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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | __FORCETOC__ |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma|diaschisma]]. Its monzo is |11 -4 -2>, and flipping that yields <<2 -4 -11|| for the wedgie for 5-limit diaschismic, or '''srutal''', temperament. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo|34edo]] is a good tuning choice, with [[46edo|46edo]], [[56edo|56edo]], [[58edo|58edo]] or [[80edo|80edo]] being other possibilities. Both [[12edo|12edo]] and [[22edo|22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities. |
| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-08-24 21:33:59 UTC</tt>.<br>
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| : The original revision id was <tt>519497090</tt>.<br>
| | [[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [600.000 to 720.000] (2 to 5) |
| : 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">[[toc]]
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| The 5-limit parent comma for the **diaschismic family** is 2048/2025, the [[diaschisma]]. Its monzo is |11 -4 -2>, and flipping that yields <<2 -4 -11|| for the wedgie for 5-limit diaschismic, or **srutal**, temperament. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]] or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities. | |
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| [[Tuning Ranges of Regular Temperaments|valid range]]: [600.000 to 720.000] (2 to 5)
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| nice range: [701.955, 706.843] | | nice range: [701.955, 706.843] |
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| strict range: [701.955, 706.843] | | strict range: [701.955, 706.843] |
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| [[POTE tuning|POTE generator]]: ~3/2 = 704.898 | | [[POTE_tuning|POTE generator]]: ~3/2 = 704.898 |
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| Map: [<2 0 11|, <0 1 -2|] | | Map: [<2 0 11|, <0 1 -2|] |
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| EDOs: 34, 46, 80, 206c, 286bc | | EDOs: 34, 46, 80, 206c, 286bc |
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| ==Seven limit children== | | ==Seven limit children== |
| The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36_35|36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. Adding 4375/4374 does no significant tuning damage, so for that we keep the 5-limit label srutal. | | The second comma of the [[Normal_lists|normal comma list]] defines which 7-limit family member we are looking at. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36/35|36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. Adding 4375/4374 does no significant tuning damage, so for that we keep the 5-limit label srutal. |
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| =Srutal= | | =Srutal= |
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| valid range: [703.448, 705.882] (58 to 34d) | | valid range: [703.448, 705.882] (58 to 34d) |
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| nice range: [701.955, 706.843] | | nice range: [701.955, 706.843] |
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| strict range: [703.448, 705.882] | | strict range: [703.448, 705.882] |
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| Map: [<2 0 11 -42|, <0 1 -2 15|] | | Map: [<2 0 11 -42|, <0 1 -2 15|] |
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| Wedgie: <<2 -4 30 -11 42 81|| | | Wedgie: <<2 -4 30 -11 42 81|| |
