Archytas clan
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[[toc|flat]] The //archytas clan// tempers out the [[http://en.wikipedia.org/wiki/Septimal_comma|Archytas comma]], 64/63. This leads to tunings with 3s and 7s quite sharp, such as those of [[22edo]]. Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, and 2430/2401 gives quasisuper. Pajara is discussed under [[diaschismic family]], dominant under [[meantone family]], augene under [[augmented family]] and porcupine under [[porcupine family]]. The rest are considered below. =Mother= Commas: 16/15, 21/20 [[POTE tuning|POTE generator]]: 721.569 Map: [<1 0 4 6|, <0 1 -1 -2|] Wedgie: <<1 -1 -2 -4 -6 -2|| EDOs: 5, 148, 153 Badness: 0.0242 =Blacksmith= Commas: 28/27, 49/48 [[POTE tuning|POTE generator]]: ~5/4 = 392.767 Map: [<5 8 0 14|, <0 0 1 0|] Wedgie: <<0 5 0 8 0 -14|| EDOs: 5, 10, 15, 25, 55 Badness: 0.0256 ==11-limit== Commas: 28/27, 49/48, 55/54 POTE generator: ~5/4 = 394.948 Map: [<5 8 0 14 29|, <0 0 1 0 -1|] EDOs: 5, 10, 15, 40be, 55be, 70bde, 85bcde Badness: 0.0246 ==13-limit== Commas: 28/27, 40/39, 49/48, 55/54 POTE generator: ~5/4 = 391.0367 Map: [<5 8 0 14 29 7|, <0 0 1 0 -1 1|] EDOs: 5, 10, 15, 25e, 40bef Badness: 0.0205 ==Farrier== Commas: 28/27, 49/48, 77/75 POTE generator: ~5/4 = 398.070 Map: [<5 8 0 14 -6|, <0 0 1 0 2|] EDOs: 5e, 15 Badness: 0.0292 [[image:blacksmith10.jpg]] ==Blackwood== //Blackwood (5-limit)// Comma: 256/243 [[POTE tuning|POTE generator]]: 399.594 Map: [<5 8 0|, <0 0 1|] EDOs: 5, 10, 15 Badness: 0.0638 =Superpyth= [[Superpyth|Main article]] Commas: 64/63, 245/243 [[POTE tuning|POTE generator]]: 710.291 Map: [<1 0 -12 6|, <0 1 9 -2|] Wedgie: <<1 9 -2 12 -6 -30|| EDOs: 5, 17, 22, 27, 49 Badness: 0.0323 ==11-limit== Commas: 64/63, 100/99, 245/243 [[POTE tuning|POTE generator]]: 710.175 Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|] EDOs: 22, 49 Badness: 0.0250 ==Suprapyth== Commas: 55/54, 64/63, 99/98 POTE generator: ~3/2 = 709.495 Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|] EDOs: 5, 7, 12, 17, 22 Badness: 0.0328 =Beatles= Commas: 64/63, 686/675 [[POTE tuning|POTE generator]]: 355.904 Map: [<1 1 5 4|,<0 2 -9 -4|] Wedgie: <<2 -9 -4 -19 -12 16|| EDOs: 10, 27, 118 Badness: 0.0459 Music: [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3|Beatles Improv]] by Herman Miller ==11-limit== Commas: 64/63, 100/99, 686/675 POTE generator: ~49/40 = 356.140 Map: [<1 1 5 4 10|,<0 2 -9 -4 -22|] EDOs: 27e, 37, 64be, 91bcde Badness: 0.0456 ==Ringo== Commas: 56/55, 64/63, 540/539 POTE generator: ~11/9 = 355.419 Map: [<1 1 5 4 2|,<0 2 -9 -4 5|] EDOs: 10, 17c, 27e Badness: 0.0329 ===13-limit=== Commas: 56/55, 64/63, 78/77, 91/90 POTE generator: ~11/9 = 355.456 Map: [<1 1 5 4 2 4|,<0 2 -9 -4 5 -1|] EDOs: 10, 17c, 27e Badness: 0.0226 =Schism= Commas: 64/63, 360/343 [[POTE tuning|POTE generator]]: 701.556 Map: [<1 0 15 6|, <0 1 -8 -2|] Wedgie: <<1 -8 -2 -15 -6 18|| EDOs: 12, 53, 65 Badness: 0.0566 =Passion= Commas: 64/63, 3125/3087 [[POTE tuning|POTE generator]]: 98.153 Map: [<1 2 2 2|, <0 -5 4 10|] Wedgie: <<5 -4 -10 -18 -30 -12|| Generators: 2, 16/15 EDOs: 12, 37, 49, 110, 379 Badness: 0.0623 =Quasisuper= Commas: 64/63, 2430/2401 [[POTE tuning|POTE