Distribution of Prime Numbers in the Tone Circle

first published 13 February 2018

The ancient notion of tuning matrices, intuited by Ernest G. McClain in the 1970s, was based on the cross-multiples of the powers of prime numbers three and five, placed in an table where the two primes define two dimensions, where the powers are ordinal (0,1,2,3,4, etc…) and the dimension for prime number 5, an upward diagonal over a horizontal extent of the powers of prime number 3. Whilst harmonic numbers have been found in the ancient world as cuneiform lists (e.g. the Nippur List circa 2,200 BCE), these “regular” numbers would have been known to only have factors of the first three prime numbers 2, 3 and 5 (amenable to their base-60 arithmetic). Furthermore, the prime number two would have been seen as not instrumental in placing where, on such harmonic matrices, each harmonic number can be seen on a harmonic matrix (in religious terms perhaps a holy mountain), as

  • “right” according to its powers of 3.
  • “above” according to its powers of 5.

The role of odd primes within octaves

An inherent duality of perspective was established, between seeing each regular number as a whole integer number and seeing it as made up of powers of the two odd two prime numbers, their harmonic composition of the powers of 3 and 5 (see figure 1). It was obvious then as now that regular numbers were the product of three different prime numbers, each raised to different powers of itself, and that the primes 3 and 5 had the special power of both (a) creating musical intervals within octaves between numerical tones and (b) uniquely locating each numerical tone upon a mountain of numerical powers of 3 and 5.

Figure 1 Viewing the harmonic primes 3 and 5 as a mountain of their products, seen as integer numers or as to these harmonic primes

The familiar form of an octave is created by prime-2 doubling of the tone used as an octave’s tonic. Inside an octave, intervals provide pathways between the notes, these usually seen as modal scales, created by the primes of 3 and 5 because only a larger prime number, employed as a denominator, can break open the octave interval range equal to 2; into three parts or five parts, or multiples thereof such as 15 parts, etc. Since each brick in the right hand mountain of figure 1 is a unique composition of powers of 3 and 5, then all the possible tone numbers, translated onto a tone circle, will carry with them a unique composition of those primes. It is interesting here to follow visually how this works within a Tone Circle.

If we locate the tonic “do” as the modern note class (or letter) D: the symmetrical disposition of white and black keys around it on the keyboard reveal the symmetrical scale formed as the modern Dorian scale. The black chromatic keys indicate where the primes 3 and 5 are transported to within key signatures and scales other than C-major and Dorian.

The lowest regular number capable of forming five of the Greek scales (as well as some chromatic tones) is D equal to a limiting number of 720. The powers of 3 and 5 are unaffected by raising the harmonic root of 45 by 2^4 (=16) and the populated octave will then form around 45 as the darker bricks, on the first three rows in figure 1. These rows are shown in figure 2 and all the three rows in figure 1 have merely been brought into the range 360:720 as integer numbers, using as many powers of 2 as it takes, but their locations, on this “hill of primes” (figure 2), of each tone number, are fixed by the powers of 3 and 5 they embody.

Figure 2 The regular tone numbers of the mountain’s first three rows, capable of forming modal scales. (screen clipping from HarmonicExplorer.org   (https://harmonicexplorer.org#720))

All the tones in figure 2 can be transposed, as factors of 3 and 5, to the tone circle for limit 720 as follows,

Figure 3 The Limit = 720 for the octave with the tones shown as composite powers of 3 and 5

Between the primes are the component intervals (figure 4), and these are entirely due to the inevitable exchanges, in powers of three and five, between the tonal numbers adjacent to each other on the tone circle.

The flow of primes within modal intervals

The tone circle for a limit such as 720 (figure 4) produces little of much direct use to modal music and hence, in the past, this tone circle appeared to be unrelated to practical music. But if one attends to the prime number transfers between adjacent tones one can, by re-hydrating the resulting ratios using prime number 2, figure out the intervals. The interval low-D to e-flat loses a three and a five giving 1/15 which, times 2^4 (=16), makes the interval 16/15, the just semitone. The next interval (e- flat to e) loses one three and gains two fives giving 25/3, which divided by 2^3 (=8), makes the interval 25/24, the chromatic semitone.

In figure 5 one sees that these two intervals combine, as 16/15 times 25/24 equals 10/9, the just tone – then an interval used in modal scales and music-making. The interval between e and E in figure 4 gains four threes whilst losing one five, an interval of 81/5 which becomes 81/80, the syntonic comma. The syntonic comma links Pythagorean and Just tones, and adding it to the just tone of 10/9 leaves 9/8 (as two threes one five and a two cancel out), the Pythagorean tone (figure 5).

Figure 5 The emergence of modal intervals as combination of the atomic intervals between adjacent tones on the tone circle

Curt Sachs’ notion of the Indian srutis was used in my book HarmonicOrigins of the World (2018), using a similar approach: finding 22 srutis could arrive at all of our modal intervals that are made up of just three types of interval worth 1, 2 and 3 srutis of different size, forming the more familiar tones and semitones of our modal scales. The results of figure 6 exactly correspond with the work of K.B.Deval which must be where Sachs gained his data (see end note**).

Figure 6 The matrix (left), tone circle (right) and linear scale (bottom) of a limit of 12 x 720 = 4320. On the linear scale the only three component intervals are 22 cents (the synodic comma of 81/80), 70 cents (chromatic semitone of 25/24) and 90 cents (the greater chromatic semitone of 135/128). (figure 6.1 of Harmonic Origins)

Based upon the prime number composition of tone numbers, a change of tonic will involve the movement of the location of D upon the mountain so that, A in the above limit of 720 will install D at the current location of A. Doing that places the tonic (4320 in figure 6) amongst the harmonic numbers associated with ancient Indian cosmology, such as those beginning with a “head number” 432, multiplied by prime 2 such as 864 and 1728 or by prime 5, then raising its location to, for example,the number of the flood heros: 8,640,000,000. In this way, as Ernest McClain proposed in The Myth of Invariance (1976) and other writings, an ancient music theory was integrated with ancient cosmological ideas, found in texts as references to “harmonic numbers” containing primes 2, 3 and 5.

**Curt Sachs, The Rise of Music in the Ancient World, New York: Norton 1943. 165. The standard work on this is Mark Levy’s Intonation in North Indian Music. Chapter 3 starts with K.B. Deval’s work (1910, 1921) in which the Sa, Ri, Ga, Ma, Pa, Dha, Ni, Sa scale is given the ascending intervals in cents of 182, 112, 204, 204, 182, 112, 204, where 112 is the semitone of 16/15, 182 is the just tone of 10/9, 204 is the Pythagorean tone of 9/8, and the scale is modern Dorian.