Step variety: Difference between revisions
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== Mathematical facts == | == Mathematical facts == | ||
=== Counting scales of a given size on a given number of letters === | === Counting scales of a given size on a given number of letters === | ||
For ''r'' ≥ 1, the number of possible patterns (up to rotation) for periodic scales of size ''n'' ≥ ''r'' | For ''r'' ≥ 1, the number of possible patterns (up to rotation) for periodic scales of size ''n'' ≥ ''r'' on ''r'' ordered step sizes ''x''<sub>1</sub> > ''x''<sub>2</sub> > ... > ''x''<sub>''r''</sub> is | ||
<math>\displaystyle{\dfrac{1}{n} \sum_{d\mid n} \phi(d) \sum_{j=1}^r (-1)^{r-j} {r \choose j} j^{n/d},}</math> | <math>\displaystyle{\dfrac{1}{n} \sum_{d\mid n} \phi(d) \sum_{j=1}^r (-1)^{r-j} {r \choose j} j^{n/d},}</math> | ||