88cET: Difference between revisions
Wikispaces>genewardsmith **Imported revision 151468937 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 165734929 - Original comment: added table - incomplete as of now** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-09-27 13:05:37 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>165734929</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>added table - incomplete as of now</tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<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> | <h4>Original Wikitext content:</h4> | ||
| Line 15: | Line 15: | ||
Continuing on, twenty steps of 88 cents gives 1760 cents, whioh we may compare to the 1751.3 cents of 11/4 and suggests 100/99 being tempered out, and four steps gives 352 cents, which may be compared to the 359.5 cents of 16/13, and suggestes 325/324 being tempered out. These would give an extended octacot, for which 88 cents would be an excellent generator tuning. | Continuing on, twenty steps of 88 cents gives 1760 cents, whioh we may compare to the 1751.3 cents of 11/4 and suggests 100/99 being tempered out, and four steps gives 352 cents, which may be compared to the 359.5 cents of 16/13, and suggestes 325/324 being tempered out. These would give an extended octacot, for which 88 cents would be an excellent generator tuning. | ||
==Intervals== | |||
88cET is considered a very consonant tuning, and you will find that many of its intervals fall very close to simple ratios in 7- and 11-limit just intonation. It is also extremely close to [[41edo]], which is itself extremely close to the 8th root of 3:2 (a perfect fifth divided into exactly 8 logarithmically equal steps). See chart: | |||
||~ Degree ||~ 88cET ||~ 41edo ||~ 8th Root of 3:2 ||~ Some Nearby JI Intervals || | |||
|| 0 || 88 || 87.805 || || || | |||
|| 1 || 176 || 175.610 || || || | |||
|| 2 || 264 || 263.415 || || || | |||
|| 3 || 352 || 351.220 || || || | |||
|| 4 || 440 || 439.024 || || || | |||
|| 5 || 528 || 526.829 || || || | |||
|| 6 || 616 || 614.634 || || || | |||
|| 7 || 704 || 702.439 || 701.955 || || | |||
|| 8 || 792 || 790.244 || || || | |||
|| 9 || 880 || 878.049 || || || | |||
|| 10 || 968 || 965.854 || || || | |||
|| 11 || 1056 || 1053.659 || || || | |||
|| 12 || 1144 || 1141.463 || || || | |||
||||||||||~ second octave || | |||
|| 13 || 32 || 29.268 || || || | |||
|| 14 || 120 || || || || | |||
|| 15 || 208 || || || || | |||
|| 16 || 296 || || || || | |||
|| 17 || 384 || || || || | |||
|| 18 || 472 || || || || | |||
|| 19 || 560 || || || || | |||
|| 20 || 648 || || || || | |||
|| 21 || 736 || || || || | |||
|| 22 || 824 || || || || | |||
|| 23 || 912 || || || || | |||
|| 24 || 1000 || || || || | |||
|| 25 || 1088 || || || || | |||
|| 26 || 1176 || || || || | |||
||||||||||~ third octave || | |||
|| 27 || 64 || || || || | |||
|| 28 || 152 || || || || | |||
|| 29 || 240 || || || || | |||
|| 30 || 328 || || || || | |||
|| 31 || 416 || || || || | |||
|| 32 || 504 || || || || | |||
|| 33 || 592 || || || || | |||
|| 34 || 680 || || || || | |||
|| 35 || 768 || || || || | |||
|| 36 || 856 || || || || | |||
|| 37 || 944 || || || || | |||
|| 38 || 1032 || || || || | |||
|| 39 || 1120 || || || || | |||
||||||||||~ fourth octave (near match) || | |||
|| 40 || 8 || || || || | |||
==Compositions== | ==Compositions== | ||
| Line 32: | Line 82: | ||
Continuing on, twenty steps of 88 cents gives 1760 cents, whioh we may compare to the 1751.3 cents of 11/4 and suggests 100/99 being tempered out, and four steps gives 352 cents, which may be compared to the 359.5 cents of 16/13, and suggestes 325/324 being tempered out. These would give an extended octacot, for which 88 cents would be an excellent generator tuning.<br /> | Continuing on, twenty steps of 88 cents gives 1760 cents, whioh we may compare to the 1751.3 cents of 11/4 and suggests 100/99 being tempered out, and four steps gives 352 cents, which may be compared to the 359.5 cents of 16/13, and suggestes 325/324 being tempered out. These would give an extended octacot, for which 88 cents would be an excellent generator tuning.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x88cET-Compositions"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x88cET-Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h2> | ||
<br /> | |||
88cET is considered a very consonant tuning, and you will find that many of its intervals fall very close to simple ratios in 7- and 11-limit just intonation. It is also extremely close to <a class="wiki_link" href="/41edo">41edo</a>, which is itself extremely close to the 8th root of 3:2 (a perfect fifth divided into exactly 8 logarithmically equal steps). See chart:<br /> | |||
<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<th>Degree<br /> | |||
