1ed88c

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This revision was by author xenwolf and made on 2010-09-28 02:46:41 UTC.

The original revision id was 165931725.

The revision comment was: some typo for table, links added (a few only)

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Original Wikitext content:

=88cET= 

==Theory== 

88 cent equal temperament uses 88 cents, or 11/150 of an octave, to generate a nonoctave rank one scale. Since 88 cents is an excellent generator for [[Tetracot family|octacot temperament]], it can be viewed as the generator chain of octacot, stripped of octaves. However viewed, octacot and 88 cents equal temperament are very closely related.

Eight steps of 88 cents gives 704 cents, two cents sharp of 3/2, and eighteen gives 1584 cents, two cents flat of 5/2. Taken together this tells us that (5/2)^4/(3/2)^9 = 20000/19683, the minimal diesis or tetracot comma, must be being tempered out. Eleven steps of 88 cents gives 968 cents, less than a cent flat of 7/4, and this tells us that (7/4)^8/(3/2)^11 = 5764801/5668704 must be tempered out also. Taking this, multiplying it by the tetracot comma and taking the fourth root yields 245/243, which therefore must be tempered out also. The tetracot comma and 245/243 taken together define 7-limit octacot. 

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 ||~ Some Nearby  ||
||~        ||~        ||~ 3-steps ||~ of 3:2 ||~ JI Intervals ||
||||||||||= **//first octave//** ||
|| 0 || 0 || 0 || 0 || 1/1=0 ||
|| 1 || 88 || 87.805 || 87.744 || 22/21=80.537, 21/20=84.467, 20/19=88.801, 19/18=93.603 ||
|| 2 || 176 || 175.610 || 175.489 || [[11_10|11/10]]=165.004, 21/19=173.268, [[10_9|10/9]]=182.404  ||
|| 3 || 264 || 263.415 || 263.233 || [[7_6|7/6]]=266.871  ||
|| 4 || 352 || 351.220 || 350.978 || [[11_9|11/9]]= 347.408, 27/22=354.547, 16/13=359.472  ||
|| 5 || 440 || 439.024 || 438.722 || 32/25=427.373, [[9_7|9/7]]=435.084, 22/17	446.363  ||
|| 6 || 528 || 526.829 || 526.466 || 19/14=528.687, 49/36=533.742, [[15_11|15/11]]=536.95  ||
|| 7 || 616 || 614.634 || 614.211 ||  [[10_7|10/7]]=617.488 ||
|| 8 || 704 || 702.439 || 701.955 || [[3_2|3/2]]=701.955  ||
|| 9 || 792 || 790.244 || 789.699 ||  [[11_7|11/7]]=782.492, 30/19=790.756, 128/81=792.180, 19/12=795.558, 27/17=800.910, [[8_5|8/5]]=813.686 ||
|| 10 || 880 || 878.049 || 878.444 || [[5_3|5/3]]=884.359  ||
|| 11 || 968 || 965.854 || 965.188 || [[7_4|7/4]]=968.826 ||
|| 12 || 1056 || 1053.659 || 1052.933 || [[11_6|11/6]]=1049.363, 35/19=1057.627, 24/13=1061.427  ||
|| 13 || 1144 || 1141.463 || 1140.677 || 27/14=1137.039, 31/16=1145.036  ||
||||||||||= **//second octave//** ||
|| 14 || 32 || 29.268 || 28.421 ||   ||
|| 15 || 120 || 117.073 || 116.166 ||   ||
|| 16 || 208 || 204.878 || 203.910 ||   ||
|| 17 || 296 || 292.683 || 291.654 ||   ||
|| 18 || 384 || 380.488 || 379.399 ||   ||
|| 19 || 472 || 468.293 || 467.143 ||   ||
|| 20 || 560 || 556.098 || 554.888 ||   ||
|| 21 || 648 || 643.902 || 642.632 ||   ||
|| 22 || 736 || 731.707 || 730.376 ||   ||
|| 23 || 824 || 819.512 || 818.121 ||   ||
|| 24 || 912 || 907.317 || 905.865 ||   ||
|| 25 || 1000 || 995.122 || 993.609 ||   ||
|| 26 || 1088 || 1082.927 || 1081.354 ||   ||
|| 27 || 1176 || 1170.732 || 1169.098 ||   ||
||||||||||= **//third octave//** ||
|| 28 || 64 || 58.537 || 56.843 ||   ||
|| 29 || 152 || 146.341 || 144.587 ||   ||
|| 30 || 240 || 234.146 || 232.331 ||   ||
|| 31 || 328 || 321.951 || 320.076 ||   ||
|| 32 || 416 || 409.756 || 407.820 ||   ||
|| 33 || 504 || 497.561 || 495.564 ||   ||
|| 34 || 592 || 585.366 || 583.309 ||   ||
|| 35 || 680 || 673.171 || 671.053 ||   ||
|| 36 || 768 || 760.976 || 758.798 ||   ||
|| 37 || 856 || 848.780 || 846.542 ||   ||
|| 38 || 944 || 936.585 || 934.286 ||   ||
|| 39 || 1032 || 1024.390 || 1022.031 ||   ||
|| 40 || 1120 || 1112.195 || 1109.775 ||   ||
||||||||||= **//fourth octave//** (near match) ||
|| 41 || 8 || 0 || 1197.59 ||   ||

