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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:genewardsmith|genewardsmith]] and made on <tt>2011-03-18 18:35:48 UTC</tt>.<br>
| |
| : The original revision id was <tt>211899520</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: #750063; font-size: 103%;">65 tone equal temperament</span>=
| |
| //65edo// divides the [[octave]] into 65 equal parts of 18.462 cents each. It can be characterized as the temperament which tempers out the [[schisma]], 32805/32768, the [[sensipent comma]], 78732/78125, and the [[wuerschmidt comma]], 393216/390625. In the [[7-limit]], there are two different maps; the first is <65 103 151 182|, tempering out 126/125, 245/243 and 686/675, so that 65edo supports sensi temperament, and the second is <65 103 151 183|, tempering out 225/224, 3125/3097, 4000/3969 and 5120/5103, so that 65edo supports garibaldi temperament. In both cases, the tuning privileges the [[5-limit]] over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit [[wuerschmidt temperament]] (wurschmidt and worschmidt) these two mappings provide.
| |
|
| |
|
| 65edo approximates the intervals 3/2, 5/4, 11/8 and 19/16 well, so that it does a good job representing the 2.3.5.11.19 [[just intonation subgroup]]. To this one may want to add 13/8 and 17/16, giving the [[19-limit]] no-sevens subgroup 2.3.5.11.13.17.19.
| | == Theory == |
| | 65et can be characterized as the temperament which [[tempering out|tempers out]] 32805/32768 ([[schisma]]), 78732/78125 ([[sensipent comma]]), 393216/390625 ([[würschmidt comma]]), and {{monzo| -13 17 -6 }} ([[graviton]]). In the [[7-limit]], there are two different maps; the first is {{val| 65 103 151 '''182''' }} (65), tempering out [[126/125]], [[245/243]] and [[686/675]], so that it [[support]]s [[sensi]], and the second is {{val| 65 103 151 '''183''' }} (65d), tempering out [[225/224]], [[3125/3087]], [[4000/3969]] and [[5120/5103]], so that it supports [[garibaldi]]. In both cases, the tuning privileges the [[5-limit]] over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit [[würschmidt]] temperament (wurschmidt and worschmidt) these two mappings provide. |
|
| |
|
| ==Intervals==
| | 65edo approximates the intervals [[3/2]], [[5/4]], [[11/8]], [[19/16]], [[23/16]], [[31/16]] and [[47/32]] well, so that it does a good job representing the 2.3.5.11.19.23.31.47 [[just intonation subgroup]]. To this one may want to add [[17/16]], [[29/16]] and [[43/32]], giving the [[47-limit]] no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of [[schismic]]/[[nestoria]] that focuses on the very primes that [[53edo]] neglects (which instead elegantly connects primes 7, 13, 37, and 41 to nestoria). Also of interest is the [[19-limit]] [[k*N subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the [[zeta]] edo [[130edo]]. |
| || Degrees of 65-EDO || Cents value ||
| |
| || 0 || 0 ||
| |
| || 1 || 18,4615 ||
| |
| || 2 || 36,9231 ||
| |
| || 3 || 55,3846 ||
| |
| || 4 || 73,8462 ||
| |
| || 5 || 92,3077 ||
| |
| || 6 || 110,7692 ||
| |
| || 7 || 129,2308 ||
| |
| || 8 || 147,6923 ||
| |
| || 9 || 166,1538 ||
| |
| || 10 || 184,6154 ||
| |
| || 11 || 203,0769 ||
| |
| || 12 || 221,5385 ||
| |
| || 13 || 240 ||
| |
| || 14 || 258,4615 ||
| |
| || 15 || 276,9231 ||
| |
| || 16 || 295,3846 ||
| |
| || 17 || 313,8462 ||
| |
| || 18 || 332,3077 ||
| |
| || 19 || 350,7692 ||
