User:CompactStar/Overtone scale: Difference between revisions

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+An '''overtone scale''' is an octave-long subset of the [[harmonic series]] repeating at the octave. An overtone scale has also be referred to as '''ADO''' (arithmetic divisions of the octave) due to an overtone scale being an arithmetically equal division 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]].
+If the first division is <math>R_1</math> (which is ratio of C/C) and the last , <math>R_n</math> (which is ratio of 2C/C), with common difference of d. 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.
+(which is 1/C), we have :
+<math>
+R_2 = R_1 + d \\
+R_3= R_1 + 2d \\
+R_4 = R_1 + 3d \\
+\vdots \\
+R_n = R_1 + (n-1)d
+</math>