, more

    specifically a stringed instrument. The same thing could be said of the Anthropologie,

    where sensibility, as the first property of the mind, is presented as a bodily organ on

    which something like music is played. (Parret, 1998, p.262)

    Kant thus imagines the mind as a sort of musical instrument, more precisely a

    stringed instrument (or perhaps the psychological equivalent of the vocal chords), the

    principle strings being understanding (shown vibrating here), reason, and imagination.

    That is why in the same paragraph he calls a state of mind a Gemtsstimmung,

    literally a pitch or tone of mind, and speaks elsewhere of the beide Gemtzustnde

    zusammenstimmen (two mental tones harmonizing, in accord) ([]16), or a

    representation that places the cognitive faculties in a proportionierte Stimmung with

    each other through its harmony (Zusammenstimmung) with the conditions of

    universality supplied by the understanding ([]9). The activity of our cognitive

    faculties generally is thus called a Spiel, a play of mental strings. Within this

    structure Kant defines beauty (that is, the subjective experience of something as

    beautiful) as the mental tone that results from the harmonious interplay of the string of

    imagination and the string of understanding. (Reed, 1980, pp.579-580)

    It is improbably that Kants intention was to hint on this interpretation deliberately.

    However, are then the proportion and the functionality of human mind, determined by

    general rules of harmony, an a priori attribute of pure reason? Is the harmony an a priori


  • unconscious presumption of mind in like manner as time and space are conscious? The

    question Kant asked himself is: what way we are conscious of a mutual subjective harmony of the

    cognitive powers with one another in the judgment of taste - is it aesthetically by mere

    internal sense and sensation, or is it intellectually by the consciousness of our designed

    activity, by which we bring them into play? (Critique of Judgment, 9)

    If the given representation activating through our senses the judgment of taste, was

    concept uniting understanding and imagination, the consciousness of this relation would be

    intellectual. But, as Kant later mentioned, this kind of judgment is not that one of the taste,

    because it lacks reference to pleasure and pain. On the contrary, if the subjective universal

    communicability of the mode of representation in a judgment of taste is possible to

    presuppose without particular concept, then it should refer to nothing else than the state of

    mind in the free play of the imagination and the understanding. Therefore the basis for

    aesthetic judgment is not a concept but harmony:

    This merely subjective (aesthetical) judging of the object, or of the

    representation by which it is given, precedes the pleasure in the same and is the

    ground of this pleasure in the harmony the cognitive faculties. (Kant, Critique of

    Judgment, 9)

    In the original text, Kant used in many places in stead of the term harmony -

    Harmonie, other, quite synonymic terms such as Zusammenstimmung, Einstimmung,

    bereinstimmung, or Beistimmung. (see also Reed 1980, pp.578-579) Although these terms

    have apparently no reference to music, the meaning is almost the same as in music,

    representing particular dynamic correlation according to particular rules.

    Forasmuch as it is not possible to prove the intelligibility of musical perception, Kant

    suggested different point on this issue in 54 of his Critique of Judgment. He described the

    psychological mechanism of laughter with relation to musical experience. He stated that both

    laughter and musical experience are based on corporeal activity with cathartic effect and

    therefore cause physical pleasure generated by releasing of a state of tension: Laughter is an

    affection arising from the sudden transformation of a strained expectation into nothing.

    (Kants italics, 54) Accordingly, Parret (1998, p.263) suggests that musical experience can

    also be described in terms of tenseness (expectation, tension, release).

    One might ask why music represents such methodological problem in Kants text. For

    this purpose may be useful to design a paradigm with reference to Critique of pure reason.



    noumenon thing of itself


    senses, physical equilibrium (principle of laughter)


    cognitive process, phenomena (thinking about objects)


    a priori knowledge, time & space

    Table 8.

    The pure reason is situated in the center representing the ability of pure knowledge,

    and of a priori judgments. The problem occurs if this paradigm is adapted to Critique of

    judgment, namely to the aesthetic judgment, and more specific to the judgments about musi