Distributional evenness: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Distinguish|Maximal evenness}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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.
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-11-14 13:30:09 UTC</tt>.<br>
: The original revision id was <tt>275251658</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A scale is **distributionally even** if it has [[maximum variety]] 2; that is, each class of interval ("seconds", "thirds", and so on) contains **no more than** two specific intervals.


In practice, such scales are often referred to as "[[MOSScales|MOS]]" scales, but this usage is technically incorrect because a MOS is defined to have **exactly** two specific intervals for each class other than multiples of the octave. When [[Erv Wilson]] discovered MOS scales and found numerous examples, DE scales that are not technically MOS such as [[pajara]], [[Augmented family|augmented]], [[diminished]], etc. were not among them.</pre></div>
== Technical definition ==
<h4>Original HTML content:</h4>
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.)
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Distributional Evenness&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A scale is &lt;strong&gt;distributionally even&lt;/strong&gt; if it has &lt;a class="wiki_link" href="/maximum%20variety"&gt;maximum variety&lt;/a&gt; 2; that is, each class of interval (&amp;quot;seconds&amp;quot;, &amp;quot;thirds&amp;quot;, and so on) contains &lt;strong&gt;no more than&lt;/strong&gt; two specific intervals.&lt;br /&gt;
 
&lt;br /&gt;
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>
In practice, such scales are often referred to as &amp;quot;&lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt;&amp;quot; scales, but this usage is technically incorrect because a MOS is defined to have &lt;strong&gt;exactly&lt;/strong&gt; two specific intervals for each class other than multiples of the octave. When &lt;a class="wiki_link" href="/Erv%20Wilson"&gt;Erv Wilson&lt;/a&gt; discovered MOS scales and found numerous examples, DE scales that are not technically MOS such as &lt;a class="wiki_link" href="/pajara"&gt;pajara&lt;/a&gt;, &lt;a class="wiki_link" href="/Augmented%20family"&gt;augmented&lt;/a&gt;, &lt;a class="wiki_link" href="/diminished"&gt;diminished&lt;/a&gt;, etc. were not among them.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
== 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 ==
* [[Minimum-ambiguity DE scales]]
== References ==
[[Category:Terms]]
[[Category:Scale]]

Latest revision as of 00:12, 28 June 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 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.
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