88cET: Difference between revisions
Wikispaces>genewardsmith **Imported revision 288623966 - Original comment: ** |
Wikispaces>spt3125 **Imported revision 615833841 - Original comment: added link (deorphaning)** |
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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:spt3125|spt3125]] and made on <tt>2017-07-22 22:00:29 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>615833841</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>added link (deorphaning)</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> | ||
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==Theory== | ==Theory== | ||
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, and the chords of 88 cents temperament are listed on the page [[chords of octacot]]. From this it may be seen that octacot, and hence 88 cents temperament, share an abundance of [[Dyadic chord|essentially tempered chords]]. | 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, and the chords of 88 cents temperament are listed on the page [[chords of octacot]]. From this it may be seen that octacot, and hence 88 cents temperament, share an abundance of [[Dyadic chord|essentially tempered chords]]. | ||
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==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== | ||
||~ Degree ||~ 11th root | ||~ Degree ||~ 11th root | ||
of 7/4 ||~ 88cET ||~ 41st root of 8 | of 7/4 ||~ 88cET ||~ 41st root of 8 | ||
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||||||||||||~ **//fourth octave//** (near match) ||~ || | ||||||||||||~ **//fourth octave//** (near match) ||~ || | ||
|| 41 || 11.078 || 8 || 0 || 1197.59 || do || 1/1=0, 2/1=1200 || | || 41 || 11.078 || 8 || 0 || 1197.59 || do || 1/1=0, 2/1=1200 || | ||
==Scales== | |||
* [[symmetrical scales of 88cET]] | |||
==Compositions== | ==Compositions== | ||
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<br /> | <br /> | ||
<!-- 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> | ||
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, and the chords of 88 cents temperament are listed on the page <a class="wiki_link" href="/chords%20of%20octacot">chords of octacot</a>. From this it may be seen that octacot, and hence 88 cents temperament, share an abundance of <a class="wiki_link" href="/Dyadic%20chord">essentially tempered chords</a>.<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, and the chords of 88 cents temperament are listed on the page <a class="wiki_link" href="/chords%20of%20octacot">chords of octacot</a>. From this it may be seen that octacot, and hence 88 cents temperament, share an abundance of <a class="wiki_link" href="/Dyadic%20chord">essentially tempered chords</a>.<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 /> | ||
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<br /> | <br /> | ||
<!-- 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> | ||
<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 /> | |||
<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> | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc4"><a name="x88cET-Compositions"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc4"><a name="x88cET-Scales"></a><!-- ws:end:WikiTextHeadingRule:9 -->Scales</h2> | ||
<ul><li><a class="wiki_link" href="/symmetrical%20scales%20of%2088cET">symmetrical scales of 88cET</a></li></ul><br /> | |||
<!-- ws:start:WikiTextHeadingRule:11:&lt;h2&gt; --><h2 id="toc5"><a name="x88cET-Compositions"></a><!-- ws:end:WikiTextHeadingRule:11 -->Compositions</h2> | |||
<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-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"><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-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 /> | ||