Distributional evenness: Difference between revisions

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-A scale is '''distributionally even''' ('''DE''') if it has [[maximum variety]] 2; that is, each generic interval class ("seconds", "thirds", and so on) contains no more than two specific intervals.
+{{Distinguish|Maximal evenness}}
+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, such 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.)
+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>
+== 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)
+=== 1 step type ===
+step type, unary: 0
+=== 2 step types ===
+step types, unary: 00
+step types, binary: 01
+=== 3 step types ===
+step types, unary: 000
+step types, binary: 001
+step types, ternary: 012
+=== 4 step types ===
+step types, unary: 0000
+step types, binary: 0001, 0101
+step types, ternary: 0102
+step types, quaternary: 0123
+=== 5 step types ===
+step types, unary: 00000
+step types, binary: 00001, 00101
+step types, ternary: 00102, 01012
+step types, quaternary: 01023
+step types, quinary: 01234
+=== 6 step types ===
+step types, unary: 000000
+step types, binary: 000001, 001001, 010101
+step types, ternary: 001002, 012012
+step types, quaternary: 010203, 012013
+step types, quinary: 012034
+step types, 6-ary: 012345
+=== 7 step types ===
+step types, unary: 0000000
+step types, binary: 0000001, 0001001, 0010101
+step types, ternary: 0001002, 0010201, 0101012, 0102012
+step types, quaternary: 0010203, 0102013, 0102032, 0120123
+step types, quinary: 0102034, 0120134, 0120314
+step types, 6-ary: 0120345
+step types, 7-ary: 0123456
+=== 8 step types ===
+step types, unary: 00000000
+step types, binary: 00000001, 00010001, 00100101, 01010101
+step types, ternary: 00010002, 01020102, 01021012
+step types, quaternary: 00100203, 01012013, 01020103, 01021013, 01230123
+step types, quinary: 01020304, 01023042, 01230124
+step types, 6-ary: 01023045, 01230145, 01230425
+step types, 7-ary: 01230456
+step types, 8-ary: 01234567
+=== 9 step types ===
+step types, unary: 000000000
+step types, binary: 000000001, 000010001, 001001001, 001010101
+step types, ternary: 000010002, 001020102, 010101012, 012012012
+step types, quaternary: 001002003, 001020103, 001020302, 010201023, 010201032, 012031023
+step types, quinary: 001020304, 010201034, 010201304, 010203042, 012013014, 012031024, 012301234
+step types, 6-ary: 010203045, 012031045, 012301245, 012301425, 012301435, 012304135
+step types, 7-ary: 012034056, 012301456, 012304156, 012304256
+step types, 8-ary: 012304567
+step types, 9-ary: 012345678
+=== 10 step types ===
+step types, unary: 0000000000
+step types, binary: 0000000001, 0000100001, 0001001001, 0010100101, 0101010101
+step types, ternary: 0000100002, 0010200102, 0101201012, 0102102012
+step types, quaternary: 0001002003, 0010200103, 0010200302, 0101201013, 0102301023, 0120120123, 0120310213
+step types, quinary: 0010200304, 0102103014, 0102301024, 0102301043, 0102304023, 0120130214, 0120310214, 0120310413, 0123401234
+step types, 6-ary: 0102030405, 0102301045, 0102304025, 0102304053, 0120130145, 0120130415, 0120310415, 0120340253, 0123401235
+step types, 7-ary: 0102304056, 0120340256, 0120340563, 0123401256, 0123401536
+step types, 8-ary: 0120340567, 0123401567, 0123405267
+step types, 9-ary: 0123405678
+step types, 10-ary: 0123456789
+== Related topics ==
+* [[Minimum-ambiguity DE scales]]
+== References ==
 [[Category:Terms]]
-[[Category:Scale theory]]
+[[Category:Scale]]