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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Infobox ET}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | {{ED intro}} |
| : This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2012-05-27 20:34:06 UTC</tt>.<br>
| |
| : The original revision id was <tt>339937392</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">=<span style="color: #790080; font-family: 'Times New Roman',Times,serif; font-size: 113%;">62 tone equal temperament</span>=
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| |
|
| 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.
| | == Theory == |
| | {{Nowrap| 62 {{=}} 2 × 31 }} and the [[patent val]] of 62edo is a [[contorsion|contorted]] [[31edo]] through the [[11-limit]], but it makes for a good tuning in the higher limits. In the 13-limit it [[tempering out|tempers out]] [[169/168]], [[1188/1183]], [[847/845]] and [[676/675]]; in the [[17-limit]] [[221/220]], [[273/272]], and [[289/288]]; in the [[19-limit]] [[153/152]], [[171/170]], [[209/208]], [[286/285]], and [[361/360]]. Unlike 31edo, which has a sharp profile for primes [[13/1|13]], [[17/1|17]], [[19/1|19]] and [[23/1|23]], 62edo has a flat profile for these, as it removes the distinction of otonal and utonal [[superparticular]] pairs of the primes (e.g. 13/12 vs 14/13 for prime 13) by tempering out the corresponding [[square-particular]]s. This flat tendency extends to higher primes too, as the first prime harmonic that is tuned sharper than its [[5/4]] is its [[59/32]]. Interestingly, the size differences between consecutive harmonics are monotonically decreasing for all first 24 harmonics, and 62edo is one of the few [[meantone]] edos that achieve this, great for those who seek higher-limit meantone harmony. |
|
| |
|
| 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. | | It provides the [[optimal patent val]] for [[gallium]], [[semivalentine]] and [[hemimeantone]] temperaments. |
|
| |
|
| ===**62-EDO Intervals**===
| | Using the 35\62 generator, which leads to the {{val| 62 97 143 173 }} val, 62edo is also an excellent tuning for septimal [[mavila]] temperament; alternatively {{val| 62 97 143 172 }} [[support]]s [[hornbostel]]. |
|
| |
|
| || **ARMODUE NOMENCLATURE 8;3 RELATION** ||
| | === Odd harmonics === |
| || * **Ɨ** = Thick (1/8-tone up)
| | {{Harmonics in equal|62}} |
| * **‡** = 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‡) || ||</pre></div>
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| <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>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 />
| |
|
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|
| | === Subsets and supersets === |
| | Since 62 factors into 2 × 31, 62edo does not contain nontrivial subset edos other than [[2edo]] and 31edo. [[186edo]] and [[248edo]] are notable supersets. |
