Minimum-ambiguity DE scales - Revision history

ArrowHead294 at 02:23, 12 October 2025

2025-10-12T02:23:31Z

← Older revision		Revision as of 02:23, 12 October 2025
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	The scale that has minimum ambiguity between notes is always an [[equal]] scale.		The scale that has minimum ambiguity between notes is always an [[equal]] scale.

	The DE scale of ''N'' notes that has strictly minimum ambiguity between intervals is always a subscale of ({{nowrap\| 2''N'' - ''P'' }})-edo, where ''P'' is the number of periods per octave. (Octave-repeating scales are assumed here, but in fact the minimum ambiguity property is insensitive to any stretching or compressing, so the same scale patterns apply as equal divisions of any interval whatsoever.)		The DE scale of ''N'' notes that has strictly minimum ambiguity between intervals is always a subscale of ({{nowrap\| 2''N'' − ''P'' }})-edo, where ''P'' is the number of periods per octave. (Octave-repeating scales are assumed here, but in fact the minimum ambiguity property is insensitive to any stretching or compressing, so the same scale patterns apply as equal divisions of any interval whatsoever.)

	Scales are also listed that have some totally ambiguous (coinciding) intervals, but have minimum ambiguity over all other intervals.		Scales are also listed that have some totally ambiguous (coinciding) intervals, but have minimum ambiguity over all other intervals.
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	One ambiguous interval allowed:		One ambiguous interval allowed:
	*1 period per octave in [[6edo]]		* 1 period per octave in [[6edo]]
	** 1 1 1 3		** 1 1 1 3

BudjarnLambeth: /* 9 notes */ {{Navbox scale gallery}}

2025-09-28T03:11:53Z

9 notes: {{Navbox scale gallery}}

← Older revision		Revision as of 03:11, 28 September 2025
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	** 1 1 1 1 1 1 1 1 7		** 1 1 1 1 1 1 1 1 7

	~~[[Category:Lists of scales]]~~		{{Navbox scale gallery}}

FloraC: Cleanup

2025-09-11T11:38:21Z

Cleanup

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Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

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Wikispaces>keenanpepper: Imported revision 279783254 - Original comment:

2011-11-28T14:12:27Z

**Imported revision 279783254 - Original comment: **

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-11-28 13:45:21 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-11-28 14:12:27 UTC</tt>.<br>
-: The original revision id was <tt>279769642</tt>.<br>
+: The original revision id was <tt>279783254</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>
@@ Line 89: / Line 89: @@
 * 1 period per octave in [[13edo]]
 ** 1 2 1 2 2 1 2 2 ([[13edo#Igliashon%27s%2013-EDO%20diatonic%20approaches|oneirotonic]])
-** 1 1 1 1 1 1 1 6</pre></div>
+** 1 1 1 1 1 1 1 6
 <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Minimum-ambiguity DE scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;These &lt;a class="wiki_link" href="/distributionally%20even"&gt;distributionally even&lt;/a&gt; scales have maximum distinguishability, where &amp;quot;distinguishability&amp;quot; is defined as the minimum separation between different notes or intervals. If &amp;quot;ambiguity&amp;quot; (confusability) is defined as the inverse of distinguishability, then they have minimum ambiguity.&lt;br /&gt;
@@ Line 126: / Line 146: @@
 &lt;ul&gt;&lt;li&gt;4 periods per octave in &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;&lt;ul&gt;&lt;li&gt;1 2 1 2 1 2 1 2 (familiar &lt;a class="wiki_link" href="/diminished"&gt;diminished&lt;/a&gt;-like octatonic scale)&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;li&gt;2 periods per octave in &lt;a class="wiki_link" href="/14edo"&gt;14edo&lt;/a&gt;&lt;ul&gt;&lt;li&gt;1 2 2 2 1 2 2 2 (&lt;a class="wiki_link" href="/hedgehog"&gt;hedgehog&lt;/a&gt;-like)&lt;/li&gt;&lt;li&gt;1 1 1 4 1 1 1 4&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;li&gt;1 period per octave in &lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&gt;&lt;ul&gt;&lt;li&gt;1 2 2 2 2 2 2 2 (&lt;a class="wiki_link" href="/porcupine"&gt;porcupine&lt;/a&gt;-like)&lt;/li&gt;&lt;li&gt;1 1 1 1 1 1 1 8&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;/ul&gt;One ambiguous interval allowed:&lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;1 period per octave in &lt;a class="wiki_link" href="/14edo"&gt;14edo&lt;/a&gt;&lt;ul&gt;&lt;li&gt;1 1 3 1 1 3 1 3&lt;/li&gt;&lt;li&gt;1 1 1 1 1 1 1 7&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;/ul&gt;Two ambiguous intervals allowed:&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;2 periods per octave in &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;&lt;ul&gt;&lt;li&gt;1 1 1 3 1 1 1 3 (&lt;a class="wiki_link" href="/pajara"&gt;pajara&lt;/a&gt;-like)&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;li&gt;1 period per octave in &lt;a class="wiki_link" href="/13edo"&gt;13edo&lt;/a&gt;&lt;ul&gt;&lt;li&gt;1 2 1 2 2 1 2 2 (&lt;a class="wiki_link" href="/13edo#Igliashon%27s%2013-EDO%20diatonic%20approaches"&gt;oneirotonic&lt;/a&gt;)&lt;/li&gt;&lt;li&gt;1 1 1 1 1 1 1 6&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;ul&gt;&lt;li&gt;2 periods per octave in &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;&lt;ul&gt;&lt;li&gt;1 1 1 3 1 1 1 3 (&lt;a class="wiki_link" href="/pajara"&gt;pajara&lt;/a&gt;-like)&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;li&gt;1 period per octave in &lt;a class="wiki_link" href="/13edo"&gt;13edo&lt;/a&gt;&lt;ul&gt;&lt;li&gt;1 2 1 2 2 1 2 2 (&lt;a class="wiki_link" href="/13edo#Igliashon%27s%2013-EDO%20diatonic%20approaches"&gt;oneirotonic&lt;/a&gt;)&lt;/li&gt;&lt;li&gt;1 1 1 1 1 1 1 6&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;