==Compositions== 
[[http://www.seraph.it/dep/det/88east.mp3|88 East]] by [[Carlo Serafini]]
[[http://www.seraph.it/dep/det/88vocoeast.mp3|88 VocoEast]] by [[Carlo Serafini]]
[[http://www.seraph.it/dep/det/88Bulgarians.mp3|88 Bulgarians]] by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/9660ca3450a996ea8b55713cbf36151f-15.html|blog entry]])
[[http://www.seraph.it/dep/int/88jinglebells.mp3|88 Jingle Bells]] by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/495ec175ce56cf38cb399d1cd24db164-17.html|blog entry]])

Original HTML content:

<html><head><title>88cET</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x88cET"></a><!-- ws:end:WikiTextHeadingRule:0 -->88cET</h1>
 <br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x88cET-Theory"></a><!-- ws:end:WikiTextHeadingRule:2 -->Theory</h2>
 <br />
88 cent equal temperament uses 88 cents, or 11/150 of an octave, to generate a nonoctave rank one scale. Since 88 cents is an excellent generator for <a class="wiki_link" href="/Tetracot%20family">octacot temperament</a>, it can be viewed as the generator chain of octacot, stripped of octaves. However viewed, octacot and 88 cents equal temperament are very closely related.<br />
<br />
Eight steps of 88 cents gives 704 cents, two cents sharp of 3/2, and eighteen gives 1584 cents, two cents flat of 5/2. Taken together this tells us that (5/2)^4/(3/2)^9 = 20000/19683, the minimal diesis or tetracot comma, must be being tempered out. Eleven steps of 88 cents gives 968 cents, less than a cent flat of 7/4, and this tells us that (7/4)^8/(3/2)^11 = 5764801/5668704 must be tempered out also. Taking this, multiplying it by the tetracot comma and taking the fourth root yields 245/243, which therefore must be tempered out also. The tetracot comma and 245/243 taken together define 7-limit octacot. <br />
<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 />
<!-- 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<br />
</th>
        <th>Some Nearby<br />
</th>
    </tr>
    <tr>
        <th><br />
</th>
        <th><br />
</th>
        <th>3-steps<br />
</th>
        <th>of 3:2<br />
</th>
        <th>JI Intervals<br />
</th>
    </tr>
    <tr>
        <td colspan="5" style="text-align: center;"><strong><em>first octave</em></strong><br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td>1/1=0<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>88<br />
</td>
        <td>87.805<br />
</td>
        <td>87.744<br />
</td>
        <td>22/21=80.537, 21/20=84.467, 20/19=88.801, 19/18=93.603<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>176<br />
</td>
        <td>175.610<br />
</td>
        <td>175.489<br />
</td>
        <td><a class="wiki_link" href="/11_10">11/10</a>=165.004, 21/19=173.268, <a class="wiki_link" href="/10_9">10/9</a>=182.404<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>264<br />
</td>
        <td>263.415<br />
</td>
        <td>263.233<br />
</td>
        <td><a class="wiki_link" href="/7_6">7/6</a>=266.871<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>352<br />
</td>
        <td>351.220<br />
</td>
        <td>350.978<br />
</td>
        <td><a class="wiki_link" href="/11_9">11/9</a>= 347.408, 27/22=354.547, 16/13=359.472<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>440<br />
</td>
        <td>439.024<br />
</td>
        <td>438.722<br />
</td>
        <td>32/25=427.373, <a class="wiki_link" href="/9_7">9/7</a>=435.084, 22/17    446.363<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>528<br />
</td>
        <td>526.829<br />
</td>
        <td>526.466<br />
</td>
        <td>19/14=528.687, 49/36=533.742, <a class="wiki_link" href="/15_11">15/11</a>=536.95<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>616<br />
</td>
        <td>614.634<br />
</td>
        <td>614.211<br />
</td>
        <td><a class="wiki_link" href="/10_7">10/7</a>=617.488<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>704<br />
</td>
        <td>702.439<br />
</td>
        <td>701.955<br />
</td>
        <td><a class="wiki_link" href="/3_2">3/2</a>=701.955<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>792<br />
</td>
        <td>790.244<br />
</td>
        <td>789.699<br />
</td>
        <td><a class="wiki_link" href="/11_7">11/7</a>=782.492, 30/19=790.756, 128/81=792.180, 19/12=795.558, 27/17=800.910, <a class="wiki_link" href="/8_5">8/5</a>=813.686<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>880<br />
</td>
        <td>878.049<br />
</td>
        <td>878.444<br />
</td>
        <td><a class="wiki_link" href="/5_3">5/3</a>=884.359<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>968<br />
</td>
        <td>965.854<br />
</td>
        <td>965.188<br />
</td>
        <td><a class="wiki_link" href="/7_4">7/4</a>=968.826<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>1056<br />
</td>
        <td>1053.659<br />
</td>
        <td>1052.933<br />
</td>
        <td><a class="wiki_link" href="/11_6">11/6</a>=1049.363, 35/19=1057.627, 24/13=1061.427<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>1144<br />
</td>
        <td>1141.463<br />
</td>
