Step variety: Difference between revisions

Line 10:

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 ==

The number of possible patterns (up to rotation) for periodic scales of size ''n'' ~~with~~ ''r'' step sizes is

The number of possible patterns (up to rotation) for periodic scales of size ''n'' over ''r'' ordered step sizes is

<math>\displaystyle{\dfrac{1}{n!} \sum_{km = n\\k,m\geq 1} \Bigg[ \sum_{j=1}^r (-1)^{r-j} {r \choose j} ~~j^m~~ \Bigg] \~~dfrac~~{~~\phi~~(k)(~~m-1~~)!}{k},}</~~math~~>

<math>\displaystyle{\dfrac{1}{n!} \sum_{km = n\\k,m\geq 1} \dfrac{\phi(k)(m-1)!}{k} \Bigg[ \sum_{j=1}^r (-1)^{r-j} j^m {r \choose j} \Bigg],}</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" (a collection that is given two different species structures simultaneously) <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}} with ''n''th coefficient 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.~~

== List of named ternary scales ==

The following is a list of (temperament-agnostic) names that have been given to ternary scales. We ignore the exact arrangement of scale words here.

@@ Line 10: / Line 10: @@
 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 ==
-The number of possible patterns (up to rotation) for periodic scales of size ''n'' with ''r'' step sizes is
+The number of possible patterns (up to rotation) for periodic scales of size ''n'' over ''r'' ordered step sizes is
-<math>\displaystyle{\dfrac{1}{n!} \sum_{km = n\\k,m\geq 1} \Bigg[ \sum_{j=1}^r (-1)^{r-j} {r \choose j} j^m \Bigg] \dfrac{\phi(k)(m-1)!}{k},}</math>
+<math>\displaystyle{\dfrac{1}{n!} \sum_{km = n\\k,m\geq 1}  \dfrac{\phi(k)(m-1)!}{k} \Bigg[ \sum_{j=1}^r (-1)^{r-j} j^m {r \choose j} \Bigg],}</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" (a collection that is given two different species structures simultaneously) <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}} with ''n''th coefficient 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.
 == List of named ternary scales ==
 The following is a list of (temperament-agnostic) names that have been given to ternary scales. We ignore the exact arrangement of scale words here.