User:Inthar/Generator variety: Difference between revisions
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If a scale has a [[generator sequence]] or is a union of multiple offset generator | If a scale has a [[generator sequence]] or is a union of multiple offset copies of the same generator sequence, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval. | ||
There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes in terms of intervals in the generator interval class (10-steps because (4 generators) * (10 steps) = 40 steps ≡ 1 mod 13): | There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes in terms of intervals in the generator interval class (10-steps because (4 generators) * (10 steps) = 40 steps ≡ 1 mod 13): | ||
Revision as of 01:13, 23 May 2024
If a scale has a generator sequence or is a union of multiple offset copies of the same generator sequence, then the generator variety is the number of generators in that sequence, not including the closing interval.
There is no known simple relationship between a scale's step variety and its generator variety. MOS scales have step variety 2 and generator variety 1, and MOS substitution scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety n have step variety at most n + 1: Consider a 13-note scale with a well-formed generator sequence GS(x, y, y, x, y) and suppose one scale step is reached via 4 generators. Then the word of stacked generators is xyyxyxyyxyxyz including the final closing generator z, and there are 4 step sizes in terms of intervals in the generator interval class (10-steps because (4 generators) * (10 steps) = 40 steps ≡ 1 mod 13):
- 2x + 2y,
- x + 3y,
- 2x + y + z,
- x + 2y + z.
More trivially, a MOS generator when not stacked so that the resulting scale is a MOS size (equal to the denominator of a semiconvergent to the generator) will not have variety 2.