| |
| || 20 || 369,2308 ||
| |
| || 21 || 387,6923 ||
| |
| || 22 || 406,1538 ||
| |
| || 23 || 424,6154 ||
| |
| || 24 || 443,0769 ||
| |
| || 25 || 461,5385 ||
| |
| || 26 || 480 ||
| |
| || 27 || 498,4615 ||
| |
| || 28 || 516,9231 ||
| |
| || 29 || 535,3846 ||
| |
| || 30 || 553,8462 ||
| |
| || 31 || 572,3077 ||
| |
| || 32 || 590,7692 ||
| |
| || 33 || 609,2308 ||
| |
| || 34 || 627,6923 ||
| |
| || 35 || 646,1538 ||
| |
| || 36 || 664,6154 ||
| |
| || 37 || 683,0769 ||
| |
| || 38 || 701,5385 ||
| |
| || 39 || 720 ||
| |
| || 40 || 738,4615 ||
| |
| || 41 || 756,9231 ||
| |
| || 42 || 775,3846 ||
| |
| || 43 || 793,8462 ||
| |
| || 44 || 812,3077 ||
| |
| || 45 || 830,7692 ||
| |
| || 46 || 849,2308 ||
| |
| || 47 || 867,6923 ||
| |
| || 48 || 886,1538 ||
| |
| || 49 || 904,6154 ||
| |
| || 50 || 923,0769 ||
| |
| || 51 || 941,5385 ||
| |
| || 52 || 960 ||
| |
| || 53 || 978,4615 ||
| |
| || 54 || 996,9231 ||
| |
| || 55 || 1015,3846 ||
| |
| || 56 || 1033,8462 ||
| |
| || 57 || 1052,3077 ||
| |
| || 58 || 1070,7692 ||
| |
| || 59 || 1089,2308 ||
| |
| || 60 || 1107,6923 ||
| |
| || 61 || 1126,1538 ||
| |
| || 62 || 1144,6154 ||
| |
| || 63 || 1163,0769 ||
| |
| || 64 || 1181,5385 ||</pre></div>
| |
| <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>65edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x65 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #750063; font-size: 103%;">65 tone equal temperament</span></h1>
| |
| <em>65edo</em> divides the <a class="wiki_link" href="/octave">octave</a> into 65 equal parts of 18.462 cents each. It can be characterized as the temperament which tempers out the <a class="wiki_link" href="/schisma">schisma</a>, 32805/32768, the <a class="wiki_link" href="/sensipent%20comma">sensipent comma</a>, 78732/78125, and the <a class="wiki_link" href="/wuerschmidt%20comma">wuerschmidt comma</a>, 393216/390625. In the <a class="wiki_link" href="/7-limit">7-limit</a>, there are two different maps; the first is &lt;65 103 151 182|, tempering out 126/125, 245/243 and 686/675, so that 65edo supports sensi temperament, and the second is &lt;65 103 151 183|, tempering out 225/224, 3125/3097, 4000/3969 and 5120/5103, so that 65edo supports garibaldi temperament. In both cases, the tuning privileges the <a class="wiki_link" href="/5-limit">5-limit</a> over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit <a class="wiki_link" href="/wuerschmidt%20temperament">wuerschmidt temperament</a> (wurschmidt and worschmidt) these two mappings provide.<br />
| |
| <br />
| |
| 65edo approximates the intervals 3/2, 5/4, 11/8 and 19/16 well, so that it does a good job representing the 2.3.5.11.19 <a class="wiki_link" href="/just%20intonation%20subgroup">just intonation subgroup</a>. To this one may want to add 13/8 and 17/16, giving the <a class="wiki_link" href="/19-limit">19-limit</a> no-sevens subgroup 2.3.5.11.13.17.19.<br /> | |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x65 tone equal temperament-Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h2>
| |
|
| |
|
| |
|
| <table class="wiki_table">
| | === Prime harmonics === |
| <tr>
| | {{Harmonics in equal|65|intervals=prime|columns=15}} |
| <td>Degrees of 65-EDO<br />
| |
| </td>
| |
| <td>Cents value<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>1<br />
| |
| </td>
| |
