Music by numbers

Music incarnates through numbers as frequency and wawelength (= time & space) proportions of the harmonic series.

Until european renaissance most tonal system where established through combination of the first tree primes 2, 3 and 5 which express themselves as the music of the harmonic series as octave (proportion 1:2, one-to-two), perfect fifth (2:3) and just major third (4:5).

Initially this creation happens within the frame of the octave 30:60 (2x3x5 : 2x2x3x5).

The interval names of the tonal systems with frequency ratios of the form 2ⁿx3ⁿx5ⁿ : 2ⁿx3ⁿx5ⁿ

may be found on this link! 

By the way the product of the following tree primes 7, 11 and 13 is 1,001!!

At the illustration the neighbouring elements placed along the axises are octaves, proportion 1:2. For example 1:2:4:.. (octaves of the fundamental/ primary tone), 3:6:12:... (octaves of perfect fifth) and 5:10:20:... (octaves of just major third).

Lesser diesis is the difference between three successive just major thirds (5/4)³ (= 125/64 = 1.953125) and the octave (2:1).

The syntonic comma is the difference between just major third, 80 (=2⁴ x 5) and the pythagorean (generated by perfect fifths) major third, 3⁴.

The values of the pythagorean comma and greater diesis, 73:74 and 27:28 respectively, are approximations, because their constituent values are situated further out in the branches of the spiral. The pythagorean comma is the difference between 12 successive perfect fifths, (3/2)¹² : 2⁷ (= 129.746338) and 7 octaves, 2⁷ (= 128).

The greater diesis is the difference between four successive just minor thirds  (6/5)⁴ (=1,296 : 625 = 2.0736) and the octave, 2:1.

... Or is it really the other way round: Numbers incarnating through music?