Distributional evenness

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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 step types are a subset of products 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 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

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

References

  1. Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.