21edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 536153340 - Original comment: ** |
Wikispaces>JosephRuhf **Imported revision 545231340 - 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: | : 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> | : 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** ||= <span style="display: block; text-align: center;">**Approximate****Ratios *2**</span> ||= <span style="display: block; text-align: center;">**Approximate**</span>**Ratios *3** || | **Ratios *1** ||= <span style="display: block; text-align: center;">**Approximate****Ratios *2**</span> ||= <span style="display: block; text-align: center;">**Approximate**</span>**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°<span style="background-color: #ffffff;">8'34"</span> || 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°<span style="background-color: #ffffff;">17'9"</span> || 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 || | <span style="background-color: #ffffff;">51°25'43"</span> || 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°<span style="background-color: #ffffff;">34'17"</span> || 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°<span style="background-color: #ffffff;">42'51"</span> || 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 || | <span style="background-color: #ffffff;">102°51'26"</span> || 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°<span style="background-color: #ffffff; line-height: 1.5;">8'34"</span> || 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 || | <span style="background-color: #ffffff;">154°17'9"</span> || 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°<span style="background-color: #ffffff;">25'43"</span> || 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°<span style="background-color: #ffffff;">34'17"</span> || F^^/Gv || G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 || | |||
|| 12 || 685.714 | |||
<span style="background-color: #ffffff;">205°42'51"</span> || G || G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 || | |||
|| 13 || 742.857 | |||
222°<span style="background-color: #ffffff;">51'26"</span> || 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 | |||
<span style="background-color: #ffffff;">257°8'34"</span> || A || H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 || | |||
|| 16 || 914.286 | |||
274°<span style="background-color: #ffffff;">17'9"</span> || A^/Bvv || A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3, 12/7 || | |||
|| 17 || 971.429 | |||
391°<span style="background-color: #ffffff;">25'43"</span> || A^^/Bv || A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 || | |||
|| 18 || 1028.571 | |||
<span style="background-color: #ffffff;">308°34'17"</span> || B || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 || | |||
|| 19 || 1085.714 | |||
325°<span style="background-color: #ffffff;">42'51"</span> || B^/Cvv || B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 || | |||
|| 20 || 1142.857 | |||
342°<span style="background-color: #ffffff;">8'34"</span> || 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 | ||
| Line 157: | Line 178: | ||
<td><strong>Degree</strong><br /> | <td><strong>Degree</strong><br /> | ||
</td> | </td> | ||
<td><strong>Cents</strong><br /> | <td><strong>Cents, DMS</strong><br /> | ||
<strong>Value</strong><br /> | <strong>Value</strong><br /> | ||
</td> | </td> | ||
| Line 199: | Line 220: | ||
</td> | </td> | ||
<td>57.143<br /> | <td>57.143<br /> | ||
17°<span style="background-color: #ffffff;">8'34&quot;</span><br /> | |||
</td> | </td> | ||
<td>C^/Dvv<br /> | <td>C^/Dvv<br /> | ||
| Line 217: | Line 239: | ||
</td> | </td> | ||
<td>114.286<br /> | <td>114.286<br /> | ||
34°<span style="background-color: #ffffff;">17'9&quot;</span><br /> | |||
</td> | </td> | ||
<td>C^^/Dv<br /> | <td>C^^/Dv<br /> | ||
| Line 235: | Line 258: | ||
</td> | </td> | ||
<td>171.429<br /> | <td>171.429<br /> | ||
