Grid of the Gods (43 page)

Or to put it into terms reminiscent of Rene Schwaller de Lubicz, music denotes the
functions
that numbers symbolize.

What certain numerical relationships in ancient mythological texts are actually doing, in other words, are defining systems of
tuning
in the attempt to reduce the chaos of the harmonic series to an engineerable
order.
Why this is so requires a little explanation by way of a simple illustration.

Every musical tone has a natural harmonic series of overtones extending above and below it into infinity. The problem is, that each “overtone” or “harmonic” decreases in interval relationship to its fundamental the higher one goes. If one sits at an acoustic piano, presses the note “C” down silently, and then hits the note “C” an octave lower, one will hear this silently pressed “C” vibrating in sympathetic resonance with the struck note. Now, go up a
fifth
to the note “G”, and repeat the experiment. Silently press down “G” while hitting the note “C”. Again, one will hear the silently held note “G” vibrating sympathetically with the struck note “C”. Again, go up a
fourth
— note that the interval is diminishing — to another note “C,” press it silently, and hit the note “C” two octaves below it. Again, the silently held “C” will vibrate.

Repeating this process will inevitably lead to a note that does not exist on the piano keyboard, but that
does
exist in the natural harmonic series of “C”, and that note lies in the crack between A and B
^b
. And as the intervals continue to diminish, more and more notes are added that do not exist on the keyboard. Our modern music, and particularly keyboard instruments, have been
tempered
or tuned to an artificially engineered overtone series that allows all twelve notes within an octave on the keyboard to function in any key of music, without having to stop to retune the whole instrument in order to do so.

Plato, in his
Republic
, encoded this “equal tempering” as a political philosophy: “The necessity of tempering the
pure
intervals, defined by the ratios of
integers
” that is to say, the harmonic series as it
naturally
occurs with its infinite overtones for each note, “is one of the great themes of Plato’s
Republic
. In his allegorical form, ‘citizens’ modeled on the tones of the scale must not demand ‘exactly what they are owed,’ but must keep in mind ‘what is best for the city.’”
¹²
That is to say, they must be willing to submit to a slight mathematical
adjustment to allow them to function with each other in a harmonious fashion.

The number twelve here is a key, for it points directly to Mesopotamia and to its “sexagesimal” numerical system, a system that as we shall see was taylor-made to solve the tuning problem, and to a secret held by those cultures and passed down through the Pythagoreans. It points also to the twelve houses of the zodiac, and with that, the connection between music and astronomy comes a little closer into view.

B. Music, the Alchemical Medium, and Astronomy

The Vedic tradition is full of allusions to the “luminous nature of sound”. Indeed, the word for light (
svar
) is similar to the word for sound (
svara
).
¹³
This, as I have observed elsewhere, is another clue that perhaps we are viewing a legacy of a lost sophistication in physics, for the idea of a “sound-light” can also be understood as “electro-acoustic,” i.e., as longitudinal electrical waves in the medium, recalling the “Sound-Eye” spoken of in the Egyptian Edfu texts.
¹⁴

The regularity of that physical medium was, for the ancient mind, expressed in the regularity of the cyclic nature of the heavens, and the precession of the equinoxes through a long cycle through the twelve houses of the Zodiac. Twelve, then, became the number of choice for expressing the musical harmony of the spheres. We may illustrate this by again sitting at an acoustic piano, and playing through the circle of perfect fourths and fifths:

Circle of Fourths (Asdending)

1) C

2) F

3) B
^b

4) E
^b

5) A
^b

6) D
^b

7) G
^b
/F
^#

8) B

9)E

10) A

11) D

12) G

13) (C)

Circle of Fifths, Descending (Reading the above chart bottom to top)

Again, the notes we play are not the result of the natural overtone series, but of a
tempering
or “tampering with” by means of a mathematical adjustment to harmonize with the celestial motions of precession through the twelve houses of the Zodiac. As McClain states it, “Ancient cosmology required just enough number theory and just enough musical theory to harmonize the heavens with the scale and the calendar.”
¹⁵
The musical allegory is expressed in poetic terms in the
Rig Veda

Formed with twelve spokes, by length of

    time, unweakened, rolls round the

    heaven, this wheel of during order.

Twelve are the fellies, and the wheel is single.
¹⁶

This astronomical and calendrical correspondence is even evident if one plays the white keys of the C major scale: C, D, E, F, G, A, B, for these seven tones were understood to represent not only the seven day week, but, in Vedic tradition, the Seven Rishis or Seven Sages.
¹⁷

The
Rig Veda
also knows and speaks of the primary “tripartation” that we discovered in connection with Angkor Wat and Vishnu’s triadic self-manifestation:

The Gods are later than this world’s production,

Who knows then whence it first came into being?
¹⁸

That is to say, all the multiplicities of the god-notes follow upon that first tripartation.

But what happens, then, if we take thirds, or the powers of three, and follow those around “a circle of thirds”? The result is perplexing, for if we start again at C, and follow intervals of a major third, we get the following:

1) C

2) E

3) G
^#
/A
^b

4) C

We are far short of generating the twelve equidistant tones of the circles of fourths and fifths. In other words, the primary mathematical and musical problem faced by ancient civilizations, was how to reconcile the two “sets” of god-tones into a harmonious whole.

