Distributional evenness - Revision history

Inthar at 16:31, 4 January 2026

2026-01-04T16:31:33Z

← Older revision		Revision as of 16:31, 4 January 2026
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	{{Distinguish\|Maximal evenness}}		{{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 which preserves the literal meaning of the term.		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 generalization which preserves the literal meaning of the term.

	== Technical definition ==		== Technical definition ==

Inthar at 20:26, 1 January 2026

2026-01-01T20:26:13Z

← Older revision		Revision as of 20:26, 1 January 2026
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	{{Distinguish\|Maximal evenness}}		{{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.		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 which preserves the literal meaning of the term.

	== Technical definition ==		== Technical definition ==

VectorGraphics at 00:12, 28 June 2025

2025-06-28T00:12:56Z

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Hkm at 00:54, 5 May 2025

2025-05-05T00:54:10Z

← Older revision		Revision as of 00:54, 5 May 2025
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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 ==

	== ~~Extended~~ 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.)

	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'' − 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.		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 ==

Inthar: /* List of distributionally even circular words */

2025-03-03T01:02:39Z

List of distributionally even circular words

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Contribution at 08:54, 6 November 2024

2024-11-06T08:54:37Z

← Older revision		Revision as of 08:54, 6 November 2024
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	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 ==
			Below is the complete list of distributionally even circular words up to 10 letters.

			=== 1 Letter ===
			1 letter, MV1: 0

			=== 2 Letters ===
			2 letters, MV1: 00

			2 letters, MV2: 01

			=== 3 Letters ===
			3 letters, MV1: 000

			3 letters, MV2: 001

			3 letters, MV3: 012

			=== 4 Letters ===
			4 letters, MV1: 0000

			4 letters, MV2: 0001, 0101

			4 letters, MV3: 0102

			4 letters, MV4: 0123

			=== 5 Letters ===
			5 letters, MV1: 00000

			5 letters, MV2: 00001, 00101

			5 letters, MV3: 00102, 01012

			5 letters, MV4: 01023

			5 letters, MV5: 01234

			=== 6 Letters ===
			6 letters, MV1: 000000

			6 letters, MV2: 000001, 001001, 010101

			6 letters, MV3: 001002, 012012

			6 letters, MV4: 010203, 012013

			6 letters, MV5: 012034

			6 letters, MV6: 012345

			=== 7 Letters ===
			7 letters, MV1: 0000000

			7 letters, MV2: 0000001, 0001001, 0010101

			7 letters, MV3: 0001002, 0010201, 0101012, 0102012

			7 letters, MV4: 0010203, 0102013, 0102032, 0120123

			7 letters, MV5: 0102034, 0120134, 0120314

			7 letters, MV6: 0120345

			7 letters, MV7: 0123456

			=== 8 Letters ===
			8 letters, MV1: 00000000

			8 letters, MV2: 00000001, 00010001, 00100101, 01010101

			8 letters, MV3: 00010002, 01020102, 01021012

			8 letters, MV4: 00100203, 01012013, 01020103, 01021013, 01230123

			8 letters, MV5: 01020304, 01023042, 01230124

			8 letters, MV6: 01023045, 01230145, 01230425

			8 letters, MV7: 01230456

			8 letters, MV8: 01234567

			=== 9 Letters ===
			9 letters, MV1: 000000000

			9 letters, MV2: 000000001, 000010001, 001001001, 001010101

			9 letters, MV3: 000010002, 001020102, 010101012, 012012012

			9 letters, MV4: 001002003, 001020103, 001020302, 010201023, 010201032, 012031023

			9 letters, MV5: 001020304, 010201034, 010201304, 010203042, 012013014, 012031024, 012301234

			9 letters, MV6: 010203045, 012031045, 012301245, 012301425, 012301435, 012304135

			9 letters, MV7: 012034056, 012301456, 012304156, 012304256

			9 letters, MV8: 012304567

			9 letters, MV9: 012345678

			=== 10 Letters ===
			10 letters, MV1: 0000000000

			10 letters, MV2: 0000000001, 0000100001, 0001001001, 0010100101, 0101010101

			10 letters, MV3: 0000100002, 0010200102, 0101201012, 0102102012

			10 letters, MV4: 0001002003, 0010200103, 0010200302, 0101201013, 0102301023, 0120120123, 0120310213

			10 letters, MV5: 0010200304, 0102103014, 0102301024, 0102301043, 0102304023, 0120130214, 0120310214, 0120310413, 0123401234

			10 letters, MV6: 0102030405, 0102301045, 0102304025, 0102304053, 0120130145, 0120130415, 0120310415, 0120340253, 0123401235

			10 letters, MV7: 0102304056, 0120340256, 0120340563, 0123401256, 0123401536

			10 letters, MV8: 0120340567, 0123401567, 0123405267

			10 letters, MV9: 0123405678

			10 letters, MV10: 0123456789

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

Inthar: /* Extended definition */

2024-02-01T23:51:02Z

Extended definition

← Older revision		Revision as of 23:51, 1 February 2024
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	== Extended definition ==		== Extended 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>) is a [[maximally even]] subset of <math>\mathbb{Z}.</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.)

	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'' − 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.		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'' − 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.

Inthar: /* Extended definition */

2024-02-01T23:50:00Z

Extended definition

← Older revision		Revision as of 23:50, 1 February 2024
Line 5:		Line 5:

	== Extended definition ==		== Extended 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>) is a [[maximally even]] ~~MOS in~~ <math>~~\mathbb{Z}/n~~\mathbb{Z}.</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>) is a [[maximally even]] subset of <math>\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'' − 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.		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'' − 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.

Inthar: /* Extended definition */

2024-02-01T23:49:04Z

Extended definition

← Older revision		Revision as of 23:49, 1 February 2024
Line 5:		Line 5:

	== Extended definition ==		== Extended 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>) is a [[maximally even]] MOS in <math>\mathbb{Z}/n\mathbb{Z}</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>) 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'' − 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.		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'' − 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.

Inthar: /* Extended definition */

2024-02-01T23:48:47Z

Extended definition

← Older revision		Revision as of 23:48, 1 February 2024
Line 5:		Line 5:

	== Extended definition ==		== Extended 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>) is a [[maximally even]] MOS in ~~'''~~Z~~'''~~/''n~~'''''~~Z~~'''.~~ (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>) 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'' − 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.		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'' − 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.