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| EDOs: 46, 80, 126, 206cd, 332bcd | | EDOs: 46, 80, 126, 206cd, 332bcd |
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| Badness: 0.0915 | | Badness: 0.0915 |
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| valid range: [704.348, 705.882] (46 to 34d) | | valid range: [704.348, 705.882] (46 to 34d) |
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| nice range: [701.955, 706.843] | | nice range: [701.955, 706.843] |
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| strict range: [704.348, 705.882] | | strict range: [704.348, 705.882] |
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| Map: [<2 0 11 -42 -28|, <0 1 -2 15 11|] | | Map: [<2 0 11 -42 -28|, <0 1 -2 15 11|] |
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| EDOs: 46, 80, 126, 206cd | | EDOs: 46, 80, 126, 206cd |
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| Badness: 0.0353 | | Badness: 0.0353 |
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| valid range: [704.348, 705.882] (46 to 34d) | | valid range: [704.348, 705.882] (46 to 34d) |
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| nice range: [701.955, 706.843] | | nice range: [701.955, 706.843] |
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| strict range: [704.348, 705.882] | | strict range: [704.348, 705.882] |
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| Map: [<2 0 11 -42 -28 -18|, <0 1 -2 15 11 8|] | | Map: [<2 0 11 -42 -28 -18|, <0 1 -2 15 11 8|] |
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| EDOs: 34d, 46, 80, 206cd, 286bcde | | EDOs: 34d, 46, 80, 206cd, 286bcde |
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| Badness: 0.0253 | | Badness: 0.0253 |
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| =Pajara= | | =Pajara= |
| Main article: [[Pajara]] | | Main article: [[pajara|Pajara]] |
| Pajara, with wedgie <<2 -4 -4 -11 -12 2|| is closely associated with 22et (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10_7|10/7]] and a [[7_5|7/5]]. Aside from 22et, 34 with the val <34 54 79 96| and 56 with the val <56 89 130 158| are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12et and of common practice Western music in general, while retaining the distictiveness of a sharp fifth. | | |
| | Pajara, with wedgie <<2 -4 -4 -11 -12 2|| is closely associated with 22et (not to mention [[Paul_Erlich|Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10/7|10/7]] and a [[7/5|7/5]]. Aside from 22et, 34 with the val <34 54 79 96| and 56 with the val <56 89 130 158| are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12et and of common practice Western music in general, while retaining the distictiveness of a sharp fifth. |
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| Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out. | | Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out. |
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| valid range: [700.000, 720.000] (12 to 10) | | valid range: [700.000, 720.000] (12 to 10) |
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| nice range: [701.955, 715.587] | | nice range: [701.955, 715.587] |
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| strict range: [701.955, 715.587] | | strict range: [701.955, 715.587] |
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| [[POTE tuning|POTE generator]]: 707.048 | | [[POTE_tuning|POTE generator]]: 707.048 |
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| Map: [<2 0 11 12|, <0 1 -2 -2|] | | Map: [<2 0 11 12|, <0 1 -2 -2|] |
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| EDOs: 22, 34, 56 | | EDOs: 22, 34, 56 |
