Prime Produced Interval Sizes
When one uses the expression "3/2" to define the size of a perfect fifth,
two interval generators are already being used : Prime 3 for the fifth ; and Prime 2 for the octave removed from its full size of "3/1"
which would be more accurately and specifically described by using the term "Harmonic 3", also Prime produced, as are all pitch distances (intervals). (Nothing but Primes here, no Fibs)
Natural Harmonics, the building blocks of the structure of Timbre and Sound,
are typically all the integer multiples of a fundamental tone, called Harmonic 1. However, the
basic acoustic ratios in the structure of Pitch limit themselves to the first three Prime Numbers (2, 3, 5).
And while it is interesting and useful to consider
these basic acoustic ratios
with a representation similar to the one used for natural harmonics,
it does not imply that the musical structure of Pitch
might come directly from the Natural Harmonics themselves,
but only that it is mathematically convenient to place them graphically
as one would place the harmonics,
with the appropriate pitch-distance between them.
The Octave - Harmonic 2 - proportion 2/1
between Harmonics 1 and 2, Harmonics 2 and 4, Harmonics 4 and 8
The Perfect Fifth - Harmonic 3 - proportion 3/1
between Harmonics 1 and 3, Harmonics 3 and 9.
The Major Third - Harmonic 5 - proportion 5/1
between Harmonics 1 and 5.
The Process of Cartesian Division
The first octave (Harmonics 1-2) remains undivided, the product of Prime 2.
The second octave (Harmonics 2-4) is divided by Prime 3 into two uneven parts, a main part of perfect fifth, 3/2, and a left-over part of perfect fourth 4/3, with a proportion of
9/8 between the two parts, the Pythagorian tone, the essence of diatonic scales (heptatonic, pentatonic, tritonic).
In the third octave (Harmonics 4-8), the main part perfect fifth (now between Harmonics 4 and 6) is divided by Prime 5 into two uneven parts, a main part of major third, 5/4, and a left-over part of minor third 6/5, with a proportion of 25/24
between the two parts, the chromatic distance between the two MEDIANs (thirds) in any chord FRAME.
In the fourth octave, the main part major third (now between Harmonics 8 and 10) is divided by 9, the square of Prime 3 (typical prime behavior of breaking a sequence), into two uneven parts, a main part of Pythagorian major second, 9/8, and
a left-over part of natural major second 10/9, with a proportion of 81/80 between the two parts, the syntonic coma, between
Pythagorian (Prime 3) and natural (Prime 5) tuning.
There seems to be no need of further primes,
including Harmonic 7 which is in a left-over part,
and the policy of remaining within the simplest ratios is duly respected.
These three fundamental interval sizes (the octave, the perfect fifth, and the major third)
will not be used indiscriminately in the Structure of Pitch.
The perfect fifth (product of Prime 3) is by far the most fruitful,
and what we call the series of 21 fifths, all the way from the Fb to the B#,
will be its back-bone with
Harmony on one side, by "dividing the fifths", and
Scales on the other side, by "multiplying the fifths".
The perfect fifth produces all the notes
and all the structures of Pitch,
both vertical and horizontal.