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 sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval. | If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, 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*10 = 40 ≡ 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): | ||
# 2'''x''' + 2'''y''', | # 2'''x''' + 2'''y''', | ||
# '''x''' + 3'''y''', | # '''x''' + 3'''y''', | ||
Revision as of 22:57, 22 May 2024
If a scale has a generator sequence or is a union of multiple offset generator sequences, 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.