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| Badness: 0.0200 | | Badness: 0.0200 |
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| ==11-limit== | | ==11-limit== |
| Commas: 50/49, 64/63, 99/98 | | Commas: 50/49, 64/63, 99/98 |
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| valid range: [700.000, 709.091] (12 to 22) | | valid range: [700.000, 709.091] (12 to 22) |
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| nice range: [701.955, 715.587] | | nice range: [701.955, 715.587] |
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| strict range: [701.955, 709.091] | | strict range: [701.955, 709.091] |
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| [[POTE tuning|POTE generator]]: 706.885 | | [[POTE_tuning|POTE generator]]: 706.885 |
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| Map: [<2 0 11 12 26|, <0 1 -2 -2 -6|] | | Map: [<2 0 11 12 26|, <0 1 -2 -2 -6|] |
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| EDOs: 22, 34, 56, 146 | | EDOs: 22, 34, 56, 146 |
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| Badness: 0.0203 | | Badness: 0.0203 |
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| valid range: [] | | valid range: [] |
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| nice range: [701.955, 738.573] | | nice range: [701.955, 738.573] |
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| strict range: [] | | strict range: [] |
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| POTE generator: ~3/2 = 708.919 | | POTE generator: ~3/2 = 708.919 |
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| Map: [<2 0 11 12 26 1|, <0 1 -2 -2 -6 2|] | | Map: [<2 0 11 12 26 1|, <0 1 -2 -2 -6 2|] |
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| EDOs: 12, 22 | | EDOs: 12, 22 |
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| Badness: 0.0276 | | Badness: 0.0276 |
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| ==Pajarous== | | ==Pajarous== |
| Commas: 50/49, 55/54, 64/63 | | Commas: 50/49, 55/54, 64/63 |
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| valid range: 709.091 (22) | | valid range: 709.091 (22) |
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| nice range: [701.955, 715.803] | | nice range: [701.955, 715.803] |
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| strict range: 709.091 | | strict range: 709.091 |
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| Map: [<2 0 11 12 -9|, <0 1 -2 -2 5|] | | Map: [<2 0 11 12 -9|, <0 1 -2 -2 5|] |
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| EDOs: 10, 12e, 22, 120bce, 142bce | | EDOs: 10, 12e, 22, 120bce, 142bce |
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| Badness: 0.0283 | | Badness: 0.0283 |
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| valid range: [] | | valid range: [] |
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| nice range: [701.955, 738.573] | | nice range: [701.955, 738.573] |
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| strict range: [] | | strict range: [] |
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| Map: [<2 0 11 12 -9 1|, <0 1 -2 -2 5 2|] | | Map: [<2 0 11 12 -9 1|, <0 1 -2 -2 5 2|] |
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| EDOs: 10, 22, 54f, 76bdf | | EDOs: 10, 22, 54f, 76bdf |
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| Badness: 0.0252 | | Badness: 0.0252 |
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| Map: [<2 0 11 12 7|, <0 1 -2 -2 0|] | | Map: [<2 0 11 12 7|, <0 1 -2 -2 0|] |