|
| |
|
| <table class="wiki_table">
| | === Miscellany === |
| <tr>
| | 62 years is the amount of years in a leap week calendar cycle which corresponds to a year of 365 days 5 hours 48 minutes 23 seconds, meaning it is both a simple cycle for a calendar, and 62 being a multiple of 31 makes it a harmonically useful and playable cycle. The corresponding maximal evenness scales are 15 & 62 and 11 & 62. |
| <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>
| |
|
| |
|
| </body></html></pre></div> | | The 11 & 62 temperament is called mabon, named so because its associated year length corresponds to an autumnal equinoctial year. In the 2.9.7 subgroup tempers out 44957696/43046721, and the three generators of 17\62 correspond to [[16/9]]. It is possible to extend this to the 11-limit with comma basis {896/891, 1331/1296}, where 17\62 is mapped to [[11/9]] and two of them make [[16/11]]. In addition, three generators make the patent val 9/8, which is also created by combining the flat patent val fifth from 31edo with the sharp 37\62 fifth. |
| | |
| | The 15 & 62 temperament, corresponding to the leap day cycle, is [[demivalentine]] in the 13-limit. |
| | |
| | == Intervals == |
| | {| class="wikitable center-all right-2 left-3" |
| | |- |
| | ! Steps |
| | ! Cents |
| | ! Approximate ratios* |
| | ! [[Ups and downs notation]] |
| | |- |
| | | 0 |
| | | 0.00 |
| | | 1/1 |
| | | {{UDnote|step=0}} |
| | |- |
| | | 1 |
| | | 19.35 |
| | | 65/64, 66/65, 78/77, 91/90, 105/104 |
| | | {{UDnote|step=1}} |
| | |- |
| | | 2 |
| | | 38.71 |
| | | ''33/32'', 36/35, 45/44, 49/48, 50/49, 55/54, 56/55, ''64/63'' |
| | | {{UDnote|step=2}} |
| | |- |
| | | 3 |
| | | 58.06 |
| | | ''26/25'', 27/26 |
| | | {{UDnote|step=3}} |
| | |- |
| | | 4 |
| | | 77.42 |
| | | 21/20, 22/21, 23/22, 24/23, 25/24, ''28/27'' |
| | | {{UDnote|step=4}} |
| | |- |
| | | 5 |
| | | 96.77 |
| | | 17/16, 18/17, 19/18, 20/19 |
| | | {{UDnote|step=5}} |
| | |- |
| | | 6 |
| | | 116.13 |
| | | 15/14, 16/15 |
| | | {{UDnote|step=6}} |
| | |- |
| | | 7 |
| | | 135.48 |
| | | 13/12, 14/13 |
| | | {{UDnote|step=7}} |
| | |- |
| | | 8 |
| | | 154.84 |
| | | ''11/10'', 12/11, 23/21 |
| | | {{UDnote|step=8}} |
| | |- |
| | | 9 |
| | | 174.19 |
| | | 21/19 |
| | | {{UDnote|step=9}} |
| | |- |
| | | 10 |
| | | 193.55 |
| | | ''9/8'', ''10/9'', 19/17, 28/25 |
| | | {{UDnote|step=10}} |
| | |- |
| | | 11 |
| | | 212.90 |
| | | 17/15 |
| | | {{UDnote|step=11}} |
| | |- |
| | | 12 |
| | | 232.26 |
| | | 8/7 |
| | | {{UDnote|step=12}} |
| | |- |
| | | 13 |
| | | 251.61 |
| | | 15/13, 22/19 |
| | | {{UDnote|step=13}} |
| | |- |
| | | 14 |
| | | 270.97 |
| | | 7/6 |
| | | {{UDnote|step=14}} |
| | |- |
| | | 15 |
| | | 290.32 |
| | | 13/11, 19/16, 20/17 |
| | | {{UDnote|step=15}} |
| | |- |
| | | 16 |
| | | 309.68 |
| | | 6/5 |
