Xenharmonic Wiki

Keenanismic chords: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 243639127 - Original comment: **
m Improve readability; recategorize
 
(16 intermediate revisions by 6 users not shown)
Line 1: Line 1:
<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>2011-07-31 20:06:56 UTC</tt>.<br>
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.  
: The original revision id was <tt>243639127</tt>.<br>
 
: The revision comment was: <tt></tt><br>
There are six keenanismic triads, consisting of three pairs of chords in inverse relationship:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
* 1–5/4–12/7 with steps 5/4, 11/8, 7/6, and its inverse
<h4>Original Wikitext content:</h4>
* 1–5/4–16/11 with steps 5/4, 7/6, 11/8;
<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 tetrad is any of five 11-limit [[Dyadic chord|essentially tempered tetrads]] which are tempered by the keenanisma, 385/384. These are the temperings the 7-limit tetrads with steps 5/4-6/5-8/7-7/6 and 5/4-8/7-6/5-7/6 and their inversions 6/5-5/4-7/6-8/7 and 5/4-7/6-6/5-8/7, leading to 385/384-tempered versions of 1-5/4-3/2-12/7, 1-5/4-10/7-12/7, 1-6/5-3/2-7/4 and 1-5/4-16/11-7/4, plus the tempering of 1-14/11-16/11-7/4, with steps of size 14/11-8/7-6/5-8/7. Equal temperaments with keenanismic tetrads include 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212 and 284. </pre></div>
* 1–8/7–11/8 with steps 8/7, 6/5, 16/11, and its inverse
<h4>Original HTML content:</h4>
* 1–6/5–11/8 with steps 6/5, 8/7, 16/11;
<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 keenanismic tetrad is any of five 11-limit &lt;a class="wiki_link" href="/Dyadic%20chord"&gt;essentially tempered tetrads&lt;/a&gt; which are tempered by the keenanisma, 385/384. These are the temperings the 7-limit tetrads with steps 5/4-6/5-8/7-7/6 and 5/4-8/7-6/5-7/6 and their inversions 6/5-5/4-7/6-8/7 and 5/4-7/6-6/5-8/7, leading to 385/384-tempered versions of 1-5/4-3/2-12/7, 1-5/4-10/7-12/7, 1-6/5-3/2-7/4 and 1-5/4-16/11-7/4, plus the tempering of 1-14/11-16/11-7/4, with steps of size 14/11-8/7-6/5-8/7. Equal temperaments with keenanismic tetrads include 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212 and 284.&lt;/body&gt;&lt;/html&gt;</pre></div>
* 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 [[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]]
Retrieved from "https://en.xen.wiki/w/Keenanismic_chords"