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| EDOs: 10, 12, 22e, 34de | | EDOs: 10, 12, 22e, 34de |
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| Badness: 0.0238 | | Badness: 0.0238 |
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| Map: [<2 0 11 12 7 17|, <0 1 -2 -2 0 -3|] | | Map: [<2 0 11 12 7 17|, <0 1 -2 -2 0 -3|] |
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| EDOs: 10, 12f, 22ef, 34def | | EDOs: 10, 12f, 22ef, 34def |
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| Badness: 0.0205 | | Badness: 0.0205 |
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| Map: [<2 0 11 12 -9 17|, <0 1 -2 -2 5 -3|] | | Map: [<2 0 11 12 -9 17|, <0 1 -2 -2 5 -3|] |
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| EDOs: 10, 22f, 32f, 54f | | EDOs: 10, 22f, 32f, 54f |
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| Badness: 0.0274 | | Badness: 0.0274 |
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| Map: [<2 1 9 10 8|, <0 2 -4 -4 -1|] | | Map: [<2 1 9 10 8|, <0 2 -4 -4 -1|] |
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| EDOs: 20, 22, 68d, 90d | | EDOs: 20, 22, 68d, 90d |
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| Badness: 0.0389 | | Badness: 0.0389 |
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| =Diaschismic= | | =Diaschismic= |
| A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46&58. However described, diaschismic has wedgie <<2 -4 -16 -11 -31 -26||, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo|58et]] provides an excellent tuning, but an alternative is to make [[7_4|7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et. | | A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46&58. However described, diaschismic has wedgie <<2 -4 -16 -11 -31 -26||, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo|58et]] provides an excellent tuning, but an alternative is to make [[7/4|7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et. |
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| Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363. The 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58. | | Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363. The 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58. |
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| Commas: 126/125, 2048/2025 | | Commas: 126/125, 2048/2025 |
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| [[POTE tuning|POTE generator]]: 703.681 | | [[POTE_tuning|POTE generator]]: 703.681 |
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| Map: [<2 0 11 31|, <0 1 -2 -8|] | | Map: [<2 0 11 31|, <0 1 -2 -8|] |
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| EDOs: 46, 58, 104c, 162c | | EDOs: 46, 58, 104c, 162c |
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| ==11-limit== | | ==11-limit== |
| Commas: 126/125, 176/175, 896/891 | | Commas: 126/125, 176/175, 896/891 |
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| [[POTE tuning|POTE generator]]: 703.714 | | [[POTE_tuning|POTE generator]]: 703.714 |
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| Map: [<2 0 11 31 45|, <0 1 -2 -8 -12|] | | Map: [<2 0 11 31 45|, <0 1 -2 -8 -12|] |
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| EDOs: 46, 58, 104c, 162ce | | EDOs: 46, 58, 104c, 162ce |
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| ==13-limit== | | ==13-limit== |
| Commas: 126/125, 196/195, 364/363, 2048/2025 | | Commas: 126/125, 196/195, 364/363, 2048/2025 |
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| [[POTE tuning|POTE generator]]: 703.704 | | [[POTE_tuning|POTE generator]]: 703.704 |
