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by Dane Rudhyar

Chapter Five
The Geometry of Music

While Hindu music is founded upon and made up of tones which are living members and therefore souls, Western music in its ideal aspect is made up essentially of intervals. Thus is constituted the great and eternal musical dualism which can be resolved eventually into unity, only as East and West, each having reached the center of its own sphere and fully understood its own dharma, realize that they are complementary, being two aspects of the same Energy which is Sound itself, two paths to the one goal of human world service through the revelation of Tone. The individual and the group, the free man and the brotherhood or state, self-liberation and the entering into relationship; all these dualities are various aspects of the great duality of Spirit and matter, of the One and the many. In music we have thus, on the one hand, the single tone and its pilgrimage through the many stations of the Harmonic Series, of the Path of incarnation and liberation; on the other hand, we have brotherhoods of tones, resonances which are like ova of sound, and what we might call tuneful space space filled in its fullness by tones, or, as the old Gnostics called it, the Pleroma.
      With this last word we come to a basic conception of the true Music of the West, or Music of Prakriti (as by West we do not mean only a geographical location), a conception which, however, we have never seen expressed so far, save perhaps in the intuitive strivings of a young Californian composer, Henry Cowell. The ordinary European mind in its sterile intellectualism conceives space as emptiness, as that void which extends between objects of the stellar universe. Likewise it conceives the interval between the units of the musical scale as mere nothingness, as the abyss of "wrong notes". Intervals are not units of space but intellectual relations between two absolutely separate musical units; a mere proportion between algebraic entities.
      To the true student of life such a standpoint is essentially false. Space is not emptiness but fullness of being. Space is the highest possible conception of the Absolute, of THAT which is the eternal reality of Spirit and matter as one. Space conceived as a fullness of being and therefore as a Host of cosmic beings is the Pleroma (which means fullness). The Pleroma of souls manifests in music as the pleroma of tones, that is as a fullness of conjoined tones within certain limits.
      We realize at once the fundamental difference existing between interval and pleroma. When Europeans speak of the interval of fifth they speak of a certain relation between two undefined musical notes. A fifth is the proportion 3:2, and nothing else. If realized as actual sounds it will mean striking together or successively two notes, the pitch or quality of which does not matter in the least. A chord is formed when two or more notes are thus struck together. What is a chord? A group of two or more notes related the one to the other in a specific manner, i.e., properly spaced. But the space between these notes is not considered; it is mere emptiness. Each note is in a sense an autonomous and separate entity. In classical music a certain magnetic or tonal attraction is understood to exist between these separate entities; but in modern music, especially in Schoenberg's "atonal" school, there is no attraction or repulsion considered whatsoever. Every note is a perfect, self-sufficient, anarchistic unit but an intellectual unit, let us not forget!
      If, on the contrary, we conceive a fifth as a pleroma instead of as an interval, we get a radically different entity. We deal then with a portion of musical space which is a fullness of tone, with a host of tones theoretically infinite in number; no longer with two abstract notes with emptiness in between, but with a pleroma of compact, homogeneous sound-substance. Sound is substantial energy, and thus a music based on substantial fullness works with the living power of sound which is the very power of Space. Space is really a fullness of akasha whose characteristic is sound.
      The music of pleromas is thus the music dealing with masses of akasha, with portions of sonorous space. It is the music of Substance, of prakriti. The science on which such a music is based can be thus truly called the Geometry of Music using the term geometry, however, not in its modern European sense, but with its archaic and Pythagorean meaning: the science of parceling the fullness of Space which is life.
      Geometry, we are told by H. P. Blavatsky, was the fifth key out of the seven keys to the cosmic Mysteries. Among the others we may mention the physiological, historical and numerical keys. Geometry, in its mystical sense, was the fifth key because of its relation to the fifth principle of the cosmos which is really Daivaprakriti, the synthesis of the six great saktis. It deals, therefore, with substantial energy in Space, with the parceling and surveying of Space, the first operation in the building of any cosmos or organism and not with the mere intellectual computation of angles, figures, volumes, which are without spiritual or vital contents, which are only applied formulas.
      Mystic geometry became intellectual geometry, and Pythagorean arithmosophy mere mathematics, after the failure of the Greek civilization to keep true to the Platonic Spirit and its adoption of the Aristotelean sensorio-intellectual method. So did the music of pleromas of sound (which may have never been much more than an ideal in Greece) become the music of intervals, the music of pure intellectuality, as exemplified in the early Flemish polyphonists and, to a large degree, in Bach.
      But before studying further the principles of such a Geometry of Music, let us see here that pleromas of sounds in their purest form are not mere dreams. They have played a very important part in the racial sense of tone, especially perhaps since the beginning of the era opened in the East by Gautama the Buddha, and in the West, by Christianity. This refers to gongs and bells and also to all types of magical drums and tomtoms built by men who understood the resonances of matter.

