Magic of Tone and the Art of Music by the late Dane Rhudyar
This is a very interesting extract about the Harmonic series from the now out of print book
Part One The Harmonic Series
considered as a series of fundamental and overtones, each of which has a
strictly defined frequency, the harmonic series is an arithmetic series. The
archetypal arithmetic series is the series of whole numbers, 1, 2, 3, 4, 5, and
so on. Such a series is formed by the endlessly repeated addition of number one
to itself. However, no instrumental or vocal fundamental tone actually produces
a complete and endless series of harmonics, and the overtones it produces do
not all vibrate with the same intensity.
The harmonic series is thus an archetypal model. It is either the process according to which the cosmogenic energy of Sound operates as it radiates or emanates from a creative spiritual source and descends step-by-step into progressively denser fields of objective and material existence, or it is an ideal concept abstracted from the experience of hearing some overtones when a variety of fundamental tones are sounded out by setting material instruments (including the human vocal organs) in motion.
In this sense the harmonic series is a myth the myth of number translated into musical terms. It is based on the association of two factors: an arithmetic progression of whole numbers and a geometric progression which at first apparently takes the form of a series of octave intervals (see figure 1). The octave is a musical interpretation of the ratio two to one (2: 1); two notes are in octave relationship when the frequency of the higher is twice that of the lower). (Hereafter, octave refers to the interval, octave-sound to an overtone. ) A series of octave-sounds follows a geometric progression and the frequencies of the sounds can be expressed exponentially: 2, 22, 23, 24, 25, and so on (or 2, 4, 8, 16, 32, and so on).
Any series of repeated intervals is a
geometric series. All series of intervals can be compared with the series of
other intervals. The interval of fifth (3:2) has been considered especially
important during the last five millennia, and a series of twelve fifths
includes slightly more than seven octaves. One can also compare twelve fourths
and five octaves. The meaning of these comparisons is discussed later in this
chapter. Comparisons between other series of intervals may or may not be
An arithmetic series refers to the transmission of power which, released from a creative source, becomes differentiated. For instance, in the government or a large corporation the power wielded by the top executive descends through several levels of authority before it reaches the realm of concrete, material results. In music, this level is the actual vibration of a resonant instrument or voice. On the other hand, a geometrical series refers to consciousness, because consciousness is implied in or is the product of the relationship between a self and another. Consciousness develops through the progressive complexification of relationships. It expands by including an ever greater number of differentiated relations in music, an increasing number of different intervals.
Thus, in the pattern of a harmonic series of fundamental and overtones, the relation between two successive harmonics the interval between them diminishes in scope; the ratios 2:1, 3:2, 4:3, 5:4, and so forth, become increasingly smaller. On the other hand, if we focus on the geometric series of octaves starting with the fundamental, we see that each successive octave contains more overtones than the preceding one. Thus, while the number of overtones increases per octave, the intervals between the overtones become smaller (see figure 1). Before the eighth octave is reached the intervals between successive overtones become so small that the human ear can no longer distinguish them clearly; the harmonic series becomes a rising continuum of sonic vibrations.
The first octave contains no intermediary harmonics; the second contains one; the third, three; the fourth, seven; the fifth (16 to 32), fifteen; the sixth (32 to 64), thirty-one; the seventh (64 to 128), sixty-three. The last interval within the seventh octave is the expression of the ratio 128:127; it is so small an interval that the ear cannot distinguish it from the following interval, 129:128. The last interval of the fifth octave (the ratio 32:31) was used in
An octave divided into equal intervals approximately the size of a comma would contain forty-eight eighth tones. No instrument, except electronic ones, could be tuned to finer intervals. Thus for most practical purposes music spans seven octaves of vibrations, approximately the extension of a piano keyboard (that is, from about 27 to 2,456 vibrations a second plus three half tones at the top of the keyboard).
The traditional esoteric world view divides the universe into seven levels of being. The lowest is physical matter, the foundation of all activities and of all the changes we perceive with our senses. It is the level at which the resonance of material entities (including musical instruments) to the descending current of power released by a creative will-emanating source becomes audible as a tone that is, as the complex vibration of the material body or vocal organ.
