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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

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

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]