| <td>18,4615<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>36,9231<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>55,3846<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>73,8462<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>92,3077<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6<br />
| |
| </td>
| |
| <td>110,7692<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7<br />
| |
| </td>
| |
| <td>129,2308<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8<br />
| |
| </td>
| |
| <td>147,6923<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9<br />
| |
| </td>
| |
| <td>166,1538<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10<br />
| |
| </td>
| |
| <td>184,6154<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11<br />
| |
| </td>
| |
| <td>203,0769<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12<br />
| |
| </td>
| |
| <td>221,5385<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13<br />
| |
| </td>
| |
| <td>240<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14<br />
| |
| </td>
| |
| <td>258,4615<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15<br />
| |
| </td>
| |
| <td>276,9231<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16<br />
| |
| </td>
| |
| <td>295,3846<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17<br />
| |
| </td>
| |
| <td>313,8462<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18<br />
| |
| </td>
| |
| <td>332,3077<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>19<br />
| |
| </td>
| |
| <td>350,7692<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>20<br />
| |
| </td>
| |
| <td>369,2308<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21<br />
| |
| </td>
| |
| <td>387,6923<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>22<br />
| |
| </td>
| |
| <td>406,1538<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>23<br />
| |
| </td>
| |
| <td>424,6154<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>24<br />
| |
| </td>
| |
| <td>443,0769<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>25<br />
| |
| </td>
| |
| <td>461,5385<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>26<br />
| |
| </td>
| |
| <td>480<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>27<br />
| |
| </td>
| |
| <td>498,4615<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>28<br />
| |
| </td>
| |
| <td>516,9231<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>29<br />
| |
| </td>
| |
| <td>535,3846<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>30<br />
| |
| </td>
| |
| <td>553,8462<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>31<br />
| |
| </td>
| |
| <td>572,3077<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>32<br />
| |
| </td>
| |
| <td>590,7692<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>33<br />
| |
| </td>
| |
| <td>609,2308<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>34<br />
| |
| </td>
| |
| <td>627,6923<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>35<br />
| |
| </td>
| |
| <td>646,1538<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>36<br />
| |
| </td>
| |
| <td>664,6154<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>37<br />
| |
| </td>
| |
| <td>683,0769<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>38<br />
| |
| </td>
| |
| <td>701,5385<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>39<br />
| |
| </td>
| |
| <td>720<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>40<br />
