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Gallery of omnitetrachordal scales: Difference between revisions

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Wikispaces>xenwolf
**Imported revision 588881512 - Original comment: **
Wikispaces>spt3125
**Imported revision 589273442 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-08-07 04:27:56 UTC</tt>.<br>
: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2016-08-12 21:23:18 UTC</tt>.<br>
: The original revision id was <tt>588881512</tt>.<br>
: The original revision id was <tt>589273442</tt>.<br>
: The revision comment was: <tt></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>
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>
<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">[[toc]]
<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">[[toc]]
An (incomplete) list of [[Omnitetrachordality|omnitetrachordal]] (OTC) scale patterns.


=Scales with two step sizes=
=Scales with two step sizes=
(not necessarily [[MOS]] or [[Distributional Evenness|DE]])
(not necessarily [[MOS]] or [[Distributional Evenness|DE]])
[[Omnitetrachordal scale|Omnitetrachordal scales]] are mostly tempered (irrational) scales.  In some cases (usually where the generator is close to a perfect 4/3 or 3/2), a Pythagorean (3-limit [[Just intonation|JI]]) version is possible.
These are mostly tempered (irrational) scales.  In some cases (usually where the generator is close to a perfect 4/3 or 3/2), a Pythagorean (3-limit [[Just intonation|JI]]) version is possible.


==5 tones (2+1+2)==
==5 tones (2+1+2)==
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Pythagorean (3-limit JI): L=[[32_27|32/27]], s=9/8
Pythagorean (3-limit JI): L=[[32_27|32/27]], s=9/8
generator = L+s
generator = L+s (480.0 to 600.0 cents)
[[{$page}#perfect|P]] = 1.4424745962
[[{$page}#perfect|P]] = 1.4424745962
notable [[EDO]]s: 7, 9, 12, 16, 17, 19, 22, 26, 29, 31
notable [[EDO]]s: 7, 9, 12, 16, 17, 19, 22, 26, 29, 31
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Father? MOS; [[{$page}#perfect|imperfect]]; [[{$page}#ref1|(1)]]
Father? MOS; [[{$page}#perfect|imperfect]]; [[{$page}#ref1|(1)]]


generator = L+s
generator = L+s (400.0 to 480.0 cents)
notable EDOs: 8, 11b, 13, 18, 23b
notable EDOs: 8, 11b, 13, 18, 23b


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[[Mavila Temperament Modal Harmony|Mavila]] diatonic MOS.  [[{$page}#perfect|Imperfect]].  [[{$page}#ref1|(1)]]
[[Mavila Temperament Modal Harmony|Mavila]] diatonic MOS.  [[{$page}#perfect|Imperfect]].  [[{$page}#ref1|(1)]]


generator = L+2s
generator = L+2s (514.3 to 600.0 cents)
notable EDOs: 9, 11, 16, 23
notable EDOs: 9, 11, 16, 23


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Pythagorean (3-limit JI): L=9/8, s=[[256_243|256/243]]
Pythagorean (3-limit JI): L=9/8, s=[[256_243|256/243]]
P = 2.2600167527
P = 2.2600167527
generator = 2L+s
generator = 2L+s (480.0 to 514.3 cents)
notable EDOs: 12, 17, 19, 22, 26, 29, 31, etc.
notable EDOs: 12, 17, 19, 22, 26, 29, 31, etc.


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* the (imperfect) tuning of 3/2 is independent of L/s ratio (fixed at exactly 720 cents)
* the (imperfect) tuning of 3/2 is independent of L/s ratio (fixed at exactly 720 cents)


generator = s
generator = s (0 to 120 cents)
[[periods and generators|period]] = 1/5 octave
[[periods and generators|period]] = 1/5 octave
notable EDOs: 15, 20, 25
notable EDOs: 15, 20, 25
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P = 2.5872804832
P = 2.5872804832
generator = 2L+s
generator = 2L+s (342.9 to 360.0 cents)
notable EDOs: 17, 24, 27, 31, 41
notable EDOs: 17, 24, 27, 31, 41


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P = 1.8849491924
P = 1.8849491924
generator = s
generator = s (0 to 120 cents)
[[periods and generators|period]] = 1/2 octave
[[periods and generators|period]] = 1/2 octave
notable EDOs: 12, 14, 16, 22, 26
notable EDOs: 12, 14, 16, 22, 26
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Pythagorean (3-limit JI): L=2187/2048, s=256/243
Pythagorean (3-limit JI): L=2187/2048, s=256/243
P = 1.2600167527
P = 1.2600167527
generator = 2L+3s
generator = 2L+3s (480.0 to 500.0 cents)
notable EDOs: 17, 22, 27, 29, 41, 53, etc.
notable EDOs: 17, 22, 27, 29, 41, 53, etc.