This tone is intended to communicate the purpose of the originating source in releasing such a creative or transforming power. But the audible tone not only contains the original creative or informative purpose implied in the descending power of Sound; it also is conditioned by and reflects the physical limitations and special characteristics of the resonant instrument. These characteristics result from the molecular nature and shape of the instrument. Because of them the ascending harmonic series produced by the audible resonant tone (the fundamental) is never a perfect arithmetic series of overtones. Only some of these overtones can be heard, and of those a few are more intense than others. The result is the specific timbre (or quality) of the tone.
Thus we never hear a complete (theoretically endless) series of equally intense overtones in any instrumental or vocal tones, because all we hear are sounds produced by the vibration of material entities. Moreover, we do not actually hear Sound, but the resonance aroused in material instruments by the impact of inaudible currents of energy and will or psychic activity (emotions). Nevertheless, the ascending series of overtones, incomplete and uneven as it is, is symmetrical to the descending series. Overtones can only occur as component parts of an ideal arithmetic series, of which the fundamental of the instrumental tone is number one.
This statement may seem arbitrary and illogical, but it would not seem so to a modern physicist thinking in terms of quanta (that is, of the discontinuous release of energy) and of the specific orbits in which electrons moving around a proton must revolve. The harmonic series thus appears to be an inherent structural factor both in the dynamic process of the release of descending Sound (or will power) and in ascending overtones generated by the symmetrical reflection of the material instrument's resonance to the impact of the Sound current.
The ascending and descending series are (in principle) symmetrical if considered as series of ever-decreasing intervals (octave, fifth, fourth, major and minor thirds, and so on). But if considered as the musical notes of Western scales, the notes of the descending series are not the same as those of the ascending series (see figure 2). An ascending progression starting from the note C as fundamental will produce a G at the place of the third partial; but a descending series starting from the same note C produces an F at this place in the ascending series.
The harmonic series considered as a series of intervals is like a ladder whose rungs occur at always diminishing distances from one another. If you place the ladder against a blank wall and mark on the wall the places at which the rungs touch the wall, you will have one series of marks if the ladder is placed with the smaller distances between the rungs toward the top of the wall, and another series if the ladder is reversed.
This relationship can be demonstrated audibly using a monochord, the didactic instrument of Pythagoras's teachings. If one plucks successively the entire string of the monochord, then one half of it, one third, one fourth, one fifth, and so on, one hears an ascending series of harmonics; these are explained by the physical fact that when the string as a whole is plucked it not only vibrates as a whole, but its aliquot parts also vibrate thus the vibrations of half of the string, a third, a quarter, and a fifth are also perceptible, at least in theory.
If, on the other hand, the hand plucks one inch of the monochord's string, then two inches, three inches, four, five, and six inches, and so on, a descending progression of sounds is produced, which gives the hearer a symbolic experience of the path followed by the descent of creative and wilful Sound. It is only a symbolic experience, because no part of a descending series of harmonics is audible. What seem to be "undertones" are combination tones (or resulting tones). These are complex auditory phenomena that acousticians consider to be subjective, in the sense that they are apparently produced in the inner ear because of the way the 25,000 extremely delicate hair cells of the cochlea vibrate. Combination tones, however, are produced only when two or more loud tones are heard. In complex and non-harmonic tones, like those of Japanese gongs or church bells, such low combination tones are often very strong. They also can be heard in a piano under certain conditions. (The frequency of a combination tone is the difference of the sum of the frequencies of the two loud tones giving rise to them, or of multiples of these frequencies. Combination tones are characteristically lower than the two original tones, but they may be higher. Two tones of frequencies 1,200 and 500 can produce a differential tone of frequency 700, or a summation tone of frequency 1,700, and still other combinations (see the entry on combination tones in the Harvard Dictionary of Music, p. 185). )
The Seven Levels of Being and the Symbolism of Number
According to the great religions of the last five millennia, and to most of the metaphysical systems dealing with the origin of the universe (or more generally with the origin of being), all there is began in unity. The religious mind usually personalizes the concept of unity as God. Metaphysicians speak of "the One." Sankaracharya, founder of
The remarkable fact is that two sounds an octave apart are given the same name, even though one of their frequencies is twice that of the other. They have for our ears an identical nature. They are the same note, even though they are obviously not the same sound. Are we conditioned by our culture to feel that two sounds at an octave interval are the same note, or is the feeling of their identity innate that is, rooted in an intuitive grasp of the nature of a metaphysical-spiritual process, which is none other than the basic process of cosmic existence and the primary manifestation of what we call life?
