Step variety: Difference between revisions
m →Mathematical facts: I don't know why this formula doesn't work for n = 4 |
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* Certain chroma-altered MOS scales, which are contained in the group generated by the period and the generator of the unaltered MOS are ternary. An example is harmonic minor in any non-edo diatonic tuning, a chroma-alteration of the diatonic MOS with step pattern msmmsLs. | * Certain chroma-altered MOS scales, which are contained in the group generated by the period and the generator of the unaltered MOS are ternary. An example is harmonic minor in any non-edo diatonic tuning, a chroma-alteration of the diatonic MOS with step pattern msmmsLs. | ||
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'' ≥ | 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 | ||
<math>\displaystyle{\dfrac{1}{n | <math>\displaystyle{\dfrac{1}{n} \sum_{km = n\\k,m\geq 1} \phi(k) \sum_{j=1}^r (-1)^{r-j} {r \choose j} j^m,}</math> | ||
where <math>\phi</math> is the Euler totient function. The formula follows from writing the {{w|combinatorial species}} (finite structure) of [[necklace]]s over ''r'' letters as the so-called "superposition" <math>\mathrm{Bal}^{[r]} \times \mathcal{C}</math> of two species: the species <math>\mathrm{Bal}^{[r]}</math> of ballots with ''r'' parts (partition where the parts are ordered) and the species <math>\mathcal{C}</math> of ordered cycles, and computing the resulting {{w|generating function}} whose ''n''th coefficient is the desired formula.<ref>Bergeron, F., Labelle, G., & Leroux, P. (1998). Combinatorial species and tree-like structures (No. 67). Cambridge University Press.</ref> | where <math>\phi</math> is the Euler totient function. The formula follows from writing the {{w|combinatorial species}} (finite structure) of [[necklace]]s over ''r'' letters as the so-called "superposition" <math>\mathrm{Bal}^{[r]} \times \mathcal{C}</math> of two species: the species <math>\mathrm{Bal}^{[r]}</math> of ballots with ''r'' parts (partition where the parts are ordered) and the species <math>\mathcal{C}</math> of ordered cycles, and computing the resulting {{w|generating function}} whose ''n''th coefficient is the desired formula.<ref>Bergeron, F., Labelle, G., & Leroux, P. (1998). Combinatorial species and tree-like structures (No. 67). Cambridge University Press.</ref> | ||
== List of named ternary scales == | == List of named ternary scales == | ||