| |
| </td>
| |
| <td>738,4615<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>41<br />
| |
| </td>
| |
| <td>756,9231<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>42<br />
| |
| </td>
| |
| <td>775,3846<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>43<br />
| |
| </td>
| |
| <td>793,8462<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>44<br />
| |
| </td>
| |
| <td>812,3077<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>45<br />
| |
| </td>
| |
| <td>830,7692<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>46<br />
| |
| </td>
| |
| <td>849,2308<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>47<br />
| |
| </td>
| |
| <td>867,6923<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>48<br />
| |
| </td>
| |
| <td>886,1538<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>49<br />
| |
| </td>
| |
| <td>904,6154<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>50<br />
| |
| </td>
| |
| <td>923,0769<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>51<br />
| |
| </td>
| |
| <td>941,5385<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>52<br />
| |
| </td>
| |
| <td>960<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>53<br />
| |
| </td>
| |
| <td>978,4615<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>54<br />
| |
| </td>
| |
| <td>996,9231<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>55<br />
| |
| </td>
| |
| <td>1015,3846<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>56<br />
| |
| </td>
| |
| <td>1033,8462<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>57<br />
| |
| </td>
| |
| <td>1052,3077<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>58<br />
| |
| </td>
| |
| <td>1070,7692<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>59<br />
| |
| </td>
| |
| <td>1089,2308<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>60<br />
| |
| </td>
| |
| <td>1107,6923<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>61<br />
| |
| </td>
| |
| <td>1126,1538<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>62<br />
| |
| </td>
| |
| <td>1144,6154<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>63<br />
| |
| </td>
| |
| <td>1163,0769<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>64<br />
| |
| </td>
| |
| <td>1181,5385<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| </body></html></pre></div> | | === Subsets and supersets === |
| | 65edo contains [[5edo]] and [[13edo]] as subsets. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see [[Andrew Heathwaite]]'s composition [https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded ''Rubble: a Xenuke Unfolded'']. |
| | |
| | [[130edo]], which doubles its, corrects its approximation to harmonics 7 and 13. |
| | |
| | == Intervals == |
| | {| class="wikitable center-all right-2 left-3" |
| | |- |
| | ! # |
| | ! [[Cent]]s |
| | ! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13/7.17.19.23.29.31.47 subgroup}}</ref> |
| | ! colspan="2" | [[Ups and downs notation]] |
| | |- |
| | | 0 |
| | | 0.00 |
| | | 1/1 |
| | | P1 |
| | | D |
| | |- |
| | | 1 |
| | | 18.46 |
| | | 81/80, 88/87, 93/92, 94/93, 95/94, 96/95, 100/99, 121/120, 115/114, 116/115, 125/124 |
| | | ^1 |
| | | ^D |
| | |- |
| | | 2 |
| | | 36.92 |
| | | 45/44, 46/45, 47/46, 48/47, 55/54, 128/125 |
| | | ^^1 |
| | | ^^D |
| | |- |
| | | 3 |
| | | 55.38 |