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12-tone MOS of the [[meantone]] and [[Meantone family#Flattone|flattone]] temperaments (slightly flat 3/2).  [[{$page}#perfect|Imperfect]], but very usable at any L/s ratio.  Many modes may be pentachordally split in two ways.
12-tone MOS of the [[meantone]] and [[Meantone family#Flattone|flattone]] temperaments (slightly flat 3/2).  [[{$page}#perfect|Imperfect]], but very usable at any L/s ratio.  Many modes may be pentachordally split in two ways.


generator = 3L+2s
generator = 3L+2s (500.0 to 514.3 cents)
notable EDOs: 19, 26, 31, 33, 43, 45, 50, etc.
notable EDOs: 19, 26, 31, 33, 43, 45, 50, etc.


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P = 1.1300083763
P = 1.1300083763
generator = L
generator = L (100.0 to 120.0 cents)
[[periods and generators|period]] = 1/2 octave
[[periods and generators|period]] = 1/2 octave
notable EDOs: 22, 32, 34, 46
notable EDOs: 22, 32, 34, 46
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P = 5.1745609664
P = 5.1745609664
generator = L+s
generator = L+s (171.4 to 184.6 cents)
notable EDOs: 20, 27, 34, 41
notable EDOs: 20, 27, 34, 41


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P = 3.5200335053
P = 3.5200335053
generator = L+2s
generator = L+2s (240.0 to 257.1 cents)
notable EDOs: 19, 24, 29, 33, 34, 53
notable EDOs: 19, 24, 29, 33, 34, 53


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[[Injera]] MODMOS.  Imperfect (P&lt;0); best for L/s &gt;= 1.5 [[{$page}#ref1|(1)]]
[[Injera]] MODMOS.  Imperfect (P&lt;0); best for L/s &gt;= 1.5 [[{$page}#ref1|(1)]]


generator = L
generator = L (85.7 to 100.0 cents)
[[periods and generators|period]] = 1/2 octave
[[periods and generators|period]] = 1/2 octave
notable EDOs: 26, 38, 40, 50
notable EDOs: 26, 38, 40, 50
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[[{$page}#perfect|P]] = 3.3809207248
[[{$page}#perfect|P]] = 3.3809207248
generator = L+s
generator = L+s (160.0 to 171.4 cents)
notable EDOs: 22, 29, 36, 37, 43, 65
notable EDOs: 22, 29, 36, 37, 43, 65


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P = 12.5377181246
P = 12.5377181246
generator = L
generator = L (80.0 to 100.0 cents)
period = 1/3 octave
period = 1/3 octave
notable EDOs: 27, 39, 51
notable EDOs: 27, 39, 51
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[[Superpyth]] / ultrapyth MOS.  Imperfect, best at low L/s.  Many modes may be split two ways.
[[Superpyth]] / ultrapyth MOS.  Imperfect, best at low L/s.  Many modes may be split two ways.


generator = 2L+5s
generator = 2L+5s (480.0 to 494.1 cents)
notable EDOs: 22, 27, 32, 37, 39
notable EDOs: 22, 27, 32, 37, 39


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Pythagorean (3-limit JI): L=256/243, s=531441/524288
Pythagorean (3-limit JI): L=256/243, s=531441/524288
P = 3.8459060415
P = 3.8459060415
generator = 5L+2s
generator = 5L+2s (494.1 to 500.0 cents)
notable EDOs: 29, 41, 46, 53, 65
notable EDOs: 29, 41, 46, 53, 65


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P = 1.5872804832
P = 1.5872804832
generator = 2L+3s
generator = 2L+3s (342.9 to 352.9 cents)
notable EDOs: 24, 31, 41, 65
notable EDOs: 24, 31, 41, 65


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Beatles? / [[Dicot family|dicot]]? MODMOS; imperfect; full L/s range mostly good
Beatles? / [[Dicot family|dicot]]? MODMOS; imperfect; full L/s range mostly good


generator = 3L+2s
generator = 3L+2s (352.9 to 360.0 cents)
notable EDOs: 27, 37, 44, 61
notable EDOs: 27, 37, 44, 61


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P = 5.968201208
P = 5.968201208
generator = 2L+3s
generator = 2L+3s (333.3 to 342.9 cents)
notable EDOs: 25, 32, 39, 46, 53
notable EDOs: 25, 32, 39, 46, 53


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[[Meantone family#Flattone|Flattone]] MOS; [[{$page}#perfect|P]]&lt;0; [[{$page}#ref1|(1)]]
[[Meantone family#Flattone|Flattone]] MOS; [[{$page}#perfect|P]]&lt;0; [[{$page}#ref1|(1)]]


generator = 3L+5s
generator = 3L+5s (505.3 to 514.3 cents)
notable EDOs: 26, 33, 40, 45
notable EDOs: 26, 33, 40, 45