generator]]: 708.328 Map: [<1 0 23 6|, <0 1 -13 -2|] Wedgie: <<1 -13 -2 -23 -2 -6 32|| EDOs: 22, 61 Badness: 0.0638 ==Quasisupra== Commas: 64/63, 99/98, 121/120 Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament [[supra]], with the quasisuper mapping of 5 thrown in (rather than the superpyth mapping of 5, which results in suprapyth). POTE generator: 708.205 Map: [<1 2 -3 2 1|, <0 -1 13 2 6|] EDOs: 17c, 22, 27c, 39d, 61d Badness: 0.0322 =Progress= Commas: 64/63, 392/375 POTE generator: ~7/5 = 562.122 Map: [<1 0 5 6|, <0 3 -5 -6|] Wedgie: <<3 -5 -6 -15 -18 0|| EDOs: 13, 15, 32c, 79bcc, 111bcc Badness: 0.0664 ==11-limit== Commas: 56/55, 64/63, 77/75 POTE generator: ~7/5 = 562.085 Map: [<1 0 5 6 4|, <0 3 -5 -6 -1|] EDOs: 13, 15, 32c, 47bc, 79bcce Badness: 0.0310
Original HTML content:
<html><head><title>Archytas clan</title></head><body><!-- ws:start:WikiTextTocRule:38:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><a href="#Mother">Mother</a><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | <a href="#Blacksmith">Blacksmith</a><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --> | <a href="#Superpyth">Superpyth</a><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --> | <a href="#Beatles">Beatles</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: --> | <a href="#Schism">Schism</a><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --> | <a href="#Passion">Passion</a><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --> | <a href="#Quasisuper">Quasisuper</a><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --> | <a href="#Progress">Progress</a><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --> <!-- ws:end:WikiTextTocRule:58 --><br /> The <em>archytas clan</em> tempers out the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow">Archytas comma</a>, 64/63. This leads to tunings with 3s and 7s quite sharp, such as those of <a class="wiki_link" href="/22edo">22edo</a>. Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, and 2430/2401 gives quasisuper. Pajara is discussed under <a class="wiki_link" href="/diaschismic%20family">diaschismic family</a>, dominant under <a class="wiki_link" href="/meantone%20family">meantone family</a>, augene under <a class="wiki_link" href="/augmented%20family">augmented family</a> and porcupine under <a class="wiki_link" href="/porcupine%20family">porcupine family</a>. The rest are considered below.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Mother"></a><!-- ws:end:WikiTextHeadingRule:0 -->Mother</h1> Commas: 16/15, 21/20<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 721.569<br /> <br /> Map: [<1 0 4 6|, <0 1 -1 -2|]<br /> Wedgie: <<1 -1 -2 -4 -6 -2||<br /> EDOs: 5, 148, 153<br /> Badness: 0.0242<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Blacksmith"></a><!