62edo: Difference between revisions

Revision as of 20:34, 27 May 2012

IMPORTED REVISION FROM WIKISPACES

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This revision was by author Osmiorisbendi and made on 2012-05-27 20:34:06 UTC.

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Original Wikitext content:

=<span style="color: #790080; font-family: 'Times New Roman',Times,serif; font-size: 113%;">62 tone equal temperament</span>= 

62edo divides the octave into 62 equal parts of 19.35484 cents each. 62 = 2 * 31 and the patent val is a contorted 31edo through the 11-limit; in the 13-limit it tempers out 169/168, 1188/1183, 847/845 and 676/675. It provides the optimal patent val for [[31 comma temperaments#Gallium|gallium]], [[Starling temperaments#Valentine%20temperament-Semivalentine|semivalentine]] and [[Meantone family#Septimal%20meantone-Unidecimal%20meantone%20aka%20Huygens-Hemimeantone|hemimeantone]] temperaments.

It is also strong as an 1/8-tone [[Armodue-Hornbostel]] system, with the 6th being 35 steps. However, 31 is a "false" quarter-tone system with respect to the same temperament since the Armodue-Hornbostel whole tone is simplest when achieved by a chain of 5 of the septimal minor third at 7\31, which is a good generator for [[Orwell]]. This makes 8\62 an Orwell whole tone as well, so 62 is twice an 1/8-tone system; but this is no real surprise since 1/8-tone systems will have 40 to 80 divisions in the octave as a matter of course.

===**62-EDO Intervals**=== 

|| **ARMODUE NOMENCLATURE 8;3 RELATION** ||
|| * **Ɨ** = Thick (1/8-tone up)
* **‡** = Semisharp (1/4-tone up)
* **b** = Flat (5/8-tone down)
* **◊** = Node (blindspot sharp/flat 1/2-tone)
* **#** = Sharp (5/8-tone up)
* **v** = Semiflat (1/4-tone down)
* **⌐** = Thin (1/8-tone down) ||
|| Degrees || Cents size || Armodue notation ||   ||
|| 0 || 0 || 1 ||   ||
|| 1 || 19.35484 || 1Ɨ ||   ||
|| 2 || 38.70968 || 1‡ (9#) ||   ||
|| 3 || 58.06452 || 2b ||   ||
|| 4 || 77.41935 || 1◊2 ||   ||
|| 5 || 96.77419 || 1# ||   ||
|| 6 || 116.12903 || 2v ||   ||
|| 7 || 135.48387 || 2⌐ ||   ||
|| 8 || 154.83871 || 2 ||   ||
|| 9 || 174.19355 || 2Ɨ ||   ||
|| 10 || 193.54839 || 2‡ ||   ||
|| 11 || 212.90323 || 3b ||   ||
|| 12 || 232.25806 || 2◊3 ||   ||
|| 13 || 251.6129 || 2# ||   ||
|| 14 || 270.96774 || 3v ||   ||
|| 15 || 290.32258 || 3⌐ ||   ||
|| 16 || 309.67742 || 3 ||   ||
|| 17 || 329.03226 || 3Ɨ ||   ||
|| 18 || 348.3871 || 3‡ ||   ||
|| 19 || 367.74194 || 4b ||   ||
|| 20 || 387.09677 || 3◊4 ||   ||
|| 21 || 406.45161 || 3# ||   ||
|| 22 || 425.80645 || 4v (5b) ||   ||
|| 23 || 445.16129 || 4⌐ ||   ||
|| 24 || 464.51613 || 4 ||   ||
|| 25 || 483.87097 || 4Ɨ (5v) ||   ||
|| 26 || 503.22581 || 5⌐ (4‡) ||   ||
|| 27 || 522.58065 || 5 ||   ||
|| 28 || 541.93548 || 5Ɨ ||   ||
|| 29 || 561.29032 || 5‡ (4#) ||   ||
|| 30 || 580.64516 || 6b ||   ||
|| 31 || 600 || 5◊6 ||   ||
|| 32 || 619.35484 || 5# ||   ||
|| 33 || 638.70968 || 6v ||   ||
|| 34 || 658.06452 || 6⌐ ||   ||
|| 35 || 677.41935 || 6 ||   ||
|| 36 || 696.77419 || 6Ɨ ||   ||
|| 37 || 716.12903 || 6‡ ||   ||
|| 38 || 735.48387 || 7b ||   ||
|| 39 || 754.83871 || 6◊7 ||   ||
|| 40 || 774.19355 || 6# ||   ||
|| 41 || 793.54839 || 7v ||   ||
|| 42 || 812.90323 || 7⌐ ||   ||
|| 43 || 832.25806 || 7 ||   ||
|| 44 || 851.6129 || 7Ɨ ||   ||
|| 45 || 870.96774 || 7‡ ||   ||
|| 46 || 890.32258 || 8b ||   ||
|| 47 || 909.67742 || 7◊8 ||   ||
|| 48 || 929.03226 || 7# ||   ||
|| 49 || 948.3871 || 8v ||   ||
|| 50 || 967.74194 || 8⌐ ||   ||
|| 51 || 987.09677 || 8 ||   ||
|| 52 || 1006.45161 || 8Ɨ ||   ||
|| 53 || 1025.80645 || 8‡ ||   ||
|| 54 || 1045.16129 || 9b ||   ||
|| 55 || 1064.51613 || 8◊9 ||   ||
|| 56 || 1083.87097 || 8# ||   ||
|| 57 || 1103.22581 || 9v (1b) ||   ||
|| 58 || 1122.58065 || 9⌐ ||   ||
|| 59 || 1141.93548 || 9 ||   ||
|| 60 || 1161.29032 || 9Ɨ (1v) ||   ||
|| 61 || 1180.64516 || 1⌐ (9‡) ||   ||

