The Harmonic Series
THE THIRD OCTAVE
The third octave of the harmonic series contains four diminishing intervals. Each of the two intervals of the second octave are now divided into two smaller ones. The fifth is divided into two thirds, major and minor, the fourth into two intervals which one might call ultraminor third and large second, or whole tone.
In a descending harmonic series the third octave may be called the realm of archetypes. It is the realm where the basic forms, the models for a quasi-infinite variety of physical embodiments, are visualized by the creative imagination. It refers to the four great purposes of the life will: the will to be a particular entity, to maintain its form, to expand it, and to reproduce it. In the evolution of human consciousness it represents the level at which the processes of mind (interval of fifth) and those of the feeling nature (interval of fourth) are given a personal form.
The seventh harmonic occurs in the third octave, and number seven is especially important in occult philosophy, astrology, and geometry. But the seven cosmic forces or "rays" esoteric philosophies mention should not be associated with any one overtone. They refer to seven aspects of the original unitary power of the One. In music they are represented by the seven Fundamentals (in some cases five) which, in their togetherness, constitute the foundation of most musical cultures. Chapter 8 discusses these Fundamentals and their role in the formation of scales.
In the harmonic series, number seven represents the archetypal source or the symbol of life processes involving an irrational or transcendent factor. This factor becomes more insistent in the succeeding octaves of the harmonic series, manifesting as number fifteen, thirty-one, sixty-three, and so on. At the close of any cycle, a period of transition occurs which always contains an element of indeterminacy an overwhelming longing for completion (a return to the mother) or an unexplainable and irrational impulse to transcend one's limitations and to lose oneself in the process of rebirth at a new level of being.
THE FOURTH OCTAVE
The fourth octave extends from the eighth to the sixteenth harmonics. Eight is a solar number. In Hindu mythology the sun, Surya, travels in a chariot drawn by eight white horses. Number eleven also has a solar character as it measures the sunspot cycle which, according to esoteric tradition, is the heartbeat rhythm of the solar energy that circulates through the solar system; astrophysicists call this energy the solar wind. In Gnostic symbolism, eight thrice repeated (888) is the number of Christ who, according to the scientist, philosopher and seer, Rudolf Steiner, was a solar archangel who gave of his spiritual substance to our planet. The fourth octave, then, is the realm in which descending spiritual forces and ascending biological forces are brought together to accomplish their essential work.
The eleventh overtone of the ascending harmonic series starting with C is approximately an F sharp (see figure 1). In Western tonality the interval C to F sharp is the tritone (it contains three whole tones).
In the European Middle Ages it was called the devil in music, being considered highly dissonant. In Franz Liszt's Sonata after a Lecture of Dante (1839) it is sounded repeatedly in its descending aspect which, in the descending harmonic series, should rather be noted C to G flat. The interval presumably symbolized in the composer's mind the descent into hell. Since Liszt's time the interval has been used often because of the dramatic feeling it conveys.
The last overtone of the fourth octave is harmonic fifteen; the fifteenth tarot card represents the devil, but this is what occultists call a blind, a symbol hiding a secret. Satan is an anagram for Sanat Kumara, who in the esoteric philosophy of India is the promethean being who gave mankind the fire of self-conscious and independent, individual selfhood. This gift (number fifteen) leads in the ascending harmonic series to the realm of the fifth octave.
THE FIFTH, SIXTH, AND SEVENTH OCTAVES
Number five (and the five-pointed star) is the hieratic symbol of individualized man. The fifth octave starts with number sixteen, which is two raised to the fourth power (24 or 2 X 2 X 2 X 2). From the point of view of the descent of spirit-radiated energy into material conditions, this level marks the full incorporation of the mother force, number two. It is the level of existence in physical bodies, spirit involved in material organization. From the point of view of the ascending evolution of the resonance of matter that is, of the capacity to act in response to the impact of the image-making faculty and the will the fifth octave marks the first stage of the process of individualization. It is the stage at which culture wholes are formed; their mental-emotional fields provide collective models as foundations upon which temples for the celebration of the individualized power and consciousness of man can rise.
What then happens to the individual? And what is the quality of his or her individual acts? The questions are symbolically asked by number fifteen. The transition between fifteen and sixteen has conditioned the answer, which leads either to the divine mother (the eternal feminine that draws one on to one's individual stature) or to the dark mother (who binds one to the realm of passion and the sins of pride and ambition).
In the first instance the spiritually individualized person reaches the level of the sixth octave, which begins with number thirty-two, the fifth power of two. Esoteric philosophy refers to the thirty-two paths to wisdom; real wisdom can only be reached through intuition. Intuition is a mode of supersensible perception, a spiritual "seeing." The intellectual mind cogitates, discusses, and argues about what might be, and can only come to a conclusion it already knows. But intuition directly perceives what is. Far more than knowledge, it is understanding. Understanding is thus symbolically related to number thirty-three (the highest grade in Freemasonry). Understanding often leads to a symbolic Crucifixion, which should be understood as the liberation of the soul from the memory of its bondage to matter. Number forty, as in the forty weeks of pregnancy, symbolizes the preparation for rebirth.