<span style="background-color: #ffffff;">51°25'43&quot;</span><br /> | |||
</td> | </td> | ||
<td>D<br /> | <td>D<br /> | ||
| Line 253: | Line 277: | ||
</td> | </td> | ||
<td>228.571<br /> | <td>228.571<br /> | ||
68°<span style="background-color: #ffffff;">34'17&quot;</span><br /> | |||
</td> | </td> | ||
<td>D^/Evv<br /> | <td>D^/Evv<br /> | ||
| Line 271: | Line 296: | ||
</td> | </td> | ||
<td>285.714<br /> | <td>285.714<br /> | ||
85°<span style="background-color: #ffffff;">42'51&quot;</span><br /> | |||
</td> | </td> | ||
<td>D^^/Ev<br /> | <td>D^^/Ev<br /> | ||
| Line 289: | Line 315: | ||
</td> | </td> | ||
<td>342.857<br /> | <td>342.857<br /> | ||
<span style="background-color: #ffffff;">102°51'26&quot;</span><br /> | |||
</td> | </td> | ||
<td>E<br /> | <td>E<br /> | ||
| Line 307: | Line 334: | ||
</td> | </td> | ||
<td>400<br /> | <td>400<br /> | ||
120°<br /> | |||
</td> | </td> | ||
<td>E^/Fvv<br /> | <td>E^/Fvv<br /> | ||
| Line 325: | Line 353: | ||
</td> | </td> | ||
<td>457.143<br /> | <td>457.143<br /> | ||
137°<span style="background-color: #ffffff; line-height: 1.5;">8'34&quot;</span><br /> | |||
</td> | </td> | ||
<td>E^^/Fv<br /> | <td>E^^/Fv<br /> | ||
| Line 343: | Line 372: | ||
</td> | </td> | ||
<td>514.286<br /> | <td>514.286<br /> | ||
<span style="background-color: #ffffff;">154°17'9&quot;</span><br /> | |||
</td> | </td> | ||
<td>F<br /> | <td>F<br /> | ||
| Line 361: | Line 391: | ||
</td> | </td> | ||
<td>571.429<br /> | <td>571.429<br /> | ||
171°<span style="background-color: #ffffff;">25'43&quot;</span><br /> | |||
</td> | </td> | ||
<td>F^/Gvv<br /> | <td>F^/Gvv<br /> | ||
| Line 379: | Line 410: | ||
</td> | </td> | ||
<td>628.571<br /> | <td>628.571<br /> | ||
188°<span style="background-color: #ffffff;">34'17&quot;</span><br /> | |||
</td> | </td> | ||
<td>F^^/Gv<br /> | <td>F^^/Gv<br /> | ||
| Line 397: | Line 429: | ||
</td> | </td> | ||
<td>685.714<br /> | <td>685.714<br /> | ||
<span style="background-color: #ffffff;">205°42'51&quot;</span><br /> | |||
</td> | </td> | ||
<td>G<br /> | <td>G<br /> | ||
| Line 415: | Line 448: | ||
</td> | </td> | ||
<td>742.857<br /> | <td>742.857<br /> | ||
222°<span style="background-color: #ffffff;">51'26&quot;</span><br /> | |||
</td> | </td> | ||
<td>G^/Avv<br /> | <td>G^/Avv<br /> | ||
| Line 433: | Line 467: | ||
</td> | </td> | ||
<td>800<br /> | <td>800<br /> | ||
240°<br /> | |||
</td> | </td> | ||
<td>G^^/Av<br /> | <td>G^^/Av<br /> | ||
| Line 451: | Line 486: | ||
</td> | </td> | ||
<td>857.143<br /> | <td>857.143<br /> | ||
<span style="background-color: #ffffff;">257°8'34&quot;</span><br /> | |||
</td> | </td> | ||
<td>A<br /> | <td>A<br /> | ||
| Line 469: | Line 505: | ||
</td> | </td> | ||
<td>914.286<br /> | <td>914.286<br /> | ||
274°<span style="background-color: #ffffff;">17'9&quot;</span><br /> | |||
</td> | </td> | ||
<td>A^/Bvv<br /> | <td>A^/Bvv<br /> | ||
| Line 487: | Line 524: | ||
</td> | </td> | ||
<td>971.429<br /> | <td>971.429<br /> | ||
391°<span style="background-color: #ffffff;">25'43&quot;</span><br /> | |||
</td> | </td> | ||
<td>A^^/Bv<br /> | <td>A^^/Bv<br /> | ||
| Line 505: | Line 543: | ||
</td> | </td> | ||
<td>1028.571<br /> | <td>1028.571<br /> | ||
<span style="background-color: #ffffff;">308°34'17&quot;</span><br /> | |||
</td> | </td> | ||
<td>B<br /> | <td>B<br /> | ||
| Line 523: | Line 562: | ||
</td> | </td> | ||
<td>1085.714<br /> | <td>1085.714<br /> | ||
325°<span style="background-color: #ffffff;">42'51&quot;</span><br /> | |||
</td> | </td> | ||
<td>B^/Cvv<br /> | <td>B^/Cvv<br /> | ||
| Line 541: | Line 581: | ||
</td> | </td> | ||
<td>1142.857<br /> | <td>1142.857<br /> | ||
342°<span style="background-color: #ffffff;">8'34&quot;</span><br /> | |||
</td> | </td> | ||
<td>B^^/Cv<br /> | <td>B^^/Cv<br /> | ||
| Line 559: | Line 600: | ||
</td> | </td> | ||
<td>1200<br /> | <td>1200<br /> | ||
360°<br /> | |||
</td> | </td> | ||
<td>C<br /> | <td>C<br /> | ||