21edo: Difference between revisions

Wikispaces>genewardsmith
**Imported revision 536153340 - Original comment: **
Wikispaces>JosephRuhf
**Imported revision 545231340 - Original comment: **
Line 1: Line 1:
<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:genewardsmith|genewardsmith]] and made on <tt>2015-01-03 22:03:53 UTC</tt>.<br>
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2015-03-25 00:36:38 UTC</tt>.<br>
: The original revision id was <tt>536153340</tt>.<br>
: The original revision id was <tt>545231340</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>
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The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit [[Laconic Family|laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic, spartan and gorgo temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.
The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit [[Laconic Family|laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic, spartan and gorgo temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.


|| **Degree** || **Cents**
|| **Degree** || **Cents, DMS**
**Value** || **7-EDO**
**Value** || **7-EDO**
**Notation** || **5L3s**
**Notation** || **5L3s**
Line 24: Line 24:
**Ratios *1** ||= &lt;span style="display: block; text-align: center;"&gt;**Approximate****Ratios *2**&lt;/span&gt; ||= &lt;span style="display: block; text-align: center;"&gt;**Approximate**&lt;/span&gt;**Ratios *3** ||
**Ratios *1** ||= &lt;span style="display: block; text-align: center;"&gt;**Approximate****Ratios *2**&lt;/span&gt; ||= &lt;span style="display: block; text-align: center;"&gt;**Approximate**&lt;/span&gt;**Ratios *3** ||
|| 0 || 0 || C || C ||= Unison ||= 1/1 ||= 1/1 ||= 1/1 ||
|| 0 || 0 || C || C ||= Unison ||= 1/1 ||= 1/1 ||= 1/1 ||
|| 1 || 57.143 || C^/Dvv || C# ||= Subminor 2nd ||= 28/27, 30/29 ||= 35/34, 36/35 ||= 64/63 ||
|| 1 || 57.143
|| 2 || 114.286 || C^^/Dv || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 ||
17°&lt;span style="background-color: #ffffff;"&gt;8'34"&lt;/span&gt; || C^/Dvv || C# ||= Subminor 2nd ||= 28/27, 30/29 ||= 35/34, 36/35 ||= 64/63 ||
|| 3 || 171.429 || D || D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||
|| 2 || 114.286
|| 4 || 228.571 || D^/Evv || D# ||= Supermajor 2nd ||= 8/7 ||= 8/7 ||= 8/7, 10/9, 11/10 ||
34°&lt;span style="background-color: #ffffff;"&gt;17'9"&lt;/span&gt; || C^^/Dv || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 ||
|| 5 || 285.714 || D^^/Ev || Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5, 7/6 ||
|| 3 || 171.429
|| 6 || 342.857 || E || E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||
&lt;span style="background-color: #ffffff;"&gt;51°25'43"&lt;/span&gt; || D || D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||
|| 7 || 400 || E^/Fvv || E#/Fb ||= Major 3rd ||= 29/23 ||= 44/35 ||= 5/4, 9/7, 11/9, 14/11 ||
|| 4 || 228.571
|| 8 || 457.143 || E^^/Fv || F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 ||
68°&lt;span style="background-color: #ffffff;"&gt;34'17"&lt;/span&gt; || D^/Evv || D# ||= Supermajor 2nd ||= 8/7 ||= 8/7 ||= 8/7, 10/9, 11/10 ||
|| 9 || 514.286 || F || F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||
|| 5 || 285.714
|| 10 || 571.429 || F^/Gvv || Gb ||= Narrow Tritone ||= 32/23 ||= 18/13 ||= 7/5, 11/8 ||
85°&lt;span style="background-color: #ffffff;"&gt;42'51"&lt;/span&gt; || D^^/Ev || Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5, 7/6 ||
|| 11 || 628.571 || F^^/Gv || G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 ||
|| 6 || 342.857
|| 12 || 685.714 || G || G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||
&lt;span style="background-color: #ffffff;"&gt;102°51'26"&lt;/span&gt; || E || E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||
|| 13 || 742.857 || G^/Avv || Hb ||= Fifth-Sixth ||= 23/15 ||= 17/11, 20/13, 26/17 ||= 20/13 ||
|| 7 || 400
|| 14 || 800 || G^^/Av || H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 11/7, 14/9, 18/11 ||
120° || E^/Fvv || E#/Fb ||= Major 3rd ||= 29/23 ||= 44/35 ||= 5/4, 9/7, 11/9, 14/11 ||
|| 15 || 857.143 || A || H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||
|| 8 || 457.143
|| 16 || 914.286 || A^/Bvv || A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3, 12/7 ||
