Step variety: Difference between revisions

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~~An ''n'''-ary scale''''' is a scale with exactly ''n'' distinct step sizes. '''''Unary''''', '''''binary''''' and '''''ternary''''' scales are scales with exactly 1, 2 and 3 step sizes, respectively.~~

~~A unary~~ scale is an [[equal tuning]]. The class of binary scales consists of all [[MOS]] ~~scales~~ and every alteration-by-permutation of a MOS scale. Ternary scales are much less well-understood than binary ones, but one well-studied type of ternary scales is the class of [[generator-offset]] scales. Most known facts about ternary scales on the wiki can be found on the ~~page~~ [[rank-3 scale]].

The '''step variety''' (or '''arity''') of a [[scale]] is the number of distinct [[step]] sizes it has. '''Unary''', '''binary''', '''ternary''', and '''quaternary''' scales are scales with exactly 1, 2, 3, and 4 step sizes, respectively. An ''n'''''-ary scale''' is a scale with exactly ''n'' distinct step sizes.

== ~~History of the term~~ ==

The terms ''binary'' and ''ternary'' are already used in some academic literature in reference to 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]".

Unary scales are [[equal tuning]]s. The class of binary scales consists of all [[MOS scale]]s and every alteration-by-permutation of an MOS scale, but do not include [[altered MOS scale]]s such as the harmonic minor scale (abstract [[step pattern]]: MsMMsLs), which gain additional step sizes from the alteration. Ternary scales are much less well-understood than binary ones, but one well-studied type of ternary scales is the class of [[generator-offset]] scales. Most known facts about ternary scales on the wiki can be found on the pages [[rank-3 scale]] and [[ternary scale theorems]].

== Etymology ==

The terms ''binary'' and ''ternary'' are already used in some academic literature in reference to [[word]]s 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]". The 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; combinatorics-on-words literature often instead uses "word on ''n'' letters" or "alphabet with ''n'' letters" in the arbitrary-''n'' case.

The term ''step variety'', coined by [[Frédéric Gagné]], is in analogy with ''[[interval variety]]'' for the number of distinct interval sizes in each [[interval class]].

== Difference from scale rank ==

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

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:

* Equal tunings contain MOS scales and ternary scales, but the group generated by the step sizes in these tunings of the scales must be rank 1.

* 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''i'' > 0, ''i'' = 1, ..., ''n'', has rank ''n'', not lower, for ''almost all'' choices of X''i'', 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, in a probabilistic sense, ''generically'' rank-''n'' (the group generated by the ''n'' step sizes X''i'' > 0, ''i'' = 1, ..., ''n'', has rank ''n'', not lower, for ''almost all'' choices of X''i'', in the same sense that almost all real numbers between 0 and 1 are irrational).

== Mathematical facts ==

=== 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'' on ''r'' ordered step sizes ''x''1 > ''x''2 > ... > ''x''''r'' 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}} \\

=\displaystyle{\dfrac{r!}{n} \sum_{d\mid n} \phi(d) S(n/d, r)}</math>

where <math>\phi</math> is the Euler totient function and <math>S(n, r)</math> is the Stirling number of the second kind which counts ways to partition an ''n''-element set into ''r'' distinguished parts.

== See also ==

* [[List of ternary scales]]

[[Category:Scale]]

[[Category:Terms]]

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-An ''n'''-ary scale''''' is a scale with exactly ''n'' distinct step sizes. '''''Unary''''', '''''binary''''' and '''''ternary''''' scales are scales with exactly 1, 2 and 3 step sizes, respectively.
+{{Redirect|Ternary|the temperament|Ternary (temperament)}}
-A unary scale is an [[equal tuning]]. The class of binary scales consists of all [[MOS]] scales and every alteration-by-permutation of a MOS scale. Ternary scales are much less well-understood than binary ones, but one well-studied type of ternary scales is the class of [[generator-offset]] scales. Most known facts about ternary scales on the wiki can be found on the page [[rank-3 scale]].
+The '''step variety''' (or '''arity''') of a [[scale]] is the number of distinct [[step]] sizes it has. '''Unary''', '''binary''', '''ternary''', and '''quaternary''' scales are scales with exactly 1, 2, 3, and 4 step sizes, respectively. An ''n'''''-ary scale''' is a scale with exactly ''n'' distinct step sizes.
-== History of the term ==
-The terms ''binary'' and ''ternary'' are already used in some academic literature in reference to 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]".
+Unary scales are [[equal tuning]]s. The class of binary scales consists of all [[MOS scale]]s and every alteration-by-permutation of an MOS scale, but do not include [[altered MOS scale]]s such as the harmonic minor scale (abstract [[step pattern]]: MsMMsLs), which gain additional step sizes from the alteration. Ternary scales are much less well-understood than binary ones, but one well-studied type of ternary scales is the class of [[generator-offset]] scales. Most known facts about ternary scales on the wiki can be found on the pages [[rank-3 scale]] and [[ternary scale theorems]].
+== Etymology ==
+The terms ''binary'' and ''ternary'' are already used in some academic literature in reference to [[word]]s 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]". The 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; combinatorics-on-words literature often instead uses "word on ''n'' letters" or "alphabet with ''n'' letters" in the arbitrary-''n'' case.
+The term ''step variety'', coined by [[Frédéric Gagné]], is in analogy with ''[[interval variety]]'' for the number of distinct interval sizes in each [[interval class]].
 == Difference from scale rank ==
-Certain abstract scale theorists in the xen community have taken to using the ''n-ary'' terminology, to respect the subtlety of the notion of a scale's [[rank]]. Examples of this subtlety are:
+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:
 * Equal tunings contain MOS scales and ternary scales, but the group generated by the step sizes in these tunings of the scales must be rank 1.
 * 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, in a probabilistic sense, ''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 ==
+=== Counting scales of a given size on a given number of letters ===
+For ''r'' &ge; 1, the number of possible patterns (up to rotation) for periodic scales of size ''n'' &ge; ''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}} \\
+=\displaystyle{\dfrac{r!}{n} \sum_{d\mid n} \phi(d) S(n/d, r)}</math>
+where <math>\phi</math> is the Euler totient function and <math>S(n, r)</math> is the Stirling number of the second kind which counts ways to partition an ''n''-element set into ''r'' distinguished parts.
+== See also ==
+* [[List of ternary scales]]
+[[Category:Scale]]
 [[Category:Terms]]