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[[Meantone]] MOS; [[{$page}#perfect|P]]&lt;0 (oddly, neither this scale nor its [[{$page}#dual|dual]] can be tuned with a perfect 3/2!)  [[{$page}#ref1|(1)]]
[[Meantone]] MOS; [[{$page}#perfect|P]]&lt;0 (oddly, neither this scale nor its [[{$page}#dual|dual]] can be tuned with a perfect 3/2!)  [[{$page}#ref1|(1)]]


generator = 5L+3s
generator = 5L+3s (500.0 to 505.3 cents)
notable EDOs: 31, 43, 50, 55
notable EDOs: 31, 43, 50, 55


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P = 2.5200335053
P = 2.5200335053
generator = L+3s
generator = L+3s (240.0 to 252.6 cents)
notable EDOs: 24, 29, 34, 39, 43, 53
notable EDOs: 24, 29, 34, 39, 43, 53


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[[Semaphore and godzilla|Semaphore / godzilla]] MODMOS; imperfect; full L/s range good
[[Semaphore and godzilla|Semaphore / godzilla]] MODMOS; imperfect; full L/s range good


generator = 3L+s
generator = 3L+s (252.6 to 257.1 cents)
notable EDOs: 33, 47, 52
notable EDOs: 33, 47, 52


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P = 2.3809207248
P = 2.3809207248
generator = L+2s
generator = L+2s (163.6 to 171.4 cents)
notable EDOs: 29, 36, 43, 50, 51
notable EDOs: 29, 36, 43, 50, 51


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[[Porcupine]] MODMOS; imperfect; full L/s range mostly good
[[Porcupine]] MODMOS; imperfect; full L/s range mostly good


generator = 2L+s
generator = 2L+s (160.0 to 163.6 cents)
notable EDOs: 37, 52b, 59
notable EDOs: 37, 52b, 59


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P = 7.6918120831
P = 7.6918120831
generator = L+s
generator = L+s (100.0 to 109.1 cents)
period = 1/2 octave
period = 1/2 octave
notable EDOs: 34, 46, 56, 58, 70
notable EDOs: 34, 46, 56, 58, 70
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notable EDOs: 32, 42, 54
notable EDOs: 32, 42, 54
generator = L+s
generator = L+s (10.91 to 120.0 cents)
period = 1/2 octave
period = 1/2 octave


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=Definitions and formulas=
=Definitions and formulas=
**"Omnitetrachordal"** (OTC for short): see [[Omnitetrachordality]].