Wikispaces>keenanpepper: Imported revision 279769642 - Original comment:

2011-11-28T13:45:21Z

**Imported revision 279769642 - Original comment: **

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-11-28 13:34:01 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-11-28 13:45:21 UTC</tt>.<br>
-: The original revision id was <tt>279763906</tt>.<br>
+: The original revision id was <tt>279769642</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>
@@ Line 68: / Line 68: @@
 * 1 2 1 2 1 2 2 ([[orgone]]-like)
 * 1 1 1 3 1 1 3
-* 1 1 1 1 1 1 5</pre></div>
+* 1 1 1 1 1 1 5
 <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Minimum-ambiguity DE scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;These &lt;a class="wiki_link" href="/distributionally%20even"&gt;distributionally even&lt;/a&gt; scales have maximum distinguishability, where &amp;quot;distinguishability&amp;quot; is defined as the minimum separation between different notes or intervals. If &amp;quot;ambiguity&amp;quot; (confusability) is defined as the inverse of distinguishability, then they have minimum ambiguity.&lt;br /&gt;
@@ Line 100: / Line 121: @@
 &lt;ul&gt;&lt;li&gt;1 2 2 2 2 2 2 (&lt;a class="wiki_link" href="/13edo#Igliashon%27s%2013-EDO%20diatonic%20approaches"&gt;archeotonic&lt;/a&gt;)&lt;/li&gt;&lt;li&gt;1 1 3 1 3 1 3&lt;/li&gt;&lt;li&gt;1 1 1 4 1 1 4&lt;/li&gt;&lt;li&gt;1 1 1 1 1 1 7&lt;/li&gt;&lt;/ul&gt;One ambiguous interval allowed: &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;&lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;1 2 2 1 2 2 2 (familiar &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt;-like diatonic scale)&lt;/li&gt;&lt;li&gt;1 1 1 1 1 1 6&lt;/li&gt;&lt;/ul&gt;Two ambiguous intervals allowed: &lt;a class="wiki_link" href="/11edo"&gt;11edo&lt;/a&gt;)&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;1 2 1 2 1 2 2 (&lt;a class="wiki_link" href="/orgone"&gt;orgone&lt;/a&gt;-like)&lt;/li&gt;&lt;li&gt;1 1 1 3 1 1 3&lt;/li&gt;&lt;li&gt;1 1 1 1 1 1 5&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;ul&gt;&lt;li&gt;1 2 1 2 1 2 2 (&lt;a class="wiki_link" href="/orgone"&gt;orgone&lt;/a&gt;-like)&lt;/li&gt;&lt;li&gt;1 1 1 3 1 1 3&lt;/li&gt;&lt;li&gt;1 1 1 1 1 1 5&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;

Wikispaces>keenanpepper: Imported revision 279763906 - Original comment:

2011-11-28T13:34:01Z

**Imported revision 279763906 - Original comment: **

New page

<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-11-28 13:34:01 UTC</tt>.<br>
: The original revision id was <tt>279763906</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">These [[distributionally even]] scales have maximum distinguishability, where "distinguishability" is defined as the minimum separation between different notes or intervals. If "ambiguity" (confusability) is defined as the inverse of distinguishability, then they have minimum ambiguity.

