Distributional evenness: Difference between revisions
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{{Distinguish|Maximal evenness}} | {{Distinguish|Maximal evenness}} | ||
A scale | 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. | ||
== 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>−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'' letters are a subset of [[product word]]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> | |||
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 circular words == | == List of distributionally even circular words == | ||
Revision as of 00:54, 5 May 2025
- Not to be confused with 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.
Technical definition
Let r ≥ 2 and let [math]\displaystyle{ 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 x1, ..., xr, i.e. such that [math]\displaystyle{ \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}, (ΔS)−1(xi) mod n is a maximally even subset of [math]\displaystyle{ \mathbb{Z}/n. }[/math] (For the original definition of DE, simply set r = 2.)
Distributionally even scales over r letters are a subset of product words of r − 1 MOS scales, which can be thought of as temperament-agnostic Fokker blocks. All DE scales in this extended sense are also billiard scales.[1]
List of distributionally even circular words
Below is the complete list of distributionally even circular words up to 10 letters, up to equivalence under reassignment of letters.
1 Letter
1 letter, unary: 0
2 Letters
2 letters, unary: 00
2 letters, binary: 01
3 Letters
3 letters, unary: 000
3 letters, binary: 001
3 letters, ternary: 012
4 Letters
4 letters, unary: 0000
4 letters, binary: 0001, 0101
4 letters, ternary: 0102
4 letters, quaternary: 0123
5 Letters
5 letters, unary: 00000
5 letters, binary: 00001, 00101
5 letters, ternary: 00102, 01012
5 letters, quaternary: 01023
5 letters, quinary: 01234
6 Letters
6 letters, unary: 000000
6 letters, binary: 000001, 001001, 010101
6 letters, ternary: 001002, 012012
6 letters, quaternary: 010203, 012013
6 letters, quinary: 012034
6 letters, 6-ary: 012345
7 Letters
7 letters, unary: 0000000
7 letters, binary: 0000001, 0001001, 0010101
7 letters, ternary: 0001002, 0010201, 0101012, 0102012
7 letters, quaternary: 0010203, 0102013, 0102032, 0120123
7 letters, quinary: 0102034, 0120134, 0120314
7 letters, 6-ary: 0120345
7 letters, 7-ary: 0123456
8 Letters
8 letters, unary: 00000000
8 letters, binary: 00000001, 00010001, 00100101, 01010101
8 letters, ternary: 00010002, 01020102, 01021012
8 letters, quaternary: 00100203, 01012013, 01020103, 01021013, 01230123
8 letters, quinary: 01020304, 01023042, 01230124
8 letters, 6-ary: 01023045, 01230145, 01230425
8 letters, 7-ary: 01230456
8 letters, 8-ary: 01234567
9 Letters
9 letters, unary: 000000000
9 letters, binary: 000000001, 000010001, 001001001, 001010101
9 letters, ternary: 000010002, 001020102, 010101012, 012012012
9 letters, quaternary: 001002003, 001020103, 001020302, 010201023, 010201032, 012031023
9 letters, quinary: 001020304, 010201034, 010201304, 010203042, 012013014, 012031024, 012301234
9 letters, 6-ary: 010203045, 012031045, 012301245, 012301425, 012301435, 012304135
9 letters, 7-ary: 012034056, 012301456, 012304156, 012304256
9 letters, 8-ary: 012304567
9 letters, 9-ary: 012345678
10 Letters
10 letters, unary: 0000000000
10 letters, binary: 0000000001, 0000100001, 0001001001, 0010100101, 0101010101
10 letters, ternary: 0000100002, 0010200102, 0101201012, 0102102012
10 letters, quaternary: 0001002003, 0010200103, 0010200302, 0101201013, 0102301023, 0120120123, 0120310213
10 letters, quinary: 0010200304, 0102103014, 0102301024, 0102301043, 0102304023, 0120130214, 0120310214, 0120310413, 0123401234
10 letters, 6-ary: 0102030405, 0102301045, 0102304025, 0102304053, 0120130145, 0120130415, 0120310415, 0120340253, 0123401235
10 letters, 7-ary: 0102304056, 0120340256, 0120340563, 0123401256, 0123401536
10 letters, 8-ary: 0120340567, 0123401567, 0123405267
10 letters, 9-ary: 0123405678
10 letters, 10-ary: 0123456789
Related topics
References
- ↑ Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.