| | | 30/29, 31/30, 32/31, 33/32, 34/33 |
| | | vvm2 |
| | | vvEb |
| | |- |
| | | 4 |
| | | 73.85 |
| | | 23/22, 24/23, 25/24, 47/45 |
| | | vm2 |
| | | vEb |
| | |- |
| | | 5 |
| | | 92.31 |
| | | 18/17, 19/18, 20/19, 58/55, 135/128, 256/243 |
| | | m2 |
| | | Eb |
| | |- |
| | | 6 |
| | | 110.77 |
| | | 16/15, 17/16, 33/31 |
| | | A1/^m2 |
| | | D#/^Eb |
| | |- |
| | | 7 |
| | | 129.23 |
| | | 14/13, 27/25, 55/51 |
| | | v~2 |
| | | ^^Eb |
| | |- |
| | | 8 |
| | | 147.69 |
| | | 12/11, 25/23 |
| | | ~2 |
| | | vvvE |
| | |- |
| | | 9 |
| | | 166.15 |
| | | 11/10, 32/29 |
| | | ^~2 |
| | | vvE |
| | |- |
| | | 10 |
| | | 184.62 |
| | | 10/9, 19/17 |
| | | vM2 |
| | | vE |
| | |- |
| | | 11 |
| | | 203.08 |
| | | 9/8, 64/57 |
| | | M2 |
| | | E |
| | |- |
| | | 12 |
| | | 221.54 |
| | | 17/15, 25/22, 33/29, 58/51 |
| | | ^M2 |
| | | ^E |
| | |- |
| | | 13 |
| | | 240.00 |
| | | 23/20, 31/27, 38/33, 54/47, 55/48 |
| | | ^^M2 |
| | | ^^E |
| | |- |
| | | 14 |
| | | 258.46 |
| | | 22/19, 29/25, 36/31, 64/55 |
| | | vvm3 |
| | | vvF |
| | |- |
| | | 15 |
| | | 276.92 |
| | | 20/17, 27/23, 34/29, 75/64 |
| | | vm3 |
| | | vF |
| | |- |
| | | 16 |
| | | 295.38 |
| | | 19/16, 32/27 |
| | | m3 |
| | | F |
| | |- |
| | | 17 |
| | | 313.85 |
| | | 6/5, 55/46 |
| | | ^m3 |
| | | ^F |
| | |- |
| | | 18 |
| | | 332.31 |
| | | 23/19, 40/33 |
| | | v~3 |
| | | ^^F |
| | |- |
| | | 19 |
| | | 350.77 |
| | | 11/9, 27/22, 38/31 |
| | | ~3 |
| | | ^^^F |
| | |- |
| | | 20 |
| | | 369.23 |
| | | 26/21, 47/38, 68/55 |
| | | ^~3 |
| | | vvF# |
| | |- |
| | | 21 |
| | | 387.69 |
| | | 5/4, 64/51 |
| | | vM3 |
| | | vF# |
| | |- |
| | | 22 |
| | | 406.15 |
| | | 19/15, 24/19, 29/23, 34/27, 81/64 |
| | | M3 |
| | | F# |
| | |- |
| | | 23 |
| | | 424.62 |
| | | 23/18, 32/25 |
| | | ^M3 |
| | | ^F# |
| | |- |
| | | 24 |
| | | 443.08 |
| | | 22/17, 31/24, 40/31, 128/99 |
| | | ^^M3 |
| | | ^^F# |
| | |- |
| | | 25 |
| | | 461.54 |
| | | 30/23, 47/36, 72/55 |
| | | vv4 |
| | | vvG |
| | |- |
| | | 26 |
| | | 480.00 |
| | | 29/22, 33/25, 62/47 |
| | | v4 |
| | | vG |
| | |- |
| | | 27 |
| | | 498.46 |
| | | 4/3 |
| | | P4 |
| | | G |
| | |- |
| | | 28 |
| | | 516.92 |
| | | 23/17, 27/20, 31/23 |
| | | ^4 |
| | | ^G |
| | |- |
| | | 29 |
| | | 535.38 |
| | | 15/11, 34/25, 64/47 |
| | | v~4 |
| | | ^^G |
| | |- |
| | | 30 |
| | | 553.85 |
| | | 11/8, 40/29, 62/45 |
| | | ~4 |
| | | ^^^G |
| | |- |
| | | 31 |
| | | 572.31 |
| | | 25/18, 32/23 |
| | | ^~4/vd5 |
| | | vvG#/vAb |
| | |- |
| | | 32 |
| | | 590.77 |
| | | 24/17, 31/22, 38/27, 45/32 |
| | | vA4/d5 |
| | | vG#/Ab |
| | |- |
| | | 33 |
| | | 609.23 |
| | | 17/12, 27/19, 44/31, 64/45 |
| | | A4/^d5 |
| | | G#/^Ab |
| | |- |
| | | 34 |
| | | 627.69 |
| | | 36/25, 23/16 |
| | | ^A4/v~5 |
| | | ^G#/^^Ab |
| | |- |
| | | 35 |
| | | 646.15 |
| | | 16/11, 29/20, 45/31 |
| | | ~5 |
| | | vvvA |
| | |- |
| | | 36 |
| | | 664.62 |
| | | 22/15, 25/17, 47/32 |
| | | ^~5 |
| | | vvA |
| | |- |
| | | 37 |
| | | 683.08 |
| | | 34/23, 40/27, 46/31 |
| | | v5 |
| | | vA |
| | |- |
| | | 38 |
| | | 701.54 |
| | | 3/2 |
| | | P5 |
| | | A |
| | |- |
| | | 39 |
| | | 720.00 |
| | | 44/29, 50/33, 47/31 |
| | | ^5 |
| | | ^A |
| | |- |
| | | 40 |
| | | 738.46 |
| | | 23/15, 55/36, 72/47 |
| | | ^^5 |
| | | ^^A |
| | |- |
| | | 41 |
| | | 756.92 |
| | | 17/11, 48/31, 31/20, 99/64 |
| | | vvm6 |
| | | vvBb |
| | |- |
| | | 42 |
| | | 775.38 |