-- ws:end:WikiTextHeadingRule:2 -->Blacksmith</h1> Commas: 28/27, 49/48<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 392.767<br /> <br /> Map: [<5 8 0 14|, <0 0 1 0|]<br /> Wedgie: <<0 5 0 8 0 -14||<br /> EDOs: 5, 10, 15, 25, 55<br /> Badness: 0.0256<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Blacksmith-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h2> Commas: 28/27, 49/48, 55/54<br /> <br /> POTE generator: ~5/4 = 394.948<br /> <br /> Map: [<5 8 0 14 29|, <0 0 1 0 -1|]<br /> EDOs: 5, 10, 15, 40be, 55be, 70bde, 85bcde<br /> Badness: 0.0246<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Blacksmith-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h2> Commas: 28/27, 40/39, 49/48, 55/54<br /> <br /> POTE generator: ~5/4 = 391.0367<br /> <br /> Map: [<5 8 0 14 29 7|, <0 0 1 0 -1 1|]<br /> EDOs: 5, 10, 15, 25e, 40bef<br /> Badness: 0.0205<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Blacksmith-Farrier"></a><!-- ws:end:WikiTextHeadingRule:8 -->Farrier</h2> Commas: 28/27, 49/48, 77/75<br /> <br /> POTE generator: ~5/4 = 398.070<br /> <br /> Map: [<5 8 0 14 -6|, <0 0 1 0 2|]<br /> EDOs: 5e, 15<br /> Badness: 0.0292<br /> <br /> <!-- ws:start:WikiTextLocalImageRule:59:<img src="/file/view/blacksmith10.jpg/271383136/blacksmith10.jpg" alt="" title="" /> --><img src="/file/view/blacksmith10.jpg/271383136/blacksmith10.jpg" alt="blacksmith10.jpg" title="blacksmith10.jpg" /><!-- ws:end:WikiTextLocalImageRule:59 --><br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Blacksmith-Blackwood"></a><!-- ws:end:WikiTextHeadingRule:10 -->Blackwood</h2> <em>Blackwood (5-limit)</em><br /> <br /> Comma: 256/243<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 399.594<br /> <br /> Map: [<5 8 0|, <0 0 1|]<br /> EDOs: 5, 10, 15<br /> Badness: 0.0638<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h1> --><h1 id="toc6"><a name="Superpyth"></a><!-- ws:end:WikiTextHeadingRule:12 -->Superpyth</h1> <a class="wiki_link" href="/Superpyth">Main article</a><br /> Commas: 64/63, 245/243<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 710.291<br /> <br /> Map: [<1 0 -12 6|, <0 1 9 -2|]<br /> Wedgie: <<1 9 -2 12 -6 -30||<br /> EDOs: 5, 17, 22, 27, 49<br /> Badness: 0.0323<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="Superpyth-11-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->11-limit</h2> Commas: 64/63, 100/99, 245/243<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 710.175<br /> <br /> Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|]<br /> EDOs: 22, 49<br /> Badness: 0.0250<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="Superpyth-Suprapyth"></a><!-- ws:end:WikiTextHeadingRule:16 -->Suprapyth</h2> Commas: 55/54, 64/63, 99/98<br /> <br /> POTE generator: ~3/2 = 709.495<br /> <br /> Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|]<br /> EDOs: 5, 7, 12, 17, 22<br /> Badness: 0.0328<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h1> --><h1 id="toc9"><a name="Beatles"></a><!-- ws:end:WikiTextHeadingRule:18 -->Beatles</h1> Commas: 64/63, 686/675<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 355.904<br /> <br /> Map: [<1 1 5 4|,<0 2 -9 -4|]<br /> Wedgie: <<2 -9 -4 -19 -12 16||<br /> EDOs: 10, 27, 118<br /> Badness: 0.0459<br /> <br /> Music: <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3" rel="nofollow">Beatles Improv</a> by Herman Miller<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h2> --><h2 id="toc10"><a name="Beatles-11-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->11-limit</h2> Commas: 64/63, 100/99, 686/675<br /> <br /> POTE generator: ~49/40 = 356.140<br /> <br /> Map: [<1 1 5 4 10|,<0 2 -9 -4 -22|]<br /> EDOs: 27e, 37, 64be, 91bcde<br /> Badness: 0.0456<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h2> --><h2 id="toc11"><a name="Beatles-Ringo"></a><!