Although the harmonic series of fundamental and overtones is an arithmetic series, and the archetype of all such series is the series of whole numbers created by the endless addition of number one to itself, the term addition may be misleading. Philosophically the series refers to the self-reproduction, self-multiplication, or self-replication of the One. All numbers are born out of number one. The birth process begins with the characteristic act of self-duplication. Duality emerges out of unity: the One produces the Other, which is identical to itself a mirror image (as it were) yet itself in a new role. This duplication process can be repeated; its repetition produces a geometrical series: two, when duplicated, produces four, which when duplicated produces eight; the eight duplicates into the sixteen, and so on.
This series might be considered a repeated process of reflection, giving rise to a series of mirror images. But sonically the series produces octave-sounds. They are not merely reflections of one another, for each one is the source of series of overtones, and each new octave of the harmonic series contains more overtones than the preceding one. Each new octave symbolizes a level of being one more step removed from the original unity sonically the fundamental, philosophically the One.
Some of the religions of
The initial process of duplication of the One into the second has its source in a release of power from the One. This is what religions interpret as God's "desire" for creation, for self-revelation, or for the exteriorization of the immense potentialities of his infinite being the One desiring to be many. More impersonally, this desire is the motion that operates throughout the universe. Motion is everywhere. Matter is the incredibly rapid motion of subatomic particles, which themselves are nothing more than whirlpools of motion. All life is God's self-revelation through rhythmic, self-duplicating motion. In its primary spiritual aspect, this motion is Sound the descending power of the One into multiplicity, the descending harmonic series.
Is this descent endless? The rational intellect can find no reason for an end to the process of self-replication. The harmonic series issued from the fundamental One can theoretically extend ad infinitum. But infinity is only an intellectual concept; it is the negation of limit. The concept of existence, however, must include limitation. Existence can only be conceived in terms of wholes, and all wholes must have limits or boundaries, metaphysical if not physical.
Thus the octave symbolizes and defines a whole of sound-the limits within which motion, as a creative factor, cyclically operates. Metaphysically, the octave is the most fundamental whole because it originates in the first act of self-duplication, of which all further acts are replications. But the momentum of the creative release of the power of the One does not stop with number two. The power release acting through number two produces number three, the symbolic child. A new relationship is established between the mother principle and the child principle. This relationship also replicates itself, generating in music a series of fifths (the ratio 3:2) a new limiting and cyclic factor.
The geometric series of octaves and the geometric series of fifths interact; and a time comes when both series reach almost the same vibratory frequency: twelve fifths extend a little farther than seven octaves, the difference between them being the Pythagorean comma. What this means becomes clear once we understand the character of the interval of fifth and experience the psychic response it normally produces in human beings. To do so, however, we have first to deal in some detail with the field of cosmic and psychic activity and consciousness represented by each successive octave of the harmonic series. What we will discuss is a series of abstract numbers and proportions, the harmonic series as an archetypal pattern that can be interpreted numerologically, descending or ascending as the occasion requires.
THE FIRST OCTAVE
The first octave, considered
in terms of a descending harmonic series, refers to the pre-cosmic realm of
being. The human mind cannot conceptualise or formulate the quality of this
realm, and the word realm is obviously inadequate because space does not yet
exist. To say that it is the realm of changeless being also means very little,
for time does not exist yet, either. The human mind can only conceive of it as
pure void, nothingness; yet in that void the potentiality of all existence is
implicit. Motion inheres in it, but only in the sense that God's desire is in
itself the cause of motion. When anthropomorphized, it is the emotion
inevitably leading to objective muscular movements. This first octave in the
descending series symbolizes the purely subjective relationship of the One and
the other that is its image. In the terminology of the Tantric systems, it
symbolizes the mysterious love of Shiva and Shakti before any manifested forms
of existence appear. It is the realm of God's desire for seeing himself
reflected in a multiplicity of potentially creative and individualized centres.
