Pythagorean family
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[[toc|flat]] The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = |-19 12>, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo]]. While the tuning of the fifth will be that of 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it. [[POTE tuning|POTE generator]]: 15.116 Map: [<12 19 0|, <0 0 1|] EDOs: [[12edo|12]], [[72edo|72]], [[84edo|84]], 156, 240, 396 =Compton temperament= In terms of the normal list, compton adds 413343/409600 = |-14 10 -2 1> to the Pythagorean comma; however it can also be characterized by saying it adds 225/224. Compton, however, does not need to be used as a 7-limit temperament; in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80. In the either the 5 or 7-limit, [[240edo]] is an excellent tuning, with 81/80 coming in at 15 cents exactly. The major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune. In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds 441/440. For this [[72edo]] can be recommended as a tuning. Commas: 225/224, 250047/250000 [[POTE tuning|POTE generator]]: ~5/4 = 383.775 (16.225) Map: [<12 19 0 -22|, <0 0 1 2|] EDOs: 12, [[60edo|60]], 72, 228, 300c, 372bc, 444bc ==11-limit== Commas: 225/224, 441/440, 4375/4356 [[POTE tuning|POTE generator]]: ~5/4 = 383.266 (16.734) Map: [<12 19 0 -22 -42|, <0 0 1 2 3|] EDOs: 12, 60e, 72 ==13-limit== Commas: 225/224, 441/440, 351/350, 364/363 POTE generator: ~5/4 = 383.963 (16.037) Map: [<12 19 0 -22 -42 -67|, <0 0 1 2 3 4|] EDOs: 72, 228f, 300cf Badness: 0.0219 ==Comptone== Commas: 225/224, 441/440, 325/324, 1001/1000 POTE generator: ~5/4 = 382.612 (17.388) Map: [<12 19 0 -22 -42 100|, <0 0 1 2 3 -2|] EDOs: 12, 60e, 72, 204cdef, 276cdef ==11-limit== Commas: 81/80, 99/98, 128/125 POTE generator: ~36/35 = 22.723 Map: [<12 19 28 0 -26|, <0 0 0 1 2|] EDOs: 12, 48c, 108cd Badness: 0.0582 ==Catlat== Commas: 81/80, 128/125, 540/539 POTE generator: ~36/35 = 27.864 Map: [<12 19 28 0 109|, <0 0 0 1 -2|] EDOs: 36, 48c, 84c Badness: 0.0819 ==Catcall== Commas: 56/55, 81/80, 128/125 POTE generator: ~36/35 = 32.776 Map: [<12 19 28 0 8|, <0 0 0 1 1|] EDOs: 12, 24, 36, 72ce Badness: 0.0345 =Omicronbeta temperament= Commas: 225/224, 243/242, 441/440, 4375/4356 Generator: ~13/8 = 837.814 Map: [<72 114 167 202 249 266|, <0 0 0 0 0 1|] EDOs: 72, 144, 216c, 288cdf, 504bcdef Badness: 0.0300 =Hours= Commas: 19683/19600, 33075/32768 POTE generator: ~225/224 = 2.100 Map: [<24 38 0 123 83|, <0 0 1 -1 0|] Wedgie: <0 24 -24 38 -38 -123| EDOs: 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd Badness: 0.1161 ==11-limit== Commas: 243/242, 385/384, 9801/9800 POTE generator: ~225/224 = 2.161 Map: [<24 38 0 123 83|, <0 0 1 -1 0|] EDOs: 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde Badness: 0.0362 ==13-limit== Commas: 243/242, 351/350, 364/363, 385/384 POTE generator: ~225/224 = 3.955 Map: [<24 38 0 123 83 33|, <0 0 1 -1 0 1|] EDOs: 24, 48f, 72, 168df, 240df Badness: 0.0269
Original HTML content:
<html><head><title>Pythagorean family</title></head><body><!-- ws:start:WikiTextTocRule:22:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><a href="#Compton temperament">Compton temperament</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Omicronbeta temperament">Omicronbeta temperament</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Hours">Hours</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> <!-- ws:end:WikiTextTocRule:34 -->The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = |-19 12>, and hence the fifths form a closed 12-note circle of fifths, identical to <a class="wiki_link" href="/12edo">12edo</a>. While the tuning of the fifth will be that of 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 15.116<br /> <br /> Map: [<12 19 0|, <0 0 1|]<br /> EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/84edo">84</a>, 156, 240, 396<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Compton temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->Compton temperament</h1> In terms of the normal list, compton adds 413343/409600 = |-14 10 -2 1> to the Pythagorean comma; however it can also be characterized by saying it adds 225/224. Compton, however, does not need to be used as a 7-limit temperament; in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and <a class="wiki_link" href="/72edo">72edo</a>, <a class="wiki_link" href="/84edo">84edo</a> or <a class="wiki_link" href="/240edo">240edo</a> make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80.