| | | {{UDnote|step=16}} |
| | |- |
| | | 17 |
| | | 329.03 |
| | | 17/14, 23/19 |
| | | {{UDnote|step=18}} |
| | |- |
| | | 18 |
| | | 348.39 |
| | | 11/9, 27/22, 28/23 |
| | | {{UDnote|step=18}} |
| | |- |
| | | 19 |
| | | 367.74 |
| | | 16/13, 21/17, 26/21 |
| | | {{UDnote|step=19}} |
| | |- |
| | | 20 |
| | | 387.10 |
| | | 5/4 |
| | | {{UDnote|step=20}} |
| | |- |
| | | 21 |
| | | 406.45 |
| | | 19/15, 24/19 |
| | | {{UDnote|step=21}} |
| | |- |
| | | 22 |
| | | 425.81 |
| | | 9/7, 14/11, 23/18, 32/25 |
| | | {{UDnote|step=22}} |
| | |- |
| | | 23 |
| | | 445.16 |
| | | 13/10, 22/17 |
| | | {{UDnote|step=23}} |
| | |- |
| | | 24 |
| | | 464.52 |
| | | 17/13, 21/16, 30/23 |
| | | {{UDnote|step=24}} |
| | |- |
| | | 25 |
| | | 483.87 |
| | | 25/19 |
| | | {{UDnote|step=25}} |
| | |- |
| | | 26 |
| | | 503.23 |
| | | 4/3 |
| | | {{UDnote|step=26}} |
| | |- |
| | | 27 |
| | | 522.58 |
| | | 19/14, 23/17 |
| | | {{UDnote|step=27}} |
| | |- |
| | | 28 |
| | | 541.94 |
| | | 11/8, 15/11, 26/19 |
| | | {{UDnote|step=28}} |
| | |- |
| | | 29 |
| | | 561.29 |
| | | 18/13 |
| | | {{UDnote|step=29}} |
| | |- |
| | | 30 |
| | | 580.65 |
| | | 7/5, ''25/18'', 32/23 |
| | | {{UDnote|step=30}} |
| | |- |
| | | 31 |
| | | 600.00 |
| | | 17/12, 24/17 |
| | | {{UDnote|step=10}} |
| | |- |
| | | 32 |
| | | 619.35 |
| | | 10/7, 23/16, ''36/25'' |
| | | {{UDnote|step=32}} |
| | |- |
| | | 33 |
| | | 638.71 |
| | | 13/9 |
| | | {{UDnote|step=33}} |
| | |- |
| | | 34 |
| | | 658.06 |
| | | 16/11, 19/13, 22/15 |
| | | {{UDnote|step=34}} |
| | |- |
| | | 35 |
| | | 677.42 |
| | | 28/19, 34/23 |
| | | {{UDnote|step=35}} |
| | |- |
| | | 36 |
| | | 696.77 |
| | | 3/2 |
| | | {{UDnote|step=36}} |
| | |- |
| | | 37 |
| | | 716.13 |
| | | 38/25 |
| | | {{UDnote|step=37}} |
| | |- |
| | | 38 |
| | | 735.48 |
| | | 23/15, 26/17, 32/21 |
| | | {{UDnote|step=38}} |
| | |- |
| | | 39 |
| | | 754.84 |
| | | 17/11, 20/13 |
| | | {{UDnote|step=39}} |
| | |- |
| | | 40 |
| | | 774.19 |
| | | 11/7, 14/9, 25/16, 36/23 |
| | | {{UDnote|step=40}} |
| | |- |
| | | 41 |
| | | 793.55 |
| | | 19/12, 30/19 |
| | | {{UDnote|step=41}} |
| | |- |
| | | 42 |
| | | 812.90 |
| | | 8/5 |
| | | {{UDnote|step=42}} |
| | |- |
| | | 43 |
| | | 832.26 |
| | | 13/8, 21/13, 34/21 |
| | | {{UDnote|step=43}} |
| | |- |
| | | 44 |
| | | 851.61 |
| | | 18/11, 23/14, 44/27 |
| | | {{UDnote|step=44}} |
| | |- |
| | | 45 |
| | | 870.97 |
| | | 28/17, 38/23 |
| | | {{UDnote|step=45}} |
| | |- |
| | | 46 |
| | | 890.32 |
| | | 5/3 |
| | | {{UDnote|step=46}} |
| | |- |
| | | 47 |
| | | 909.68 |
| | | 17/10, 22/13, 32/19 |
| | | {{UDnote|step=47}} |
| | |- |
| | | 48 |
| | | 929.03 |
| | | 12/7 |
| | | {{UDnote|step=48}} |
| | |- |
| | | 49 |
| | | 948.39 |
| | | 19/11, 26/15 |
| | | {{UDnote|step=49}} |
| | |- |
| | | 50 |
| | | 967.74 |
| | | 7/4 |
| | | {{UDnote|step=50}} |
| | |- |
