In Modular (Mod) 12, Messiaen’s octatonic (0,1) scale is essentially the cyclical repetition of 4 simultaneous Mod 3 scales; (C, C#, D) < (Eb, E, F) < (F#, G, G#) < (A, Bb, B)… derived from the intervallic relationship 0+1+2, resulting PC’s = C, C#, C + Eb, E, Eb + F#, G, F# + A, Bb, A.
By applying modular arithmetic to the runsum (1+2+3+4…), where the Mods are defined as chromatic PC sets/groups (where C=1, C#=2, Bb=10…, and Mod 2 = C to C#, Mod 4 = C to Eb, Mod 8 = C to G, etc.), the following results occur (see fig I & 2):
The number of utilised PC’s in each modular set corresponds to the OEIS integer sequence of triangular numbers mod n – A117484. This sequence is counter intuitive; in that the simplest means we have of defining a concept of infinite linear expansion (1+2+3+4…) reveals embedded levels/layers of oscillating “wave-like” patterns (see fig iii.)
The isolated “crests” are immediately apparent (the Aii7484(n) sequence from 1 – 10,000 can be viewed here). Taking the values of every 1/16 integer produces the fractal like diagram in fig iv, where the extreme differences occur every 128th integer (10+ octaves, 128 PC’s similar to the range of human hearing).
A direct correspondence between (integer position) and n number of PC’s in a set (integer value) is revealed by consecutively doubling the result of n x 2, with n starting at 128, see fig V.
The structural (musicological & philosophical) implications of these results are that, within even a simplistic concept of linear expansion (1+2+3+4…), equal tempered PC’s & IC’s may carry alternative proportional potential, and extended serial identities, beyond their octave equivalent boundaries.
The set of even powers of n2 (https://oeis.org/A000079) crop up everywhere.
Further to this (again using C=0, C#=+1 etc.). Fig VI (below) expands the second positive integer/interval class (IC) to explore extended linear intervallic relationships and their resolution within a non-singular (multiple) octave equivalence; C < C, i.e., 0+1+2… 0+1+3… 0+1+4… etc.
The Octave resolution sequence in fig VI (above) is directly correspondent to OEIS A051724 – Numerator of n/12 (https://oeis.org/A051724 ). Thus revealing (a) further levels of eidetic and emergent scalic relationships beyond the singular chromatic (notated in fig vii), (b) a dodecaphonic/chromatic proportional sequence (A051724) that (as with the Fibonacci sequence) might be employed within any structural aspect of composition, and (c) another example of mirrored symmetry, where the number of occurring PC’s in the 9 sets from 0+1+1 to 0+1+9 = 12-8-6-12-4-12-6-8-12.