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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>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-13 23:33:32 UTC</tt>.<br>
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| : The original revision id was <tt>285750310</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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| <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.
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| 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. |
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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.
| | 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. |
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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.</pre></div>
| | Keenanismic tetrads are especially interesting; there are fifteen in total. These include three palindromic tetrads: |
| <h4>Original HTML content:</h4>
| | * 1–5/4–11/8–12/7 with steps 5/4, 11/10, 5/4, 7/6; |
| <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"><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 />
| | * 1–6/5–11/8–7/4 with steps 6/5, 8/7, 14/11, 8/7; |
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| | * 1–7/5–8/5–7/4 with steps 7/5, 8/7, 12/11, 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 <a class="wiki_link" href="/Tablets#x4et tablets-The keenanismic tablet">keenanismic tablet</a> section of the article on tablets.<br /> | | |
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| | 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 |
| 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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| | * 1–5/4–3/2–12/7 with steps 5/4, 6/5, 8/7, 7/6, and its inverse |
| 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> | | * 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. |
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| | 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. |
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| | 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. |
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| | 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. |
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| | 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. |
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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 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. |
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| | 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. |
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| | 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 }}. |
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| | [[Category:11-odd-limit chords]] |
| | [[Category:Essentially tempered chords]] |
| | [[Category:Triads]] |
| | [[Category:Tetrads]] |
| | [[Category:Pentads]] |
| | [[Category:Keenanismic]] |