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
When 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 significant.
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 Greece as the characteristic
enharmonic interval. It was slightly larger than a quarter tone in the modern
Western scale. The Pythagorean comma — the difference between the musical space
of seven octaves and twelve fifths — is about one-eighth of a tone.
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). )
Part Two
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 India's Advaita (or non dual) system of
metaphysics, speaks of "the One without a second." The great (and
actually insoluble) problem of metaphysics and religious cosmogonies is how to
explain or interpret the passage from unity to duality, and from duality to the
quasi-infinite multiplicity of entities active in the universe. Religions speak
of God's desire to create, of the primordial Eros that moves the One to produce
out of its unity a second, perhaps a mirror, image. Hindu metaphysics
interprets such a process of doublement or replication as the great illusion,
the essential Maya, root of all existence. The harmonic series of fundamental
and overtones provides a very significant and experienceable realization of the
relation between pure unity (the One) and duality (the second) by referring to
the unique and mysterious character of the octave in music.
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 India speak of the One
as Shiva, and the second as Shiva's shakti — his power, which is personified as
his feminine counterpart, Shakti, the beloved. In the Tantric cosmogony, once
Shiva has created Shakti and consummates his union with her, he retires (as it
were) and becomes a mysterious presence beyond the cosmic manifestation to
which Shakti gives birth. The mother, having produced the child, becomes the
manager of the generative process and of its results. She rules over the
universe of concrete and multiple entities — over all subsequent generations,
each of which begins with a reflection of the primordial mother. Each of these
mothers (like each of the subsequent octave-sounds of number two) also manages
and rules over her own progeny and her own level of being.
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.
Part Three
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 versus 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).( 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 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, G2 , D3
, A4, and so forth). (This can be called the avataric series,
symbolically the direct series of manifestations of the One (see the footnote
on page 67). )
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.