137°&lt;span style="background-color: #ffffff; line-height: 1.5;"&gt;8'34"&lt;/span&gt; || E^^/Fv || F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 ||
|| 17 || 971.429 || A^^/Bv || A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 ||
|| 9 || 514.286
|| 18 || 1028.571 || B || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||
&lt;span style="background-color: #ffffff;"&gt;154°17'9"&lt;/span&gt; || F || F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||
|| 19 || 1085.714 || B^/Cvv || B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 ||
|| 10 || 571.429
|| 20 || 1142.857 || B^^/Cv || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||= 35/18, 68/35 ||= 63/32 ||
171°&lt;span style="background-color: #ffffff;"&gt;25'43"&lt;/span&gt; || F^/Gvv || Gb ||= Narrow Tritone ||= 32/23 ||= 18/13 ||= 7/5, 11/8 ||
|| 21 || 1200 || C || C ||= Octave ||= 2/1 ||= 2/1 ||= 2/1 ||
|| 11 || 628.571
188°&lt;span style="background-color: #ffffff;"&gt;34'17"&lt;/span&gt; || F^^/Gv || G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 ||
|| 12 || 685.714
&lt;span style="background-color: #ffffff;"&gt;205°42'51"&lt;/span&gt; || G || G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||
|| 13 || 742.857
222°&lt;span style="background-color: #ffffff;"&gt;51'26"&lt;/span&gt; || G^/Avv || Hb ||= Fifth-Sixth ||= 23/15 ||= 17/11, 20/13, 26/17 ||= 20/13 ||
|| 14 || 800
240° || G^^/Av || H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 11/7, 14/9, 18/11 ||
|| 15 || 857.143
&lt;span style="background-color: #ffffff;"&gt;257°8'34"&lt;/span&gt; || A || H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||
|| 16 || 914.286
274°&lt;span style="background-color: #ffffff;"&gt;17'9"&lt;/span&gt; || A^/Bvv || A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3, 12/7 ||
|| 17 || 971.429
391°&lt;span style="background-color: #ffffff;"&gt;25'43"&lt;/span&gt; || A^^/Bv || A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 ||
|| 18 || 1028.571
&lt;span style="background-color: #ffffff;"&gt;308°34'17"&lt;/span&gt; || B || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||
|| 19 || 1085.714
325°&lt;span style="background-color: #ffffff;"&gt;42'51"&lt;/span&gt; || B^/Cvv || B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 ||
|| 20 || 1142.857
342°&lt;span style="background-color: #ffffff;"&gt;8'34"&lt;/span&gt; || B^^/Cv || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||= 35/18, 68/35 ||= 63/32 ||
|| 21 || 1200
360° || C || C ||= Octave ||= 2/1 ||= 2/1 ||= 2/1 ||


*1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament
*1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament
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         &lt;td&gt;&lt;strong&gt;Degree&lt;/strong&gt;&lt;br /&gt;
         &lt;td&gt;&lt;strong&gt;Degree&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;&lt;strong&gt;Cents&lt;/strong&gt;&lt;br /&gt;
         &lt;td&gt;&lt;strong&gt;Cents, DMS&lt;/strong&gt;&lt;br /&gt;
&lt;strong&gt;Value&lt;/strong&gt;&lt;br /&gt;
&lt;strong&gt;Value&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;57.143&lt;br /&gt;
         &lt;td&gt;57.143&lt;br /&gt;
17°&lt;span style="background-color: #ffffff;"&gt;8'34&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;C^/Dvv&lt;br /&gt;
         &lt;td&gt;C^/Dvv&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;114.286&lt;br /&gt;
         &lt;td&gt;114.286&lt;br /&gt;
34°&lt;span style="background-color: #ffffff;"&gt;17'9&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;C^^/Dv&lt;br /&gt;
         &lt;td&gt;C^^/Dv&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;171.429&lt;br /&gt;
         &lt;td&gt;171.429&lt;br /&gt;
&lt;span style="background-color: #ffffff;"&gt;51°25'43&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;D&lt;br /&gt;
         &lt;td&gt;D&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;228.571&lt;br /&gt;
         &lt;td&gt;228.571&lt;br /&gt;
68°&lt;span style="background-color: #ffffff;"&gt;34'17&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;D^/Evv&lt;br /&gt;
         &lt;td&gt;D^/Evv&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;285.714&lt;br /&gt;
         &lt;td&gt;285.714&lt;br /&gt;