The ratio 32:31 measured the smallest theoretical interval of Greek music, the enharmonic quarter tone. The seventh octave begins with number sixty-four, and its intervals are increasingly difficult to appreciate or to consider as steps in melodic sequences or chord combinations. At the end of the seventh octave the harmonics 127 and 128 can hardly be distinguished from each other. Further differentiation of the resonant energy issued from a material instrumentality is no longer possible.
At this point (the seventh power of two) the octave-sound occurs at a frequency which a geometric series of twelve equal fifths has already reached, as it extends the very small Pythagorean comma beyond it. This Pythagorean comma (there are other kinds of commas in the theory of Greek music) is the small interval by which twelve fifths are larger than seven octaves about the eighth of a whole tone. If the series of octaves and fifths begin at number two, the frequency of the end of the octave series is 256 (the eighth power of two), while the end of the fifth series is 259.48.
Mind verus Nature
The relationship between the series of seven octaves and twelve fifths is analogous to the relationship of nature to mind. There is a similar traditional relationship between agriculture (man's intimate participation in the seasonal activity of nature) and industry (the use of machines and of processes coming under the general heading of fire).(1
) Natural intonation in music refers to the intervals of the harmonic series and its seven octaves of vibrations. Tones produced by the voice and man-made musical instruments "naturally" contain harmonics. (Gongs, bells, and machines, for example, make nonharmonic sounds, but these require fire to melt and shape their metals.)
A series of perfect fifths is also a series of natural intervals (the harmonic ratio 3:2), but except for the initial ones, the tones of the series produce overtones which do not reinforce one another; they do not refer to whole numbers (see figure 3).
The series of seven octaves is both
arithmetic and geometric. Its overall structure is geometric, for it is a sequence of equal intervals, but all its terms are parts of a more inclusive arithmetic series whose prototype is the series of whole numbers. Series of equal fifths, fourths, thirds, and so on, are only geometric; each of them therefore represents the development of only one aspect of nature that is, of the cosmos as a whole. If the fifth equates with mind, then mind is only one aspect of nature. But it is the first and most basic function of the COSMOS.
This cosmic, superintellectual mind is the foundation of all mental processes. It is the root of mental activity, the noumenon of all mental phenomena. It is mind still totally infused with divine love. In the symbolism of the twelve disciples of Jesus, it is represented by John the Beloved, who represents the pure mental expression of human consciousness as it develops throughout a complete cycle of evolution (symbolized in astrology as the Piscean Age).
The twelfth fifth of the series symbolizes Judas, the betrayer, because he incarnates the will to go beyond nature; the twelfth fifth going beyond the octave-sound that ends the seven octave series. This amount is the Pythagorean comma. Judas represents what the German historian, Oswald Spengler, called "the Faustian spirit," the restless dissatisfaction with any natural fulfillment, the eternal quest for the beyond. His self-destruction by hanging on a tree the tree of nature has its musical counterpart in equal temperament, that is the reduction of each of the twelve perfect fifths by the twelfth part of a comma, so that the last note of the twelvefold series corresponds exactly to the vibration of the last octave-sound in the harmonic series of nature. What is implied in equal temperament is that every one of the twelve apostles participated in the sin of Judas the sin of egotistic pride and spiritual ambition.
When the series of twelve fifths is reduced to the span of an octave, a chromatic scale is produced. The piano keyboard, with its white and black keys, is a chromatic scale in its most fixed and tempered form. The pianist cannot deviate from it. He or she can, however, blend (or allow to interpenetrate) the resonances of the metal strings struck by felt-covered hammers attached to the keys, producing complex, nonharmonic tones some of which may resemble gong tones.
The great gongs of Buddhist countries are vibrant symbols of the root of cosmic existence which Buddhism calls the Buddha mind. The typical Buddhist monument (stupa), is shaped like a bell, a resonator ready to vibrate as a channel for the descent of spiritual power incarnated out of supreme compassion for all living beings. The great bells of Christian cathedrals which called the people to prayer to communion with divine love were also manifestations of this root reality of mind pervaded with love.
The beautiful Baha'i temple near Chicago is shaped like a huge, nine-sided bell calling the faithful to the birth of a new age. It is to be an age of power, whose generally misunderstood astrological symbol, Aquarius, symbolizes the descent of a cosmic power released by the mind if this mind is attuned to the Buddha mind and Christ love. The number of this coming age is nine. Number nine is the second power of three. It is the second term in a geometric series of whole numbers based not on duplication, but on triplication, thus 1, 3, 9, 27, 81, 243, and so forth a series of intervals of twelfths (C1
, and so forth). (2
The interval C to G reduced to the octave is the fifth; the interval C to D is the whole tone. The following chapter discusses how they formed the foundation of the Pythagorean scale and what the sevennote (diatonic) scale means philosophically and cosmogonically.
See Rudhyar, We Can Begin Again Together
(Garberville, Ca.: Seed Center, 1974). Return
This can be called the avataric series, symbolically the direct
series of manifestations of the One (see the footnote on page 67). Return
By permission of Leyla Rudhyar Hill
Copyright © 1982; by Dane Rudhyar
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