Keenanismic chords: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
A '''keenanismic chord''' is an [[essentially tempered chord]] which is tempered by the keenanisma, [[385/384]].  
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>2011-12-14 23:21:47 UTC</tt>.<br>
: The original revision id was <tt>286272678</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>
<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">A //keenanismic chord// is an 11-odd-limit [[Dyadic chord|essentially tempered chord]] which is tempered by the keenanisma, 385/384. There are six keenanismic triads, consisting of three pairs of chords in inverse relationship: 1-5/4-12/7 with steps 5/4-11/8-7/6 and 1-7/6-8/5 with steps 7/6-11/8-5/4; 1-6/5-7/4 with steps 6/5-16/11-8/7 and 1-6/5-11/8 with steps 6/5-8/7-16/11; and 1-8/5-7/4 with steps 8/5-12/11-8/7 and 1-8/5-11/6 with steps 8/5-8/7-12/11.  


Keenanismic tetrads are especially interesting; there are fifteen in total. These include three palindromic tetrads: 1-5/4-11/8-12/7 with steps 5/4-11/10-5/4-7/6; 1-6/5-11/8-7/4 with steps 6/5-8/7-14/11-8/7; and 1-7/5-8/5-7/4 with steps 7/5-8/7-12/11-8/7. Then there are four remarkable tetrads in two inversely related pairs which are the remaining (circular) permutations of the intervals 5/4, 6/5, 7/6, 8/7 making up the otonal and utonal tetrad after those two chords have been removed. These are 1-5/4-3/2-12/7 with steps 5/4-6/5-8/7-7/6 and 1-6/5-3/2-7/4 with steps 6/5-5/4-7/6-8/7; and 1-5/4-10/7-12/7 with steps 5/4-8/7-6/5-7/6 and 1-5/4-16/11-7/4 with steps 5/4-7/6-6/5-8/7. Another set of four tetrads coming in two inverse pairs are 1-11/8-3/2-12/7 with steps 11/8-12/11-8/7-7/6 and 1-12/11-3/2-7/4 with steps 12/11-11/8-7/6-8/7; and 1-7/6-8/5-11/6 with steps 7/6-11/8-8/7-12/11 and 1-11/8-8/5-7/4 with steps 11/8-7/6-12/11-8/7. Another pair of inversely related tetrads are 1-6/5-11/8-3/2 with steps 6/5-8/7-12/11-4/3 and 1-6/5-8/5-7/4 with steps 6/5-4/3-12/11-8/7. Finally, there is the pair 1-8/7-5/4-11/8 with steps 8/7-12/11-11/10-16/11 and 1-16/11-8/5-7/4 with steps 16/11-11/10-12/11-8/7. Five of these tetrads, the four which permute the steps of the otonal tetrad and one of a palindromic tetrads, have all step sizes larger than 9/8, and it is these five which are featured in the [[Tablets#x4et tablets-The keenanismic tablet|keenanismic tablet]] section of the article on tablets.
Keenanismic chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 2]] in the [[11-odd-limit]], meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures.  


Finally, there are six keenanismic pentads coming in three inverse pairs. These are 1-5/4-11/8-3/2-12/7 with steps 5/4-11/10-12/11-8/7-7/6 and 1-8/7-5/4-11/8-12/7 with steps 8/7-12/11-11/10-5/4-7/6; and the pair of pairs 1-6/5-11/8-3/2-7/4 with steps 6/5-8/7-12/11-7/6-8/7 and 1-12/11-5/4-3/2-12/7 with steps 12/11-8/7-6/5-8/7-7/6 plus 1-6/5-11/8-3/2-12/7 with steps 6/5-8/7-12/11-8/7-7/6 and 1-12/11-5/4-3/2-7/4 with steps 12/11-8/7-6/5-7/6-8/7.
There are six keenanismic triads, consisting of three pairs of chords in inverse relationship:
* 1–5/4–12/7 with steps 5/4, 11/8, 7/6, and its inverse
* 1–5/4–16/11 with steps 5/4, 7/6, 11/8;
* 1–8/7–11/8 with steps 8/7, 6/5, 16/11, and its inverse
* 1–6/5–11/8 with steps 6/5, 8/7, 16/11;
* 1–8/5–7/4 with steps 8/5, 12/11, 8/7, and its inverse
* 1–12/11–7/4 with steps 12/11, 8/5, 8/7.  


