25edo
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author jdfreivald and made on 2011-05-31 22:12:58 UTC.
- The original revision id was 233346970.
- The revision comment was: Added comma table.
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Original Wikitext content:
=<span style="color: #006b2e;">25 tone equal temperament</span>= 25EDO divides the octave in 25 equal steps of exact size 48 cents each. It is a good way to tune the Blackwood temperament, which takes the very sharp fifths of [[5EDO]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 and 7. 25EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 [[Just intonation subgroups|subgroup]] tuning. Looking just at 2, 5, and 7, it equates five 8/7s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a 128/125 diesis and two septimal tritones of 7/5 with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50EDO]]. If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the [[k*N subgroups|2*25 subgroup]] 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for wide range of harmony. Some example of a keyboard in 25-EDO [[image:mm25.PNG]] ==Intervals== || Degrees of 25-EDO || Cents value || || 0 || 0 || || 1 || 48 || || 2 || 96 || || 3 || 144 || || 4 || 192 || || 5 || 240 || || 6 || 288 || || 7 || 336 || || 8 || 384 || || 9 || 432 || || 10 || 480 || || 11 || 528 || || 12 || 576 || || 13 || 624 || || 14 || 672 || || 15 || 720 || || 16 || 768 || || 17 || 816 || || 18 || 864 || || 19 || 912 || || 20 || 960 || || 21 || 1008 || || 22 || 1056 || || 23 || 1104 || || 24 || 1152 || ==Commas== 25 EDO tempers out the following commas. (Note: This assumes the val < 25 40 58 70 86 93 |.) ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 || || 256/243 || | 8 -5 > || 90.22 || Limma || Pythagorean Minor 2nd || || || 3125/3072 || | -10 -1 5 > || 29.61 || Small Diesis || Magic Comma || || || 6719816/6714445 || | 38 -2 -15 > || 1.38 || Hemithirds Comma || || || || 49/48 || | -4 -1 0 2 > || 35.70 || Slendro Diesis || || || || 64/63 || | 6 -2 0 -1 > || 27.26 || Septimal Comma || Archytas' Comma || Leipziger Komma || || 3125/3087 || | 0 -2 5 -3 > || 21.18 || Gariboh || || || || 50421/50000 || | -4 1 -5 5 > || 14.52 || Trimyna || || || || 1029/1024 || | -10 1 0 3 > || 8.43 || Gamelisma || || || || 3136/3125 || | 6 0 -5 2 > || 6.08 || Hemimean || || || || 65625/65536 || | -16 1 5 1 > || 2.35 || Horwell || || || || 100/99 || | 2 -2 2 0 -1 > || 17.40 || Ptolemisma || || || || 176/175 || | 4 0 -2 -1 1 > || 9.86 || Valinorsma || || || || 16807/16384 || | -14 0 0 5 0 0 > || 44.13 || || || || || 91/90 || | -1 -2 -1 1 0 1 > || 19.13 || Superleap || || || || 676/675 || | 2 -3 -2 0 0 2 > || 2.56 || Parizeksma || || ||
Original HTML content:
<html><head><title>25edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x25 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #006b2e;">25 tone equal temperament</span></h1>
<br />
25EDO divides the octave in 25 equal steps of exact size 48 cents each. It is a good way to tune the Blackwood temperament, which takes the very sharp fifths of <a class="wiki_link" href="/5EDO">5EDO</a> as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 and 7.<br />
<br />
25EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> tuning. Looking just at 2, 5, and 7, it equates five 8/7s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a 128/125 diesis and two septimal tritones of 7/5 with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is <a class="wiki_link" href="/50EDO">50EDO</a>.<br />
<br />
If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the <a class="wiki_link" href="/k%2AN%20subgroups">2*25 subgroup</a> 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for wide range of harmony.<br />
<br />
Some example of a keyboard in 25-EDO<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:390:<img src="/file/view/mm25.PNG/179204243/mm25.PNG" alt="" title="" /> --><img src="/file/view/mm25.PNG/179204243/mm25.PNG" alt="mm25.PNG" title="mm25.PNG" /><!-- ws:end:WikiTextLocalImageRule:390 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x25 tone equal temperament-Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h2>
<br />
<table class="wiki_table">
<tr>
<td>Degrees of 25-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>48<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>96<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>144<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>192<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>240<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>288<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>336<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>384<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>432<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>480<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>528<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>576<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>624<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>672<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>720<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>768<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>816<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>864<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>912<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>960<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>1008<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>1056<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>1104<br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td>1152<br />
</td>
</tr>
</table>
<!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x25 tone equal temperament-Commas"></a><!-- ws:end:WikiTextHeadingRule:4 -->Commas</h2>
25 EDO tempers out the following commas. (Note: This assumes the val < 25 40 58 70 86 93 |.)<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>256/243<br />
</td>
<td>| 8 -5 ><br />
</td>
<td>90.22<br />
</td>
<td>Limma<br />
</td>
<td>Pythagorean Minor 2nd<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3125/3072<br />
</td>
<td>| -10 -1 5 ><br />
</td>
<td>29.61<br />
</td>
<td>Small Diesis<br />
</td>
<td>Magic Comma<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>6719816/6714445<br />
</td>
<td>| 38 -2 -15 ><br />
</td>
<td>1.38<br />
</td>
<td>Hemithirds Comma<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>49/48<br />
</td>
<td>| -4 -1 0 2 ><br />
</td>
<td>35.70<br />
</td>
<td>Slendro Diesis<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>64/63<br />
</td>
<td>| 6 -2 0 -1 ><br />
</td>
<td>27.26<br />
</td>
<td>Septimal Comma<br />
</td>
<td>Archytas' Comma<br />
</td>
<td>Leipziger Komma<br />
</td>
</tr>
<tr>
<td>3125/3087<br />
</td>
<td>| 0 -2 5 -3 ><br />
</td>
<td>21.18<br />
</td>
<td>Gariboh<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>50421/50000<br />
</td>
<td>| -4 1 -5 5 ><br />
</td>
<td>14.52<br />
</td>
<td>Trimyna<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1029/1024<br />
</td>
<td>| -10 1 0 3 ><br />
</td>
<td>8.43<br />
</td>
<td>Gamelisma<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3136/3125<br />
</td>
<td>| 6 0 -5 2 ><br />
</td>
<td>6.08<br />
</td>
<td>Hemimean<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>65625/65536<br />
</td>
<td>| -16 1 5 1 ><br />
</td>
<td>2.35<br />
</td>
<td>Horwell<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>100/99<br />
</td>
<td>| 2 -2 2 0 -1 ><br />
</td>
<td>17.40<br />
</td>
<td>Ptolemisma<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>176/175<br />
</td>
<td>| 4 0 -2 -1 1 ><br />
</td>
<td>9.86<br />
</td>
<td>Valinorsma<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>16807/16384<br />
</td>
<td>| -14 0 0 5 0 0 ><br />
</td>
<td>44.13<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>91/90<br />
</td>
<td>| -1 -2 -1 1 0 1 ><br />
</td>
<td>19.13<br />
</td>
<td>Superleap<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>676/675<br />
</td>
<td>| 2 -3 -2 0 0 2 ><br />
</td>
<td>2.56<br />
</td>
<td>Parizeksma<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
</table>
</body></html>