Original HTML content:

<html><head><title>62edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x62 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #790080; font-family: 'Times New Roman',Times,serif; font-size: 113%;">62 tone equal temperament</span></h1>
 <br />
62edo divides the octave into 62 equal parts of 19.35484 cents each. 62 = 2 * 31 and the patent val is a contorted 31edo through the 11-limit; in the 13-limit it tempers out 169/168, 1188/1183, 847/845 and 676/675. It provides the optimal patent val for <a class="wiki_link" href="/31%20comma%20temperaments#Gallium">gallium</a>, <a class="wiki_link" href="/Starling%20temperaments#Valentine%20temperament-Semivalentine">semivalentine</a> and <a class="wiki_link" href="/Meantone%20family#Septimal%20meantone-Unidecimal%20meantone%20aka%20Huygens-Hemimeantone">hemimeantone</a> temperaments.<br />
<br />
It is also strong as an 1/8-tone <a class="wiki_link" href="/Armodue-Hornbostel">Armodue-Hornbostel</a> system, with the 6th being 35 steps. However, 31 is a &quot;false&quot; quarter-tone system with respect to the same temperament since the Armodue-Hornbostel whole tone is simplest when achieved by a chain of 5 of the septimal minor third at 7\31, which is a good generator for <a class="wiki_link" href="/Orwell">Orwell</a>. This makes 8\62 an Orwell whole tone as well, so 62 is twice an 1/8-tone system; but this is no real surprise since 1/8-tone systems will have 40 to 80 divisions in the octave as a matter of course.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x62 tone equal temperament--62-EDO Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 --><strong>62-EDO Intervals</strong></h3>
 <br />