If we are willing to go as high as the 15 limit, there are many more keenanismic chords, all of which have a step of size 16/15. There are six additional keenanismic triads in the 15 limit, 32 additional keenanismic tetrads and 44 additional pentads.
Keenanismic tetrads are especially interesting; there are fifteen in total. These include three palindromic tetrads:
* 1–5/4–11/8–12/7 with steps 5/4, 11/10, 5/4, 7/6;
* 1–6/5–11/8–7/4 with steps 6/5, 8/7, 14/11, 8/7;
* 1–7/5–8/5–7/4 with steps 7/5, 8/7, 12/11, 8/7.  


Equal temperaments with keenanismic tetrads include 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212, 284 and 402de.</pre></div>
Then there are four remarkable tetrads in two inversely related pairs which are the remaining (circular) permutations of the intervals 5/4, 6/5, 7/6, 8/7 making up the otonal and utonal tetrad after those two chords have been removed. These are  
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;keenanismic chords&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;em&gt;keenanismic chord&lt;/em&gt; is an 11-odd-limit &lt;a class="wiki_link" href="/Dyadic%20chord"&gt;essentially tempered chord&lt;/a&gt; which is tempered by the keenanisma, 385/384. There are six keenanismic triads, consisting of three pairs of chords in inverse relationship: 1-5/4-12/7 with steps 5/4-11/8-7/6 and 1-7/6-8/5 with steps 7/6-11/8-5/4; 1-6/5-7/4 with steps 6/5-16/11-8/7 and 1-6/5-11/8 with steps 6/5-8/7-16/11; and 1-8/5-7/4 with steps 8/5-12/11-8/7 and 1-8/5-11/6 with steps 8/5-8/7-12/11. &lt;br /&gt;
* 1–5/4–3/2–12/7 with steps 5/4, 6/5, 8/7, 7/6, and its inverse
&lt;br /&gt;
* 1–6/5–3/2–7/4 with steps 6/5, 5/4, 7/6, 8/7;  
Keenanismic tetrads are especially interesting; there are fifteen in total. These include three palindromic tetrads: 1-5/4-11/8-12/7 with steps 5/4-11/10-5/4-7/6; 1-6/5-11/8-7/4 with steps 6/5-8/7-14/11-8/7; and 1-7/5-8/5-7/4 with steps 7/5-8/7-12/11-8/7. Then there are four remarkable tetrads in two inversely related pairs which are the remaining (circular) permutations of the intervals 5/4, 6/5, 7/6, 8/7 making up the otonal and utonal tetrad after those two chords have been removed. These are 1-5/4-3/2-12/7 with steps 5/4-6/5-8/7-7/6 and 1-6/5-3/2-7/4 with steps 6/5-5/4-7/6-8/7; and 1-5/4-10/7-12/7 with steps 5/4-8/7-6/5-7/6 and 1-5/4-16/11-7/4 with steps 5/4-7/6-6/5-8/7. Another set of four tetrads coming in two inverse pairs are 1-11/8-3/2-12/7 with steps 11/8-12/11-8/7-7/6 and 1-12/11-3/2-7/4 with steps 12/11-11/8-7/6-8/7; and 1-7/6-8/5-11/6 with steps 7/6-11/8-8/7-12/11 and 1-11/8-8/5-7/4 with steps 11/8-7/6-12/11-8/7. Another pair of inversely related tetrads are 1-6/5-11/8-3/2 with steps 6/5-8/7-12/11-4/3 and 1-6/5-8/5-7/4 with steps 6/5-4/3-12/11-8/7. Finally, there is the pair 1-8/7-5/4-11/8 with steps 8/7-12/11-11/10-16/11 and 1-16/11-8/5-7/4 with steps 16/11-11/10-12/11-8/7. Five of these tetrads, the four which permute the steps of the otonal tetrad and one of a palindromic tetrads, have all step sizes larger than 9/8, and it is these five which are featured in the &lt;a class="wiki_link" href="/Tablets#x4et tablets-The keenanismic tablet"&gt;keenanismic tablet&lt;/a&gt; section of the article on tablets.&lt;br /&gt;
* 1–5/4–10/7–12/7 with steps 5/4, 8/7, 6/5, 7/6, and its inverse
&lt;br /&gt;
* 1–5/4–16/11–7/4 with steps 5/4, 7/6, 6/5, 8/7.  
Finally, there are six keenanismic pentads coming in three inverse pairs. These are 1-5/4-11/8-3/2-12/7 with steps 5/4-11/10-12/11-8/7-7/6 and 1-8/7-5/4-11/8-12/7 with steps 8/7-12/11-11/10-5/4-7/6; and the pair of pairs 1-6/5-11/8-3/2-7/4 with steps 6/5-8/7-12/11-7/6-8/7 and 1-12/11-5/4-3/2-12/7 with steps 12/11-8/7-6/5-8/7-7/6 plus 1-6/5-11/8-3/2-12/7 with steps 6/5-8/7-12/11-8/7-7/6 and 1-12/11-5/4-3/2-7/4 with steps 12/11-8/7-6/5-7/6-8/7.&lt;br /&gt;
 