| | | 51 |
| | | 987.10 |
| | | 30/17 |
| | | {{UDnote|step=51}} |
| | |- |
| | | 52 |
| | | 1006.45 |
| | | ''9/5'', ''16/9'', 25/14, 34/19 |
| | | {{UDnote|step=52}} |
| | |- |
| | | 53 |
| | | 1025.81 |
| | | 38/21 |
| | | {{UDnote|step=53}} |
| | |- |
| | | 54 |
| | | 1045.16 |
| | | 11/6, ''20/11'', 42/23 |
| | | {{UDnote|step=54}} |
| | |- |
| | | 55 |
| | | 1064.52 |
| | | 13/7, 24/13 |
| | | {{UDnote|step=55}} |
| | |- |
| | | 56 |
| | | 1083.87 |
| | | 15/8, 28/15 |
| | | {{UDnote|step=56}} |
| | |- |
| | | 57 |
| | | 1103.23 |
| | | 17/9, 19/10, 32/17, 36/19 |
| | | {{UDnote|step=57}} |
| | |- |
| | | 58 |
| | | 1122.58 |
| | | 21/11, 23/12, ''27/14'', 40/21, 44/23, 48/25 |
| | | {{UDnote|step=58}} |
| | |- |
| | | 59 |
| | | 1141.94 |
| | | ''25/13'', 52/27 |
| | | {{UDnote|step=59}} |
| | |- |
| | | 60 |
| | | 1161.29 |
| | | 35/18, 49/25, 55/28, ''63/32'', ''64/33'', 88/45, 96/49, 108/55 |
| | | {{UDnote|step=60}} |
| | |- |
| | | 61 |
| | | 1180.65 |
| | | 65/33, 77/39, 128/65, 180/91, 208/105 |
| | | {{UDnote|step=61}} |
| | |- |
| | | 62 |
| | | 1200.00 |
| | | 2/1 |
| | | {{UDnote|step=62}} |
| | |} |
| | <nowiki />* 23-limit patent val, inconsistent intervals in ''italic'' |
| | |
| | == Notation == |
| | === Ups and downs notation === |
| | 62edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat. |
| | {{Sharpness-sharp4a}} |
| | [[Alternative symbols for ups and downs notation]] uses sharps and flats and quarter-tone accidentals combined with arrows, borrowed from extended [[Helmholtz–Ellis notation]]: |
| | {{Sharpness-sharp4}} |
| | === Sagittal notation === |
| | This notation uses the same sagittal sequence as EDOs [[69edo#Sagittal notation|69]] and [[76edo#Sagittal notation|76]], and is a superset of the notation for [[31edo#Sagittal notation|31-EDO]]. |
| | |
| | ==== Evo flavor ==== |
| | <imagemap> |
| | File:62-EDO_Evo_Sagittal.svg |
| | desc none |
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 170 106 [[1053/1024]] |
| | rect 170 80 290 106 [[33/32]] |
| | default [[File:62-EDO_Evo_Sagittal.svg]] |
| | </imagemap> |
| | |
| | ==== Revo flavor ==== |
| | <imagemap> |
| | File:62-EDO_Revo_Sagittal.svg |
| | desc none |
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | rect 300 0 687 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 170 106 [[1053/1024]] |
| | rect 170 80 290 106 [[33/32]] |
| | default [[File:62-EDO_Revo_Sagittal.svg]] |
| | </imagemap> |
| | |
| | ==== Evo-SZ flavor ==== |
| | <imagemap> |
| | File:62-EDO_Evo-SZ_Sagittal.svg |
| | desc none |
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 170 106 [[1053/1024]] |
| | rect 170 80 290 106 [[33/32]] |
| | default [[File:62-EDO_Evo-SZ_Sagittal.svg]] |
| | </imagemap> |
| | |
| | In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO. |
| | |
| | === Armodue notation === |
