1ed88c: Difference between revisions
Wikispaces>Andrew_Heathwaite **Imported revision 252536858 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 278372892 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011- | : This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-11-22 23:20:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>278372892</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></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> | ||
| Line 10: | Line 10: | ||
==Theory== | ==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. | 88 cent [[Equal|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. | 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. | ||
| Line 18: | Line 18: | ||
==The 88cET family== | ==The 88cET family== | ||
Gary Morrison originally conceived of 88cET as composed of steps of exactly 88¢. Nonetheless, composers have recognized a kinship between strict 88cET and some other scales -- in particular, the 41st root of 8 (equivalent to taking three steps of [[41edo]] as a generator with no octaves), the 8th root of 3/2, and the 11th root of 7/4, the latter being a preferred variant of composer and software designer X. J. Scott. These three cousins of strict 88cET have single steps of approximately 87.805¢, 87.744¢, and 88.075¢, respectively. These small differences add up, as can be seen by examining the interval list below. | [[Gary Morrison]] originally conceived of 88cET as composed of steps of exactly 88¢. Nonetheless, composers have recognized a kinship between strict 88cET and some other scales -- in particular, the 41st root of 8 (equivalent to taking three steps of [[41edo]] as a generator with no octaves), the 8th root of 3/2, and the 11th root of 7/4, the latter being a preferred variant of composer and software designer [[X. J. Scott]]. These three cousins of strict 88cET have single steps of approximately 87.805¢, 87.744¢, and 88.075¢, respectively. These small differences add up, as can be seen by examining the interval list below. | ||
==Intervals== | ==Intervals== | ||
| Line 75: | Line 75: | ||
==Compositions== | ==Compositions== | ||
<span class="ymp-btn-page-play ymp-media-495785593852bb775b5c611348abe945">[[http://www.seraph.it/dep/det/88east.mp3|88 East]] </span> by [[Carlo Serafini]] | <span class="ymp-btn-page-play ymp-media-495785593852bb775b5c611348abe945"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover">[[http://www.seraph.it/dep/det/88east.mp3|88 East]]</span> </span> by [[Carlo Serafini]] | ||
<span class="ymp-btn-page-play ymp-media-0eec9409132570e087858a748b80b969">[[http://www.seraph.it/dep/det/88vocoeast.mp3|88 VocoEast]] </span> by [[Carlo Serafini]] | <span class="ymp-btn-page-play ymp-media-0eec9409132570e087858a748b80b969"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover">[[http://www.seraph.it/dep/det/88vocoeast.mp3|88 VocoEast]]</span> </span> by [[Carlo Serafini]] | ||
<span class="ymp-btn-page-play ymp-media-a43b46e2ba54959d045772c4e452232c">[[http://www.seraph.it/dep/det/88Bulgarians.mp3|88 Bulgarians]] </span> by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/9660ca3450a996ea8b55713cbf36151f-15.html|blog entry]]) | <span class="ymp-btn-page-play ymp-media-a43b46e2ba54959d045772c4e452232c"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover">[[http://www.seraph.it/dep/det/88Bulgarians.mp3|88 Bulgarians]]</span> </span> by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/9660ca3450a996ea8b55713cbf36151f-15.html|blog entry]]) | ||
<span class="ymp-btn-page-play ymp-media-db5d569ced088a975707d1a1ba01c739">[[http://www.seraph.it/dep/int/88jinglebells.mp3|88 Jingle Bells]] </span> by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/495ec175ce56cf38cb399d1cd24db164-17.html|blog entry]]) | <span class="ymp-btn-page-play ymp-media-db5d569ced088a975707d1a1ba01c739"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover">[[http://www.seraph.it/dep/int/88jinglebells.mp3|88 Jingle Bells]]</span> </span> by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/495ec175ce56cf38cb399d1cd24db164-17.html|blog entry]]) | ||
