User:CompactStar/Overtone scale: Difference between revisions

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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]]. 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.

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

For a Mode C system, the m-th degree is equal to the ratio (C+m)/C. 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

(which is 1/C), we have :

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 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]]. 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.
-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
+For a Mode C system, the m-th degree is equal to the ratio (C+m)/C. 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
 (which is 1/C), we have :