| | ; Armodue nomenclature 8;3 relation |
| | * '''Ɨ''' = Thick (1/8-tone up) |
| | * '''‡''' = Semisharp (1/4-tone up) |
| | * '''b''' = Flat (5/8-tone down) |
| | * '''◊''' = Node (sharp/flat blindspot 1/2-tone) |
| | * '''#''' = Sharp (5/8-tone up) |
| | * '''v''' = Semiflat (1/4-tone down) |
| | * '''⌐''' = Thin (1/8-tone down) |
| | |
| | {| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed" |
| | |- |
| | ! colspan="2" | # |
| | ! Cents |
| | ! Armodue notation |
| | ! Associated ratio |
| | |- |
| | | 0 |
| | | |
| | | 0.0 |
| | | 1 |
| | | |
| | |- |
| | | 1 |
| | | |
| | | 19.4 |
| | | 1Ɨ |
| | | |
| | |- |
| | | 2 |
| | | |
| | | 38.7 |
| | | 1‡ (9#) |
| | | |
| | |- |
| | | 3 |
| | | |
| | | 58.1 |
| | | 2b |
| | | |
| | |- |
| | | 4 |
| | | |
| | | 77.4 |
| | | 1◊2 |
| | | |
| | |- |
| | | 5 |
| | | |
| | | 96.8 |
| | | 1# |
| | | |
| | |- |
| | | 6 |
| | | |
| | | 116.1 |
| | | 2v |
| | | |
| | |- |
| | | 7 |
| | | |
| | | 135.5 |
| | | 2⌐ |
| | | |
| | |- |
| | | 8 |
| | | |
| | | 154.8 |
| | | 2 |
| | | 11/10~12/11 |
| | |- |
| | | 9 |
| | | |
| | | 174.2 |
| | | 2Ɨ |
| | | |
| | |- |
| | | 10 |
| | | |
| | | 193.5 |
| | | 2‡ |
| | | |
| | |- |
| | | 11 |
| | | |
| | | 212.9 |
| | | 3b |
| | | 8/7 |
| | |- |
| | | 12 |
| | | |
| | | 232.3 |
| | | 2◊3 |
| | | |
| | |- |
| | | 13 |
| | | |
| | | 251.6 |
| | | 2# |
| | | |
| | |- |
| | | 14 |
| | | |
| | | 271.0 |
| | | 3v |
| | | |
| | |- |
| | | 15 |
| | | |
| | | 290.3 |
| | | 3⌐ |
| | | |
| | |- |
| | | 16 |
| | | |
| | | 309.7 |
| | | 3 |
| | | 6/5~7/6 |
| | |- |
| | | 17 |
| | | |
| | | 329.0 |
| | | 3Ɨ |
| | | |
| | |- |
| | | 18 |
| | | |
| | | 348.4 |
| | | 3‡ |
| | | |
| | |- |
| | | 19 |
| | | · |
| | | 367.7 |
| | | 4b |
| | | 5/4 |
| | |- |
| | | 20 |
| | | |
| | | 387.1 |
| | | 3◊4 |
| | | |
| | |- |
| | | 21 |
| | | |
| | | 406.5 |
| | | 3# |
| | | |
| | |- |
| | | 22 |
| | | |
| | | 425.8 |
| | | 4v (5b) |
| | | |
| | |- |
| | | 23 |
| | | |
| | | 445.2 |
| | | 4⌐ |
| | | |
| | |- |
| | | 24 |
| | | |
| | | 464.5 |
| | | 4 |
| | | |
| | |- |
| | | 25 |
| | | |
| | | 483.9 |
| | | 4Ɨ (5v) |
| | | |
| | |- |
| | | 26 |
| | | |
| | | 503.2 |
| | | 5⌐ (4‡) |
| | | |
| | |- |
| | | 27 |
| | | · |
| | | 522.6 |
| | | 5 |
| | | 4/3~11/8 |
| | |- |
| | | 28 |
| | | |
| | | 541.9 |
| | | 5Ɨ |
| | | |
| | |- |
| | | 29 |
| | | |
| | | 561.3 |
| | | 5‡ (4#) |
| | | |
| | |- |
| | | 30 |
| | | |
| | | 580.6 |
| | | 6b |
| | | 10/7 |
| | |- |
| | | 31 |
| | | |
| | | 600.0 |
| | | 5◊6 |
| | | |
| | |- |
| | | 32 |
| | | |
| | | 619.4 |
| | | 5# |
| | | 7/5 |
| | |- |
| | | 33 |
| | | |
| | | 638.7 |
| | | 6v |
| | | |
| | |- |
| | | 34 |
| | | |
| | | 658.1 |
| | | 6⌐ |
| | | |
| | |- |
| | | 35 |
| | | · |
| | | 677.4 |
| | | 6 |
| | | 3/2~16/11 |
| | |- |
| | | 36 |
| | | |
| | | 696.8 |
| | | 6Ɨ |
| | | |
| | |- |
| | | 37 |
| | | |
| | | 716.1 |
| | | 6‡ |