[[#dual]]**"Dual"** refers to the "inverse" of a L+s scale pattern, where every L is replaced by s, and vice versa.  For example, {{ sLssL }} and {{ LsLLs }} are duals.
[[#dual]]**"Dual"** refers to the "inverse" of a L+s scale pattern, where every L is replaced by s, and vice versa.  For example, {{ sLssL }} and {{ LsLLs }} are duals.
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&lt;!-- ws:end:WikiTextTocRule:183 --&gt;&lt;!-- ws:start:WikiTextTocRule:184: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:183 --&gt;&lt;!-- ws:start:WikiTextTocRule:184: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:184 --&gt;&lt;br /&gt;
&lt;!-- ws:end:WikiTextTocRule:184 --&gt;&lt;br /&gt;
An (incomplete) list of &lt;a class="wiki_link" href="/Omnitetrachordality"&gt;omnitetrachordal&lt;/a&gt; (OTC) scale patterns.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Scales with two step sizes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;Scales with two step sizes&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Scales with two step sizes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;Scales with two step sizes&lt;/h1&gt;
(not necessarily &lt;a class="wiki_link" href="/MOS"&gt;MOS&lt;/a&gt; or &lt;a class="wiki_link" href="/Distributional%20Evenness"&gt;DE&lt;/a&gt;)&lt;br /&gt;
(not necessarily &lt;a class="wiki_link" href="/MOS"&gt;MOS&lt;/a&gt; or &lt;a class="wiki_link" href="/Distributional%20Evenness"&gt;DE&lt;/a&gt;)&lt;br /&gt;
&lt;a class="wiki_link" href="/Omnitetrachordal%20scale"&gt;Omnitetrachordal scales&lt;/a&gt; are mostly tempered (irrational) scales.  In some cases (usually where the generator is close to a perfect 4/3 or 3/2), a Pythagorean (3-limit &lt;a class="wiki_link" href="/Just%20intonation"&gt;JI&lt;/a&gt;) version is possible.&lt;br /&gt;
These are mostly tempered (irrational) scales.  In some cases (usually where the generator is close to a perfect 4/3 or 3/2), a Pythagorean (3-limit &lt;a class="wiki_link" href="/Just%20intonation"&gt;JI&lt;/a&gt;) version is possible.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Scales with two step sizes-5 tones (2+1+2)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;5 tones (2+1+2)&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Scales with two step sizes-5 tones (2+1+2)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;5 tones (2+1+2)&lt;/h2&gt;
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&lt;br /&gt;
&lt;br /&gt;
Pythagorean (3-limit JI): L=&lt;a class="wiki_link" href="/32_27"&gt;32/27&lt;/a&gt;, s=9/8&lt;br /&gt;
Pythagorean (3-limit JI): L=&lt;a class="wiki_link" href="/32_27"&gt;32/27&lt;/a&gt;, s=9/8&lt;br /&gt;
generator = L+s&lt;br /&gt;
generator = L+s (480.0 to 600.0 cents)&lt;br /&gt;
&lt;!-- ws:start:WikiTextVariableComponentRule:05:[[{$page}#perfect|P]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;P&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:05 --&gt; = 1.4424745962&lt;br /&gt;
&lt;!-- ws:start:WikiTextVariableComponentRule:05:[[{$page}#perfect|P]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;P&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:05 --&gt; = 1.4424745962&lt;br /&gt;
notable &lt;a class="wiki_link" href="/EDO"&gt;EDO&lt;/a&gt;s: 7, 9, 12, 16, 17, 19, 22, 26, 29, 31&lt;br /&gt;
notable &lt;a class="wiki_link" href="/EDO"&gt;EDO&lt;/a&gt;s: 7, 9, 12, 16, 17, 19, 22, 26, 29, 31&lt;br /&gt;
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Father? MOS; &lt;!-- ws:start:WikiTextVariableComponentRule:07:[[{$page}#perfect|imperfect]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;imperfect&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:07 --&gt;; &lt;!-- ws:start:WikiTextVariableComponentRule:09:[[{$page}#ref1|(1)]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#ref1"&gt;(1)&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:09 --&gt;&lt;br /&gt;
Father? MOS; &lt;!-- ws:start:WikiTextVariableComponentRule:07:[[{$page}#perfect|imperfect]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;imperfect&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:07 --&gt;; &lt;!-- ws:start:WikiTextVariableComponentRule:09:[[{$page}#ref1|(1)]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#ref1"&gt;(1)&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:09 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
generator = L+s&lt;br /&gt;
generator = L+s (400.0 to 480.0 cents)&lt;br /&gt;
notable EDOs: 8, 11b, 13, 18, 23b&lt;br /&gt;