<span class="ymp-btn-page-play ymp-media-e32c31e0585b669853d287adb8753ae8">[[http://micro.soonlabel.com/88cent_nonoctave/STE-004_88_cent_guitar.mp3|88 cent guitar improvisation]]</span> by [[@http://www.chrisvaisvil.com|Chris Vaisvil]] | <span class="ymp-btn-page-play ymp-media-e32c31e0585b669853d287adb8753ae8"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover">[[http://micro.soonlabel.com/88cent_nonoctave/STE-004_88_cent_guitar.mp3|88 cent guitar improvisation]]</span></span> by [[@http://www.chrisvaisvil.com|Chris Vaisvil]] | ||
<span class="ymp-btn-page-play ymp-media-bb747291b4a026593ee10b7c928912e4">[[http://micro.soonlabel.com/88cent_nonoctave/Prelude_in_88_Cent_Tuning.mp3|A Simple Prelude for 88 Cent Piano]]</span> by [[http://chrisvaisvil.com/?p=951|Chris Vaisvil]] ([[http://micro.soonlabel.com/88cent_nonoctave/A_Simple_Prelude_in_88_Cent_Tuning.pdf|scordata]])</pre></div> | <span class="ymp-btn-page-play ymp-media-bb747291b4a026593ee10b7c928912e4"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover">[[http://micro.soonlabel.com/88cent_nonoctave/Prelude_in_88_Cent_Tuning.mp3|A Simple Prelude for 88 Cent Piano]]</span></span> by [[http://chrisvaisvil.com/?p=951|Chris Vaisvil]] ([[http://micro.soonlabel.com/88cent_nonoctave/A_Simple_Prelude_in_88_Cent_Tuning.pdf|scordata]])</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>88cET</title></head><body><!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="x88cET"></a><!-- ws:end:WikiTextHeadingRule:1 --><!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458032?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;9458032&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>88cET</title></head><body><!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="x88cET"></a><!-- ws:end:WikiTextHeadingRule:1 --><!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458032?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;9458032&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"> | ||
| Line 87: | Line 87: | ||
<!-- ws:start:WikiTextHeadingRule:3:&lt;h2&gt; --><h2 id="toc1"><a name="x88cET-Theory"></a><!-- ws:end:WikiTextHeadingRule:3 -->Theory</h2> | <!-- ws:start:WikiTextHeadingRule:3:&lt;h2&gt; --><h2 id="toc1"><a name="x88cET-Theory"></a><!-- ws:end:WikiTextHeadingRule:3 -->Theory</h2> | ||
<br /> | <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 /> | 88 cent <a class="wiki_link" href="/Equal">equal temperament</a> uses 88 cents, or 11/150 of an octave, to generate a <a class="wiki_link" href="/nonoctave">nonoctave</a> 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 /> | <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 /> | 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 /> | ||
| Line 95: | Line 95: | ||
<!-- ws:start:WikiTextHeadingRule:5:&lt;h2&gt; --><h2 id="toc2"><a name="x88cET-The 88cET family"></a><!-- ws:end:WikiTextHeadingRule:5 -->The 88cET family</h2> | <!-- ws:start:WikiTextHeadingRule:5:&lt;h2&gt; --><h2 id="toc2"><a name="x88cET-The 88cET family"></a><!-- ws:end:WikiTextHeadingRule:5 -->The 88cET family</h2> | ||
<br /> | <br /> | ||
Gary Morrison originally conceived of 88cET as composed of steps of exactly 88¢. Nonetheless, composers have recognized a kinship between strict 88cET and some other scales -- in particular, the 41st root of 8 (equivalent to taking three steps of <a class="wiki_link" href="/41edo">41edo</a> as a generator with no octaves), the 8th root of 3/2, and the 11th root of 7/4, the latter being a preferred variant of composer and software designer X. J. Scott. These three cousins of strict 88cET have single steps of approximately 87.805¢, 87.744¢, and 88.075¢, respectively. These small differences add up, as can be seen by examining the interval list below.<br /> | <a class="wiki_link" href="/Gary%20Morrison">Gary Morrison</a> originally conceived of 88cET as composed of steps of exactly 88¢. Nonetheless, composers have recognized a kinship between strict 88cET and some other scales -- in particular, the 41st root of 8 (equivalent to taking three steps of <a class="wiki_link" href="/41edo">41edo</a> as a generator with no octaves), the 8th root of 3/2, and the 11th root of 7/4, the latter being a preferred variant of composer and software designer <a class="wiki_link" href="/X.%20J.%20Scott">X. J. Scott</a>. These three cousins of strict 88cET have single steps of approximately 87.805¢, 87.744¢, and 88.075¢, respectively. These small differences add up, as can be seen by examining the interval list below.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc3"><a name="x88cET-Intervals"></a><!-- ws:end:WikiTextHeadingRule:7 -->Intervals</h2> | <!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc3"><a name="x88cET-Intervals"></a><!-- ws:end:WikiTextHeadingRule:7 -->Intervals</h2> | ||
