Distributional evenness: Difference between revisions
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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>−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.) | 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>−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'' | Distributionally even scales over ''r'' step types are a subset of [[product word|product]]s of ''r'' − 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 | == List of distributionally even scale patterns == | ||
Below is the complete list of distributionally even | 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 | === 1 step type === | ||
1 | 1 step type, unary: 0 | ||
=== 2 | === 2 step types === | ||
2 | 2 step types, unary: 00 | ||
2 | 2 step types, binary: 01 | ||
=== 3 | === 3 step types === | ||
3 | 3 step types, unary: 000 | ||
3 | 3 step types, binary: 001 | ||
3 | 3 step types, ternary: 012 | ||
=== 4 | === 4 step types === | ||
4 | 4 step types, unary: 0000 | ||
4 | 4 step types, binary: 0001, 0101 | ||
4 | 4 step types, ternary: 0102 | ||
4 | 4 step types, quaternary: 0123 | ||
=== 5 | === 5 step types === | ||
5 | 5 step types, unary: 00000 | ||
5 | 5 step types, binary: 00001, 00101 | ||
5 | 5 step types, ternary: 00102, 01012 | ||
5 | 5 step types, quaternary: 01023 | ||
5 | 5 step types, quinary: 01234 | ||
=== 6 | === 6 step types === | ||
6 | 6 step types, unary: 000000 | ||
6 | 6 step types, binary: 000001, 001001, 010101 | ||
6 | 6 step types, ternary: 001002, 012012 | ||
6 | 6 step types, quaternary: 010203, 012013 | ||
6 | 6 step types, quinary: 012034 | ||
6 | 6 step types, 6-ary: 012345 | ||
=== 7 | === 7 step types === | ||
7 | 7 step types, unary: 0000000 | ||
7 | 7 step types, binary: 0000001, 0001001, 0010101 | ||
7 | 7 step types, ternary: 0001002, 0010201, 0101012, 0102012 | ||
7 | 7 step types, quaternary: 0010203, 0102013, 0102032, 0120123 | ||
7 | 7 step types, quinary: 0102034, 0120134, 0120314 | ||
7 | 7 step types, 6-ary: 0120345 | ||
7 | 7 step types, 7-ary: 0123456 | ||
=== 8 | === 8 step types === | ||
8 | 8 step types, unary: 00000000 | ||
8 | 8 step types, binary: 00000001, 00010001, 00100101, 01010101 | ||
8 | 8 step types, ternary: 00010002, 01020102, 01021012 | ||
8 | 8 step types, quaternary: 00100203, 01012013, 01020103, 01021013, 01230123 | ||
8 | 8 step types, quinary: 01020304, 01023042, 01230124 | ||
8 | 8 step types, 6-ary: 01023045, 01230145, 01230425 | ||
8 | 8 step types, 7-ary: 01230456 | ||
8 | 8 step types, 8-ary: 01234567 | ||
=== 9 | === 9 step types === | ||
9 | 9 step types, unary: 000000000 | ||
9 | 9 step types, binary: 000000001, 000010001, 001001001, 001010101 | ||
9 | 9 step types, ternary: 000010002, 001020102, 010101012, 012012012 | ||
9 | 9 step types, quaternary: 001002003, 001020103, 001020302, 010201023, 010201032, 012031023 | ||
9 | 9 step types, quinary: 001020304, 010201034, 010201304, 010203042, 012013014, 012031024, 012301234 | ||
9 | 9 step types, 6-ary: 010203045, 012031045, 012301245, 012301425, 012301435, 012304135 | ||
9 | 9 step types, 7-ary: 012034056, 012301456, 012304156, 012304256 | ||
9 | 9 step types, 8-ary: 012304567 | ||
9 | 9 step types, 9-ary: 012345678 | ||
=== 10 | === 10 step types === | ||
10 | 10 step types, unary: 0000000000 | ||
10 | 10 step types, binary: 0000000001, 0000100001, 0001001001, 0010100101, 0101010101 | ||
10 | 10 step types, ternary: 0000100002, 0010200102, 0101201012, 0102102012 | ||
10 | 10 step types, quaternary: 0001002003, 0010200103, 0010200302, 0101201013, 0102301023, 0120120123, 0120310213 | ||
10 | 10 step types, quinary: 0010200304, 0102103014, 0102301024, 0102301043, 0102304023, 0120130214, 0120310214, 0120310413, 0123401234 | ||
10 | 10 step types, 6-ary: 0102030405, 0102301045, 0102304025, 0102304053, 0120130145, 0120130415, 0120310415, 0120340253, 0123401235 | ||
10 | 10 step types, 7-ary: 0102304056, 0120340256, 0120340563, 0123401256, 0123401536 | ||
10 | 10 step types, 8-ary: 0120340567, 0123401567, 0123405267 | ||
10 | 10 step types, 9-ary: 0123405678 | ||
10 | 10 step types, 10-ary: 0123456789 | ||
== Related topics == | == Related topics == | ||