User:Inthar/Generator variety: Difference between revisions

Latest revision as of 03:36, 23 May 2024

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-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.
+#redirect [[Generator sequence#Generator variety]]
-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''',
-# '''x''' + 3'''y''',
-# 2'''x''' + '''y''' + '''z''',
-# '''x''' + 2'''y''' + '''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.