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{{Distinguish|Maximal evenness}}
{{Distinguish|Maximal evenness}}
A scale with two step sizes is '''distributionally even''' ('''DE''') if it has its two step sizes distributed as evenly as possible (i.e. each step size is distributed in a [[maximal evenness|maximally even]] pattern among the steps of the scale). This turns out to be equivalent to the property of having [[maximum variety]] 2; that is, each [[interval class]] ("seconds", "thirds", and so on) contains no more than two sizes. Though the term as originally defined is limited to scales with two step sizes, distributional evenness has an obvious generalization to scales of arbitrary [[arity]]: we simply require that each of the three or more step sizes be evenly distributed.
A scale is '''distributionally even''' if equating all step sizes except one will always result in a MOS. MOSses are the only distributionally even binary scales. The term was originally defined as a generalization of [[maximal evenness]] specifically for binary scales; this is the most convenient generalization.


In practice, binary DE scales are often referred to as "[[MOS scale]]s", but some consider this usage to be technically incorrect because a MOS as defined by [[Erv Wilson]] was to have ''exactly'' two specific intervals for each class other than multiples of the octave. When Wilson discovered MOS scales and found numerous examples, DE scales with period a fraction of an octave such as [[pajara]], [[augmented]], [[diminished]], etc. were not among them.
== Technical definition ==
Let ''r'' ≥ 2 and let <math>S: \mathbb{Z}\to\mathbb{R}</math> be an ''r''-ary [[periodic scale]] with length ''n'' (i.e. ''S''(''kn'') = ''kP'' where ''P'' is the period), with step sizes ''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>, i.e. such that <math>\Delta S(i) := S(i+1)-S(i)\in \{x_1, ..., x_r\} \forall i \in \mathbb{Z}.</math> The scale ''S'' is ''distributionally even'' if for every ''i'' ∈ {1, ..., ''r''}, (Δ''S'')<sup>&minus;1</sup>(''x''<sub>''i''</sub>) mod ''n'' is a [[maximally even]] subset of <math>\mathbb{Z}/n.</math> (For the original definition of DE, simply set ''r'' = 2.)


== Extended definition ==
Distributionally even scales over ''r'' step types are a subset of [[product word|product]]s of ''r'' &minus; 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s. All DE scales in this extended sense are also [[billiard scales]].<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref>
Let ''r'' ≥ 2 and let <math>S: \mathbb{Z}\to\mathbb{R}</math> be an ''r''-ary [[periodic scale]] with length ''n'' (i.e. ''S''(''kn'') = ''kP'' where ''P'' is the period), with step sizes ''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>, i.e. such that <math>\Delta S(i) := S(i+1)-S(i)\in \{x_1, ..., x_r\} \forall i \in \mathbb{Z}.</math> The scale ''S'' is ''distributionally even'' if for every ''i'' ∈ {1, ..., ''r''},  (Δ''S'')<sup>&minus;1</sup>(''x''<sub>''i''</sub>) is a [[maximally even]] MOS in <math>\mathbb{Z}/n\mathbb{Z}.</math> (For the original definition of DE, simply set ''r'' = 2.)


Using this definition, a scale word on ''r'' letters ''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub> is DE if and only if for every ''i'' ∈ {1, ..., ''r''}, the binary scale obtained by equating all step sizes except ''x''<sub>''i''</sub> is DE. This generalization of DE is thus an extraordinarily strong property: distributionally even scales over ''r'' letters are a subset of [[product word]]s of ''r'' &minus; 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.
== List of distributionally even scale patterns ==
Below is the complete list of distributionally even scale patterns up to 10 kinds of steps, without information on their relative sizes (so that these can each be seen as collections of [[sister]] scales)


All DE scales in this extended sense are also [[billiard scales]].<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref>
=== 1 step type ===
1 step type, unary: 0
 
=== 2 step types ===
2 step types, unary: 00
 
2 step types, binary: 01
 
=== 3 step types ===
3 step types, unary: 000
 
3 step types, binary: 001
 
3 step types, ternary: 012
 
=== 4 step types ===
4 step types, unary: 0000
 
4 step types, binary: 0001, 0101
 
4 step types, ternary: 0102
 
4 step types, quaternary: 0123
 
=== 5 step types ===
5 step types, unary: 00000
 
5 step types, binary: 00001, 00101
 
5 step types, ternary: 00102, 01012
 
5 step types, quaternary: 01023
 
5 step types, quinary: 01234
 
=== 6 step types ===
6 step types, unary: 000000
 
6 step types, binary: 000001, 001001, 010101
 
6 step types, ternary: 001002, 012012
 
6 step types, quaternary: 010203, 012013
 
6 step types, quinary: 012034
 
6 step types, 6-ary: 012345
 
=== 7 step types ===
7 step types, unary: 0000000
 
7 step types, binary: 0000001, 0001001, 0010101
 
7 step types, ternary: 0001002, 0010201, 0101012, 0102012
 
7 step types, quaternary: 0010203, 0102013, 0102032, 0120123
 
7 step types, quinary: 0102034, 0120134, 0120314
 
7 step types, 6-ary: 0120345
 
7 step types, 7-ary: 0123456
 
=== 8 step types ===
8 step types, unary: 00000000
 
8 step types, binary: 00000001, 00010001, 00100101, 01010101
 
8 step types, ternary: 00010002, 01020102, 01021012
 
8 step types, quaternary: 00100203, 01012013, 01020103, 01021013, 01230123
 
8 step types, quinary: 01020304, 01023042, 01230124
 
8 step types, 6-ary: 01023045, 01230145, 01230425
 
8 step types, 7-ary: 01230456
 
8 step types, 8-ary: 01234567
 
=== 9 step types ===
9 step types, unary: 000000000
 
9 step types, binary: 000000001, 000010001, 001001001, 001010101
 
9 step types, ternary: 000010002, 001020102, 010101012, 012012012
 
9 step types, quaternary: 001002003, 001020103, 001020302, 010201023, 010201032, 012031023
 
9 step types, quinary: 001020304, 010201034, 010201304, 010203042, 012013014, 012031024, 012301234
 
9 step types, 6-ary: 010203045, 012031045, 012301245, 012301425, 012301435, 012304135
 
9 step types, 7-ary: 012034056, 012301456, 012304156, 012304256
 
9 step types, 8-ary: 012304567
 
9 step types, 9-ary: 012345678
 
=== 10 step types ===
10 step types, unary: 0000000000
 
10 step types, binary: 0000000001, 0000100001, 0001001001, 0010100101, 0101010101
 
10 step types, ternary: 0000100002, 0010200102, 0101201012, 0102102012
 
10 step types, quaternary: 0001002003, 0010200103, 0010200302, 0101201013, 0102301023, 0120120123, 0120310213
 
10 step types, quinary: 0010200304, 0102103014, 0102301024, 0102301043, 0102304023, 0120130214, 0120310214, 0120310413, 0123401234
 
10 step types, 6-ary: 0102030405, 0102301045, 0102304025, 0102304053, 0120130145, 0120130415, 0120310415, 0120340253, 0123401235
 
10 step types, 7-ary: 0102304056, 0120340256, 0120340563, 0123401256, 0123401536
 
10 step types, 8-ary: 0120340567, 0123401567, 0123405267
 
10 step types, 9-ary: 0123405678
 
10 step types, 10-ary: 0123456789


== Related topics ==
== Related topics ==
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