notable EDOs: 8, 11b, 13, 18, 23b&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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&lt;a class="wiki_link" href="/Mavila%20Temperament%20Modal%20Harmony"&gt;Mavila&lt;/a&gt; diatonic MOS.  &lt;!-- ws:start:WikiTextVariableComponentRule:011:[[{$page}#perfect|Imperfect]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;Imperfect&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:011 --&gt;.  &lt;!-- ws:start:WikiTextVariableComponentRule:013:[[{$page}#ref1|(1)]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#ref1"&gt;(1)&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:013 --&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/Mavila%20Temperament%20Modal%20Harmony"&gt;Mavila&lt;/a&gt; diatonic MOS.  &lt;!-- ws:start:WikiTextVariableComponentRule:011:[[{$page}#perfect|Imperfect]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;Imperfect&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:011 --&gt;.  &lt;!-- ws:start:WikiTextVariableComponentRule:013:[[{$page}#ref1|(1)]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#ref1"&gt;(1)&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:013 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
generator = L+2s&lt;br /&gt;
generator = L+2s (514.3 to 600.0 cents)&lt;br /&gt;
notable EDOs: 9, 11, 16, 23&lt;br /&gt;
notable EDOs: 9, 11, 16, 23&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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Pythagorean (3-limit JI): L=9/8, s=&lt;a class="wiki_link" href="/256_243"&gt;256/243&lt;/a&gt;&lt;br /&gt;
Pythagorean (3-limit JI): L=9/8, s=&lt;a class="wiki_link" href="/256_243"&gt;256/243&lt;/a&gt;&lt;br /&gt;
P = 2.2600167527&lt;br /&gt;
P = 2.2600167527&lt;br /&gt;
generator = 2L+s&lt;br /&gt;
generator = 2L+s (480.0 to 514.3 cents)&lt;br /&gt;
notable EDOs: 12, 17, 19, 22, 26, 29, 31, etc.&lt;br /&gt;
notable EDOs: 12, 17, 19, 22, 26, 29, 31, etc.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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Blackwood MOS.  This omnitetrachordal scale has some unusual properties:&lt;br /&gt;
Blackwood MOS.  This omnitetrachordal scale has some unusual properties:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;it has only two modes&lt;/li&gt;&lt;li&gt;either mode may be split in any of the three possible omnitetrachordal ways (9/8-4/3-4/3, 4/3-9/8-4/3, 4/3-4/3-9/8)&lt;/li&gt;&lt;li&gt;it is its own dual (changing all L to s, and vice versa, gives the same scale)&lt;/li&gt;&lt;li&gt;the (imperfect) tuning of 3/2 is independent of L/s ratio (fixed at exactly 720 cents)&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;it has only two modes&lt;/li&gt;&lt;li&gt;either mode may be split in any of the three possible omnitetrachordal ways (9/8-4/3-4/3, 4/3-9/8-4/3, 4/3-4/3-9/8)&lt;/li&gt;&lt;li&gt;it is its own dual (changing all L to s, and vice versa, gives the same scale)&lt;/li&gt;&lt;li&gt;the (imperfect) tuning of 3/2 is independent of L/s ratio (fixed at exactly 720 cents)&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
generator = s&lt;br /&gt;
generator = s (0 to 120 cents)&lt;br /&gt;
&lt;a class="wiki_link" href="/periods%20and%20generators"&gt;period&lt;/a&gt; = 1/5 octave&lt;br /&gt;
&lt;a class="wiki_link" href="/periods%20and%20generators"&gt;period&lt;/a&gt; = 1/5 octave&lt;br /&gt;
notable EDOs: 15, 20, 25&lt;br /&gt;
notable EDOs: 15, 20, 25&lt;br /&gt;
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&lt;br /&gt;
&lt;br /&gt;
P = 2.5872804832&lt;br /&gt;
P = 2.5872804832&lt;br /&gt;
generator = 2L+s&lt;br /&gt;
generator = 2L+s (342.9 to 360.0 cents)&lt;br /&gt;
notable EDOs: 17, 24, 27, 31, 41&lt;br /&gt;
notable EDOs: 17, 24, 27, 31, 41&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 1,383: Line 1,387:
&lt;br /&gt;
&lt;br /&gt;
P = 1.8849491924&lt;br /&gt;
P = 1.8849491924&lt;br /&gt;
generator = s&lt;br /&gt;
generator = s (0 to 120 cents)&lt;br /&gt;
&lt;a class="wiki_link" href="/periods%20and%20generators"&gt;period&lt;/a&gt; = 1/2 octave&lt;br /&gt;
&lt;a class="wiki_link" href="/periods%20and%20generators"&gt;period&lt;/a&gt; = 1/2 octave&lt;br /&gt;
notable EDOs: 12, 14, 16, 22, 26&lt;br /&gt;
notable EDOs: 12, 14, 16, 22, 26&lt;br /&gt;
Line 1,515: Line 1,519:
Pythagorean (3-limit JI): L=2187/2048, s=256/243&lt;br /&gt;
Pythagorean (3-limit JI): L=2187/2048, s=256/243&lt;br /&gt;
P = 1.2600167527&lt;br /&gt;
P = 1.2600167527&lt;br /&gt;
generator = 2L+3s&lt;br /&gt;
generator = 2L+3s (480.0 to 500.0 cents)&lt;br /&gt;
notable EDOs: 17, 22, 27, 29, 41, 53, etc.&lt;br /&gt;
notable EDOs: 17, 22, 27, 29, 41, 53, etc.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 1,625: Line 1,629:
12-tone MOS of the &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; and &lt;a class="wiki_link" href="/Meantone%20family#Flattone"&gt;flattone&lt;/a&gt; temperaments (slightly flat 3/2).  &lt;!-- ws:start:WikiTextVariableComponentRule:021:[[{$page}#perfect|Imperfect]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;Imperfect&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:021 --&gt;, but very usable at any L/s ratio.  Many modes may be pentachordally split in two ways.&lt;br /&gt;
12-tone MOS of the &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; and &lt;a class="wiki_link" href="/Meantone%20family#Flattone"&gt;flattone&lt;/a&gt; temperaments (slightly flat 3/2).  &lt;!-- ws:start:WikiTextVariableComponentRule:021:[[{$page}#perfect|Imperfect]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;Imperfect&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:021 --&gt;, but very usable at any L/s ratio.  Many modes may be pentachordally split in two ways.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
generator = 3L+2s&lt;br /&gt;
generator = 3L+2s (500.0 to 514.3 cents)&lt;br /&gt;
notable EDOs: 19, 26, 31, 33, 43, 45, 50, etc.&lt;br /&gt;
notable EDOs: 19, 26, 31, 33, 43, 45, 50, etc.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 1,734: Line 1,738:
&lt;br /&gt;
&lt;br /&gt;
P = 1.1300083763&lt;br /&gt;
P = 1.1300083763&lt;br /&gt;
generator = L&lt;br /&gt;
generator = L (100.0 to 120.0 cents)&lt;br /&gt;
&lt;a class="wiki_link" href="/periods%20and%20generators"&gt;period&lt;/a&gt; = 1/2 octave&lt;br /&gt;
&lt;a class="wiki_link" href="/periods%20and%20generators"&gt;period&lt;/a&gt; = 1/2 octave&lt;br /&gt;
notable EDOs: 22, 32, 34, 46&lt;br /&gt;
notable EDOs: 22, 32, 34, 46&lt;br /&gt;
Line 1,860: Line 1,864:
&lt;br /&gt;
&lt;br /&gt;
P = 5.1745609664&lt;br /&gt;
P = 5.1745609664&lt;br /&gt;
generator = L+s&lt;br /&gt;
generator = L+s (171.4 to 184.6 cents)&lt;br /&gt;
notable EDOs: 20, 27, 34, 41&lt;br /&gt;
notable EDOs: 20, 27, 34, 41&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 1,986: Line 1,990:
&lt;br /&gt;
&lt;br /&gt;
P = 3.5200335053&lt;br /&gt;
P = 3.5200335053&lt;br /&gt;
generator = L+2s&lt;br /&gt;
generator = L+2s (240.0 to 257.1 cents)&lt;br /&gt;
notable EDOs: 19, 24, 29, 33, 34, 53&lt;br /&gt;
notable EDOs: 19, 24, 29, 33, 34, 53&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 2,117: Line 2,121:
&lt;a class="wiki_link" href="/Injera"&gt;Injera&lt;/a&gt; MODMOS.  Imperfect (P&amp;lt;0); best for L/s &amp;gt;= 1.5 &lt;!-- ws:start:WikiTextVariableComponentRule:025:[[{$page}#ref1|(1)]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#ref1"&gt;(1)&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:025 --&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/Injera"&gt;Injera&lt;/a&gt; MODMOS.  Imperfect (P&amp;lt;0); best for L/s &amp;gt;= 1.5 &lt;!-- ws:start:WikiTextVariableComponentRule:025:[[{$page}#ref1|(1)]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#ref1"&gt;(1)&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:025 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
generator = L&lt;br /&gt;
generator = L (85.7 to 100.0 cents)&lt;br /&gt;
&lt;a class="wiki_link" href="/periods%20and%20generators"&gt;period&lt;/a&gt; = 1/2 octave&lt;br /&gt;
&lt;a class="wiki_link" href="/periods%20and%20generators"&gt;period&lt;/a&gt; = 1/2 octave&lt;br /&gt;
notable EDOs: 26, 38, 40, 50&lt;br /&gt;
notable EDOs: 26, 38, 40, 50&lt;br /&gt;
Line 2,251: Line 2,255:
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextVariableComponentRule:027:[[{$page}#perfect|P]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;P&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:027 --&gt; = 3.3809207248&lt;br /&gt;
&lt;!-- ws:start:WikiTextVariableComponentRule:027:[[{$page}#perfect|P]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;P&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:027 --&gt; = 3.3809207248&lt;br /&gt;
generator = L+s&lt;br /&gt;
generator = L+s (160.0 to 171.4 cents)&lt;br /&gt;
notable EDOs: 22, 29, 36, 37, 43, 65&lt;br /&gt;
notable EDOs: 22, 29, 36, 37, 43, 65&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 2,390: Line 2,394:
&lt;br /&gt;
&lt;br /&gt;
P = 12.5377181246&lt;br /&gt;
P = 12.5377181246&lt;br /&gt;
generator = L&lt;br /&gt;
generator = L (80.0 to 100.0 cents)&lt;br /&gt;
period = 1/3 octave&lt;br /&gt;
period = 1/3 octave&lt;br /&gt;
notable EDOs: 27, 39, 51&lt;br /&gt;
notable EDOs: 27, 39, 51&lt;br /&gt;
Line 2,531: Line 2,535:
&lt;a class="wiki_link" href="/Superpyth"&gt;Superpyth&lt;/a&gt; / ultrapyth MOS.  Imperfect, best at low L/s.  Many modes may be split two ways.&lt;br /&gt;