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| Map: [<2 0 11 31 45 55|, <0 1 -2 -8 -12 -15|] | | Map: [<2 0 11 31 45 55|, <0 1 -2 -8 -12 -15|] |
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| EDOs: [[46edo|46]], [[58edo|58]], [[104edo|104c]], [[162edo|162cef]] | | EDOs: [[46edo|46]], [[58edo|58]], [[104edo|104c]], [[162edo|162cef]] |
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| ==17-limit== | | ==17-limit== |
| Commas: 126/125, 136/135, 176/175, 196/195, 256/255 | | Commas: 126/125, 136/135, 176/175, 196/195, 256/255 |
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| [[POTE tuning|POTE generator]]: 703.812 | | [[POTE_tuning|POTE generator]]: 703.812 |
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| Map: [<2 0 11 31 45 55 5|, <0 1 -2 -8 -12 -15 1|] | | Map: [<2 0 11 31 45 55 5|, <0 1 -2 -8 -12 -15 1|] |
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| EDOs: 46, 58, 104c | | EDOs: 46, 58, 104c |
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| =Keen= | | =Keen= |
| Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie <<2 -4 18 -11 23 53||. It may also be described as the 22&56 temperament. [[78edo|78et]] is a good tuning choice, and remains a good one in the 11-limit, where keen, <<2 -4 18 -12 ...||, is really more interesting, adding 100/99 and 385/384 to the commas. | | Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie <<2 -4 18 -11 23 53||. It may also be described as the 22&56 temperament. [[78edo|78et]] is a good tuning choice, and remains a good one in the 11-limit, where keen, <<2 -4 18 -12 ...||, is really more interesting, adding 100/99 and 385/384 to the commas. |
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| Commas: 2048/2025, 875/864 | | Commas: 2048/2025, 875/864 |
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| [[POTE tuning|POTE generator]]: 707.571 | | [[POTE_tuning|POTE generator]]: 707.571 |
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| Map: [<2 0 11 -23|, <0 1 -2 9|] | | Map: [<2 0 11 -23|, <0 1 -2 9|] |
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| EDOs: 22, 56, 78, 134b, 212b, 290b | | EDOs: 22, 56, 78, 134b, 212b, 290b |
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| ==11-limit== | | ==11-limit== |
| Commas: 100/99, 385/384, 1232/1215 | | Commas: 100/99, 385/384, 1232/1215 |
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| [[POTE tuning|POTE generator]]: 707.609 | | [[POTE_tuning|POTE generator]]: 707.609 |
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| Map: [<2 0 11 -23 26|, <0 1 -2 9 -6|] | | Map: [<2 0 11 -23 26|, <0 1 -2 9 -6|] |
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| EDOs: 22, 56, 78, 212bf, 290bf | | EDOs: 22, 56, 78, 212bf, 290bf |
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| =Bidia= | | =Bidia= |
| Bidia adds 3136/3125 to the commas, splitting the period into 1/4 octave. It may be called the 12&56 temperament. | | Bidia adds 3136/3125 to the commas, splitting the period into 1/4 octave. It may be called the 12&56 temperament. |
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| Map: [<4 0 22 43|,<0 1 -2 -5|] | | Map: [<4 0 22 43|,<0 1 -2 -5|] |
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| Wedgie: <<4 -8 -20 -22 -43 -24|| | | Wedgie: <<4 -8 -20 -22 -43 -24|| |
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| EDOs: 12, 56, 68, 80, 148d | | EDOs: 12, 56, 68, 80, 148d |
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| Badness: 0.0565 | | Badness: 0.0565 |
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| ==11-limit== | | ==11-limit== |
