From a chromatic perspective, Srinivasa Ramanujan’s summation that an infinite runsum = –1/12 and the subsequent prevalence of the numbers 12 & 24 to string theory is intriguing. However, of primary interest here is the correspondence between the Interval Classes (IC’s) of Messian’s 2nd mode of limited transposition (Oct 0,1) and the infinite sequence of triangular numbers; derived from applying the simple Gauss formula n (n+1) / 2 to the runsum of natural numbers (1+2+3+4…).
By assigning each natural number to a pitch class (PC), where PC c=0 (12, 24,36…), c#=1(13, 25, 37…), and b=(11,23.35…), i.e., C+1=C#, C#+2=Eb, Eb+3=F#, F#+4=Bb, Bb+5=Eb… etc, the following infinite, cyclical & symmetrical sequence of PC’s is produced.
The structural use of triangular numbers and the octatonic is of course not unique within compositional process & practice, what is here unique is the subliminal and inherent, cyclical & symmetrical patterns of linear intervallic expansion (LIE) upon which my research is (will be) based. The conjecture being that the encultured (and consequently intuitive) Pythagorean notion of harmony and proportion is only one dimension/level of a multi-dimensional (consciously integrated and emergent) auditory informational structure.
The chromatic / dodecaphonic system = mod 12. Further interesting modular patterns emerge when we utilize differing radix/basis.