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

MOS scales have step variety 2 and generator variety 1 and MOS substitution 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: '''xyyxyxyyxyxyz'''. Then there are 4 step sizes:

* MOS substitution scales have step variety 3 and generator variety 2

# 2'''x''' + 2'''y''',

* 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: '''xyyxyxyyxyxyz'''. Then there are 4 step sizes: 2'''x''' + 2'''y''', '''x''' + 3'''y''', 2'''x''' + '''y'' + '''z''', ~~and~~ '''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.

# '''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.

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 == Relationship between step variety and generator variety ==
-* MOS scales have step variety 2 and generator variety 1
+MOS scales have step variety 2 and generator variety 1 and MOS substitution 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: '''xyyxyxyyxyxyz'''. Then there are 4 step sizes:
-* MOS substitution scales have step variety 3 and generator variety 2
+# 2'''x''' + 2'''y''',
-* 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: '''xyyxyxyyxyxyz'''. Then there are 4 step sizes: 2'''x''' + 2'''y''', '''x''' + 3'''y''', 2'''x''' + '''y'' + '''z''', and '''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.
+# '''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.