| Commas: 176/175, 896/891, 1375/1372 | | Commas: 176/175, 896/891, 1375/1372 |
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| Map: [<4 0 22 43 71|,<0 1 -2 -5 -9|] | | Map: [<4 0 22 43 71|,<0 1 -2 -5 -9|] |
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| EDOs: 12, 68, 80 | | EDOs: 12, 68, 80 |
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| Badness: 0.0402 | | Badness: 0.0402 |
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| ==13-limit== | | ==13-limit== |
| Commas: 176/175, 325/324, 640/637, 896/891 | | Commas: 176/175, 325/324, 640/637, 896/891 |
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| Map: [<4 0 22 43 71|,<0 1 -2 -5 -9|] | | Map: [<4 0 22 43 71|,<0 1 -2 -5 -9|] |
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| EDOs: 12, 68, 80, 148d, 228bcd, 376bcdf | | EDOs: 12, 68, 80, 148d, 228bcd, 376bcdf |
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| Badness: 0.0411 | | Badness: 0.0411 |
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| =Echidna= | | =Echidna= |
| Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie <<6 -12 10 -33 -1 57|| and may be called the 22&58 temperament. [[58edo|58et]] or [[80edo|80et]] make for good tunings, or their vals can be add to <138 219 321 388|. | | Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie <<6 -12 10 -33 -1 57|| and may be called the 22&58 temperament. [[58edo|58et]] or [[80edo|80et]] make for good tunings, or their vals can be add to <138 219 321 388|. |
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| Commas: 2048/2025, 1728/1715 | | Commas: 2048/2025, 1728/1715 |
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| [[POTE tuning|POTE generator]]: 434.856 | | [[POTE_tuning|POTE generator]]: 434.856 |
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| Map: [<2 1 9 2|, <0 3 -6 5|] | | Map: [<2 1 9 2|, <0 3 -6 5|] |
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| EDOs: 22, 58, 80, 138cd, 218cd | | EDOs: 22, 58, 80, 138cd, 218cd |
| | |
| Badness: 0.0580 | | Badness: 0.0580 |
|
| |
|
| ==11-limit== | | ==11-limit== |
| Commas: 176/175, 896/891, 540/539 | | Commas: 176/175, 896/891, 540/539 |
|
| |
|
| 11-limit minimax | | 11-limit minimax |
| | |
| [|1 0 0 0 0>, |7/4 0 0 1/4 -1/4>, |2 0 0 -1/2 1/2>, | | [|1 0 0 0 0>, |7/4 0 0 1/4 -1/4>, |2 0 0 -1/2 1/2>, |
| |37/12 0 0 5/12 -5/12>, |37/12 0 0 -7/12 7/12>] | | |37/12 0 0 5/12 -5/12>, |37/12 0 0 -7/12 7/12>] |
| | |
| Eigenmonzos: 2, 11/7 | | Eigenmonzos: 2, 11/7 |
|
| |
|
| Minimax generator: (224/11)^(1/12) = 434.792 | | Minimax generator: (224/11)^(1/12) = 434.792 |
| [[POTE tuning|POTE generator]]: 434.852 | | |
| | [[POTE_tuning|POTE generator]]: 434.852 |
|
| |
|
| Map: [<2 1 9 2 12|, <0 3 -6 5 -7|] | | Map: [<2 1 9 2 12|, <0 3 -6 5 -7|] |
| | |
| EDOs: 22, 58, 80, 138cde, 218cde | | EDOs: 22, 58, 80, 138cde, 218cde |
| | |
| Badness: 0.0260 | | Badness: 0.0260 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 176/175, 351/350, 364/363, 540/539 | | Commas: 176/175, 351/350, 364/363, 540/539 |
|
| |
|
| [[POTE tuning|POTE generator]]: 434.756 | | [[POTE_tuning|POTE generator]]: 434.756 |
|
| |
|
| Map: [<2 1 9 2 12 19|, <0 3 -6 5 -7 -16|] | | Map: [<2 1 9 2 12 19|, <0 3 -6 5 -7 -16|] |
| | |
| EDOs: 22, 58, 80, 138cde | | EDOs: 22, 58, 80, 138cde |
| | |
| Badness: 0.0237 | | Badness: 0.0237 |
|
| |
|
| ==17-limit== | | ==17-limit== |
| Commas: 136/135, 176/175, 221/220, 256/255, 540/539 | | Commas: 136/135, 176/175, 221/220, 256/255, 540/539 |
|
| |
|
| [[POTE tuning|POTE generator]]: 434.816 | | [[POTE_tuning|POTE generator]]: 434.816 |