This leads us inevitably to a consideration of the “female” and “male” numbers of the ancient tuning systems.

C. The Male and Female Numbers and the Physical Medium

The image of the musical circle was one that haunted the Vedic literature.

The chariot of the gods is actually a “wheel-less car… fashioned mentally” (10.135.3). The Celestial Race itself “was made by singers with their lips,” and the same singers “with their mind formed horses harnessed by a word” to drive the chariot,” a light car moving every way (1.20.1–3). Of the car itself, in what I assume to be an allusion to rotation and counter-rotation in the tone-circle, we hear that it “works on either side,” the car-pole to which the horses are harnessed thus “turning every way” (10.102.1 and 10.135.3).
¹⁹

In other words, the primary feature of the physical medium in view in the ancient cosmology was its
rotational frequency
, and the resulting “music” therefrom.

Frequency is the key here, for it shows how and why the ancients approached the subject of tuning via whole number integers and ratios. Suppose you are sitting at the massive console of a pipe organ. Before you are five octaves on its various keyboards, beginning with the lowest note, “C” on your left. You pull an eight foot stop, say, the stop
Prinzipal
. On the stop knob there is a number 8 followed by a foot sign, thusly: 8’.
²⁰
This means that the pipe, from the mouth to the tip, is exactly eight feet long, and will sound the note “C”. Go up an octave, and the pipe will be exactly four feet long, and so on.

We thus have the ratio 8:4, which reduces to 2:1, which reduces even further, to 2. Two thus becomes the ancient number assigned to the octave of any note. It is a “female” number, and with this, occurs another problem:

The number 2 is “female” in the sense that it creates the matrix, the octave, in which all other tones are born. By itself, however, it can only create “cycles of barrenness.” In Socrates’ metaphor, for multipication and division by 2 can never
introduce
new tones into our tone-mandala. In musical arithmetic, the powers of 2 (2±n) generate cyclic identities; that is, they leave the musical relationship of the octave cycle invariant.
²¹

Previously we saw how the primordial Nothing tripartitioned itself into Vishnu’s primordial trinitarian manifestation.

McClain speaks of this same thing, and now we come to the musical application of our previous topological analysis. McClain obserfved that “the starting point of the
Rg Veda’s
intentional life is the
Asat
, the non-existent, “the whole undifferentiated primordial chaos…”
²²
Thus, the sexual metaphor also enters the picture:

It is a theme of much ancient mythology that the Divine Unity is a hermaphrodite, producing a daughter, “2,” by a process of division without benefit of of a mother. God is “1”, but he cannot
preocreate except via his daughter, “2,” the female principle and mother of all.
²³

But this, as we saw in our topological commentary on Angkor Wat, is not strictly true. It is not the number
two
which is first produced, but the number
three
, corresponding to the two regions of “differentiated Nothing” sharing a common surface. The common surface of the two regions becomes the topological symbol of the octave matrix, the regions defined by it.

To put it in terms of the numerical ratios that would have been used by the ancients, a primordial unity, once differentiated, gave rise to a primordial “triad,” and two is the arithmetic mean between the two: 1+3=4/2=2. In other words, our “ancient topological metaphor” is also a
musical
metaphor. The
odd
numbers are thus “male” numbers, including 1, 3, and especially 5, since these numbers, 3 and 5, generate the
differentiation
of tones within the octave “mother” or matrix.
²⁴
In this, the “female principle was deified, but it was exclusively the male element on which the world developed and differentiated itself.”
²⁵
Thus, Plato could define a father as “‘the model in whose likeness that which becomes is born.’ The only fathers we need from here on are ‘3’ and ‘5,’ which appear to have meant in the
Rg Veda
exactly what they meant long afterward in the
Republic
.”
²⁶
The powers of three, and the multiples of three and five, define the eleven tones of our modern equally-tempered chromatic scale.
²⁷

But why would ancient cosmological-musical systems equate femininity with the octave, and masculinity with the differentiating process that gave rise to (tonal and topological) diversity? We are perhaps here in the presence of another clue to the degree of sophisticated scientific knowledge bequeathed as a legacy to these ancient civilizations, for as modern genetics now knows, females carry only the female sexual chromosome, but males carry
both
sexual chromosomes and are capable of biologically reproducing either a
male or female offspring. There is, as it were, a kind of inbuilt androgyny to the male that does not exist in the female. This might account for the assignation of “maleness” to those arithmetic ratios of odd numbers that produced
differentiation
in the generation of musical tones.

Why give such prominence to the “male” numbers? And what other numbers do they generate. Again, the answer is cosmological in nature:

The male odd numbers take precedence over their female octave “doubles” not only because they lie closed to god = 1, but presumably because they permit the divine unity to be subdivided rigorously according to the principle of unity; if we limit generative ratios, as Greek musical theory did traditionally, to those between two consecutive intergers… then each odd number (oddness itself being due to an element of unity) functions as the arithmetic mean for an earlier superparticular ratio, and subdivides it according to the same principle. We can schematize this subdivision as follows: