User:CompactStar/Overtone scale

An overtone scale' is an octave-long subset of the harmonic series repeating at the octave. Alternative terms for overtone scale include ADO (arithmetic divisions of the octave) due to an overtone scale being an arithmetically equal division of the octave, and ODO (otonal divisions of the octave).

An overtone scale with n notes maybe referred to as mode n of the harmonic series or n-ADO. For example, Mode 5 is a pentatonic scale with the intervals 1/1-6/5-7/5-8/5-9/5-2/1:

harmonic	5	6	7	8	9	10
JI ratio	1/1	6/5	7/5	8/5	9/5	2/1

A "mode" in other musical contexts is usually a different rotation of the same intervals. In the case of different harmonic modes, that's not exactly the case. However, in some sense it's a reasonable comparison, because as you slide the subset of harmonics around, you're essentially sampling different segments of integers whose prime factorizations follow simple, constant patterns (every 2nd number has a 2, every 3rd number has a 3, every 5th number has a 5) and therefore the full internal interval set (all dyads, triads, tetrads, etc.) from one mode to the next is more alike than it is different.

For a Mode C system, the m-th degree is equal to the ratio (C+m)/C. If the first division is [math]\displaystyle{ R_1 }[/math] (which is ratio of C/C) and the last , [math]\displaystyle{ R_n }[/math] (which is ratio of 2C/C), with common difference of d

(which is 1/C), we have :

[math]\displaystyle{ R_2 = R_1 + d \\ R_3= R_1 + 2d \\ R_4 = R_1 + 3d \\ \vdots \\ R_n = R_1 + (n-1)d }[/math]