|
| |
|
| Map: [<2 1 9 2 12 19 6|, <0 3 -6 5 -7 -16 3|] | | Map: [<2 1 9 2 12 19 6|, <0 3 -6 5 -7 -16 3|] |
| | |
| EDOs: 22, 58, 80, 138cde | | EDOs: 22, 58, 80, 138cde |
| | |
| Badness: 0.0203 | | Badness: 0.0203 |
|
| |
|
| =Echidnic= | | =Echidnic= |
| Commas: 686/675, 1029/1024 | | Commas: 686/675, 1029/1024 |
|
| |
|
| [[POTE tuning|POTE generator]]: 234.492 | | [[POTE_tuning|POTE generator]]: 234.492 |
|
| |
|
| Map: [<2 2 7 6|, <0 3 -6 -1|] | | Map: [<2 2 7 6|, <0 3 -6 -1|] |
| | |
| EDOs: 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bcd | | EDOs: 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bcd |
| | |
| Badness: 0.0722 | | Badness: 0.0722 |
|
| |
|
| ==11-limit== | | ==11-limit== |
| Commas: 385/384, 441/440, 686/675 | | Commas: 385/384, 441/440, 686/675 |
|
| |
|
| [[POTE tuning|POTE generator]]: 235.096 | | [[POTE_tuning|POTE generator]]: 235.096 |
|
| |
|
| Map: [<2 2 7 6 3|, <0 3 -6 -1 10|] | | Map: [<2 2 7 6 3|, <0 3 -6 -1 10|] |
| | |
| EDOs: 10, 46, 102, 148, 342bcd | | EDOs: 10, 46, 102, 148, 342bcd |
| | |
| Badness: 0.0451 | | Badness: 0.0451 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 91/90, 169/168, 385/384, 441/440 | | Commas: 91/90, 169/168, 385/384, 441/440 |
|
| |
|
| [[POTE tuning|POTE generator]]: 235.088 | | [[POTE_tuning|POTE generator]]: 235.088 |
|
| |
|
| Map: [<2 2 7 6 3 7|, <0 3 -6 -1 10 1|] | | Map: [<2 2 7 6 3 7|, <0 3 -6 -1 10 1|] |
| | |
| EDOs: 10, 46, 102, 148f, 194bcdf | | EDOs: 10, 46, 102, 148f, 194bcdf |
| | |
| Badness: 0.0289 | | Badness: 0.0289 |
|
| |
|
| Compositions: | | Compositions: |
|
| |
|
| http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 | | [http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3] |
| | |
| (the description says "lemba" which has a similar sale structure but different mapping for 5) | | (the description says "lemba" which has a similar sale structure but different mapping for 5) |
|
| |
|
| =Shrutar= | | =Shrutar= |
| Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. With wedgie <<4 -8 14 -22 11 55||, it can also be described as 22&46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another and excellent choice is a generator of 14^(1/7), making 7s just. | | Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. With wedgie <<4 -8 14 -22 11 55||, it can also be described as 22&46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo|68edo]] makes for a good tuning, but another and excellent choice is a generator of 14^(1/7), making 7s just. |
|
| |
|
| By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14^(1/7) generator can again be used as tunings. | | By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14^(1/7) generator can again be used as tunings. |
| Line 336: |
Line 405: |
| Commas: 2048/2025, 245/243 | | Commas: 2048/2025, 245/243 |
|
| |
|
| [[POTE tuning|POTE generator]]: 52.811 | | [[POTE_tuning|POTE generator]]: 52.811 |
|
| |
|
| Map: [<2 1 9 -2|, <0 2 -4 7|] | | Map: [<2 1 9 -2|, <0 2 -4 7|] |
| | |
| EDOs: 22, 46, 68, 182b, 250bc | | EDOs: 22, 46, 68, 182b, 250bc |
|
| |
|
| ==11-limit== | | ==11-limit== |
| Commas: 2048/2025, 245/243, 121/120 | | Commas: 2048/2025, 245/243, 121/120 |
|
| |
|
| [[POTE tuning|POTE generator]]: 52.680 | | [[POTE_tuning|POTE generator]]: 52.680 |
|
| |
|
| Map: [<2 1 9 -2 8|, <0 2 -4 7 -1|] | | Map: [<2 1 9 -2 8|, <0 2 -4 7 -1|] |
| | |
| EDOs: 22, 46, 68, 114, 296bce, 410bce | | EDOs: 22, 46, 68, 114, 296bce, 410bce |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 121/120, 176/175, 196/195, 245/243 | | Commas: 121/120, 176/175, 196/195, 245/243 |