&lt;a class="wiki_link" href="/Superpyth"&gt;Superpyth&lt;/a&gt; / ultrapyth MOS.  Imperfect, best at low L/s.  Many modes may be split two ways.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
generator = 2L+5s&lt;br /&gt;
generator = 2L+5s (480.0 to 494.1 cents)&lt;br /&gt;
notable EDOs: 22, 27, 32, 37, 39&lt;br /&gt;
notable EDOs: 22, 27, 32, 37, 39&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 2,683: Line 2,687:
Pythagorean (3-limit JI): L=256/243, s=531441/524288&lt;br /&gt;
Pythagorean (3-limit JI): L=256/243, s=531441/524288&lt;br /&gt;
P = 3.8459060415&lt;br /&gt;
P = 3.8459060415&lt;br /&gt;
generator = 5L+2s&lt;br /&gt;
generator = 5L+2s (494.1 to 500.0 cents)&lt;br /&gt;
notable EDOs: 29, 41, 46, 53, 65&lt;br /&gt;
notable EDOs: 29, 41, 46, 53, 65&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 2,834: Line 2,838:
&lt;br /&gt;
&lt;br /&gt;
P = 1.5872804832&lt;br /&gt;
P = 1.5872804832&lt;br /&gt;
generator = 2L+3s&lt;br /&gt;
generator = 2L+3s (342.9 to 352.9 cents)&lt;br /&gt;
notable EDOs: 24, 31, 41, 65&lt;br /&gt;
notable EDOs: 24, 31, 41, 65&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 2,988: Line 2,992:
Beatles? / &lt;a class="wiki_link" href="/Dicot%20family"&gt;dicot&lt;/a&gt;? MODMOS; imperfect; full L/s range mostly good&lt;br /&gt;
Beatles? / &lt;a class="wiki_link" href="/Dicot%20family"&gt;dicot&lt;/a&gt;? MODMOS; imperfect; full L/s range mostly good&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
generator = 3L+2s&lt;br /&gt;
generator = 3L+2s (352.9 to 360.0 cents)&lt;br /&gt;
notable EDOs: 27, 37, 44, 61&lt;br /&gt;
notable EDOs: 27, 37, 44, 61&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 3,145: Line 3,149:
&lt;br /&gt;
&lt;br /&gt;
P = 5.968201208&lt;br /&gt;
P = 5.968201208&lt;br /&gt;
generator = 2L+3s&lt;br /&gt;
generator = 2L+3s (333.3 to 342.9 cents)&lt;br /&gt;
notable EDOs: 25, 32, 39, 46, 53&lt;br /&gt;
notable EDOs: 25, 32, 39, 46, 53&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 3,309: Line 3,313:
&lt;a class="wiki_link" href="/Meantone%20family#Flattone"&gt;Flattone&lt;/a&gt; MOS; &lt;!-- ws:start:WikiTextVariableComponentRule:031:[[{$page}#perfect|P]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;P&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:031 --&gt;&amp;lt;0; &lt;!-- ws:start:WikiTextVariableComponentRule:033:[[{$page}#ref1|(1)]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#ref1"&gt;(1)&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:033 --&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/Meantone%20family#Flattone"&gt;Flattone&lt;/a&gt; MOS; &lt;!-- ws:start:WikiTextVariableComponentRule:031:[[{$page}#perfect|P]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;P&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:031 --&gt;&amp;lt;0; &lt;!-- ws:start:WikiTextVariableComponentRule:033:[[{$page}#ref1|(1)]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#ref1"&gt;(1)&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:033 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
generator = 3L+5s&lt;br /&gt;
generator = 3L+5s (505.3 to 514.3 cents)&lt;br /&gt;
notable EDOs: 26, 33, 40, 45&lt;br /&gt;
notable EDOs: 26, 33, 40, 45&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 3,475: Line 3,479:
&lt;a class="wiki_link" href="/Meantone"&gt;Meantone&lt;/a&gt; MOS; &lt;!-- ws:start:WikiTextVariableComponentRule:035:[[{$page}#perfect|P]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;P&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:035 --&gt;&amp;lt;0 (oddly, neither this scale nor its &lt;!-- ws:start:WikiTextVariableComponentRule:037:[[{$page}#dual|dual]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#dual"&gt;dual&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:037 --&gt; can be tuned with a perfect 3/2!)  &lt;!-- ws:start:WikiTextVariableComponentRule:039:[[{$page}#ref1|(1)]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#ref1"&gt;(1)&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:039 --&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/Meantone"&gt;Meantone&lt;/a&gt; MOS; &lt;!-- ws:start:WikiTextVariableComponentRule:035:[[{$page}#perfect|P]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#perfect"&gt;P&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:035 --&gt;&amp;lt;0 (oddly, neither this scale nor its &lt;!-- ws:start:WikiTextVariableComponentRule:037:[[{$page}#dual|dual]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#dual"&gt;dual&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:037 --&gt; can be tuned with a perfect 3/2!)  &lt;!-- ws:start:WikiTextVariableComponentRule:039:[[{$page}#ref1|(1)]] --&gt;&lt;a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales#ref1"&gt;(1)&lt;/a&gt;&lt;!-- ws:end:WikiTextVariableComponentRule:039 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
