User:Inthar/Generator variety: Difference between revisions

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== Relationship between step variety and 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:

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:

# 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 (denominator of semiconvergent to the generator) will not have variety 2.

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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 == Relationship between step variety and 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:
+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:
 # 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 (denominator of semiconvergent to the generator) will not have variety 2.
+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.