80edo
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author guest and made on 2012-08-30 19:51:04 UTC.
- The original revision id was 361115194.
- The revision comment was: scala says otherwise
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Original Wikitext content:
The //80 equal temperament//, often abbreviated 80-tET, 80-EDO, or 80-ET, is the scale derived by dividing the octave into 80 equally-sized steps. Each step represents a frequency ratio of exactly 15 [[xenharmonic/cent|cent]]s. 80et is the first equal temperament that represents the [[xenharmonic/19-limit|19-limit]] [[xenharmonic/tonality diamond|tonality diamond]] [[xenharmonic/consistent|consistent]]ly (it barely manages to do so). 80 et [[xenharmonic/tempering out|tempers out]] 136/135, 169/168, 176/175, 190/189, 221/220, 256/255, 286/285, 289/288, 325/324, 351/350, 352/351, 361/360, 364/363, 400/399, 456/455, 476/475, 540/539, 561/560, 595/594, 715/714, 936/935, 969/968, 1001/1000, 1275/1274, 1331/1330, 1445/1444, 1521/1520, 1540/1539 and 1729/1728, not to mention such important non-superparticular commas as 2048/2025, 4000/3969, 1728/1715 and 3136/3125. 80 supports a profusion of 19-limit (and lower) rank two temperaments which have mostly not been explored. We might mention: 31&80 <<7 6 15 27 -24 -23 -20 ... || 72&80 <<24 30 40 24 32 24 0 ... || 34&80 <<2 -4 -50 22 16 2 -40 ... || 46&80 <<2 -4 30 22 16 2 40 ... || 29&80 <<3 34 45 33 24 -37 20 ... || 12&80 <<4 -8 -20 -36 32 4 0 ... || 22&80 <<6 -10 12 -14 -32 6 -40 ... || 58&80 <<6 -10 12 -14 -32 6 40 ... || 41&80 <<7 26 25 -3 -24 -33 20 ... || In each case, the numbers joined by an ampersand represent 19-limit [[xenharmonic/Patent val|patent vals]] (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given. =Intervals of 80edo= ||~ degrees ||~ cents ||~ ratios* || || 0 || 0 || 1/1 || || 1 || 15 || 64/63 || || 2 || 30 || 81/80 || || 3 || 45 || 34/33, 36/35 || || 4 || 60 || 26/25, 28/27, 33/32, 35/34 || || 5 || 75 || 22/21, 25/24, 27/26 || || 6 || 90 || 19/18, 20/19, 21/20 || || 7 || 105 || 16/15, 17/16, 18/17 || || 8 || 120 || 14/13, 15/14 || || 9 || 135 || 13/12 || || 10 || 150 || 12/11 || || 11 || 165 || 11/10 || || 12 || 180 || 10/9, 21/19 || || 13 || 195 || 19/17 || || 14 || 210 || 9/8, 17/15 || || 15 || 225 || 8/7 || || 16 || 240 || || || 17 || 255 || 15/13, 22/19 || || 18 || 270 || 7/6 || || 19 || 285 || 13/11, 20/17 || || 20 || 300 || 19/16, 25/21 || || 21 || 315 || 6/5 || || 22 || 330 || 17/14 || || 23 || 345 || 11/9 || || 24 || 360 || 16/13, 21/17 || || 25 || 375 || || || 26 || 390 || 5/4 || || 27 || 405 || 19/15, 24/19 || || 28 || 420 || 14/11 || || 29 || 435 || 9/7 || || 30 || 450 || 13/10, 22/17 || || 31 || 465 || 17/13 || || 32 || 480 || 21/16, 25/19 || || 33 || 495 || 4/3 || || 34 || 510 || || || 35 || 525 || 19/14 || || 36 || 540 || 26/19 || || 37 || 555 || 11/8 || || 38 || 570 || 18/13 || || 39 || 585 || 7/5 || || 40 || 600 || 17/12, 24/17 || *based on treating 80edo as a [[19-limit]] temperament; other approaches are possible.