generator = 5L+3s&lt;br /&gt;
generator = 5L+3s (500.0 to 505.3 cents)&lt;br /&gt;
notable EDOs: 31, 43, 50, 55&lt;br /&gt;
notable EDOs: 31, 43, 50, 55&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 3,642: Line 3,646:
&lt;br /&gt;
&lt;br /&gt;
P = 2.5200335053&lt;br /&gt;
P = 2.5200335053&lt;br /&gt;
generator = L+3s&lt;br /&gt;
generator = L+3s (240.0 to 252.6 cents)&lt;br /&gt;
notable EDOs: 24, 29, 34, 39, 43, 53&lt;br /&gt;
notable EDOs: 24, 29, 34, 39, 43, 53&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 3,812: Line 3,816:
&lt;a class="wiki_link" href="/Semaphore%20and%20godzilla"&gt;Semaphore / godzilla&lt;/a&gt; MODMOS; imperfect; full L/s range good&lt;br /&gt;
&lt;a class="wiki_link" href="/Semaphore%20and%20godzilla"&gt;Semaphore / godzilla&lt;/a&gt; MODMOS; imperfect; full L/s range good&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
generator = 3L+s&lt;br /&gt;
generator = 3L+s (252.6 to 257.1 cents)&lt;br /&gt;
notable EDOs: 33, 47, 52&lt;br /&gt;
notable EDOs: 33, 47, 52&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 3,985: Line 3,989:
&lt;br /&gt;
&lt;br /&gt;
P = 2.3809207248&lt;br /&gt;
P = 2.3809207248&lt;br /&gt;
generator = L+2s&lt;br /&gt;
generator = L+2s (163.6 to 171.4 cents)&lt;br /&gt;
notable EDOs: 29, 36, 43, 50, 51&lt;br /&gt;
notable EDOs: 29, 36, 43, 50, 51&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 4,179: Line 4,183:
&lt;a class="wiki_link" href="/Porcupine"&gt;Porcupine&lt;/a&gt; MODMOS; imperfect; full L/s range mostly good&lt;br /&gt;
&lt;a class="wiki_link" href="/Porcupine"&gt;Porcupine&lt;/a&gt; MODMOS; imperfect; full L/s range mostly good&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
generator = 2L+s&lt;br /&gt;
generator = 2L+s (160.0 to 163.6 cents)&lt;br /&gt;
notable EDOs: 37, 52b, 59&lt;br /&gt;
notable EDOs: 37, 52b, 59&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 4,374: Line 4,378:
&lt;br /&gt;
&lt;br /&gt;
P = 7.6918120831&lt;br /&gt;
P = 7.6918120831&lt;br /&gt;
generator = L+s&lt;br /&gt;
generator = L+s (100.0 to 109.1 cents)&lt;br /&gt;
period = 1/2 octave&lt;br /&gt;
period = 1/2 octave&lt;br /&gt;
notable EDOs: 34, 46, 56, 58, 70&lt;br /&gt;
notable EDOs: 34, 46, 56, 58, 70&lt;br /&gt;
Line 4,570: Line 4,574:
&lt;br /&gt;
&lt;br /&gt;
notable EDOs: 32, 42, 54&lt;br /&gt;
notable EDOs: 32, 42, 54&lt;br /&gt;
generator = L+s&lt;br /&gt;
generator = L+s (10.91 to 120.0 cents)&lt;br /&gt;
period = 1/2 octave&lt;br /&gt;
period = 1/2 octave&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 5,160: Line 5,164:
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:132:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc45"&gt;&lt;a name="Definitions and formulas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:132 --&gt;Definitions and formulas&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:132:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc45"&gt;&lt;a name="Definitions and formulas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:132 --&gt;Definitions and formulas&lt;/h1&gt;
&lt;br /&gt;
&lt;strong&gt;&amp;quot;Omnitetrachordal&amp;quot;&lt;/strong&gt; (OTC for short): see &lt;a class="wiki_link" href="/Omnitetrachordality"&gt;Omnitetrachordality&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextAnchorRule:185:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@dual&amp;quot; title=&amp;quot;Anchor: dual&amp;quot;/&amp;gt; --&gt;&lt;a name="dual"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:185 --&gt;&lt;strong&gt;&amp;quot;Dual&amp;quot;&lt;/strong&gt; refers to the &amp;quot;inverse&amp;quot; of a L+s scale pattern, where every L is replaced by s, and vice versa.  For example, &lt;tt&gt; sLssL &lt;/tt&gt; and &lt;tt&gt; LsLLs &lt;/tt&gt; are duals.&lt;br /&gt;
&lt;!-- ws:start:WikiTextAnchorRule:185:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@dual&amp;quot; title=&amp;quot;Anchor: dual&amp;quot;/&amp;gt; --&gt;&lt;a name="dual"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:185 --&gt;&lt;strong&gt;&amp;quot;Dual&amp;quot;&lt;/strong&gt; refers to the &amp;quot;inverse&amp;quot; of a L+s scale pattern, where every L is replaced by s, and vice versa.  For example, &lt;tt&gt; sLssL &lt;/tt&gt; and &lt;tt&gt; LsLLs &lt;/tt&gt; are duals.&lt;br /&gt;
Retrieved from "https://en.xen.wiki/w/Gallery_of_omnitetrachordal_scales"