85°&lt;span style="background-color: #ffffff;"&gt;42'51&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;D^^/Ev&lt;br /&gt;
         &lt;td&gt;D^^/Ev&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;342.857&lt;br /&gt;
         &lt;td&gt;342.857&lt;br /&gt;
&lt;span style="background-color: #ffffff;"&gt;102°51'26&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;E&lt;br /&gt;
         &lt;td&gt;E&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;400&lt;br /&gt;
         &lt;td&gt;400&lt;br /&gt;
120°&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;E^/Fvv&lt;br /&gt;
         &lt;td&gt;E^/Fvv&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;457.143&lt;br /&gt;
         &lt;td&gt;457.143&lt;br /&gt;
137°&lt;span style="background-color: #ffffff; line-height: 1.5;"&gt;8'34&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;E^^/Fv&lt;br /&gt;
         &lt;td&gt;E^^/Fv&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;514.286&lt;br /&gt;
         &lt;td&gt;514.286&lt;br /&gt;
&lt;span style="background-color: #ffffff;"&gt;154°17'9&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;F&lt;br /&gt;
         &lt;td&gt;F&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;571.429&lt;br /&gt;
         &lt;td&gt;571.429&lt;br /&gt;
171°&lt;span style="background-color: #ffffff;"&gt;25'43&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;F^/Gvv&lt;br /&gt;
         &lt;td&gt;F^/Gvv&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;628.571&lt;br /&gt;
         &lt;td&gt;628.571&lt;br /&gt;
188°&lt;span style="background-color: #ffffff;"&gt;34'17&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;F^^/Gv&lt;br /&gt;
         &lt;td&gt;F^^/Gv&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;685.714&lt;br /&gt;
         &lt;td&gt;685.714&lt;br /&gt;
&lt;span style="background-color: #ffffff;"&gt;205°42'51&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;G&lt;br /&gt;
         &lt;td&gt;G&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;742.857&lt;br /&gt;
         &lt;td&gt;742.857&lt;br /&gt;
222°&lt;span style="background-color: #ffffff;"&gt;51'26&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;G^/Avv&lt;br /&gt;
         &lt;td&gt;G^/Avv&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;800&lt;br /&gt;
         &lt;td&gt;800&lt;br /&gt;
240°&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;G^^/Av&lt;br /&gt;
         &lt;td&gt;G^^/Av&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;857.143&lt;br /&gt;
         &lt;td&gt;857.143&lt;br /&gt;
&lt;span style="background-color: #ffffff;"&gt;257°8'34&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;A&lt;br /&gt;
         &lt;td&gt;A&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;914.286&lt;br /&gt;
         &lt;td&gt;914.286&lt;br /&gt;
274°&lt;span style="background-color: #ffffff;"&gt;17'9&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;A^/Bvv&lt;br /&gt;
         &lt;td&gt;A^/Bvv&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;971.429&lt;br /&gt;
         &lt;td&gt;971.429&lt;br /&gt;
391°&lt;span style="background-color: #ffffff;"&gt;25'43&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;A^^/Bv&lt;br /&gt;
         &lt;td&gt;A^^/Bv&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;1028.571&lt;br /&gt;
         &lt;td&gt;1028.571&lt;br /&gt;
&lt;span style="background-color: #ffffff;"&gt;308°34'17&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;B&lt;br /&gt;
         &lt;td&gt;B&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;1085.714&lt;br /&gt;
         &lt;td&gt;1085.714&lt;br /&gt;
325°&lt;span style="background-color: #ffffff;"&gt;42'51&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;B^/Cvv&lt;br /&gt;
         &lt;td&gt;B^/Cvv&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;1142.857&lt;br /&gt;
         &lt;td&gt;1142.857&lt;br /&gt;
342°&lt;span style="background-color: #ffffff;"&gt;8'34&amp;quot;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;B^^/Cv&lt;br /&gt;
         &lt;td&gt;B^^/Cv&lt;br /&gt;
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;1200&lt;br /&gt;
         &lt;td&gt;1200&lt;br /&gt;
360°&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;C&lt;br /&gt;
         &lt;td&gt;C&lt;br /&gt;