</th> | |||
<th>88cET<br /> | |||
</th> | |||
<th>41edo<br /> | |||
</th> | |||
<th>8th Root of 3:2<br /> | |||
</th> | |||
<th>Some Nearby JI Intervals<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td>0<br /> | |||
</td> | |||
<td>88<br /> | |||
</td> | |||
<td>87.805<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1<br /> | |||
</td> | |||
<td>176<br /> | |||
</td> | |||
<td>175.610<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2<br /> | |||
</td> | |||
<td>264<br /> | |||
</td> | |||
<td>263.415<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>3<br /> | |||
</td> | |||
<td>352<br /> | |||
</td> | |||
<td>351.220<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>4<br /> | |||
</td> | |||
<td>440<br /> | |||
</td> | |||
<td>439.024<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>5<br /> | |||
</td> | |||
<td>528<br /> | |||
</td> | |||
<td>526.829<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>6<br /> | |||
</td> | |||
<td>616<br /> | |||
</td> | |||
<td>614.634<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>7<br /> | |||
</td> | |||
<td>704<br /> | |||
</td> | |||
<td>702.439<br /> | |||
</td> | |||
<td>701.955<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>8<br /> | |||
</td> | |||
<td>792<br /> | |||
</td> | |||
<td>790.244<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>9<br /> | |||
</td> | |||
<td>880<br /> | |||
</td> | |||
<td>878.049<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>10<br /> | |||
</td> | |||
<td>968<br /> | |||
</td> | |||
<td>965.854<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11<br /> | |||
</td> | |||
<td>1056<br /> | |||
</td> | |||
<td>1053.659<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>12<br /> | |||
</td> | |||
<td>1144<br /> | |||
</td> | |||
<td>1141.463<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<th colspan="5">second octave<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td>13<br /> | |||
</td> | |||
<td>32<br /> | |||
</td> | |||
<td>29.268<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>14<br /> | |||
</td> | |||
<td>120<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15<br /> | |||
</td> | |||
<td>208<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>16<br /> | |||
</td> | |||
<td>296<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>17<br /> | |||
</td> | |||
<td>384<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>18<br /> | |||
</td> | |||
<td>472<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>19<br /> | |||
</td> | |||
<td>560<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>20<br /> | |||
</td> | |||
<td>648<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>21<br /> | |||
</td> | |||
<td>736<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>22<br /> | |||
</td> | |||
<td>824<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>23<br /> | |||
</td> | |||
<td>912<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>24<br /> | |||
</td> | |||
<td>1000<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>25<br /> | |||
</td> | |||
<td>1088<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>26<br /> | |||
</td> | |||
<td>1176<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<th colspan="5">third octave<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td>27<br /> | |||
</td> | |||
<td>64<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>28<br /> | |||
</td> | |||
<td>152<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>29<br /> | |||
</td> | |||
<td>240<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>30<br /> | |||
</td> | |||
<td>328<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>31<br /> | |||
</td> | |||
<td>416<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>32<br /> | |||
</td> | |||
<td>504<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>33<br /> | |||
</td> | |||
<td>592<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>34<br /> | |||
</td> | |||
<td>680<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>35<br /> | |||
</td> | |||
<td>768<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>36<br /> | |||
</td> | |||
<td>856<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>37<br /> | |||
</td> | |||
<td>944<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>38<br /> | |||
</td> | |||
<td>1032<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>39<br /> | |||
</td> | |||
<td>1120<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<th colspan="5">fourth octave (near match)<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td>40<br /> | |||
</td> | |||
<td>8<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x88cET-Compositions"></a><!-- ws:end:WikiTextHeadingRule:6 -->Compositions</h2> | |||
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