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Mike Battaglia https://en.xen.wiki/index.php?title=Talk:Cluster_temperament/WikispacesArchive&diff=34559&oldid=prev 2018-10-01T18:01:16Z

<p>Text replacement - &quot;&#039;&#039;&#039;All discussion below is archived from the Wikispaces export in its original unaltered form.&#039;&#039;&#039;&quot; to &quot;&#039;&#039;&#039;All discussion below is archived from the Wikispaces export in its original unaltered form.&#039;&#039;&#039; &lt;span style=&quot;color:#800000&quot;&gt;&#039;&#039;...</p> <p><b>New page</b></p><div>{{WSArchiveHeader}}<br /> <br /> <br /> == amity ==<br /> Keenan, this is an interesting idea, and I&#039;m curious to see you expand on it.<br /> <br /> I&#039;m confused about your statement, &lt;tt&gt; For example, the list of amity &quot;thirds&quot; includes ...6/5 (339.5) (363.2) 5/4... where the intervals given in cents are not representable as simple JI intervals (243/200 and 100/81 are about as simple as you can get). &lt;/tt&gt;<br /> <br /> But in the 13-limit, 339.5 is 11/9 and 363.2 is 16/13, which are relatively simple. So I wonder if it would be fair to call amity a cluster temperament in the 13-limit, but not in a lower one? I understand it&#039;s a fuzzy definition right now, so maybe there are other reasons to not count amity that I don&#039;t understand.<br /> <br /> - &#039;&#039;&#039;Andrew_Heathwaite&#039;&#039;&#039; June 20, 2012, 07:37:53 AM UTC-0700<br /> ----<br /> Oh, I guess I missed this back in July. Right you are, Andrew. I guess you&#039;d call that &quot;2.3.5.11.13 hitchcock&quot; temperament. (It doesn&#039;t have the same mapping of 11 as 11-limit amity.)<br /> <br /> - &#039;&#039;&#039;keenanpepper&#039;&#039;&#039; November 24, 2012, 09:13:41 AM UTC-0800<br /> ----<br /> Hey Keenan, just saw this page, very interesting! Would you say that the schismatic temperament is a cluster temperament? If so, would 10/9 and 9/8 be a cluster, also 32/27 and 6/5, etc? This seems like an example of &quot;clustering&quot; that&#039;s been done subconsciously in the West for centuries.<br /> <br /> Some thoughts on firming things up mathematically: a cluster temperament can be found by taking two commas and tempering out the difference between them. In your slendric example, it&#039;s 49/48 and 64/63. In general, it&#039;s possible to combine these two commas to get a 3-limit comma via hermite reduction. This gives us 256/243, which implies quasi-5-edo, hence 5 clusters per octave, and pentatonic scales. The 3-limit comma is the sum of 3 7-limit commas, so there are 3 chromas from 9/8 to 32/27.<br /> <br /> I think you could take any comma of the form 2^a·3^b·P^c, with c = ±1, like 81/80 or 64/63 or 33/32, and equate some number of them to any 3-limit comma, and get a cluster temperament. Equating 64/63 and the pythagorean comma implies 12-note scales, and makes 9/8 cluster with 8/7. The amity comma equates the apotome to 5 syntonic commas, hence 7 clusters = heptatonic. Each cluster contains 5 steps between 3-limit ratios, hence 6 ratios: 32/27 - 6/5 - 11/9 - 16/13 - 5/4 - 81/64.<br /> <br /> - &#039;&#039;&#039;TallKite&#039;&#039;&#039; January 24, 2018, 06:34:40 PM UTC-0800<br /> ----</div>

WikispacesArchive>Mike Battaglia