Marvel temperaments
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[[toc|flat]] This page discusses some of the temperaments tempering out 225/224, the marvel comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august, compton, sharp, escapade and mavila. Since (5/4)^3 = 225/224 * 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess 5/4-5/4-9/7 versions of the augmented triad. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds. =Negri= Commas: 49/48, 225/224 [[POTE tuning|POTE temperament]]: 125.608 Map: [<1 2 2 3|, <0 -4 3 -2|] Wedgie: <<4 -3 2 -14 -8 13|| EDOs: 9, 10, 19, 86 ==11-limit== Commas: 45/44, 49/48, 56/55 [[POTE tuning|POTE temperament]]: 126.474 Map: [<1 2 2 3 4|, <0 -4 3 -2 -5|] EDOs: 9, 10, 19 ==13-limit== Commas: 45/44, 49/48, 56/55, 78/77 [[POTE tuning|POTE generator]]: 126.431 Map: [<1 2 2 3 4 4|, <0 -4 3 -2 -5 -3|] EDOs: 9, 10, 19 =Negric= Commas: 33/32, 49/48, 77/75 POTE generator: ~15/14 = 127.039 Map: [<1 2 2 3 3|, <0 -4 3 -2 4|] EDOs: 9, 19, 85 Badness: 0.0306 ==13-limit== Commas: 33/32, 49/48, 65/64, 91/90 POTE generator: ~15/14 = 127.039 Map: [<1 2 2 3 3 4|, <0 -4 3 -2 4 -3|] EDOs: 9, 19, 85 Badness: 0.0202 =Wizard= Commas: 225/224, 118098/117649 [[POTE tuning|POTE temperament]]: 383.256 Map: [<2 1 5 2|, <0 6 -1 10|] Wedgie: <<12 -2 20 -31 -2 52|| EDOs: 22, 50, 72, 382, 454 ==11-limit== Commas: 225/224, 385/384, 4000/3993 [[POTE tuning|POTE temperament]]: 383.232 Map: [<2 1 5 2 8|, <0 6 -1 10 -3|] EDOs: 22, 50, 72, 166, 310 ==13-limit== Commas: 225/224, 351/350, 364/363, 385/384 [[POTE tuning|POTE temperament]]: 383.389 Map: [<2 1 5 2 8 11|, <0 6 -1 10 -3 -10|] EDOs: 22, 50, 72 ==17-limit== Commas: 221/220, 273/272, 289/288, 351/350, 375/374 [[POTE tuning|POTE temperament]]: 383.381 Map: [<2 1 5 2 8 11 6|, <0 6 -1 10 -3 -10 6|] EDOs: 22, 50, 72 ==19-limit== Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342 [[POTE tuning|POTE temperament]]: 383.477 Map: [<2 1 5 2 8 11 6 2|, <0 6 -1 10 -3 -10 6 18|] EDOs: 22, 50, 72, 194, 460 =Tritonic= Commas: 225/224, 50421/50000 [[POTE tuning|POTE temperament]]: 580.286 Map: [<1 4 -3 -3|, <0 -5 11 12|] Wedgie: <<5 -11 -12 -29 -33 3|| EDOs: 29, 31, 60, 91, 122, 213 ==11-limit== Commas: 225/224, 385/384, 27783/27500 [[POTE tuning|POTE temperament]]: 580.389 Map: [<1 4 -3 -3 17|, <0 -5 11 12 -28|] EDOs: 29, 31, 60, 91, 122, 153 =Septimin= Commas: 225/224, 84035/82944 [[POTE tuning|POTE temperament]]: 263.632 Map: [<1 4 1 5|, <0 -11 6 -10|] Wedgie: <<11 -6 10 -35 -15 40|| EDOs: 9, 32, 41, 91, 132 ==11-limit== Commas: 225/224, 385/384, 2401/2376 [[POTE tuning|POTE temperament]]: 263.634 Map: [<1 4 1 5 5|, <0 -11 6 -10 -7|] EDOs: 9, 32, 41, 91 ==13-limit== Commas: 105/104, 144/143, 196/195, 245/242 [[POTE tuning|POTE temperament]]: 263.700 Map: [<1 4 1 5 5 7|, <0 -11 6 -10 -7 -15|] EDOs: 9, 41, 91 =Slender= Commas: 225/225, 589824/588245 [[POTE tuning|POTE temperament]]: ~49/48 = 38.413 Map: [<1 2 2 3|, <0 -13 10 -6|] Wedgie: <<13 -10 6 -46 -27 42|| EDOs: 31, 94, 125 Badness: 0.0569 ==11-limit== Commas: 225/224, 385/384, 1331/1323 [[POTE tuning|POTE temperament]]: ~49/48 = 38.387 Map: [<1 2 2 3 4|, <0 -13 10 -6 -17|] EDOs: 31, 63, 94, 125 Badness: 25.342 ==13-limit== Commas: 225/224, 275/273, 385/384, 1331/1323 POTE generator: ~49/48 = 38.314 Map: [<1 2 2 3 4 3|, <0 -13 10 -6 -17 22|] EDOs: 31, 63, 94 Badness: 25.913 =Triton= Commas: 225/224, 1029/1000 [[POTE tuning|POTE temperament]]: 568.865 Map: [<1 0 6 7|, <0 3 -7 -8|] Wedgie: <<3 -7 -8 -18 -21 1|| EDOs: 19, 135, 289 =Marvo= Commas: 225/224, 78125000/78121827 [[POTE tuning|POTE temperament]]: (~27/20) 516.694 Map: [<1 5 12 29|, <0 -6 -17 -46|] Wedgie: <<6, 17, 46, 13, 56 59|| EDOs: 65, 72 ==11-limit== Commas: 225/224, 243/242, 4000/3993 [[POTE tuning|POTE temperament]]: 516.699 Map: [<1 5 12 29 12|, <0 -6 -17 -46 -15|] EDOs: 65, 72, 281, 497
Original HTML content:
