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| The ''generator variety''{{idiosyncratic}} is the number of generators in the generator sequence, not including the closing interval. | | The ''generator variety''{{idiosyncratic}} is the number of generators in the generator sequence, not including the closing interval. |
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| 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 in general no simple relationship between a scale's [[step variety]] and its generator variety. For any generator variety ''p'' > 1 and for any ''k'' > 1 it is possible to construct a long WFGS so that all combinatorially possible sums of ''k'' generators (there are <math>{k + p - 1 \choose k}</math> of them) are obtained for scale steps. |
| # 2'''x''' + 2'''y''',
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| # '''x''' + 3'''y''',
| | 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. |
| # 2'''x''' + '''y''' + '''z''',
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| # '''x''' + 2'''y''' + '''z'''.
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| 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.
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| == JI scales obtained from guided generator sequences == | | == JI scales obtained from guided generator sequences == |