        <td>1140.677<br />
</td>
        <td>27/14=1137.039, 31/16=1145.036<br />
</td>
    </tr>
    <tr>
        <td colspan="5" style="text-align: center;"><strong><em>second octave</em></strong><br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>32<br />
</td>
        <td>29.268<br />
</td>
        <td>28.421<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>120<br />
</td>
        <td>117.073<br />
</td>
        <td>116.166<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>208<br />
</td>
        <td>204.878<br />
</td>
        <td>203.910<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>296<br />
</td>
        <td>292.683<br />
</td>
        <td>291.654<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>384<br />
</td>
        <td>380.488<br />
</td>
        <td>379.399<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>472<br />
</td>
        <td>468.293<br />
</td>
        <td>467.143<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>560<br />
</td>
        <td>556.098<br />
</td>
        <td>554.888<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>648<br />
</td>
        <td>643.902<br />
</td>
        <td>642.632<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>736<br />
</td>
        <td>731.707<br />
</td>
        <td>730.376<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>824<br />
</td>
        <td>819.512<br />
</td>
        <td>818.121<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>912<br />
</td>
        <td>907.317<br />
</td>
        <td>905.865<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>1000<br />
</td>
        <td>995.122<br />
</td>
        <td>993.609<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>1088<br />
</td>
        <td>1082.927<br />
</td>
        <td>1081.354<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>1176<br />
</td>
        <td>1170.732<br />
</td>
        <td>1169.098<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td colspan="5" style="text-align: center;"><strong><em>third octave</em></strong><br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>64<br />
</td>
        <td>58.537<br />
</td>
        <td>56.843<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>29<br />
</td>
        <td>152<br />
</td>
        <td>146.341<br />
</td>
        <td>144.587<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>30<br />
</td>
        <td>240<br />
</td>
        <td>234.146<br />
</td>
        <td>232.331<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>31<br />
</td>
        <td>328<br />
</td>
        <td>321.951<br />
</td>
        <td>320.076<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>32<br />
</td>
        <td>416<br />
</td>
        <td>409.756<br />
</td>
        <td>407.820<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>33<br />
</td>
        <td>504<br />
</td>
        <td>497.561<br />
</td>
        <td>495.564<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>34<br />
</td>
        <td>592<br />
</td>
        <td>585.366<br />
</td>
        <td>583.309<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>35<br />
</td>
        <td>680<br />
</td>
        <td>673.171<br />
</td>
        <td>671.053<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>36<br />
</td>
        <td>768<br />
</td>
        <td>760.976<br />
</td>
        <td>758.798<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>37<br />
</td>
        <td>856<br />
</td>
        <td>848.780<br />
</td>
        <td>846.542<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>38<br />
</td>
        <td>944<br />
</td>
        <td>936.585<br />
</td>
        <td>934.286<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>39<br />
</td>
        <td>1032<br />
</td>
        <td>1024.390<br />
</td>
        <td>1022.031<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>40<br />
</td>
        <td>1120<br />
</td>
        <td>1112.195<br />
</td>
        <td>1109.775<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td colspan="5" style="text-align: center;"><strong><em>fourth octave</em></strong> (near match)<br />
</td>
    </tr>
    <tr>
        <td>41<br />
</td>
        <td>8<br />
</td>
        <td>0<br />
</td>
        <td>1197.59<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>
 <a class="wiki_link_ext" href="http://www.seraph.it/dep/det/88east.mp3" rel="nofollow">88 East</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a><br />
<a class="wiki_link_ext" href="http://www.seraph.it/dep/det/88vocoeast.mp3" rel="nofollow">88 VocoEast</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a><br />
<a class="wiki_link_ext" href="http://www.seraph.it/dep/det/88Bulgarians.mp3" rel="nofollow">88 Bulgarians</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/9660ca3450a996ea8b55713cbf36151f-15.html" rel="nofollow">blog entry</a>)<br />
<a class="wiki_link_ext" href="http://www.seraph.it/dep/int/88jinglebells.mp3" rel="nofollow">88 Jingle Bells</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/495ec175ce56cf38cb399d1cd24db164-17.html" rel="nofollow">blog entry</a>)</body></html>