<html><head><title>Marvel temperaments</title></head><body><!-- ws:start:WikiTextTocRule:42:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><a href="#Negri">Negri</a><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --> | <a href="#Negric">Negric</a><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --> | <a href="#Wizard">Wizard</a><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --> | <a href="#Tritonic">Tritonic</a><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --> | <a href="#Septimin">Septimin</a><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --> | <a href="#Slender">Slender</a><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --> | <a href="#Triton">Triton</a><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --> | <a href="#Marvo">Marvo</a><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --> <!-- ws:end:WikiTextTocRule:64 --><br /> This page discusses some of the temperaments tempering out 225/224, the marvel comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august, compton, sharp, escapade and mavila.<br /> <br /> Since (5/4)^3 = 225/224 * 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess 5/4-5/4-9/7 versions of the augmented triad. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Negri"></a><!-- ws:end:WikiTextHeadingRule:0 -->Negri</h1> Commas: 49/48, 225/224<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 125.608<br /> <br /> Map: [<1 2 2 3|, <0 -4 3 -2|]<br /> Wedgie: <<4 -3 2 -14 -8 13||<br /> EDOs: 9, 10, 19, 86<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Negri-11-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit</h2> Commas: 45/44, 49/48, 56/55<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 126.474<br /> <br /> Map: [<1 2 2 3 4|, <0 -4 3 -2 -5|]<br /> EDOs: 9, 10, 19<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Negri-13-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->13-limit</h2> Commas: 45/44, 49/48, 56/55, 78/77<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 126.431<br /> <br /> Map: [<1 2 2 3 4 4|, <0 -4 3 -2 -5 -3|]<br /> EDOs: 9, 10, 19<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Negric"></a><!-- ws:end:WikiTextHeadingRule:6 -->Negric</h1> Commas: 33/32, 49/48, 77/75<br /> <br /> POTE generator: ~15/14 = 127.039<br /> <br /> Map: [<1 2 2 3 3|, <0 -4 3 -2 4|]<br /> EDOs: 9, 19, 85<br /> Badness: 0.0306<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Negric-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h2> Commas: 33/32, 49/48, 65/64, 91/90<br /> <br /> POTE generator: ~15/14 = 127.039<br /> <br /> Map: [<1 2 2 3 3 4|, <0 -4 3 -2 4 -3|]<br /> EDOs: 9, 19, 85<br /> Badness: 0.0202<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="Wizard"></a><!-- ws:end:WikiTextHeadingRule:10 -->Wizard</h1> Commas: 225/224, 118098/117649<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 383.256<br /> <br /> Map: [<2 1 5 2|, <0 6 -1 10|]<br /> Wedgie: <<12 -2 20 -31 -2 52||<br /> EDOs: 22, 50, 72, 382, 454<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="Wizard-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h2> Commas: 225/224, 385/384, 4000/3993<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 383.232<br /> <br /> Map: [<2 1 5 2 8|, <0 6 -1 10 -3|]<br /> EDOs: 22, 50, 72, 166, 310<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="Wizard-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h2> Commas: 225/224, 351/350, 364/363, 385/384<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 383.389<br /> <br /> Map: [<2 1 5 2 8 11|, <0 6 -1 10 -3 -10|]<br /> EDOs: 22, 50, 72<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="Wizard-17-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->17-limit</h2> Commas: 221/220, 273/272, 289/288, 351/350, 375/374<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 383.381<br /> <br /> Map: [<2 1 5 2 8 11 6|, <0 6 -1 10 -3 -10 6|]<br /> EDOs: 22, 50, 72<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="Wizard-19-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->19-limit</h2> Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 383.477<br /> <br /> Map: [<2 1 5 2 8 11 6 2|, <0 6 -1 10 -3 -10 6 18|]<br /> EDOs: 22, 50, 72, 194, 460<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h1> --><h1 id="toc10"><a name="Tritonic"></a><!