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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.

Hi all! I have a - what seems like a fairly simple - question which I am unable to answer. How would you categorize the following subset of the harmonic series as a tuning

5:7:8:10:11:12

Some observations - its pentatonic - its period is 12/5 (its a non-octave tuning) - its a subset of the harmonic series - its arithmetic - its non-equal, the (arithmetic) step sizes are 2/5, 1/5, 2/5, 1/5, 1/5 - every interval is unique (in cents: 528, 231, 386, 165, 150) - its not harmonotonic - its not a over-n scale: period is not the octave, therefore, its not a AFDO (arithmetic frequency division of octave) either - its not a OS (otonal sequence): 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 5:7:9:11:13:... - its not a OD (otonal division), but could be viewed as one specific scale/subset of 7-OD12/5 [5:6:7:8:9:10:11:12]

Is it a generator sequence? Every interval is unique, therefore it has as many generators as it does intervals (5) - does calling it a quinary GS even make sense here? I feel like with every interval being present only once it somehow defeats the 'generator' aspect of GS.

What do you think?

.scl file

! 5_7_8_10_11_12.scl ! Otonal pentatonic 5:7:8:10:11:12 ! 5 ! 7/5 8/5 10/5 11/5 12/5