A gong or bell produces a true pleroma of sounds. If you strike it at various points of its surface, various tones are released; yet all these tones converge to the one central tone, which is the key note or heart of the brotherhood of tones. Here you have a mass of vibrating substance, throbbing with multitudinous lives united in a compact group (or lodge or host). Sonorous space is thus made tangible. We can feel its constitution, its resonant Life, its form also. Its geometrical properties are intimately connected with the power and meaning of the resonance obtained. In it we can see the symbol of prakriti, the great Mother vibrating under the impact of purusha-purusha that strikes the gong from without, but that beats within the bell; symbolical facts truly. We shall see later on how the Western chorus or orchestra aims unconsciously at constituting by the power of tone alchemy pleromas of tones or magnified gongs. But while gongs are spiritual and homogeneous yet static hosts of tones, the polyphonic choruses of the West (brought to perfection by Vittoria and Palestrina) are dynamic brotherhoods of individualized and self-moving tones (i.e., melodic cycles).
      If we consider the Harmonic Series of overtones beginning with its primary (or fundamental), we see that while the frequencies of the overtones increase regularly by a constant value (which is the frequency of the primary), i.e., 100-200-300-400-500, etc., the intervals between these successive overtones decrease progressively. The interval between primary and the first overtone is an octave; then follow a fifth, a fourth, thirds, seconds, etc., until the intervals reached become so small as to be altogether indiscernible. Within the first octave of the series we find no subtones, but if we examine succeeding octaves we find that an ever increasing number of overtones are contained therein, until we come to the point where the interval between overtones becomes negligible and the octave selected seems as a compact mass of continuous sounds.
      The Harmonic Series is thus seen as the ceaselessly accelerated multiplication of one tone (primary) adding itself to itself continuously until the whole universe is filled with its progeny. The lone value (i.e., frequency) increasing by constant steps, the intervals decrease with accelerated speed. We get the reverse process if we build a series of intervals (or pleromas) increasing by the constant addition of the initial interval; then we find that the differences between the values of the two tones forming the limits of the interval increase with accelerated speed.
      In other words, music is built on one of two types of series or progressions: series of frequencies where the initial tone adds itself to itself, and series of intervals where the initial interval adds itself to itself. In the first case we get what is called an arithmetical progression (n, n+n, n+n+n, etc.). In the second case we obtain a geometrical progression of some kind (n, nXn, nXnXn, etc., that is n, n2 n3, n4, etc.), the most well known among the latter being the cycle of twelve fifths which is a series of twelve perfect fifths, the initial notes of which are:

C. G. D. A. E. B. F#. C#. G#. D#. A#. E#.

      The thirteenth fifths would logically be a B sharp; but B sharp is nearly identical to C, to that C which is found seven octaves above the initial C of the series; the difference being a very small interval named the Pythagorean comma. These twelve notes, if reduced to the octave constitute the chromatic scale of European music, or the twelve lyus cycle of Chinese music.
      Let it be repeated again that the cycle of fifths and all similar cycles, to which has been given the name of zodiacs of sound, are series of intervals, whereas what has been called so far the Harmonic Series is a series of tones. Both types of series are really "harmonic," that is reduceable to unity; but this unity is one thing or the other, either a tone monad or a brotherhood of tones, an individual or a group.
      Hindu music is based on the former, on the arithmetic of tone. Pythagorean music proper (which we trust will soon manifest as American music) and Chinese music are founded on the latter, on the geometry of tone. European music is in a way a mixture of both and yet neither one nor the other. It has perverted living tones into intellectual notes; it has seen of the pleromas of sounds but their intellectual shells, intervals. It ignored the melodic flow of sonal energy while trying to be expressionistic and melodical. It never developed any real sense of vital resonance though it was dealing with complex harmonic elements. Moreover, it destroyed the vital and magical power of music by dismissing all knowledge of cosmological correspondences and occult tone correlations, something which even degenerated Chinese musicians never thought of doing.
      Such Series of twelve equal intervals (fifths or fourths or octaves, especially) have been called zodiacs of sound for the following reasons: In our present stage of human development the idea of zodiacs is related to the number 12 which is the number of the manifested universe, the number of prakriti, and of magnetism. There are six great forces (or saktis) in Nature and as every one is dual, we obtain the number 12 as the complete number of the cycle of forces. The twelve Adityas are the twelve great aspects of Aditi, the cosmic Mother. So also the twelve apostles are the twelve cyclic phases of the Christ.
      The twelve fifths are symbols of the very same cosmic realities, and can be considered as the twelve great hierarchies of sound, as the twelve zodiacal Hosts or pleromas. The cycle of fifths represents the outer shell, or skin rather, of the Egg of Brahma the twelve gates through which the energy of sound radiates. By the proper adjustment of these radiations vital resonances are produced; the outer and the inner worlds become related and both are energized by the great Fire of the Mother, Kundalini of which the zodiacs of fifths and fourths are symbols.
      Because of this principle of adjustment of inner and outer elements there is constant interchange, and duality rules in the zodiacal fields. Intervals, groups, pleromas and every multiplicity aspect of Life are rooted in the dualism of polarities. Every manifested element or force is dual; among which being the energy of growth which manifests in the structural patterns or organisms, plants and men alike. It has been proven that the distribution of leaves on a stem follows a sort of geometrical progression, the quotient of any of its terms by the preceding one being a constant value (1:618). The same progression is found when the various parts of the human body are measured.
      In other words, while the magnetic inner life of the plant is expressed in terms of the arithmetical Harmonic Series, the morphology of all organisms is ruled by geometrical progressions. The soul or monad is a living number; but the archetypal form of organisms (be they cells or solar systems) is characterized essentially by surfaces which are portions of living Space. Thus we have two different bases for Art expressions of any kind, according as Art is founded upon either the unity of the Self or the duality of Substance. Let us not forget, however, that in the former case this unity expresses itself in two directions, i.e., in the ascending and descending modes, while in the latter, there is only one direction considered. So that in the two we have really a trinity of principles.