The first octave in an ascending harmonic series that is, in the rebounding of the descending current of will, emotion, or creative power from the material instrument that gave it a concrete, limited, and audible reality symbolizes the sexual love union of male and female as it reflects the divine love of Shiva and Shakti. God's desire for self-revelation has become the dynamic power we call life. Life acts in and through the lovers. The two sexually complementary bodies are merely instrumentalities which life sets into resonant vibration in order to perpetuate one of its specialized forms (or modes of activity and consciousness) which we categorize as homo sapiens. A modern biologist would therefore say that, in sex and in instinctual love (the glamorization of sex) the bodies of the two lovers are merely convenient apparatuses for the perpetuation of genes. (See Lifetide, Lyall Watson (New York: Simon & Schuster, 1979).)
What we call life is therefore the symmetrical reflection of the process, following which a creative release of energy from the pre-cosmic union of the One and its image sets into vibration a physical organism. Before sexual differentiation occurs in biological evolution, life perpetuates itself through mitosis, the division of one cell into two. When reproduction is sexual, two complementary factors temporarily unite to produce a third. However, in some biological species the female, after being fecundated, kills and eats the male. If we translate this into musical symbolism, the second partial (the octave-sound of the fundamental) alone remains active and gives birth to number three.
There are indeed instances when this octave-sound (often confusingly referred to as the first overtone) is the sound we actually hear instead of the fundamental. In any case, this octave-sound (number two of the harmonic series), as the mother, rules the home as well as is the source of a progeny. The prototype of all children is symbolized by number three in the series of whole numbers. In ancient tribal and in many more recent systems of family and social organization it represents the first son. However, in the cosmic scheme it has a much deeper character and meaning.
Number three is the result of the operation of the One through its image and duplication, number two. Number three represents the desire to exteriorise all that is inherent in the One producing the first result that is not merely a mirror image. Number three is the origin of a potentially quasi-infinite series of diverse yet complementary realizations, all of which were latent and implicit in the One. If number two is (metaphysically) the projected reflection of the One, number three is the projection of the love of the One for this reflection. It is the desire for exteriorization of the One operating as will, the first manifestation of cosmic motion.
THE SECOND OCTAVE
The second octave begins
with number two. If we think of the harmonic series as a descending current of
creative energy radiating from God, the second octave represents the first
realm of manifestation of the principles according to which the cosmos will be
built. In the first octave, duality was implicit; in the second octave it is
explicit. The first octave is the noumenon of space as a field of potential
activity. The potentiality is there because of the desire of the One for a
second; but this second is a mirror image of the One. The One and the second
are identical. We can hardly speak of a relationship, because identity is not
really relatedness. Yet there is an implied difference between One and two. Two
is One charged with the power to be the source of an immensely varied progeny.
This power is Sound, Nada Brahman. This power has a dual nature, and the second
octave is divided by number three into two unequal intervals, a fifth (the
ratio 3:2) and a fourth (the ratio 4:3).(These terms, fifth and fourth, are
unfortunate and may be confusing. They originated in the fact that in our
ascending diatonic scale (C, D, E, F, G, A, B) the F which ends the interval of
a fourth (C to F) is the fourth note. G is the fifth note, and thus the
interval C to G is called a fifth. )
We can state this another way: love as a subjective desire is unitary; creative power is bipolar and operates through the interplay of two principles, expansion and contraction. The interval of fifth is expansive; the fourth is contractive since it has to balance and counteract the centrifugal power of the fifth in order that number four may be the exact duplication of number two. Such an exact process of duplication reflects the pre-cosmic, primordial love of the One for the second producing a geometric series of octaves. Each new octave-sound begins a new level of cosmic manifestation, thus giving birth to a new rhythm of activity and consciousness.