-- ws:end:WikiTextHeadingRule:22 -->Ringo</h2> Commas: 56/55, 64/63, 540/539<br /> <br /> POTE generator: ~11/9 = 355.419<br /> <br /> Map: [<1 1 5 4 2|,<0 2 -9 -4 5|]<br /> EDOs: 10, 17c, 27e<br /> Badness: 0.0329<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h3> --><h3 id="toc12"><a name="Beatles-Ringo-13-limit"></a><!-- ws:end:WikiTextHeadingRule:24 -->13-limit</h3> Commas: 56/55, 64/63, 78/77, 91/90<br /> <br /> POTE generator: ~11/9 = 355.456<br /> <br /> Map: [<1 1 5 4 2 4|,<0 2 -9 -4 5 -1|]<br /> EDOs: 10, 17c, 27e<br /> Badness: 0.0226<br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h1> --><h1 id="toc13"><a name="Schism"></a><!-- ws:end:WikiTextHeadingRule:26 -->Schism</h1> Commas: 64/63, 360/343<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.556<br /> <br /> Map: [<1 0 15 6|, <0 1 -8 -2|]<br /> Wedgie: <<1 -8 -2 -15 -6 18||<br /> EDOs: 12, 53, 65<br /> Badness: 0.0566<br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h1> --><h1 id="toc14"><a name="Passion"></a><!-- ws:end:WikiTextHeadingRule:28 -->Passion</h1> Commas: 64/63, 3125/3087<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 98.153<br /> <br /> Map: [<1 2 2 2|, <0 -5 4 10|]<br /> Wedgie: <<5 -4 -10 -18 -30 -12||<br /> Generators: 2, 16/15<br /> EDOs: 12, 37, 49, 110, 379<br /> Badness: 0.0623<br /> <br /> <!-- ws:start:WikiTextHeadingRule:30:<h1> --><h1 id="toc15"><a name="Quasisuper"></a><!-- ws:end:WikiTextHeadingRule:30 -->Quasisuper</h1> Commas: 64/63, 2430/2401<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 708.328<br /> <br /> Map: [<1 0 23 6|, <0 1 -13 -2|]<br /> Wedgie: <<1 -13 -2 -23 -2 -6 32||<br /> EDOs: 22, 61<br /> Badness: 0.0638<br /> <br /> <!-- ws:start:WikiTextHeadingRule:32:<h2> --><h2 id="toc16"><a name="Quasisuper-Quasisupra"></a><!-- ws:end:WikiTextHeadingRule:32 -->Quasisupra</h2> Commas: 64/63, 99/98, 121/120<br /> <br /> Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament <a class="wiki_link" href="/supra">supra</a>, with the quasisuper mapping of 5 thrown in (rather than the superpyth mapping of 5, which results in suprapyth).<br /> <br /> POTE generator: 708.205<br /> <br /> Map: [<1 2 -3 2 1|, <0 -1 13 2 6|]<br /> EDOs: 17c, 22, 27c, 39d, 61d<br /> Badness: 0.0322<br /> <br /> <!-- ws:start:WikiTextHeadingRule:34:<h1> --><h1 id="toc17"><a name="Progress"></a><!-- ws:end:WikiTextHeadingRule:34 -->Progress</h1> Commas: 64/63, 392/375<br /> <br /> POTE generator: ~7/5 = 562.122<br /> <br /> Map: [<1 0 5 6|, <0 3 -5 -6|]<br /> Wedgie: <<3 -5 -6 -15 -18 0||<br /> EDOs: 13, 15, 32c, 79bcc, 111bcc<br /> Badness: 0.0664<br /> <br /> <!-- ws:start:WikiTextHeadingRule:36:<h2> --><h2 id="toc18"><a name="Progress-11-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->11-limit</h2> Commas: 56/55, 64/63, 77/75<br /> <br /> POTE generator: ~7/5 = 562.085<br /> <br /> Map: [<1 0 5 6 4|, <0 3 -5 -6 -1|]<br /> EDOs: 13, 15, 32c, 47bc, 79bcce<br /> Badness: 0.0310</body></html>