In the Harmonic Series we see duality arising out of unity in a graphic way. For the first octave of the Series includes no overtone, the first overtone being the octave of the fundamental; but the second octave is divided into two unequal intervals of fifth or fourth. If we start with the fundamental Sa, we find that it is Pa which brings this duality, the fifth tone. Pa therefore symbolizes the creative power, that which brings about the polarization of the asexual unity i.e., mind considered as energy. The interval of fifth is therefore the symbol of Man, of positive power. The fourth, its complement, represents the Woman in a mystical sense, i.e., Buddhi.
      I have said that the growth of the stem is the result of two different forces: the upward push given by the roots and the suction of the solar magnetism. The former is represented by the fifth and the zodiac of fifths, the latter by the fourth and the zodiac of fourths. We have a similar dualism of forces in the human organism. Thus is explained the peculiar nature of fifth and fourth, when understood as pleromas of tones the fifth being full-sounding, open, self-assertive, the sustainer of all harmonies, with its center below; the fourth being more concentrated, elusive, mystical, a reaching forth toward its center above. The fourth is a great mystery in fact; for it is usually considered as a descending interval, as the reflection above the tonic Sa of its fifth below Ma. But this is an unspiritual conception which is the result of the prevailing degenerescence of the ideal of womanhood, a symbol of male intellectual family autocracy, if not of prostitution.
      The problem is a very mystical one, fully treated in its spiritual sense in the great Gnostic works and especially in Pistis Sophia. Sophia or Wisdom is the fourth; Christ is the fifth. Out of their mystical union arises the new Pleroma, which is the octave. The importance of such a symbol is very great, both philosphically and practically. For on the relationship of fifth to fourth rests the entire structure of the Pythagorean and Chinese scales and of the larger system which I believe will be used in the future music of a regenerated Western civilization.
      To put it briefly, in China the substance of music is a series of twelve lyus which are obtained by a series of ascending fifths and descending fourths. Beginning, let us say, with the note C we get by an ascending fifth G and by a descending fourth the note D. The interval C D is a tone. It is the progeny of the masculine fifth and of the feminine fourth. The lyus reached by ascending fifths are male; those reached by descending fourths are female. The former belong to the principle yang, the latter to the principle yin heaven and earth.
      What does this mean if not that woman is the falling back of the tide which had its apex in man? That she is contained within man as the fourth within the fifth? This may be true in a personal and physiological sense and there is of course a beautiful poesy in this double motion of life, in this procreation of the child pleroma within the family circle traced by the effort of the man. Yet there is something more beautiful in a spiritual sense, that is, to realize in woman the Soul which is beyond descent, which reaches upward too, whose center of attraction is the octave tone of the initial tone of man. Man and woman as Companions, the latter adding her mystic sense to the mentality of the former, and spiritual growth resulting from this dual soaring toward the spiritual Fundamental. Thus instead of procreating a mere tone among the several tones of the scale the union of man and woman recreates the initial seed and Nirvana is reached.
      This, however, might mean spiritual selfishness if the cycle of completion were not extended and the relation of a single fifth to a single fourth were not transformed into that of zodiac of fifths to zodiac of fourths. Twelve fifths make seven octaves plus a comma; twelve fourths, five octaves less a comma. Both conjoined constitute the Great Zodiac of twelve octaves, the complete cosmic sphere. The brotherhood of fifths and the brotherhood of fourths adding their impersonalized strength and devotion form the Seed of the new humanity, the shistas who collectively represent the Body or Vahan of the Kalki Avatar, the White Horse upon which He, it is said, comes from Shamballa to usher in the new Satya Yuga.
      This Seed is the true sangha which was founded by Gautama the Buddha. When the true sangha is completed, when the resonance of the two zodiacs is perfect and fully vibrant, Gautama, the beginning, will become reflected in Maitreya, the completion, after twenty-four cycles (twelve fifths plus twelve fourths) the twenty-four Elders mentioned in St. John's "Revelation," which, in its historical sense, is a prophecy of the events which take place in the entire Kali Yuga, until the New Jerusalem (the new Race) manifests upon earth. Likewise, when the full zodiacs of sound vibrate together in perfect harmony, then occurs the new birth of Tone within as without. In the perfect resonance of the instrument the spiritual Tone incarnates, as we saw in a preceding chapter.
      From a practical standpoint what is important to grasp is the fact that, the use of such zodiacs or brotherhoods or democracies of sounds presupposes the disenthroning of the octave, the ruling element in all musicalities based on the Harmonic Series. The octave symbolizes blood-relationship and therefore the family, the perfect and complete circle of the home. It does so particularly in Europe where it is conceived as an interval, as a form, as the self-sufficient world of autocratic tonalities. In India, however, the octave is only one of the many stations on the Path of sound. It is only the magnetic circle within which tones which are spiritual entities temporarily evolve. It is either ascending or descending, perpetually in motion. In Europe the octave is so much sound-space divided into twelve equal intervals. It is pre-eminently static.
      In other words, the octave can only be spiritually conceived as the basis of music when it is considered as a fragment of the Harmonic Series. When the Harmonic Series ceases to be the substance of the musical flow, and intervals or pleromas take the place of the single tones perpetually evolving, then the octave becomes truly the symbol either of spiritual selfishness or of a purely physical system of blood relationship, which gives birth to family selfishness when the principle of temperament is accepted; temperament being the operation by which the twelve intervals of fifths are reduced in some manner so as to fit in the interval constituted by seven octaves.
      As the octave loses its ruling function the harmonic system changes entirely and tonality has no more meaning. The fifth or fourth and to some extent the thirds, which subdivide the fifth in two unequal parts, become the pivots of the harmonic system, or say rather of a new sense of resonance.