As a centrifugal power the fifth represents the will to self-exteriorisation, the power to make what is potential actual, and what is implicit explicit thus the cosmogonic, creative mind. The fifth symbolizes electricity; the fourth, magnetism. The second octave, which contains both a fifth and a fourth, is a realm in which electromagnetism is the primordial mode of motion. At a lower and human (or psychological) level the dualism is that of mind and feelings.
In Chinese philosophy two principles, yang and yin, constitute the primordial dualism of motion within the circle of Tao; but yang and yin are equal as well as opposites. In music, the fifth and the fourth are not equal intervals, even though they, too, are contained within a circular and cyclic pattern, the octave. But twelve fifths are slightly more extensive than seven octaves. Thus the universe of the creative mind is a spiral, not a circle; there is no Nietzschean eternal return, no unceasing repetition. Even though the octaves within the harmonic series repeat themselves, the contents of the octaves potentially expand infinitely, each new octave containing a greater number of overtones than the preceding one. What also expands are the ramifications and diversifications of the original power radiating from the fundamental, One.
As the interval of fourth symbolizes contraction, it balances the centrifugal, open quality of the fifth. Twelve fifths plus twelve fourths thus cover the same musical space as twelve octaves. But human beings do not have the capacity to hear twelve octaves of sound.
The harmonic series considered merely as a geometric series of octaves refers to the reflection, level after level, of what the first octave means the One's desire to be many. The geometric series of fifths and fourths refer symbolically to the many's development of consciousness; and consciousness oscillates between two poles, creative expansion and enjoyment of being. In Indian philosophy, this enjoyment of being is ananda, a word usually translated as bliss but really meaning the return of the many to the One as One can be understood and experienced by the returning consciousness of one of the many. Thus the one to which it returns is not the original One but its reflection in an octave-sound. This is why Tantric devotees worship the mother force (number two or its octave-sounds), believing the One (the hidden Father) to be unreachable. The interval of fourth thus symbolizes the return to the mother. At the human level, such a return may compensate for psychological defeat and neurosis, or it can mean that the negative aspects of mind egocentric ambition and pride have been overcome by a surrender of the ego. Ideally, the ego is surrendered to the impersonal cosmic principle of motherhood, but more often it is surrendered to a personalization of this principle, to a woman who becomes the symbol of the universal mother force the cosmic Mahashakti (the Great Mother). (This process is the foundation of the path of devotion (bhakti marga in Sanskrit). A less well known, more esoteric path, is symbolized by the direct relation between number One and number three. This implies a direct channeling of the power of One which is Sound (or creative motion) to the mind represented by number three (in the descending harmonic series). But this mind is not a product of cogitating, classifying, and generalizing brain activity; it is mind acting as a formative power through creative imagination and centralized will (kriyashakti and ichchashakti in Sanskrit). This process could be called rakti, from the Egyptian root ra, the power of the spiritual sun (sometimes called ra-orakti). its path could be called the way of the avatar, provided one realizes that besides the mythified great avatars of the Hindu tradition, who perform planetary or cosmic functions, there are many other avatars charged with various kinds of messages which they formulate in words, deeds, or other modes of spirit-impelled creativity. This avataric way would thus be symbolized by the ratio 3: 1. In the diatonic musical system this ratio defines the interval of twelfth. )
Metaphysically and metacosmically in the descending harmonic series, number three refers only to the idea of a future universe thus to what H. P. Blavatsky's Secret Doctrine calls "cosmic ideation." The second octave deals with the two great noumena of manifested being. The third octave, between harmonics four and eight, is the realm of archetypes in which number three operates through its octave-sound (its reflection) as number six. Each harmonic whose frequency is twice that of a preceding one restates the character and function of the earlier one at a more concrete level of being. Numbers twelve and twenty-four are therefore new manifestations of the creative imagination and will symbolized by number three. These manifestations actualize potentialities symbolized by the fourth and fifth octaves of 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
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
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.
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).( See Rudhyar, We Can Begin Again Together (Garberville, Ca.: Seed Center, 1974). ) 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
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.