The scale that has minimum ambiguity between notes is always an [[equal]] scale.

The DE scale of N notes that has strictly minimum ambiguity between intervals is always a subscale of (2N-P)-EDO, where P is the number of periods per octave. (Octave-repeating scales are assumed here, but in fact the minimum ambiguity property is insensitive to any stretching or compressing, so the same scale patterns apply as equal divisions of any interval whatsoever.)

Scales are also listed that have some totally ambiguous (coinciding) intervals, but have minimum ambiguity over all other intervals.

==3 notes==
Minimum ambiguity between notes: [[3edo]]
Minimum ambiguity between intervals: 3-note DE scales in [[5edo]]
* 1 2 2
* 1 1 3
One ambiguous interval allowed: 1 1 2 in [[4edo]]

==4 notes==
Minimum ambiguity between notes: [[4edo]] (always N-edo for N notes; hereafter omitted)
Minimum ambiguity between intervals:
* 2 periods per octave in [[6edo]]
** 1 2 1 2
* 1 period per octave in [[7edo]]
** 1 2 2 2 (a [[neutral third scale]])
** 1 1 1 4
One ambiguous interval allowed:
* 1 period per octave in [[6edo]]
** 1 1 1 3

==5 notes==
Minimum ambiguity between intervals: [[9edo]]
* 1 2 2 2 2 ([[superpelog]]-like)
* 1 1 3 1 3 ([[mavila]]- or [[pelog]]-like)
* 1 1 1 1 5
One ambiguous interval allowed: [[8edo]]
* 1 2 1 2 2 ([[father]]-like)
* 1 1 1 1 4
Two ambiguous intervals allowed: [[7edo]]
* 1 1 2 1 2 (2s ambiguous with L, 2L+s ambiguous with L+3s)
* 1 1 1 1 3

==6 notes==
Minimum ambiguity between intervals:
* 3 periods per octave in [[9edo]]
** 1 2 1 2 1 2
* 2 periods per octave in [[10edo]]
** 1 2 2 1 2 2 ([[decimal]]-like)
** 1 1 3 1 1 3
* 1 period per octave in [[11edo]]
** 1 2 2 2 2 2 ([[machine]]-like)
** 1 1 1 1 1 6

