1ed88c: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-28 01:47:50 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-28 01:48:38 UTC</tt>.<br>

: The original revision id was <tt>~~288623936~~</tt>.<br>

: The original revision id was <tt>288623966</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>

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==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]].

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.

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<h2 id="toc1"><a name="x88cET-Theory"></a>Theory</h2>

<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 />

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 />

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 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-28 01:47:50 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-28 01:48:38 UTC</tt>.<br>
-: The original revision id was <tt>288623936</tt>.<br>
+: The original revision id was <tt>288623966</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>
@@ Line 10: / Line 10: @@
 ==Theory==
-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]].
+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]].
 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 87: / Line 87: @@
 &lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x88cET-Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Theory&lt;/h2&gt;
   &lt;br /&gt;
-cent &lt;a class="wiki_link" href="/Equal"&gt;equal temperament&lt;/a&gt; uses 88 cents, or 11\150 of an octave, to generate a &lt;a class="wiki_link" href="/nonoctave"&gt;nonoctave&lt;/a&gt; rank one scale. Since 88 cents is an excellent generator for &lt;a class="wiki_link" href="/Tetracot%20family"&gt;octacot temperament&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/Chords%20of%20octacot"&gt;Chords of octacot&lt;/a&gt;. From this it may be seen that octacot, and hence 88 cents temperament, share an abundance of &lt;a class="wiki_link" href="/Dyadic%20chord"&gt;essentially tempered chords&lt;/a&gt;.&lt;br /&gt;
+cent &lt;a class="wiki_link" href="/Equal"&gt;equal temperament&lt;/a&gt; uses 88 cents, or 11\150 of an octave, to generate a &lt;a class="wiki_link" href="/nonoctave"&gt;nonoctave&lt;/a&gt; rank one scale. Since 88 cents is an excellent generator for &lt;a class="wiki_link" href="/Tetracot%20family"&gt;octacot temperament&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/chords%20of%20octacot"&gt;chords of octacot&lt;/a&gt;. From this it may be seen that octacot, and hence 88 cents temperament, share an abundance of &lt;a class="wiki_link" href="/Dyadic%20chord"&gt;essentially tempered chords&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 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.&lt;br /&gt;