Original HTML content:
<html><head><title>80edo</title></head><body>The <em>80 equal temperament</em>, often abbreviated 80-tET, 80-EDO, or 80-ET, is the scale derived by dividing the octave into 80 equally-sized steps. Each step represents a frequency ratio of exactly 15 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/cent">cent</a>s. 80et is the first equal temperament that represents the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/19-limit">19-limit</a> <a class="wiki_link" href="http://xenharmonic.wikispaces.com/tonality%20diamond">tonality diamond</a> <a class="wiki_link" href="http://xenharmonic.wikispaces.com/consistent">consistent</a>ly (it barely manages to do so).<br />
<br />
80 et <a class="wiki_link" href="http://xenharmonic.wikispaces.com/tempering%20out">tempers out</a> 136/135, 169/168, 176/175, 190/189, 221/220, 256/255, 286/285, 289/288, 325/324, 351/350, 352/351, 361/360, 364/363, 400/399, 456/455, 476/475, 540/539, 561/560, 595/594, 715/714, 936/935, 969/968, 1001/1000, 1275/1274, 1331/1330, 1445/1444, 1521/1520, 1540/1539 and 1729/1728, not to mention such important non-superparticular commas as 2048/2025, 4000/3969, 1728/1715 and 3136/3125.<br />
<br />
80 supports a profusion of 19-limit (and lower) rank two temperaments which have mostly not been explored. We might mention:<br />
<br />
31&80 <<7 6 15 27 -24 -23 -20 ... ||<br />
72&80 <<24 30 40 24 32 24 0 ... ||<br />
34&80 <<2 -4 -50 22 16 2 -40 ... ||<br />
46&80 <<2 -4 30 22 16 2 40 ... ||<br />
29&80 <<3 34 45 33 24 -37 20 ... ||<br />
12&80 <<4 -8 -20 -36 32 4 0 ... ||<br />
22&80 <<6 -10 12 -14 -32 6 -40 ... ||<br />
58&80 <<6 -10 12 -14 -32 6 40 ... ||<br />
41&80 <<7 26 25 -3 -24 -33 20 ... ||<br />
<br />
In each case, the numbers joined by an ampersand represent 19-limit <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Patent%20val">patent vals</a> (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Intervals of 80edo"></a><!-- ws:end:WikiTextHeadingRule:0 -->Intervals of 80edo</h1>
<table class="wiki_table">
<tr>
<th>degrees<br />
</th>
<th>cents<br />
</th>
<th>ratios*<br />
</th>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td>1/1<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>15<br />
</td>
<td>64/63<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>30<br />
</td>
<td>81/80<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>45<br />
</td>
<td>34/33, 36/35<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>60<br />
</td>
<td>26/25, 28/27, 33/32, 35/34<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>75<br />
</td>
<td>22/21, 25/24, 27/26<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>90<br />
</td>
<td>19/18, 20/19, 21/20<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>105<br />
</td>
<td>16/15, 17/16, 18/17<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>120<br />
</td>
<td>14/13, 15/14<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>135<br />
</td>
<td>13/12<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>150<br />
</td>
<td>12/11<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>165<br />
</td>
<td>11/10<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>180<br />
</td>
<td>10/9, 21/19<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>195<br />
</td>
<td>19/17<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>210<br />
</td>
<td>9/8, 17/15<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>225<br />
</td>
<td>8/7<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>240<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>255<br />
</td>
<td>15/13, 22/19<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>270<br />
</td>
<td>7/6<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>285<br />
</td>
<td>13/11, 20/17<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>300<br />
</td>
<td>19/16, 25/21<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>315<br />
</td>
<td>6/5<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>330<br />
</td>
<td>17/14<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>345<br />
</td>
<td>11/9<br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td>360<br />
</td>
<td>16/13, 21/17<br />
</td>
</tr>
<tr>
<td>25<br />
</td>
<td>375<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>26<br />
</td>
<td>390<br />
</td>
<td>5/4<br />
</td>
</tr>
<tr>
<td>27<br />
</td>
<td>405<br />
</td>
<td>19/15, 24/19<br />
</td>
</tr>
<tr>
<td>28<br />
</td>
<td>420<br />
</td>
<td>14/11<br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td>435<br />
</td>
<td>9/7<br />
</td>
</tr>
<tr>
<td>30<br />
</td>
<td>450<br />
</td>
<td>13/10, 22/17<br />
</td>
</tr>
<tr>
<td>31<br />
</td>
<td>465<br />
</td>
<td>17/13<br />
</td>
</tr>
<tr>
<td>32<br />
</td>
<td>480<br />
</td>
<td>21/16, 25/19<br />
</td>
</tr>
<tr>
<td>33<br />
</td>
<td>495<br />
</td>
<td>4/3<br />
</td>
</tr>
<tr>
<td>34<br />
</td>
<td>510<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>35<br />
</td>
<td>525<br />
</td>
<td>19/14<br />
</td>
</tr>
<tr>
<td>36<br />
</td>
<td>540<br />
</td>
<td>26/19<br />
</td>
</tr>
<tr>
<td>37<br />
</td>
<td>555<br />
</td>
<td>11/8<br />
</td>
</tr>
<tr>
<td>38<br />
</td>
<td>570<br />
</td>
<td>18/13<br />
</td>
</tr>
<tr>
<td>39<br />
</td>
<td>585<br />
</td>
<td>7/5<br />
</td>
</tr>
<tr>
<td>40<br />
</td>
<td>600<br />
</td>
<td>17/12, 24/17<br />
</td>
</tr>
</table>
*based on treating 80edo as a <a class="wiki_link" href="/19-limit">19-limit</a> temperament; other approaches are possible.</body></html>