==7 notes==
Minimum ambiguity between intervals: [[13edo]]
* 1 2 2 2 2 2 2 ([[13edo#Igliashon%27s%2013-EDO%20diatonic%20approaches|archeotonic]])
* 1 1 3 1 3 1 3
* 1 1 1 4 1 1 4
* 1 1 1 1 1 1 7
One ambiguous interval allowed: [[12edo]]
* 1 2 2 1 2 2 2 (familiar [[meantone]]-like diatonic scale)
* 1 1 1 1 1 1 6
Two ambiguous intervals allowed: [[11edo]])
* 1 2 1 2 1 2 2 ([[orgone]]-like)
* 1 1 1 3 1 1 3
* 1 1 1 1 1 1 5</pre></div>
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Minimum-ambiguity DE scales</title></head><body>These <a class="wiki_link" href="/distributionally%20even">distributionally even</a> scales have maximum distinguishability, where &quot;distinguishability&quot; is defined as the minimum separation between different notes or intervals. If &quot;ambiguity&quot; (confusability) is defined as the inverse of distinguishability, then they have minimum ambiguity.<br />
<br />
The scale that has minimum ambiguity between notes is always an <a class="wiki_link" href="/equal">equal</a> scale.<br />
<br />
The DE scale of N notes that has strictly minimum ambiguity between intervals is always a subscale of (2N-P)-EDO, where P is the number of periods per octave. (Octave-repeating scales are assumed here, but in fact the minimum ambiguity property is insensitive to any stretching or compressing, so the same scale patterns apply as equal divisions of any interval whatsoever.)<br />
<br />
Scales are also listed that have some totally ambiguous (coinciding) intervals, but have minimum ambiguity over all other intervals.<br />
<br />
<h2 id="toc0"><a name="x-3 notes"></a>3 notes</h2>
Minimum ambiguity between notes: <a class="wiki_link" href="/3edo">3edo</a><br />
Minimum ambiguity between intervals: 3-note DE scales in <a class="wiki_link" href="/5edo">5edo</a><br />
<ul><li>1 2 2</li><li>1 1 3</li></ul>One ambiguous interval allowed: 1 1 2 in <a class="wiki_link" href="/4edo">4edo</a><br />
<br />
<h2 id="toc1"><a name="x-4 notes"></a>4 notes</h2>
Minimum ambiguity between notes: <a class="wiki_link" href="/4edo">4edo</a> (always N-edo for N notes; hereafter omitted)<br />
Minimum ambiguity between intervals:<br />
<ul><li>2 periods per octave in <a class="wiki_link" href="/6edo">6edo</a><ul><li>1 2 1 2</li></ul></li><li>1 period per octave in <a class="wiki_link" href="/7edo">7edo</a><ul><li>1 2 2 2 (a <a class="wiki_link" href="/neutral%20third%20scale">neutral third scale</a>)</li><li>1 1 1 4</li></ul></li></ul>One ambiguous interval allowed:<br />
<ul><li>1 period per octave in <a class="wiki_link" href="/6edo">6edo</a><ul><li>1 1 1 3</li></ul></li></ul><br />
<h2 id="toc2"><a name="x-5 notes"></a>5 notes</h2>
Minimum ambiguity between intervals: <a class="wiki_link" href="/9edo">9edo</a><br />
<ul><li>1 2 2 2 2 (<a class="wiki_link" href="/superpelog">superpelog</a>-like)</li><li>1 1 3 1 3 (<a class="wiki_link" href="/mavila">mavila</a>- or <a class="wiki_link" href="/pelog">pelog</a>-like)</li><li>1 1 1 1 5</li></ul>One ambiguous interval allowed: <a class="wiki_link" href="/8edo">8edo</a><br />
<ul><li>1 2 1 2 2 (<a class="wiki_link" href="/father">father</a>-like)</li><li>1 1 1 1 4</li></ul>Two ambiguous intervals allowed: <a class="wiki_link" href="/7edo">7edo</a><br />
<ul><li>1 1 2 1 2 (2s ambiguous with L, 2L+s ambiguous with L+3s)</li><li>1 1 1 1 3</li></ul><br />
<h2 id="toc3"><a name="x-6 notes"></a>6 notes</h2>
Minimum ambiguity between intervals:<br />
<ul><li>3 periods per octave in <a class="wiki_link" href="/9edo">9edo</a><ul><li>1 2 1 2 1 2</li></ul></li><li>2 periods per octave in <a class="wiki_link" href="/10edo">10edo</a><ul><li>1 2 2 1 2 2 (<a class="wiki_link" href="/decimal">decimal</a>-like)</li><li>1 1 3 1 1 3</li></ul></li><li>1 period per octave in <a class="wiki_link" href="/11edo">11edo</a><ul><li>1 2 2 2 2 2 (<a class="wiki_link" href="/machine">machine</a>-like)</li><li>1 1 1 1 1 6</li></ul></li></ul><br />
<h2 id="toc4"><a name="x-7 notes"></a>7 notes</h2>
Minimum ambiguity between intervals: <a class="wiki_link" href="/13edo">13edo</a><br />
<ul><li>1 2 2 2 2 2 2 (<a class="wiki_link" href="/13edo#Igliashon%27s%2013-EDO%20diatonic%20approaches">archeotonic</a>)</li><li>1 1 3 1 3 1 3</li><li>1 1 1 4 1 1 4</li><li>1 1 1 1 1 1 7</li></ul>One ambiguous interval allowed: <a class="wiki_link" href="/12edo">12edo</a><br />
<ul><li>1 2 2 1 2 2 2 (familiar <a class="wiki_link" href="/meantone">meantone</a>-like diatonic scale)</li><li>1 1 1 1 1 1 6</li></ul>Two ambiguous intervals allowed: <a class="wiki_link" href="/11edo">11edo</a>)<br />
<ul><li>1 2 1 2 1 2 2 (<a class="wiki_link" href="/orgone">orgone</a>-like)</li><li>1 1 1 3 1 1 3</li><li>1 1 1 1 1 1 5</li></ul></body></html></pre></div>