-- ws:end:WikiTextHeadingRule:20 -->Tritonic</h1> Commas: 225/224, 50421/50000<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 580.286<br /> <br /> Map: [<1 4 -3 -3|, <0 -5 11 12|]<br /> Wedgie: <<5 -11 -12 -29 -33 3||<br /> EDOs: 29, 31, 60, 91, 122, 213<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h2> --><h2 id="toc11"><a name="Tritonic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->11-limit</h2> Commas: 225/224, 385/384, 27783/27500<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 580.389<br /> <br /> Map: [<1 4 -3 -3 17|, <0 -5 11 12 -28|]<br /> EDOs: 29, 31, 60, 91, 122, 153<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h1> --><h1 id="toc12"><a name="Septimin"></a><!-- ws:end:WikiTextHeadingRule:24 -->Septimin</h1> Commas: 225/224, 84035/82944<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 263.632<br /> <br /> Map: [<1 4 1 5|, <0 -11 6 -10|]<br /> Wedgie: <<11 -6 10 -35 -15 40||<br /> EDOs: 9, 32, 41, 91, 132<br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h2> --><h2 id="toc13"><a name="Septimin-11-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->11-limit</h2> Commas: 225/224, 385/384, 2401/2376<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 263.634<br /> <br /> Map: [<1 4 1 5 5|, <0 -11 6 -10 -7|]<br /> EDOs: 9, 32, 41, 91<br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h2> --><h2 id="toc14"><a name="Septimin-13-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->13-limit</h2> Commas: 105/104, 144/143, 196/195, 245/242<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 263.700<br /> <br /> Map: [<1 4 1 5 5 7|, <0 -11 6 -10 -7 -15|]<br /> EDOs: 9, 41, 91<br /> <br /> <!-- ws:start:WikiTextHeadingRule:30:<h1> --><h1 id="toc15"><a name="Slender"></a><!-- ws:end:WikiTextHeadingRule:30 -->Slender</h1> Commas: 225/225, 589824/588245<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: ~49/48 = 38.413<br /> <br /> Map: [<1 2 2 3|, <0 -13 10 -6|]<br /> Wedgie: <<13 -10 6 -46 -27 42||<br /> EDOs: 31, 94, 125<br /> Badness: 0.0569<br /> <br /> <!-- ws:start:WikiTextHeadingRule:32:<h2> --><h2 id="toc16"><a name="Slender-11-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->11-limit</h2> Commas: 225/224, 385/384, 1331/1323<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: ~49/48 = 38.387<br /> <br /> Map: [<1 2 2 3 4|, <0 -13 10 -6 -17|]<br /> EDOs: 31, 63, 94, 125<br /> Badness: 25.342<br /> <br /> <!-- ws:start:WikiTextHeadingRule:34:<h2> --><h2 id="toc17"><a name="Slender-13-limit"></a><!-- ws:end:WikiTextHeadingRule:34 -->13-limit</h2> Commas: 225/224, 275/273, 385/384, 1331/1323<br /> <br /> POTE generator: ~49/48 = 38.314<br /> <br /> Map: [<1 2 2 3 4 3|, <0 -13 10 -6 -17 22|]<br /> EDOs: 31, 63, 94<br /> Badness: 25.913<br /> <br /> <!-- ws:start:WikiTextHeadingRule:36:<h1> --><h1 id="toc18"><a name="Triton"></a><!-- ws:end:WikiTextHeadingRule:36 -->Triton</h1> Commas: 225/224, 1029/1000<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 568.865<br /> <br /> Map: [<1 0 6 7|, <0 3 -7 -8|]<br /> Wedgie: <<3 -7 -8 -18 -21 1||<br /> EDOs: 19, 135, 289<br /> <br /> <!-- ws:start:WikiTextHeadingRule:38:<h1> --><h1 id="toc19"><a name="Marvo"></a><!-- ws:end:WikiTextHeadingRule:38 -->Marvo</h1> Commas: 225/224, 78125000/78121827<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: (~27/20) 516.694<br /> <br /> Map: [<1 5 12 29|, <0 -6 -17 -46|]<br /> Wedgie: <<6, 17, 46, 13, 56 59||<br /> EDOs: 65, 72<br /> <br /> <!-- ws:start:WikiTextHeadingRule:40:<h2> --><h2 id="toc20"><a name="Marvo-11-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->11-limit</h2> Commas: 225/224, 243/242, 4000/3993<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE temperament</a>: 516.699<br /> <br /> Map: [<1 5 12 29 12|, <0 -6 -17 -46 -15|]<br /> EDOs: 65, 72, 281, 497</body></html>