30edo
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Original Wikitext content:
The //30 equal division// divides the octave into 30 equal steps of precisely 40 cents each. Its [[patent val]] is a doubled version of the patent val for [[15edo]] through the 11-limit, so 30 can be viewed as a [[contorted]] version of 15. In the 13-limit it supplies the optimal patent val for [[Trienstonic clan#Quindecic|quindecic temperament]]. However, 5\30 is 200 cents, which is a good (and familiar) approximation for 9/8, and hence 30 can be viewed inconsistently, as having a 9' at 95\30 as well as a 9 at 96\30. Instead of the 18\30 fifth of 720 cents, 30 also makes available a 17\30 fifth of 680 cents. This is an ideal tuning for pelogic (5-limit mavila), which tempers out 135/128. When 30 is used for pelogic, 5\30 can again be used inconsistently as a 9/8. Below is a plot of the Z function around 30: [[image:plot30.png]] =Commas= 30 EDO tempers out the following commas. (Note: This assumes the val < [[tel/30 48 70 84 104 111|30 48 70 84 104 111]] | .) ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 || ||= 256/243 ||< | 8 -5 > ||> 90.22 ||= Limma ||= Pythagorean Minor 2nd ||= || ||= 250/243 ||< | 1 -5 3 > ||> 49.17 ||= Maximal Diesis ||= Porcupine Comma ||= || ||= 128/125 ||< | 7 0 -3 > ||> 41.06 ||= Diesis ||= Augmented Comma ||= || ||= 15625/15552 ||< | -6 -5 6 > ||> 8.11 ||= Kleisma ||= Semicomma Majeur ||= || ||= 1029/1000 ||< | -3 1 -3 3 > ||> 49.49 ||= Keega ||= ||= || ||= 49/48 ||< | -4 -1 0 2 > ||> 35.70 ||= Slendro Diesis ||= ||= || ||= 64/63 ||< | 6 -2 0 -1 > ||> 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma || ||= 64827/64000 ||< | -9 3 -3 4 > ||> 22.23 ||= Squalentine ||= ||= || ||= 875/864 ||< | -5 -3 3 1 > ||> 21.90 ||= Keema ||= ||= || ||= 126/125 ||< | 1 2 -3 1 > ||> 13.79 ||= Septimal Semicomma ||= Starling Comma ||= || ||= 4000/3969 ||< | 5 -4 3 -2 > ||> 13.47 ||= Octagar ||= ||= || ||= 1029/1024 ||< | -10 1 0 3 > ||> 8.43 ||= Gamelisma ||= ||= || ||= 6144/6125 ||< | 11 1 -3 -2 > ||> 5.36 ||= Porwell ||= ||= || ||= ||< | -4 6 -6 3 > ||> 0.33 ||= Landscape Comma ||= ||= || ||= 100/99 ||< | 2 -2 2 0 -1 > ||> 17.40 ||= Ptolemisma ||= ||= || ||= 121/120 ||< | -3 -1 -1 0 2 > ||> 14.37 ||= Biyatisma ||= ||= || ||= 176/175 ||< | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||= ||= || ||= 65536/65219 ||< | 16 0 0 -2 -3 > ||> 8.39 ||= Orgonisma ||= ||= || ||= 385/384 ||< | -7 -1 1 1 1 > ||> 4.50 ||= Keenanisma ||= ||= || ||= 441/440 ||< | -3 2 -1 2 -1 > ||> 3.93 ||= Werckisma ||= ||= || ||= 4000/3993 ||< | 5 -1 3 0 -3 > ||> 3.03 ||= Wizardharry ||= ||= || ||= 3025/3024 ||< | -4 -3 2 -1 2 > ||> 0.57 ||= Lehmerisma ||= ||= || ==Intervals== || **Step** || **Size in Cents coarse/fine** **DMS** || || 0 || 0 || || 1 || 40, 48 12° || || 2 || 80, 96 24° || || 3 || 120, 144 36° || || 4 || 160, 192 48° || || 5 || 200, 240 60° || || 6 || 240, 288 72° || || 7 || 280, 336 84° || || 8 || 320, 384 96° || || 9 || 360, 432 108° || || 10 || 400, 480 120° || || 11 || 440, 528 132° || || 12 || 480, 576 144° || || 13 || 520, 624 156° || || 14 || 560, 672 168° || || 15 || 600, 720 180° || || 16 || 640, 768 192° || || 17 || 680, 816 204° || || 18 || 720, 864 216° || || 19 || 760, 912 228° || || 20 || 800, 960 240° || || 21 || 840, 1008 252° || || 22 || 880, 1056 264° || || 23 || 920, 1104 276° || || 24 || 960, 1152 288° || || 25 || 1000, 1200 300° || || 26 || 1040, 1248 312° || || 27 || 1080, 1296 324° || || 28 || 1120, 1344 336° || || 29 || 1160, 1392 348° || =Music= [[@http://spectropolrecords.bandcamp.com/track/todd-harrop-fifteen-short-pieces|Fifteen Short Pieces]] by <span class="wiki_link_ext">Todd Harrop</span>
Original HTML content:
<html><head><title>30edo</title></head><body>The <em>30 equal division</em> divides the octave into 30 equal steps of precisely 40 cents each. Its <a class="wiki_link" href="/patent%20val">patent val</a> is a doubled version of the patent val for <a class="wiki_link" href="/15edo">15edo</a> through the 11-limit, so 30 can be viewed as a <a class="wiki_link" href="/contorted">contorted</a> version of 15. In the 13-limit it supplies the optimal patent val for <a class="wiki_link" href="/Trienstonic%20clan#Quindecic">quindecic temperament</a>.<br />
<br />