| | | |
| | |- |
| | | 38 |
| | | |
| | | 735.5 |
| | | 7b |
| | | |
| | |- |
| | | 39 |
| | | |
| | | 754.8 |
| | | 6◊7 |
| | | |
| | |- |
| | | 40 |
| | | |
| | | 774.2 |
| | | 6# |
| | | |
| | |- |
| | | 41 |
| | | |
| | | 793.5 |
| | | 7v |
| | | |
| | |- |
| | | 42 |
| | | |
| | | 812.9 |
| | | 7⌐ |
| | | |
| | |- |
| | | 43 |
| | | · |
| | | 832.3 |
| | | 7 |
| | | 8/5 |
| | |- |
| | | 44 |
| | | |
| | | 851.6 |
| | | 7Ɨ |
| | | |
| | |- |
| | | 45 |
| | | |
| | | 871.0 |
| | | 7‡ |
| | | |
| | |- |
| | | 46 |
| | | |
| | | 890.3 |
| | | 8b |
| | | 5/3~12/7 |
| | |- |
| | | 47 |
| | | |
| | | 909.7 |
| | | 7◊8 |
| | | |
| | |- |
| | | 48 |
| | | |
| | | 929.0 |
| | | 7# |
| | | |
| | |- |
| | | 49 |
| | | |
| | | 948.4 |
| | | 8v |
| | | |
| | |- |
| | | 50 |
| | | |
| | | 967.7 |
| | | 8⌐ |
| | | |
| | |- |
| | | 51 |
| | | |
| | | 987.1 |
| | | 8 |
| | | 7/4 |
| | |- |
| | | 52 |
| | | |
| | | 1006.5 |
| | | 8Ɨ |
| | | |
| | |- |
| | | 53 |
| | | |
| | | 1025.8 |
| | | 8‡ |
| | | |
| | |- |
| | | 54 |
| | | |
| | | 1045.2 |
| | | 9b |
| | | 11/6~20/11 |
| | |- |
| | | 55 |
| | | |
| | | 1064.5 |
| | | 8◊9 |
| | | |
| | |- |
| | | 56 |
| | | |
| | | 1083.9 |
| | | 8# |
| | | |
| | |- |
| | | 57 |
| | | |
| | | 1103.2 |
| | | 9v (1b) |
| | | |
| | |- |
| | | 58 |
| | | |
| | | 1122.6 |
| | | 9⌐ |
| | | |
| | |- |
| | | 59 |
| | | |
| | | 1141.9 |
| | | 9 |
| | | |
| | |- |
| | | 60 |
| | | |
| | | 1161.3 |
| | | 9Ɨ (1v) |
| | | |
| | |- |
| | | 61 |
| | | |
| | | 1180.6 |
| | | 1⌐ (9‡) |
| | | |
| | |- |
| | | 62 |
| | | |
| | | 1200.0 |
| | | 1 |
| | | |
| | |} |
| | |
| | == Approximation to JI == |
| | === Zeta peak index === |
| | {{ZPI |
| | | zpi = 314 |
| | | steps = 61.9380472360525 |
| | | step size = 19.3741981471691 |
| | | tempered height = 6.262952 |
| | | pure height = 4.11259 |
| | | integral = 0.952068 |
| | | gap = 15.026453 |
| | | octave = 1201.20028512448 |
| | | consistent = 8 |
| | | distinct = 8 |
| | }} |
| | |
| | == Regular temperament properties == |
| | 62edo is contorted 31edo through the 11-limit. |
| | {| class="wikitable center-4 center-5 center-6" |
| | |- |
| | ! rowspan="2" | [[Subgroup]] |
| | ! rowspan="2" | [[Comma list]] |
| | ! rowspan="2" | [[Mapping]] |
| | ! rowspan="2" | Optimal<br>8ve stretch (¢) |
| | ! colspan="2" | Tuning error |
| | |- |
| | ! [[TE error|Absolute]] (¢) |
| | ! [[TE simple badness|Relative]] (%) |
| | |- |
| | | 2.3.5.7.11.13 |
| | | 81/80, 99/98, 121/120, 126/125, 169/168 |
| | | {{mapping| 62 98 144 174 214 229 }} |
| | | +1.38 |
| | | 1.41 |
| | | 7.28 |
| | |- |
| | | 2.3.5.7.11.13.17 |
| | | 81/80, 99/98, 121/120, 126/125, 169/168, 221/220 |
| | | {{mapping| 62 98 144 174 214 229 253 }} |
| | | +1.47 |
| | | 1.32 |
| | | 6.83 |
| | |- |
| | | 2.3.5.7.11.13.17.19 |
| | | 81/80, 99/98, 121/120, 126/125, 153/152, 169/168, 209/208 |