<br /> <br /> In the either the 5 or 7-limit, <a class="wiki_link" href="/240edo">240edo</a> is an excellent tuning, with 81/80 coming in at 15 cents exactly. The major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.<br /> <br /> In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds 441/440. For this <a class="wiki_link" href="/72edo">72edo</a> can be recommended as a tuning.<br /> <br /> Commas: 225/224, 250047/250000<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 383.775 (16.225)<br /> <br /> Map: [<12 19 0 -22|, <0 0 1 2|]<br /> EDOs: 12, <a class="wiki_link" href="/60edo">60</a>, 72, 228, 300c, 372bc, 444bc<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Compton temperament-11-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit</h2> Commas: 225/224, 441/440, 4375/4356<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 383.266 (16.734)<br /> <br /> Map: [<12 19 0 -22 -42|, <0 0 1 2 3|]<br /> EDOs: 12, 60e, 72<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Compton temperament-13-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->13-limit</h2> Commas: 225/224, 441/440, 351/350, 364/363<br /> <br /> POTE generator: ~5/4 = 383.963 (16.037)<br /> <br /> Map: [<12 19 0 -22 -42 -67|, <0 0 1 2 3 4|]<br /> EDOs: 72, 228f, 300cf<br /> Badness: 0.0219<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Compton temperament-Comptone"></a><!-- ws:end:WikiTextHeadingRule:6 -->Comptone</h2> Commas: 225/224, 441/440, 325/324, 1001/1000<br /> <br /> POTE generator: ~5/4 = 382.612 (17.388)<br /> <br /> Map: [<12 19 0 -22 -42 100|, <0 0 1 2 3 -2|]<br /> EDOs: 12, 60e, 72, 204cdef, 276cdef<br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Compton temperament-11-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit</h2> Commas: 81/80, 99/98, 128/125<br /> <br /> POTE generator: ~36/35 = 22.723<br /> <br /> Map: [<12 19 28 0 -26|, <0 0 0 1 2|]<br /> EDOs: 12, 48c, 108cd<br /> Badness: 0.0582<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Compton temperament-Catlat"></a><!-- ws:end:WikiTextHeadingRule:10 -->Catlat</h2> Commas: 81/80, 128/125, 540/539<br /> <br /> POTE generator: ~36/35 = 27.864<br /> <br /> Map: [<12 19 28 0 109|, <0 0 0 1 -2|]<br /> EDOs: 36, 48c, 84c<br /> Badness: 0.0819<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="Compton temperament-Catcall"></a><!-- ws:end:WikiTextHeadingRule:12 -->Catcall</h2> Commas: 56/55, 81/80, 128/125<br /> <br /> POTE generator: ~36/35 = 32.776<br /> <br /> Map: [<12 19 28 0 8|, <0 0 0 1 1|]<br /> EDOs: 12, 24, 36, 72ce<br /> Badness: 0.0345<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="Omicronbeta temperament"></a><!-- ws:end:WikiTextHeadingRule:14 -->Omicronbeta temperament</h1> Commas: 225/224, 243/242, 441/440, 4375/4356<br /> <br /> Generator: ~13/8 = 837.814<br /> <br /> Map: [<72 114 167 202 249 266|, <0 0 0 0 0 1|]<br /> EDOs: 72, 144, 216c, 288cdf, 504bcdef<br /> Badness: 0.0300<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h1> --><h1 id="toc8"><a name="Hours"></a><!-- ws:end:WikiTextHeadingRule:16 -->Hours</h1> Commas: 19683/19600, 33075/32768<br /> <br /> POTE generator: ~225/224 = 2.100<br /> <br /> Map: [<24 38 0 123 83|, <0 0 1 -1 0|]<br /> Wedgie: <0 24 -24 38 -38 -123|<br /> EDOs: 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd<br /> Badness: 0.1161<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="Hours-11-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit</h2> Commas: 243/242, 385/384, 9801/9800<br /> <br /> POTE generator: ~225/224 = 2.161<br /> <br /> Map: [<24 38 0 123 83|, <0 0 1 -1 0|]<br /> EDOs: 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde<br /> Badness: 0.0362<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h2> --><h2 id="toc10"><a name="Hours-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h2> Commas: 243/242, 351/350, 364/363, 385/384<br /> <br /> POTE generator: ~225/224 = 3.955<br /> <br /> Map: [<24 38 0 123 83 33|, <0 0 1 -1 0 1|]<br /> EDOs: 24, 48f, 72, 168df, 240df<br /> Badness: 0.0269</body></html>