|
| |
|
| Line 355: |
Line 426: |
|
| |
|
| Map: [<2 1 9 -2 8 -10|, <0 2 -4 7 -1 16|] | | Map: [<2 1 9 -2 8 -10|, <0 2 -4 7 -1 16|] |
| | |
| EDOs: 22, 24, 46, 68, 114 | | EDOs: 22, 24, 46, 68, 114 |
| | |
| Badness: 0.0281 | | Badness: 0.0281 |
|
| |
|
| ==17-limit== | | ==17-limit== |
| Commas: 121/120, 136/135, 154/153, 176/175, 196/195 | | Commas: 121/120, 136/135, 154/153, 176/175, 196/195 |
|
| |
|
| Line 364: |
Line 437: |
|
| |
|
| Map: [<2 1 9 -2 8 -10 6|, <0 2 -4 7 -1 16 2|] | | Map: [<2 1 9 -2 8 -10 6|, <0 2 -4 7 -1 16 2|] |
| | |
| EDOs: 22, 24, 46, 68, 114 | | EDOs: 22, 24, 46, 68, 114 |
| | |
| Badness: 0.0187 | | Badness: 0.0187 |
|
| |
|
| ==19-limit== | | ==19-limit== |
| Commas: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342 | | Commas: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342 |
|
| |
|
| Line 373: |
Line 448: |
|
| |
|
| Map: [<2 1 9 -2 8 -10 6 -10|, <0 2 -4 7 -1 16 2 17|] | | Map: [<2 1 9 -2 8 -10 6 -10|, <0 2 -4 7 -1 16 2 17|] |
| | |
| EDOs: 22, 24, 46, 68, 114, 182bef | | EDOs: 22, 24, 46, 68, 114, 182bef |
| | |
| Badness: 0.0175 | | Badness: 0.0175 |
|
| |
|
| =Sruti= | | =Sruti= |
| Commas: 2048/2025, 19683/19600 | | Commas: 2048/2025, 19683/19600 |
|
| |
|
| Line 382: |
Line 459: |
|
| |
|
| Map: [<2 0 11 -15|, <0 2 -4 13|] | | Map: [<2 0 11 -15|, <0 2 -4 13|] |
| | |
| Wedgie: <<4 -8 26 -22 30 83|| | | Wedgie: <<4 -8 26 -22 30 83|| |
| | |
| EDOs: 24, 34d, 58, 150cd, 208cd, 266cd | | EDOs: 24, 34d, 58, 150cd, 208cd, 266cd |
| | |
| Badness: 0.1174 | | Badness: 0.1174 |
|
| |
|
| ==11-limit== | | ==11-limit== |
| Commas: 176/175, 243/242, 896/891 | | Commas: 176/175, 243/242, 896/891 |
|
| |
|
| Line 392: |
Line 472: |
|
| |
|
| Map: [<2 0 11 -15 -1|, <0 2 -4 13 5|] | | Map: [<2 0 11 -15 -1|, <0 2 -4 13 5|] |
| | |
| EDOs: 24, 34d, 58, 150cde, 208cde | | EDOs: 24, 34d, 58, 150cde, 208cde |
| | |
| Badness: 0.0415 | | Badness: 0.0415 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 144/143, 176/175, 351/350, 676/675 | | Commas: 144/143, 176/175, 351/350, 676/675 |
|
| |
|
| Line 401: |
Line 483: |
|
| |
|
| Map: [<2 0 11 -15 -1 9|, <0 2 -4 13 5 -1|] | | Map: [<2 0 11 -15 -1 9|, <0 2 -4 13 5 -1|] |
| | |
| EDOs: 24, 34d, 58, 150cdef, 208cdef | | EDOs: 24, 34d, 58, 150cdef, 208cdef |
| | |
| Badness: 0.0238 | | Badness: 0.0238 |
|
| |
|
| Line 410: |
Line 494: |
|
| |
|
| Map: [<2 0 11 4|, <0 2 -4 1|] | | Map: [<2 0 11 4|, <0 2 -4 1|] |
| | |
| Wedgie: <<4 -8 2 -22 -8 27|| | | Wedgie: <<4 -8 2 -22 -8 27|| |
| | |
| EDOs: 10, 24, 34 | | EDOs: 10, 24, 34 |
| | |
| Badness: 0.0780 | | Badness: 0.0780 |
|
| |
|
| Line 420: |
Line 507: |
|
| |
|
| Map: [<2 0 11 4 -1|, <0 2 -4 1 5|] | | Map: [<2 0 11 4 -1|, <0 2 -4 1 5|] |
| | |
| EDOs: 10, 24, 34, 58d, 92de | | EDOs: 10, 24, 34, 58d, 92de |
| | |
| Badness: 0.0493 | | Badness: 0.0493 |
|
| |
|
| Line 429: |
Line 518: |
|
| |
|
| Map: [<2 0 11 4 -1 9|, <0 2 -4 1 5 -1|] | | Map: [<2 0 11 4 -1 9|, <0 2 -4 1 5 -1|] |
| | |
| EDOs: 10, 24, 34, 58d, 92def | | EDOs: 10, 24, 34, 58d, 92def |
| | |
| Badness: 0.0308 | | Badness: 0.0308 |
|
| |
|
| Line 438: |
Line 529: |
|
| |
|
| Map: [<2 1 9 11|, <0 2 -4 -5|] | | Map: [<2 1 9 11|, <0 2 -4 -5|] |
| | |
| Wedgie: <<4 -8 -10 -22 -27 -1|| | | Wedgie: <<4 -8 -10 -22 -27 -1|| |
| | |
| EDOs: 22d, 24 | | EDOs: 22d, 24 |
| | |
| Badness: 0.1576 | | Badness: 0.1576 |
|
| |
|
| Line 448: |
Line 542: |
|
| |
|
| Map: [<2 1 9 11 8|, <0 2 -4 -5 -1|] | | Map: [<2 1 9 11 8|, <0 2 -4 -5 -1|] |
| | |
| EDOs: 22d, 24 | | EDOs: 22d, 24 |
| | |
| Badness: 0.0635 | | Badness: 0.0635 |
|
| |
|
| