Breedsmic temperaments
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[[toc|flat]] Breedsmic temperaments are rank two temperaments tempering out the breedsma, |-5 -1 -2 4> = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma. It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system. =Hemififths= Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with [[99edo]] and [[140edo]] providing good tunings, and [[239edo]] an even better one; and other possible tunings are (160)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&58 temperament and has wedgie <<2 25 13 35 15 -40||, which tells us that it requires 25 generator steps to get to the class for major thirds, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS. By adding 243/242 (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. [[99edo]] is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding 144/143 brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be 16/13. 99 remains a good tuning choice. Commas: 2401/2400, 5120/5103 7 and 9-limit minimax [|1 0 0 0>, |7/5, 0, 2/25, 0>, |0 0 1 0>, |8/5 0 13/25 0>] Eigenvalues: 2, 5 Algebraic generator: (2 + sqrt(2))/2 Map: [<1 1 -5 -1|, <0 2 25 13|] EDOs: [[41edo|41]], [[58edo|58]], [[99edo|99]], [[239edo|239]], [[338edo|338]] Badness: 0.0222 ==11-limit== Commas: 243/242, 441/440, 896/891 POTE generator: ~11/9 = 351.521 Map: [<1 1 -5 -1 2|, <0 2 25 13 5|] EDOs: 7, 17, 41, 58, 99 Badness: 0.0235 ==13-limit== Commas: 144/143, 196/195, 243/242, 364/363 POTE generator: ~11/9 = 351.573 Map: [<1 1 -5 -1 2 4|, <0 2 25 13 5 -1|] EDOs: 7, 17, 41, 58, 99 Badness: 0.0191 =Tertiaseptal= Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well. Commas: 2401/2400, 65625/65536 POTE generator: ~256/245 = 77.191 Map: [<1 3 2 3|, <0 -22 5 -3|] EDOs: 15, 16, 31, 109, 140, 171 Badness: 0.0130 ==11-limit== Commas: 243/242, 441/440, 65625/65536 POTE generator: ~256/245 = 77.227 Map: [<1 3 2 3 7|, <0 -22 5 -3 -55|] EDOs: 15, 16, 31, 171, 202 Badness: 0.0356 =Harry= Commas: 2401/2400, 19683/19600 Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&72 temperament, with wedgie <<12 34 20 26 -2 -49||. The period is half an octave, and the generator 21/20, with generator tunings of 9/130 or 14/202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices. Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is <<12 34 20 30 ...||. Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with <<12 34 20 30 52 ...|| as the octave wedgie. [[130edo]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies. [[POTE tuning|POTE generator]]: ~21/20 = 83.156 Map: [<2 4 7 7|, <0 -6 -17 -10|] Wedgie: <<12 34 20 26 -2 -49|| EDOs: 14, 58, 72, 130, 202, 534, 938 Badness: 0.0341 ==11-limit== Commas: 243/242, 441/440, 4000/3993 [[POTE tuning|POTE generator]]: ~21/20 = 83.167 Map: [<2 4 7 7 9|, <0 -6 -17 -10 -15|] EDOs: 14, 58, 72, 130, 202 Badness: 0.0159 ==13-limit== Commas: 243/242, 351/350, 441/440, 676/675 [[POTE tuning|POTE generator]]: ~21/20 = 83.116 Map: [<2 4 7 7 9 11|, <0 -6 -17 -10 -15 -26|] EDOs: 14, 58, 72, 130, 462 Badness: 0.0130 =Quasiorwell= In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1>. It has a wedgie <<38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths. Adding 3025/3024 extends to the 11-limit and gives <<38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning. Commas: 2401/2400, 29360128/29296875 POTE generator: ~1024/875 = 271.107 Map: [<1 31 0 9|, <0 -38 3 -8|] EDOs: [[31edo|31]], [[177edo|177]], [[208edo|208]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]] Badness: 0.0358 ==11-limit== Commas: 2401/2400, 3025/3024, 5632/5625 POTE generator: ~90/77 = 271.111 Map: [<1 31 0 9 53|, <0 -38 3 -8 -64|] EDOs: [[31edo|31]], [[208edo|208]], [[239edo|239]], [[270edo|270]] Badness: 0.0175 ==13-limit== Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095 POTE generator: ~90/77 = 271.107 Map: [<1 31 0 9 53 -59|, <0 -38 3 -8 -64 81|] EDOs: [[31edo|31]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]] Badness: 0.0179 =Decoid= Commas: 2401/2400, 67108864/66976875 POTE generator: ~8/7 = 231.099 Map: [<10 0 47 36|, <0 2 -3 -1|] Wedgie: <<20 -30 -10 -94 -72 61|| EDOs: 10, 120, 130, 270 Badness: 0.0339 ==11-limit== Commas: 2401/2400, 5832/5825, 9801/9800 POTE generator: ~8/7 = 231.070 Map: [<10 0 47 36 98|, <0 2 -3 -1 -8|] EDOs: 130, 270, 670, 940, 1210 Badness: 0.0187 ==13-limit== Commas: 676/675, 1001/1000, 1716/1715, 4225/4224 POTE generator: ~8/7 = 231.083 Map: [<10 0 47 36 98 37|, <0 2 -3 -1 -8 0|] EDOs: 130, 270, 940, 1480 Badness: 0.0135
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<html><head><title>Breedsmic temperaments</title></head><body><!-- ws:start:WikiTextTocRule:28:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><a href="#Hemififths">Hemififths</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#Tertiaseptal">Tertiaseptal</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <a href="#Harry">Harry</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --> | <a href="#Quasiorwell">Quasiorwell</a><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | <a href="#Decoid">Decoid</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 -->Breedsmic temperaments are rank two temperaments tempering out the breedsma, |-5 -1 -2 4> = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.<br /> <br /> It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Hemififths"></a><!-- ws:end:WikiTextHeadingRule:0 -->Hemififths</h1> Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with <a class="wiki_link" href="/99edo">99edo</a> and <a class="wiki_link" href="/140edo">140edo</a> providing good tunings, and <a class="wiki_link" href="/239edo">239edo</a> an even better one; and other possible tunings are (160)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&58 temperament and has wedgie <<2 25 13 35 15 -40||, which tells us that it requires 25 generator steps to get to the class for major thirds, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS.<br /> <br /> By adding 243/242 (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. <a class="wiki_link" href="/99edo">99edo</a> is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding 144/143 brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be 16/13. 99 remains a good tuning choice.<br /> <br /> Commas: 2401/2400, 5120/5103<br /> <br /> 7 and 9-limit minimax<br /> [|1 0 0 0>, |7/5, 0, 2/25, 0>, |0 0 1 0>, |8/5 0 13/25 0>]<br /> Eigenvalues: 2, 5<br /> <br /> Algebraic generator: (2 + sqrt(2))/2<br /> <br /> Map: [<1 1 -5 -1|, <0 2 25 13|]<br /> EDOs: <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/58edo">58</a>, <a class="wiki_link" href="/99edo">99</a>, <a class="wiki_link" href="/239edo">239</a>, <a class="wiki_link" href="/338edo">338</a><br /> Badness: 0.0222<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Hemififths-11-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit</h2> Commas: 243/242, 441/440, 896/891<br /> <br /> POTE generator: ~11/9 = 351.521<br /> <br /> Map: [<1 1 -5 -1 2|, <0 2 25 13 5|]<br /> EDOs: 7, 17, 41, 58, 99<br /> Badness: 0.0235<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Hemififths-13-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->13-limit</h2> Commas: 144/143, 196/195, 243/242, 364/363<br /> <br /> POTE generator: ~11/9 = 351.573<br /> <br /> Map: [<1 1 -5 -1 2 4|, <0 2 25 13 5 -1|]<br /> EDOs: 7, 17, 41, 58, 99<br /> Badness: 0.0191<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Tertiaseptal"></a><!-- ws:end:WikiTextHeadingRule:6 -->Tertiaseptal</h1> Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. <a class="wiki_link" href="/171edo">171edo</a> makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.<br /> <br /> Commas: 2401/2400, 65625/65536<br /> <br /> POTE generator: ~256/245 = 77.191<br /> <br /> Map: [<1 3 2 3|, <0 -22 5 -3|]<br /> EDOs: 15, 16, 31, 109, 140, 171<br /> Badness: 0.0130<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Tertiaseptal-11-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit</h2> Commas: 243/242, 441/440, 65625/65536<br /> <br /> POTE generator: ~256/245 = 77.227<br /> <br /> Map: [<1 3 2 3 7|, <0 -22 5 -3 -55|]<br /> EDOs: 15, 16, 31, 171, 202<br /> Badness: 0.0356<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="Harry"></a><!-- ws:end:WikiTextHeadingRule:10 -->Harry</h1> Commas: 2401/2400, 19683/19600<br /> <br /> Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&72 temperament, with wedgie <<12 34 20 26 -2 -49||. The period is half an octave, and the generator 21/20, with generator tunings of 9/130 or 14/202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.<br /> <br /> Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is <<12 34 20 30 ...||.<br /> <br /> Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with <<12 34 20 30 52 ...|| as the octave wedgie. <a class="wiki_link" href="/130edo">130edo</a> is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~21/20 = 83.156<br /> <br /> Map: [<2 4 7 7|, <0 -6 -17 -10|]<br /> Wedgie: <<12 34 20 26 -2 -49||<br /> EDOs: 14, 58, 72, 130, 202, 534, 938<br /> Badness: 0.0341<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="Harry-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h2> Commas: 243/242, 441/440, 4000/3993<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~21/20 = 83.167<br /> <br /> Map: [<2 4 7 7 9|, <0 -6 -17 -10 -15|]<br /> EDOs: 14, 58, 72, 130, 202<br /> Badness: 0.0159<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="Harry-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h2> Commas: 243/242, 351/350, 441/440, 676/675<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~21/20 = 83.116<br /> <br /> Map: [<2 4 7 7 9 11|, <0 -6 -17 -10 -15 -26|]<br /> EDOs: 14, 58, 72, 130, 462<br /> Badness: 0.0130<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h1> --><h1 id="toc8"><a name="Quasiorwell"></a><!-- ws:end:WikiTextHeadingRule:16 -->Quasiorwell</h1> In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1>. It has a wedgie <<38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.<br /> <br /> Adding 3025/3024 extends to the 11-limit and gives <<38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning.<br /> <br /> Commas: 2401/2400, 29360128/29296875<br /> <br /> POTE generator: ~1024/875 = 271.107<br /> <br /> Map: [<1 31 0 9|, <0 -38 3 -8|]<br /> EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/177edo">177</a>, <a class="wiki_link" href="/208edo">208</a>, <a class="wiki_link" href="/239edo">239</a>, <a class="wiki_link" href="/270edo">270</a>, <a class="wiki_link" href="/571edo">571</a>, <a class="wiki_link" href="/841edo">841</a>, <a class="wiki_link" href="/1111edo">1111</a><br /> Badness: 0.0358<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="Quasiorwell-11-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit</h2> Commas: 2401/2400, 3025/3024, 5632/5625<br /> <br /> POTE generator: ~90/77 = 271.111<br /> <br /> Map: [<1 31 0 9 53|, <0 -38 3 -8 -64|]<br /> EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/208edo">208</a>, <a class="wiki_link" href="/239edo">239</a>, <a class="wiki_link" href="/270edo">270</a><br /> Badness: 0.0175<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h2> --><h2 id="toc10"><a name="Quasiorwell-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h2> Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095<br /> <br /> POTE generator: ~90/77 = 271.107<br /> <br /> Map: [<1 31 0 9 53 -59|, <0 -38 3 -8 -64 81|]<br /> EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/239edo">239</a>, <a class="wiki_link" href="/270edo">270</a>, <a class="wiki_link" href="/571edo">571</a>, <a class="wiki_link" href="/841edo">841</a>, <a class="wiki_link" href="/1111edo">1111</a><br /> Badness: 0.0179<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h1> --><h1 id="toc11"><a name="Decoid"></a><!-- ws:end:WikiTextHeadingRule:22 -->Decoid</h1> Commas: 2401/2400, 67108864/66976875<br /> <br /> POTE generator: ~8/7 = 231.099<br /> <br /> Map: [<10 0 47 36|, <0 2 -3 -1|]<br /> Wedgie: <<20 -30 -10 -94 -72 61||<br /> EDOs: 10, 120, 130, 270<br /> Badness: 0.0339<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h2> --><h2 id="toc12"><a name="Decoid-11-limit"></a><!-- ws:end:WikiTextHeadingRule:24 -->11-limit</h2> Commas: 2401/2400, 5832/5825, 9801/9800<br /> <br /> POTE generator: ~8/7 = 231.070<br /> <br /> Map: [<10 0 47 36 98|, <0 2 -3 -1 -8|]<br /> EDOs: 130, 270, 670, 940, 1210<br /> Badness: 0.0187<br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h2> --><h2 id="toc13"><a name="Decoid-13-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->13-limit</h2> Commas: 676/675, 1001/1000, 1716/1715, 4225/4224<br /> <br /> POTE generator: ~8/7 = 231.083<br /> <br /> Map: [<10 0 47 36 98 37|, <0 2 -3 -1 -8 0|]<br /> EDOs: 130, 270, 940, 1480<br /> Badness: 0.0135</body></html>