However, 5\30 is 200 cents, which is a good (and familiar) approximation for 9/8, and hence 30 can be viewed inconsistently, as having a 9' at 95\30 as well as a 9 at 96\30. Instead of the 18\30 fifth of 720 cents, 30 also makes available a 17\30 fifth of 680 cents. This is an ideal tuning for pelogic (5-limit mavila), which tempers out 135/128. When 30 is used for pelogic, 5\30 can again be used inconsistently as a 9/8.<br />
<br />
Below is a plot of the Z function around 30:<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:518:<img src="/file/view/plot30.png/220060356/plot30.png" alt="" title="" /> --><img src="/file/view/plot30.png/220060356/plot30.png" alt="plot30.png" title="plot30.png" /><!-- ws:end:WikiTextLocalImageRule:518 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:0 -->Commas</h1>
30 EDO tempers out the following commas. (Note: This assumes the val < <a class="wiki_link" href="http://tel.wikispaces.com/30%2048%2070%2084%20104%20111">30 48 70 84 104 111</a> | .)<br />
<table class="wiki_table">
<tr>
<th>Comma<br />
</th>
<th>Monzo<br />
</th>
<th>Value (Cents)<br />
</th>
<th>Name 1<br />
</th>
<th>Name 2<br />
</th>
<th>Name 3<br />
</th>
</tr>
<tr>
<td style="text-align: center;">256/243<br />
</td>
<td style="text-align: left;">| 8 -5 ><br />
</td>
<td style="text-align: right;">90.22<br />
</td>
<td style="text-align: center;">Limma<br />
</td>
<td style="text-align: center;">Pythagorean Minor 2nd<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">250/243<br />
</td>
<td style="text-align: left;">| 1 -5 3 ><br />
</td>
<td style="text-align: right;">49.17<br />
</td>
<td style="text-align: center;">Maximal Diesis<br />
</td>
<td style="text-align: center;">Porcupine Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">128/125<br />
</td>
<td style="text-align: left;">| 7 0 -3 ><br />
</td>
<td style="text-align: right;">41.06<br />
</td>
<td style="text-align: center;">Diesis<br />
</td>
<td style="text-align: center;">Augmented Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">15625/15552<br />
</td>
<td style="text-align: left;">| -6 -5 6 ><br />
</td>
<td style="text-align: right;">8.11<br />
</td>
<td style="text-align: center;">Kleisma<br />
</td>
<td style="text-align: center;">Semicomma Majeur<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">1029/1000<br />
</td>
<td style="text-align: left;">| -3 1 -3 3 ><br />
</td>
<td style="text-align: right;">49.49<br />
</td>
<td style="text-align: center;">Keega<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">49/48<br />
</td>
<td style="text-align: left;">| -4 -1 0 2 ><br />
</td>
<td style="text-align: right;">35.70<br />
</td>
<td style="text-align: center;">Slendro Diesis<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">64/63<br />
</td>
<td style="text-align: left;">| 6 -2 0 -1 ><br />
</td>
<td style="text-align: right;">27.26<br />
</td>
<td style="text-align: center;">Septimal Comma<br />
</td>
<td style="text-align: center;">Archytas' Comma<br />
</td>
<td style="text-align: center;">Leipziger Komma<br />
</td>
</tr>
<tr>
<td style="text-align: center;">64827/64000<br />
</td>
<td style="text-align: left;">| -9 3 -3 4 ><br />
</td>
<td style="text-align: right;">22.23<br />
</td>
<td style="text-align: center;">Squalentine<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">875/864<br />
</td>
<td style="text-align: left;">| -5 -3 3 1 ><br />
</td>
<td style="text-align: right;">21.90<br />
</td>
<td style="text-align: center;">Keema<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">126/125<br />
</td>
<td style="text-align: left;">| 1 2 -3 1 ><br />
</td>
<td style="text-align: right;">13.79<br />
</td>
<td style="text-align: center;">Septimal Semicomma<br />
</td>
<td style="text-align: center;">Starling Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">4000/3969<br />
</td>
<td style="text-align: left;">| 5 -4 3 -2 ><br />
</td>
<td style="text-align: right;">13.47<br />
</td>
<td style="text-align: center;">Octagar<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">1029/1024<br />
</td>
<td style="text-align: left;">| -10 1 0 3 ><br />
</td>
<td style="text-align: right;">8.43<br />
</td>
<td style="text-align: center;">Gamelisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">6144/6125<br />
</td>
<td style="text-align: left;">| 11 1 -3 -2 ><br />
</td>
<td style="text-align: right;">5.36<br />
</td>
<td style="text-align: center;">Porwell<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: left;">| -4 6 -6 3 ><br />
</td>
<td style="text-align: right;">0.33<br />
</td>
<td style="text-align: center;">Landscape Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">100/99<br />
