..."in the spiral of perfect fifths, there the major and minor scales are the reverse of each other when the arithmetic vibration frequency ratios are put into geometrical form in a spiral diagram".

The 3-4-5 triangle can be written thus:

Representing Single and Double Squares.

3^2 = 9

4^2 = 16

5^2 = 25

3 x 96 = 288 cps (3rd harmonic of 96 cps).

4 x 96 = 384 cps (4th harmonic of 96 cps).

5 x 96 = 480 cps (5th harmonic of 96 cps).

"The 12 Lynus Tonal System or the cycle of fifths of China and the 22 natural intonation Scruti scales of India, are arithmetical progressions... the musical scale systems of different peoples can be mathematically oriented into a spiral complex when translated into digits 1, 2, 3, 4 and 5. This means, as already noted, that a "spherical music" may eventually express the evolution toward a world-wide acceptance of a common tonal system, perhaps the Mercator, equal-tempered 53 tones-to-the-octave scale within the overall mathematical perimeter of the circle of fifths"...

(Correspondences, File, "30 Unique Divisions").

..."the binary aspect of the octave intervals of the tonal spiral requires a Boolean algebra".

Dr. Andrew Pikler, International Congress on Acoustics, Tokyo, Aug. 1968, Generation and Plotting of Musical Tone System with the Digital Computer.

..."possibility of a "double helix" in the domain of the musical tone system...the "sharps" and "flats' furnish helical formations, each separately split with the Pythagorean comma"...

Dr. Andrew Pikler.

"Harold's concept of giving a measure number to opposite forces (yin-yang) in order to describe them, seems to relate to music, that is:

If a note in a minor key is at once the same sound and a different sound, it must be more or less of one sound. Therefore...if...a minor key...represents negative, and a major key represents positive... Harold's theory assigns 2 as the measure number of the minor, 3 as the measure of the major, and 5 as the measure of wholeness or the 1 sound that can be played in either the major or minor key- but cannot be played in both at the same instant of time".

If the foregoing is true, we ourselves cannot make the 1 sound, with the measure number 5, until we strike a chord:

3 (+ major) and 2 (- minor).

Harold's description of the nuclear force of the hydrogen atom can be read as 3(+) / 2(-). It would say, "that a chord can be struck using a formula that could read:

2(-) / 3(+), or 3(+) / 2(-),

these 2 would not sound alike, but either would convey the sound of wholeness.

In the famous lines of Brownings Abt Vogler poem...concerning resonance..."that out of 3 sounds are framed not a 4th sound, but a "star", we find that this results from a creation by way of "acoustic bass", as this is known technically. Here the highly selective tone phenomena of resonance represents a production of energy wherein the fundamental or lowest pitched tone in a series of overtones and the 5th above it (the 2nd overtone) combine to produce the "chordiness" (The "Star").

Alexander Wood,

The Physics of Music,

1962, p. 29.

Cosmic Humanism and World Unity

By Oliver Reiser

113. R2775c

© Copyright. Robert Grace. 1999