Step variety: Difference between revisions

Line 4:

== History of the term ==

The terms ''binary'' and ''ternary'' are already used in some academic literature in reference to words over an alphabet, in particular to circular words that represent abstract scales; see e.g. Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]". Our use of the term ''arity'' borrows an {{w|Arity|existing technical term}} and generalizes from this use of ''binary'', ''ternary'', and ''n-ary'' to refer to the number of letters in an alphabet in combinatorics on words; standard academic usage often instead uses "word on ''n'' letters" or "alphabet with ''n'' letters" in the arbitrary-''n'' case.

== Difference from scale rank ==

Certain abstract scale theorists in the xen community have taken to using the ''n-ary'' terminology in view of the subtlety of the notion of a scale's [[rank]]. Examples of this subtlety are:

Line 10:

Line 9:

* 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).

== Mathematical facts ==

The number of possible patterns (up to rotation) for periodic scales of size ''n'' with ''r'' 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>

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.

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.

=== 7 notes ===

* [[nicetone]] (3L 2M 2S)

@@ Line 4: / Line 4: @@
 == History of the term ==
 The terms ''binary'' and ''ternary'' are already used in some academic literature in reference to words over an alphabet, in particular to circular words that represent abstract scales; see e.g. Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]". Our use of the term ''arity'' borrows an {{w|Arity|existing technical term}} and generalizes from this use of ''binary'', ''ternary'', and ''n-ary'' to refer to the number of letters in an alphabet in combinatorics on words; standard academic usage often instead uses "word on ''n'' letters" or "alphabet with ''n'' letters" in the arbitrary-''n'' case.
 == Difference from scale rank ==
 Certain abstract scale theorists in the xen community have taken to using the ''n-ary'' terminology in view of the subtlety of the notion of a scale's [[rank]]. Examples of this subtlety are:
@@ Line 10: / Line 9: @@
 * 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).
+== Mathematical facts ==
+The number of possible patterns (up to rotation) for periodic scales of size ''n'' with ''r'' 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>
+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.
+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.
 === 7 notes ===
 * [[nicetone]] (3L 2M 2S)