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[DRAFT] Non-octave / generalized (?) overtone scale
TODO: research if a generalized form like this already exists

COS - constrained otonal sequence<br>

in comparison to<br>

OS: COS is constrained, OS is open ended,

Non-octave overtone scales are an approach to describe overtone scales without the need of the octave as the period.
Therefore, they are non-octave-repeating scales based on a generating sequence which itself is a subset of the harmonic series.
They can also be viewed as a form of generator sequence.

Non-octave overtone scales are described by the form n...p:s

where

n ... root to which the following integers in the scale are relative to
p ... the period of the scale
s ... the step size, how many of the integers in the scale are skipped

n...p describes the integer sequence from including n to including p, for example, 4...7 gives: [4, 5, 6, 7]

for example, the scale 4...9:1 describes this 5-tone scale
4/4 - 5/4 - 6/4 - 7/4 - 8/4 - 9/4
the :1 indicates that every integer in the sequence is visited (step size of 1)

Contrast this to 4...9:2 which generates the following 3-tone scale
4/4 - 6/4 - 8/4 - 9/4
note that the :2 indicates that every other integer in the sequence from n ... p is visited (step size of 2)

Contrast this to 4...9:3 which generates the following 2-tone scale
4/4 - 7/4 - 9/4

etc.

5:7:8:10:11:12

- is pentatonic - period is 12/5 - is arithmetic - is non-equal, (arithmetic) step sizes: 2/5, 1/5, 2/5, 1/5, 1/5 - is still harmonotonic though? by nature of being a subset of the harmonic series => is actually not harmonotonic

in my own semantics, I'd refer to it by 5->12[2,1,2,1,1] (from including overtone 5 to including overtone 12 in MTS-ESP Master I'd use the same semantics

!! comparisons irrelevant because not harmonotonic !! in comparison to OS OS has one step size (interval p) and does not care about the end of the sequence/ the period, rather, it's approach is 'take the first n in the sequence' I could do 2-OS2/5 but that would generate 5:7:9

in comparison to OD could be one specific scale/subset of 6-OD5/4 [5:6:7:8:9:10:11]