| | | {{mapping| 62 98 144 174 214 229 253 263 }} |
| | | +1.50 |
| | | 1.24 |
| | | 6.40 |
| | |- |
| | | 2.3.5.7.11.13.17.19.23 |
| | | 81/80, 99/98, 121/120, 126/125, 153/152, 161/160, 169/168, 209/208 |
| | | {{mapping| 62 98 144 174 214 229 253 263 280 }} |
| | | +1.55 |
| | | 1.18 |
| | | 6.09 |
| | |} |
| | |
| | === Rank-2 temperaments === |
| | {| class="wikitable center-all left-5" |
| | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator |
| | |- |
| | ! Periods<br>per 8ve |
| | ! Generator* |
| | ! Cents* |
| | ! Associated<br>ratio* |
| | ! Temperament |
| | |- |
| | | 1 |
| | | 3\62 |
| | | 58.1 |
| | | 27/26 |
| | | [[Hemisecordite]] |
| | |- |
| | | 1 |
| | | 7\62 |
| | | 135.5 |
| | | 13/12 |
| | | [[Doublethink]] |
| | |- |
| | | 1 |
| | | 13\62 |
| | | 251.6 |
| | | 15/13 |
| | | [[Hemimeantone]] |
| | |- |
| | | 1 |
| | | 17\62 |
| | | 329.0 |
| | | 16/11 |
| | | [[Mabon]] |
| | |- |
| | | 1 |
| | | 29\62 |
| | | 561.3 |
| | | 18/13 |
| | | [[Demivalentine]] |
| | |- |
| | | 2 |
| | | 3\62 |
| | | 58.1 |
| | | 27/26 |
| | | [[Semihemisecordite]] |
| | |- |
| | | 2 |
| | | 4\62 |
| | | 77.4 |
| | | 21/20 |
| | | [[Semivalentine]] |
| | |- |
| | | 2 |
| | | 6\62 |
| | | 116.1 |
| | | 15/14 |
| | | [[Semimiracle]] |
| | |- |
| | | 2 |
| | | 26\62 |
| | | 503.2 |
| | | 4/3 |
| | | [[Semimeantone]] |
| | |- |
| | | 31 |
| | | 29\62<br>(1\62) |
| | | 561.3<br>(19.4) |
| | | 11/8<br>(196/195) |
| | | [[Kumhar]] (62e) |
| | |- |
| | | 31 |
| | | 19\62<br>(1\62) |
| | | 367.7<br>(19.4) |
| | | 16/13<br>(77/76) |
| | | [[Gallium]] |
| | |} |
| | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct |
| | |
| | == Instruments == |
| | |
| | === Lumatone === |
| | * [[Lumatone mapping for 62edo]] |
| | |
| | === Skip fretting === |
| | '''[[Skip fretting]] system 62 6 11''' has strings tuned 11\62 apart, while frets are 6\62. |
| | |
| | On a 4-string bass, here are your open strings: |
| | |
| | 0 11 22 33 |
| | |
| | A good supraminor 3rd is found on the 2nd string, 1st fret. A supermajor third is found on the open 3rd string. The major 6th can be found on the 4th string, 2nd fret. |
| | |
| | 5-string bass |
| | |
| | 51 0 11 22 33 |
| | |
| | This adds an interval of a major 7th (minus an 8ve) at the first string, 1st fret. |
| | |
| | 6-string guitar |
| | |
| | 0 11 22 33 44 55 |
| | |
| | ”Major” 020131 |
| | |
| | 7-string guitar |
| | |
| | 0 11 22 33 44 55 4 |
| | |
| | |
| | '''Skip fretting system 62 9 11''' is another 62edo skip fretting system. The 5th is on the 5th string. The major 3rd is on the 2nd string, 1st fret. |
| | {{todo|add illustration|text=Base it off of the diagram from [[User:MisterShafXen/Skip fretting system 62 9 11]]}} |
| | |
| | == Music == |
| | ; [[Bryan Deister]] |
| | * [https://www.youtube.com/shorts/UerD0NqBbng ''microtonal improvisation in 62edo''] (2025) |