| Line 837: | Line 837: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc4"><a name="x88cET-Compositions"></a><!-- ws:end:WikiTextHeadingRule:9 -->Compositions</h2> | <!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc4"><a name="x88cET-Compositions"></a><!-- ws:end:WikiTextHeadingRule:9 -->Compositions</h2> | ||
<span class="ymp-btn-page-play ymp-media-495785593852bb775b5c611348abe945"><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/88east.mp3" rel="nofollow">88 East</a> </span> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a><br /> | <span class="ymp-btn-page-play ymp-media-495785593852bb775b5c611348abe945"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/88east.mp3" rel="nofollow">88 East</a></span> </span> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a><br /> | ||
<span class="ymp-btn-page-play ymp-media-0eec9409132570e087858a748b80b969"><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/88vocoeast.mp3" rel="nofollow">88 VocoEast</a> </span> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a><br /> | <span class="ymp-btn-page-play ymp-media-0eec9409132570e087858a748b80b969"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/88vocoeast.mp3" rel="nofollow">88 VocoEast</a></span> </span> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a><br /> | ||
<span class="ymp-btn-page-play ymp-media-a43b46e2ba54959d045772c4e452232c"><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/88Bulgarians.mp3" rel="nofollow">88 Bulgarians</a> </span> 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 /> | <span class="ymp-btn-page-play ymp-media-a43b46e2ba54959d045772c4e452232c"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/88Bulgarians.mp3" rel="nofollow">88 Bulgarians</a></span> </span> 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 /> | ||
<span class="ymp-btn-page-play ymp-media-db5d569ced088a975707d1a1ba01c739"><a class="wiki_link_ext" href="http://www.seraph.it/dep/int/88jinglebells.mp3" rel="nofollow">88 Jingle Bells</a> </span> 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>)<br /> | <span class="ymp-btn-page-play ymp-media-db5d569ced088a975707d1a1ba01c739"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><a class="wiki_link_ext" href="http://www.seraph.it/dep/int/88jinglebells.mp3" rel="nofollow">88 Jingle Bells</a></span> </span> 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>)<br /> | ||
<span class="ymp-btn-page-play ymp-media-e32c31e0585b669853d287adb8753ae8"><a class="wiki_link_ext" href="http://micro.soonlabel.com/88cent_nonoctave/STE-004_88_cent_guitar.mp3" rel="nofollow">88 cent guitar improvisation</a></span> by <a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank">Chris Vaisvil</a><br /> | <span class="ymp-btn-page-play ymp-media-e32c31e0585b669853d287adb8753ae8"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><a class="wiki_link_ext" href="http://micro.soonlabel.com/88cent_nonoctave/STE-004_88_cent_guitar.mp3" rel="nofollow">88 cent guitar improvisation</a></span></span> by <a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank">Chris Vaisvil</a><br /> | ||
<span class="ymp-btn-page-play ymp-media-bb747291b4a026593ee10b7c928912e4"><a class="wiki_link_ext" href="http://micro.soonlabel.com/88cent_nonoctave/Prelude_in_88_Cent_Tuning.mp3" rel="nofollow">A Simple Prelude for 88 Cent Piano</a></span> by <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=951" rel="nofollow">Chris Vaisvil</a> (<a class="wiki_link_ext" href="http://micro.soonlabel.com/88cent_nonoctave/A_Simple_Prelude_in_88_Cent_Tuning.pdf" rel="nofollow">scordata</a>)</body></html></pre></div> | <span class="ymp-btn-page-play ymp-media-bb747291b4a026593ee10b7c928912e4"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><a class="wiki_link_ext" href="http://micro.soonlabel.com/88cent_nonoctave/Prelude_in_88_Cent_Tuning.mp3" rel="nofollow">A Simple Prelude for 88 Cent Piano</a></span></span> by <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=951" rel="nofollow">Chris Vaisvil</a> (<a class="wiki_link_ext" href="http://micro.soonlabel.com/88cent_nonoctave/A_Simple_Prelude_in_88_Cent_Tuning.pdf" rel="nofollow">scordata</a>)</body></html></pre></div> | ||