The ancient notion of holy mountains, intuited by Ernest G. McClain in the 1970s, was based on the cross-multiples of the powers of the prime numbers three and five, placed in an table where the two primes defined 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 and, furthermore, the prime number two would have been seen as not instrumental in placing where, on such holy mountains, each number must appear. Thus, an inherent duality was recognised 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
In contrast, the familiar form of the octave is created by prime-2 since it consists of a doubling in tone, and inside the octave, intervals provide pathways between the notes found within modal scales (created by primes 3 and 5) just because only larger prime number can subdivide 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 their unique composition of those primes. It is be interesting to see how this works visually, within the Tone Circle. In this we locate the tonic "do" as equal to modern note class (or note letter) D, using the symmetrical disposition of white and black keys around it on the keyboard; when symmetrical, the modern Dorian scale is formed as with the white keys, but all the twelve chromatic tones can be shown so as to indicate where the primes 3 and 5 are transported on to the twelve note classes within every tone circle.
The lowest regular number capable of forming five of the modern scales as well as some degree of chromatism 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 24 (=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 location, on this "hill of primes" (figure 2), of each tone number, remains 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)
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 number adjacent to each other, on the tone circle.
Figure 4 The sub-culture of atomic intervals within the tone circle.
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, the tone circle appeared to be unrelated to music. However, if one attends to the prime number transfers between adjacent tones one can, by rehydrating 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 24 (=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 23 (=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 equls 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
Using Curt Sachs' notion of the Indian srutis**, my book Harmonic Origins of the World finds, through a similar approach, that all modal intervals could be seen as made up of just three types of interval worth 1, 2 and 3 srutis, of which there would always be a total of 22 srutis in an octave. It is also clear that, based upon the prime number composition of tone numbers, that a change of tonic will move 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 will place the possible limits as belonging to the ancient Indian numbers associated with cosmology, such as those beginning with "head number" 432, multipled 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 and other writings, ancient music theory was integrated with ancient cosmological ideas findable in texts.
**Curt Sachs, The Rise of Music in the Ancient World, New York: Norton 1943. 165.