<table class="wiki_table">
    <tr>
        <td><strong>ARMODUE NOMENCLATURE 8;3 RELATION</strong><br />
</td>
    </tr>
    <tr>
        <td><ul><li><strong>Ɨ</strong> = Thick (1/8-tone up)</li><li><strong>‡</strong> = Semisharp (1/4-tone up)</li><li><strong>b</strong> = Flat (5/8-tone down)</li><li><strong>◊</strong> = Node (blindspot sharp/flat 1/2-tone)</li><li><strong>#</strong> = Sharp (5/8-tone up)</li><li><strong>v</strong> = Semiflat (1/4-tone down)</li><li><strong>⌐</strong> = Thin (1/8-tone down)</li></ul></td>
    </tr>
    <tr>
        <td>Degrees<br />
</td>
        <td>Cents size<br />
</td>
        <td>Armodue notation<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td>1<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>19.35484<br />
</td>
        <td>1Ɨ<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>38.70968<br />
</td>
        <td>1‡ (9#)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>58.06452<br />
</td>
        <td>2b<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>77.41935<br />
</td>
        <td>1◊2<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>96.77419<br />
</td>
        <td>1#<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>116.12903<br />
</td>
        <td>2v<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>135.48387<br />
</td>
        <td>2⌐<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>154.83871<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>174.19355<br />
</td>
        <td>2Ɨ<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>193.54839<br />
</td>
        <td>2‡<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>212.90323<br />
</td>
        <td>3b<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>232.25806<br />
</td>
        <td>2◊3<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>251.6129<br />
</td>
        <td>2#<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>270.96774<br />
</td>
        <td>3v<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>290.32258<br />
</td>
        <td>3⌐<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>309.67742<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>329.03226<br />
</td>
        <td>3Ɨ<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>348.3871<br />
</td>
        <td>3‡<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>367.74194<br />
</td>
        <td>4b<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>387.09677<br />
</td>
        <td>3◊4<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>406.45161<br />
</td>
        <td>3#<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>425.80645<br />
</td>
        <td>4v (5b)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>445.16129<br />
</td>
        <td>4⌐<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>464.51613<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>483.87097<br />
</td>
        <td>4Ɨ (5v)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>503.22581<br />
</td>
        <td>5⌐ (4‡)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>522.58065<br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>541.93548<br />
</td>
        <td>5Ɨ<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>29<br />
</td>
        <td>561.29032<br />
</td>
        <td>5‡ (4#)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>30<br />
</td>
        <td>580.64516<br />
</td>
        <td>6b<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>31<br />
</td>
        <td>600<br />
</td>
        <td>5◊6<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>32<br />
</td>
        <td>619.35484<br />
</td>
        <td>5#<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>33<br />
</td>
        <td>638.70968<br />
</td>
        <td>6v<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>34<br />
</td>
        <td>658.06452<br />
</td>
        <td>6⌐<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>35<br />
</td>
        <td>677.41935<br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>36<br />
</td>
        <td>696.77419<br />
</td>
        <td>6Ɨ<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>37<br />
</td>
        <td>716.12903<br />
</td>
        <td>6‡<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>38<br />
</td>
        <td>735.48387<br />
</td>
        <td>7b<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>39<br />
</td>
        <td>754.83871<br />
</td>
        <td>6◊7<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>40<br />
</td>
        <td>774.19355<br />
</td>
        <td>6#<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>41<br />
</td>
        <td>793.54839<br />
</td>
        <td>7v<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>42<br />
</td>
        <td>812.90323<br />
</td>
        <td>7⌐<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>43<br />
</td>
        <td>832.25806<br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>44<br />
</td>
        <td>851.6129<br />
</td>
        <td>7Ɨ<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>45<br />
</td>
        <td>870.96774<br />
</td>
        <td>7‡<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>46<br />
</td>
        <td>890.32258<br />
</td>
        <td>8b<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>47<br />
</td>
        <td>909.67742<br />
</td>
        <td>7◊8<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>48<br />
</td>
        <td>929.03226<br />
</td>
        <td>7#<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>49<br />
</td>
        <td>948.3871<br />
</td>
        <td>8v<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>50<br />
</td>
        <td>967.74194<br />
</td>
        <td>8⌐<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>51<br />
</td>
        <td>987.09677<br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>52<br />
</td>
        <td>1006.45161<br />
</td>
        <td>8Ɨ<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>53<br />
</td>
        <td>1025.80645<br />
</td>
        <td>8‡<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>54<br />
</td>
        <td>1045.16129<br />
</td>
        <td>9b<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>55<br />
</td>
        <td>1064.51613<br />
</td>
        <td>8◊9<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>56<br />
</td>
        <td>1083.87097<br />
</td>
        <td>8#<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>57<br />
</td>
        <td>1103.22581<br />
</td>
        <td>9v (1b)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>58<br />
</td>
        <td>1122.58065<br />
</td>
        <td>9⌐<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>59<br />
</td>
        <td>1141.93548<br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>60<br />
</td>
        <td>1161.29032<br />
</td>
        <td>9Ɨ (1v)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>61<br />
</td>
        <td>1180.64516<br />
</td>
        <td>1⌐ (9‡)<br />
</td>
        <td><br />
</td>
    </tr>
</table>

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62edo: Difference between revisions

Revision as of 20:34, 27 May 2012

IMPORTED REVISION FROM WIKISPACES

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Original HTML content:

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