The aim of the geometry of Tone is to disclose the law of cosmic resonance. Resonance is the very principle of the true music of the West, the Music of Prakriti. But resonance is not born of the Harmonic Series, as usually said by Western musicians. The basis for harmony (or the science of chord formation) is not the Harmonic Series, as usually believed; but the cycles of fifths and fourths. True melodies are founded on modes (or rags) which are segments of the Harmonic Series; but true chords or true polyphony (two methods of creating resonance) ought to be constructed on principles resulting from various cyclic aspects of the zodiacs of sound. Before this most important point is fully understood neither Western nor Eastern music will ever be pure, nor can they fulfill their higher magical theurgic dharma, nor find their common center.
      When Western musicians understand this they will give back to the term "harmony" the meaning it had in Greece at the time of Pythagoras; and the science of resonance will take the place of both European harmony and orchestration. The aim of Western music will be grasped. This aim is the building of resonances which are of the soul and not of the body, which are animistic and not animal, spiritual and not merely psychic resonances which Atma may illumine from within.
      Harmony is unity of resonance. It means really the same thing as Maitreya. It is the fullness of the life of prakriti homogenized by the intense, all-absorbing devotion to purusha. It is the Flower becoming the Fruit, whose death will mean the release of the Seed or food and sattvic energy to entities of a higher kingdom. The science and law of harmony is exactly the science and law of true brotherhood. It is based on the principle of real Alchemy, of the mystic and never publicly understood Rosicrucianism. For the Rose and the Cross are nothing else but symbols of the universal Flower and the universal Root. The Cross is mystically the womb of the Christs, Kro being, by the law of spiritual phonetics, the container and mother substance of Kri.
      Rosicrucianism is spiritual Alchemy. lt is the spiritual doctrine of the West.(1) Applied to music it gives the basis for the musical zodiacs and the system founded thereon; a system which was applied, more or less consciously, the first time by the great Russian composer Scriabin. Scriabin built his later works on the principle of true resonance using the cycle of fourths as his musical substance. Even this, however, is not yet understood by his biographers and by musicians in general who persist in studying his works in the false light of the old European attitude to music. In America today a few composers are unconsciously working toward a music founded on the cycle of fifths. Schoenberg's atonalism in Europe is but the shadow and utter perversion of this New Music of the West which is to come.
      Until Europeanism, as a generic attitude toward life and manifested in whatever nation or race it may be, is dead, the new West cannot fully blossom forth. European nations lost their chance of regeneration after the Great War; and America is in the throes of a birth, the outcome of which is not wholly discernible perhaps. The same thing is true of music. Until tonalism, which is musical feudalism, is dead, the true zodiacs of tones cannot fully, openly resonate throughout the new continent, Scriabin's influence is very small indeed in Europe where a deadly neo-classicism is master. European musicians chose Stravinsky instead of Scriabin. It meant spiritual failure and aesthetic death under the cloak of technical excellence. In America the dawn is hardly breaking as yet; but there is hope. To the extent to which the pioneers of the race will be able to live, brotherhood and to purify their minds of European intellectualism and materialism, to this extent will Western music be regenerated. To such an extent will the true Geometry of Music be understood as a key to the mastery of Life as well as of art, science and philosophy taught by Pythagoras who was in a very real sense the Western aspect of Gautama the Buddha, the teacher of Harmony.

1. Let us not mistake many modern groups or schools who have adopted the sacred name of "Rosicrucian" for the most occult original brotherhood founded late in the fourteenth century at the time of the great Tibetan Reformation of Tsong-Kha-Pa. No known organization bearing that name has any right to bear it, as far as we can see. Historical Rosicrucianism is dead in Europe for all aim and purpose. But the spirit of it lives forever.  Return

By permission of Leyla Rudhyar Hill
Copyright © 1979 by Dane Rudhyar
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