| | | 25/16, 36/23 |
| | | vm6 |
| | | vBb |
| | |- |
| | | 43 |
| | | 793.85 |
| | | 19/12, 27/17, 30/19, 46/29, 128/81 |
| | | m6 |
| | | Bb |
| | |- |
| | | 44 |
| | | 812.31 |
| | | 8/5, 51/32 |
| | | ^m6 |
| | | ^Bb |
| | |- |
| | | 45 |
| | | 830.77 |
| | | 21/13, 55/34, 76/47 |
| | | v~6 |
| | | ^^Bb |
| | |- |
| | | 46 |
| | | 849.23 |
| | | 18/11, 31/19, 44/27 |
| | | ~6 |
| | | vvvB |
| | |- |
| | | 47 |
| | | 867.69 |
| | | 33/20, 38/23 |
| | | ^~6 |
| | | vvB |
| | |- |
| | | 48 |
| | | 886.15 |
| | | 5/3, 92/55 |
| | | vM6 |
| | | vB |
| | |- |
| | | 49 |
| | | 904.62 |
| | | 27/16, 32/19 |
| | | M6 |
| | | B |
| | |- |
| | | 50 |
| | | 923.08 |
| | | 17/10, 29/17, 46/27, 128/75 |
| | | ^M6 |
| | | ^B |
| | |- |
| | | 51 |
| | | 941.54 |
| | | 19/11, 31/18, 50/29, 55/32 |
| | | ^^M6 |
| | | ^^B |
| | |- |
| | | 52 |
| | | 960.00 |
| | | 33/19, 40/23, 47/27, 54/31, 96/55 |
| | | vvm7 |
| | | vvC |
| | |- |
| | | 53 |
| | | 978.46 |
| | | 30/17, 44/25, 51/29, 58/33 |
| | | vm7 |
| | | vC |
| | |- |
| | | 54 |
| | | 996.92 |
| | | 16/9, 57/32 |
| | | m7 |
| | | C |
| | |- |
| | | 55 |
| | | 1015.38 |
| | | 9/5, 34/19 |
| | | ^m7 |
| | | ^C |
| | |- |
| | | 56 |
| | | 1033.85 |
| | | 20/11, 29/16 |
| | | v~7 |
| | | ^^C |
| | |- |
| | | 57 |
| | | 1052.31 |
| | | 11/6, 46/25 |
| | | ~7 |
| | | ^^^C |
| | |- |
| | | 58 |
| | | 1070.77 |
| | | 13/7, 50/27, 102/55 |
| | | ^~7 |
| | | vvC# |
| | |- |
| | | 59 |
| | | 1089.23 |
| | | 15/8, 32/17, 62/33 |
| | | vM7 |
| | | vC# |
| | |- |
| | | 60 |
| | | 1107.69 |
| | | 17/9, 19/10, 36/19, 55/29, 243/128, 256/135 |
| | | M7 |
| | | C# |
| | |- |
| | | 61 |
| | | 1126.15 |
| | | 23/12, 44/23, 48/25, 90/47 |
| | | ^M7 |
| | | ^C# |
| | |- |
| | | 62 |
| | | 1144.62 |
| | | 29/15, 31/16, 33/17, 60/31, 64/33 |
| | | ^^M7 |
| | | ^^C# |
| | |- |
| | | 63 |
| | | 1163.08 |
| | | 45/23, 47/24, 88/45, 92/47, 108/55, 125/64 |
| | | vv8 |
| | | vvD |
| | |- |
| | | 64 |
| | | 1181.54 |
| | | 87/55, 93/47, 95/48, 99/50, 115/58, 160/81, 184/93, 188/95, 228/115, 240/121, 248/125 |
| | | v8 |
| | | vD |
| | |- |
| | | 65 |
| | | 1200.00 |
| | | 2/1 |
| | | P8 |
| | | D |
| | |} |
| | <references group="note" /> |
| | |
| | == Notation == |
| | === Ups and downs notation === |
| | 65edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc. |
| | {{Sharpness-sharp6a}} |
| | |
| | Half-sharps and half-flats can be used to avoid triple arrows: |
| | {{Sharpness-sharp6b}} |
| | |
| | [[Alternative symbols for ups and downs notation#Sharp-6| Alternative ups and downs]] have arrows borrowed from extended [[Helmholtz–Ellis notation]]: |
| | {{Sharpness-sharp6}} |
| | |
| | If double arrows are not desirable, arrows can be attached to quarter-tone accidentals: |
| | {{Sharpness-sharp6-qt}} |
| | |
| | === Ivan Wyschnegradsky's notation === |
| | Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used: |
| | |
| | {{sharpness-sharp6-iw}} |
| | |
| | === Sagittal notation === |
| | This notation uses the same sagittal sequence as EDOs [[72edo#Sagittal notation|72]] and [[79edo#Sagittal notation|79]]. |
| | |
| | ==== Evo flavor ==== |
| | <imagemap> |
| | File:65-EDO_Evo_Sagittal.svg |
| | desc none |
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 120 106 [[81/80]] |
| | rect 120 80 220 106 [[64/63]] |
| | rect 220 80 340 106 [[33/32]] |
| | default [[File:65-EDO_Evo_Sagittal.svg]] |
| | </imagemap> |
