Keenanismic chords: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 285750310 - Original comment: ** |
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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:genewardsmith|genewardsmith]] and made on <tt>2011-12- | : 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> | : The original revision id was <tt>286272678</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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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. | 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. | ||
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. | |||
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> | 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> | ||
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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.<br /> | 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.<br /> | ||
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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.<br /> | |||
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Equal temperaments with keenanismic tetrads include 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212, 284 and 402de.</body></html></pre></div> | Equal temperaments with keenanismic tetrads include 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212, 284 and 402de.</body></html></pre></div> | ||
Revision as of 23:21, 14 December 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-12-14 23:21:47 UTC.
- The original revision id was 286272678.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
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. 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. 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. Equal temperaments with keenanismic tetrads include 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212, 284 and 402de.
Original HTML content:
<html><head><title>keenanismic chords</title></head><body>A <em>keenanismic chord</em> is an 11-odd-limit <a class="wiki_link" href="/Dyadic%20chord">essentially tempered chord</a> 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. <br /> <br /> 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 <a class="wiki_link" href="/Tablets#x4et tablets-The keenanismic tablet">keenanismic tablet</a> section of the article on tablets.<br /> <br /> 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.<br /> <br /> 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.<br /> <br /> Equal temperaments with keenanismic tetrads include 19, 22, 31, 41, 46, 53, 68, 72, 87, 94, 103, 118, 159, 190, 212, 284 and 402de.</body></html>