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 == | ||
For ''r'' ≥ 3, the number of possible patterns (up to rotation) for periodic scales of size ''n'' over ''r'' ordered step sizes ''x''<sub>1</sub> > ''x''<sub>2</sub> > ... > ''x''<sub>''r''</sub> is | For ''r'' ≥ 3, 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 | ||
<math>\displaystyle{\dfrac{1}{n!} \sum_{km = n\\k,m\geq 1} \dfrac{\phi(k)(m-1)!}{k} \sum_{j=1}^r (-1)^{r-j} j^m {r \choose j},}</math> | <math>\displaystyle{\dfrac{1}{n!} \sum_{km = n\\k,m\geq 1} \dfrac{\phi(k)(m-1)!}{k} \sum_{j=1}^r (-1)^{r-j} j^m {r \choose j},}</math> | ||