&lt;br /&gt;
Another set of four tetrads coming in two inverse pairs are  
If we are willing to go as high as the 15 limit, there are many more keenanismic chords, all of which have a step of size 16/15. There are six additional keenanismic triads in the 15 limit, 32 additional keenanismic tetrads and 44 additional pentads.&lt;br /&gt;
* 1–11/8–3/2–12/7 with steps 11/8, 12/11, 8/7, 7/6, and its inverse
&lt;br /&gt;
* 1–12/11–3/2–7/4 with steps 12/11, 11/8, 7/6, 8/7;  
Equal temperaments with keenanismic tetrads include 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212, 284 and 402de.&lt;/body&gt;&lt;/html&gt;</pre></div>
* 1–7/6–8/5–11/6 with steps 7/6, 11/8, 8/7, 12/11, and its inverse
* 1–11/8–8/5–7/4 with steps 11/8, 7/6, 12/11, 8/7.  
 
Another pair of inversely related tetrads are  
* 1–6/5–11/8–3/2 with steps 6/5, 8/7, 12/11, 4/3, and its inverse
* 1–12/11–5/4–3/2 with steps 12/11, 8/7, 6/5, 4/3.  
 
Finally, there is the pair  
* 1–8/7–5/4–11/8 with steps 8/7, 12/11, 11/10, 16/11, and its inverse
* 1–16/11–8/5–7/4 with steps 16/11, 11/10, 12/11, 8/7.  
 
Five of these tetrads, the four which permute the steps of the otonal tetrad and one of the palindromic tetrads, have all step sizes larger than 9/8, and it is these five which are featured in the [[Tablets #The keenanismic tablet|keenanismic tablet]] section of the article on tablets.
 
Finally, there are six keenanismic pentads coming in three inverse pairs. These are  
* 1–5/4–11/8–3/2–12/7 with steps 5/4, 11/10, 12/11, 8/7, 7/6, and its inverse
* 1–12/11–6/5–3/2–7/4 with steps 12/11, 11/10, 5/4, 7/6, 8/7;  
* 1–6/5–11/8–3/2–7/4 with steps 6/5, 8/7, 12/11, 7/6, 8/7, and its inverse
* 1–12/11–5/4–3/2–12/7 with steps 12/11, 8/7, 6/5, 8/7, 7/6;
* 1–6/5–11/8–3/2–12/7 with steps 6/5, 8/7, 12/11, 8/7, 7/6, and its inverse
* 1–12/11–5/4–3/2–7/4 with steps 12/11, 8/7, 6/5, 7/6, 8/7.
 
If we are willing to go as high as the [[15-odd-limit]], there are many more keenanismic chords, all of which have a step of size [[16/15]]. There are six additional keenanismic triads in the 15-odd-limit, 32 additional keenanismic tetrads and 44 additional pentads.
 
Equal temperaments with keenanismic tetrads include {{EDOs| 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212, 284 and 402de }}.
 
[[Category:11-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Keenanismic]]

Latest revision as of 13:32, 11 October 2024

A keenanismic chord is an essentially tempered chord which is tempered by the keenanisma, 385/384.