</td>
<td style="text-align: left;">| 2 -2 2 0 -1 ><br />
</td>
<td style="text-align: right;">17.40<br />
</td>
<td style="text-align: center;">Ptolemisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">121/120<br />
</td>
<td style="text-align: left;">| -3 -1 -1 0 2 ><br />
</td>
<td style="text-align: right;">14.37<br />
</td>
<td style="text-align: center;">Biyatisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">176/175<br />
</td>
<td style="text-align: left;">| 4 0 -2 -1 1 ><br />
</td>
<td style="text-align: right;">9.86<br />
</td>
<td style="text-align: center;">Valinorsma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">65536/65219<br />
</td>
<td style="text-align: left;">| 16 0 0 -2 -3 ><br />
</td>
<td style="text-align: right;">8.39<br />
</td>
<td style="text-align: center;">Orgonisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">385/384<br />
</td>
<td style="text-align: left;">| -7 -1 1 1 1 ><br />
</td>
<td style="text-align: right;">4.50<br />
</td>
<td style="text-align: center;">Keenanisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">441/440<br />
</td>
<td style="text-align: left;">| -3 2 -1 2 -1 ><br />
</td>
<td style="text-align: right;">3.93<br />
</td>
<td style="text-align: center;">Werckisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">4000/3993<br />
</td>
<td style="text-align: left;">| 5 -1 3 0 -3 ><br />
</td>
<td style="text-align: right;">3.03<br />
</td>
<td style="text-align: center;">Wizardharry<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">3025/3024<br />
</td>
<td style="text-align: left;">| -4 -3 2 -1 2 ><br />
</td>
<td style="text-align: right;">0.57<br />
</td>
<td style="text-align: center;">Lehmerisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Commas-Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h2>
<table class="wiki_table">
<tr>
<td><strong>Step</strong><br />
</td>
<td><strong>Size in Cents coarse/fine</strong><br />
<strong>DMS</strong><br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>40, 48<br />
12°<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>80, 96<br />
24°<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>120, 144<br />
36°<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>160, 192<br />
48°<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>200, 240<br />
60°<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>240, 288<br />
72°<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>280, 336<br />
84°<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>320, 384<br />
96°<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>360, 432<br />
108°<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>400, 480<br />
120°<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>440, 528<br />
132°<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>480, 576<br />
144°<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>520, 624<br />
156°<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>560, 672<br />
168°<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>600, 720<br />
180°<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>640, 768<br />
192°<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>680, 816<br />
204°<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>720, 864<br />
216°<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>760, 912<br />
228°<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>800, 960<br />
240°<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>840, 1008<br />
252°<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>880, 1056<br />
264°<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>920, 1104<br />
276°<br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td>960, 1152<br />
288°<br />
</td>
</tr>
<tr>
<td>25<br />
</td>
<td>1000, 1200<br />
300°<br />
</td>
</tr>
<tr>
<td>26<br />
</td>
<td>1040, 1248<br />
312°<br />
</td>
</tr>
<tr>
<td>27<br />
</td>
<td>1080, 1296<br />
324°<br />
</td>
</tr>
<tr>
<td>28<br />
</td>
<td>1120, 1344<br />
336°<br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td>1160, 1392<br />
348°<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:4 -->Music</h1>
<br />
<a class="wiki_link_ext" href="http://spectropolrecords.bandcamp.com/track/todd-harrop-fifteen-short-pieces" rel="nofollow" target="_blank">Fifteen Short Pieces</a> by <span class="wiki_link_ext">Todd Harrop</span></body></html>