| | |
| | ==== Revo flavor ==== |
| | <imagemap> |
| | File:65-EDO_Revo_Sagittal.svg |
| | desc none |
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | rect 300 0 650 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 120 106 [[81/80]] |
| | rect 120 80 220 106 [[64/63]] |
| | rect 220 80 340 106 [[33/32]] |
| | default [[File:65-EDO_Revo_Sagittal.svg]] |
| | </imagemap> |
| | |
| | ==== Evo-SZ flavor ==== |
| | <imagemap> |
| | File:65-EDO_Evo-SZ_Sagittal.svg |
| | desc none |
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | rect 300 0 639 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 120 106 [[81/80]] |
| | rect 120 80 220 106 [[64/63]] |
| | rect 220 80 340 106 [[33/32]] |
| | default [[File:65-EDO_Evo-SZ_Sagittal.svg]] |
| | </imagemap> |
| | |
| | == Approximation to JI == |
| | === Zeta peak index === |
| | {{ZPI |
| | | zpi = 334 |
| | | steps = 65.0158450885860 |
| | | step size = 18.4570391781413 |
| | | tempered height = 7.813349 |
| | | pure height = 7.642373 |
| | | integral = 1.269821 |
| | | gap = 16.514861 |
| | | octave = 1199.70754657919 |
| | | consistent = 6 |
| | | distinct = 6 |
| | }} |
| | |
| | == Regular temperament properties == |
| | {| 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 |
| | | {{monzo| -103 65 }} |
| | | {{mapping| 65 103 }} |
| | | +0.131 |
| | | 0.131 |
| | | 0.71 |
| | |- |
| | | 2.3.5 |
| | | 32805/32768, 78732/78125 |
| | | {{mapping| 65 103 151 }} |
| | | −0.110 |
| | | 0.358 |
| | | 1.94 |
| | |- |
| | | 2.3.5.11 |
| | | 243/242, 4000/3993, 5632/5625 |
| | | {{mapping| 65 103 151 225 }} |
| | | −0.266 |
| | | 0.410 |
| | | 2.22 |
| | |} |
| | |
| | === 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\65 |
| | | 55.38 |
| | | 33/32 |
| | | [[Escapade]] |
| | |- |
| | | 1 |
| | | 9\65 |
| | | 166.15 |
| | | 11/10 |
| | | [[Squirrel]] etc. |
| | |- |
| | | 1 |
| | | 12\65 |
| | | 221.54 |
| | | 25/22 |
| | | [[Hemisensi]] |
| | |- |
| | | 1 |
| | | 19\65 |
| | | 350.77 |
| | | 11/9 |
| | | [[Karadeniz]] |
| | |- |
| | | 1 |
| | | 21\65 |
| | | 387.69 |
| | | 5/4 |
| | | [[Würschmidt]] |
| | |- |
| | | 1 |
| | | 24\65 |
| | | 443.08 |
| | | 162/125 |
| | | [[Sensipent]] |
| | |- |
| | | 1 |
| | | 27\65 |
| | | 498.46 |
| | | 4/3 |
| | | [[Helmholtz (temperament)|Helmholtz]] / [[nestoria]] / [[photia]] |
| | |- |
| | | 1 |
| | | 28\65 |
| | | 516.92 |
| | | 27/20 |
| | | [[Larry]] |
| | |- |
| | | 5 |
| | | 20\65<br>(6\65) |
| | | 369.23<br>(110.77) |
| | | 99/80<br>(16/15) |
| | | [[Quintosec]] |
| | |- |
| | | 5 |
| | | 27\65<br>(1\65) |
| | | 498.46<br>(18.46) |
| | | 4/3<br>(81/80) |
| | | [[Quintile]] |
| | |- |
| | | 5 |
| | | 30\65<br>(4\65) |
| | | 553.85<br>(73.85) |
| | | 11/8<br>(25/24) |
| | | [[Countdown]] |
| | |} |
| | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct |
| | |
| | == Scales == |
| | * Amulet{{idiosyncratic}}, (approximated from [[25edo]], subset of [[würschmidt]]): 5 3 5 5 3 5 12 5 5 3 5 12 5 |
| | * [[Photia7]] |
| | * [[Photia12]] |
| | * [[Skateboard7]] |
| | |
| | == Instruments == |
| | [[Lumatone mapping for 65edo]] |
| | |
| | == Music == |
| | ; [[Bryan Deister]] |
| | * [https://www.youtube.com/shorts/W5PXWFduPco ''microtonal improvisation in 65edo''] (2025). |
| | |
| | [[Category:Listen]] |
| | [[Category:Schismic]] |
| | [[Category:Sensipent]] |
| | [[Category:Subgroup temperaments]] |
| | [[Category:Würschmidt]] |