Keenanismic chords are of pattern 2 in the 11-odd-limit, meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures.

There are six keenanismic triads, consisting of three pairs of chords in inverse relationship:

  • 1–5/4–12/7 with steps 5/4, 11/8, 7/6, and its inverse
  • 1–5/4–16/11 with steps 5/4, 7/6, 11/8;
  • 1–8/7–11/8 with steps 8/7, 6/5, 16/11, and its inverse
  • 1–6/5–11/8 with steps 6/5, 8/7, 16/11;
  • 1–8/5–7/4 with steps 8/5, 12/11, 8/7, and its inverse
  • 1–12/11–7/4 with steps 12/11, 8/5, 8/7.

Keenanismic tetrads are especially interesting; there are fifteen in total. These include three palindromic tetrads:

  • 1–5/4–11/8–12/7 with steps 5/4, 11/10, 5/4, 7/6;
  • 1–6/5–11/8–7/4 with steps 6/5, 8/7, 14/11, 8/7;
  • 1–7/5–8/5–7/4 with steps 7/5, 8/7, 12/11, 8/7.

Then there are four remarkable tetrads in two inversely related pairs which are the remaining (circular) permutations of the intervals 5/4, 6/5, 7/6, 8/7 making up the otonal and utonal tetrad after those two chords have been removed. These are

  • 1–5/4–3/2–12/7 with steps 5/4, 6/5, 8/7, 7/6, and its inverse
  • 1–6/5–3/2–7/4 with steps 6/5, 5/4, 7/6, 8/7;
  • 1–5/4–10/7–12/7 with steps 5/4, 8/7, 6/5, 7/6, and its inverse
  • 1–5/4–16/11–7/4 with steps 5/4, 7/6, 6/5, 8/7.

Another set of four tetrads coming in two inverse pairs are

  • 1–11/8–3/2–12/7 with steps 11/8, 12/11, 8/7, 7/6, and its inverse
  • 1–12/11–3/2–7/4 with steps 12/11, 11/8, 7/6, 8/7;
  • 1–7/6–8/5–11/6 with steps 7/6, 11/8, 8/7, 12/11, and its inverse
  • 1–11/8–8/5–7/4 with steps 11/8, 7/6, 12/11, 8/7.

Another pair of inversely related tetrads are

  • 1–6/5–11/8–3/2 with steps 6/5, 8/7, 12/11, 4/3, and its inverse
  • 1–12/11–5/4–3/2 with steps 12/11, 8/7, 6/5, 4/3.

Finally, there is the pair

  • 1–8/7–5/4–11/8 with steps 8/7, 12/11, 11/10, 16/11, and its inverse
  • 1–16/11–8/5–7/4 with steps 16/11, 11/10, 12/11, 8/7.

Five of these tetrads, the four which permute the steps of the otonal tetrad and one of the palindromic tetrads, have all step sizes larger than 9/8, and it is these five which are featured in the keenanismic tablet section of the article on tablets.

Finally, there are six keenanismic pentads coming in three inverse pairs. These are

  • 1–5/4–11/8–3/2–12/7 with steps 5/4, 11/10, 12/11, 8/7, 7/6, and its inverse
  • 1–12/11–6/5–3/2–7/4 with steps 12/11, 11/10, 5/4, 7/6, 8/7;
  • 1–6/5–11/8–3/2–7/4 with steps 6/5, 8/7, 12/11, 7/6, 8/7, and its inverse
  • 1–12/11–5/4–3/2–12/7 with steps 12/11, 8/7, 6/5, 8/7, 7/6;
  • 1–6/5–11/8–3/2–12/7 with steps 6/5, 8/7, 12/11, 8/7, 7/6, and its inverse
  • 1–12/11–5/4–3/2–7/4 with steps 12/11, 8/7, 6/5, 7/6, 8/7.

If we are willing to go as high as the 15-odd-limit, there are many more keenanismic chords, all of which have a step of size 16/15. There are six additional keenanismic triads in the 15-odd-limit, 32 additional keenanismic tetrads and 44 additional pentads.

Equal temperaments with keenanismic tetrads include 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212, 284 and 402de.

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