Step variety: Difference between revisions
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The term ''n-ary'' disregards the rank of the group generated by the step sizes, although an ''n''-ary scale is still ''generically'' rank-''n'' (the group generated by the ''n'' step sizes X<sub>''i''</sub> > 0, ''i'' = 1, ..., ''n'', has rank ''n'', not lower, for ''almost all'' choices of X<sub>''i''</sub>, in the same sense that almost all real numbers between 0 and 1 are irrational). | The term ''n-ary'' disregards the rank of the group generated by the step sizes, although an ''n''-ary scale is still ''generically'' rank-''n'' (the group generated by the ''n'' step sizes X<sub>''i''</sub> > 0, ''i'' = 1, ..., ''n'', has rank ''n'', not lower, for ''almost all'' choices of X<sub>''i''</sub>, in the same sense that almost all real numbers between 0 and 1 are irrational). | ||
== Mathematical facts == | == Mathematical facts == | ||
=== Counting scales of given arity and size === | |||
For ''r'' ≥ 1, the number of possible patterns (up to rotation) for periodic scales of size ''n'' ≥ ''r'' over ''r'' ordered step sizes ''x''<sub>1</sub> > ''x''<sub>2</sub> > ... > ''x''<sub>''r''</sub> is | For ''r'' ≥ 1, the number of possible patterns (up to rotation) for periodic scales of size ''n'' ≥ ''r'' over ''r'' ordered step sizes ''x''<sub>1</sub> > ''x''<sub>2</sub> > ... > ''x''<sub>''r''</sub> is | ||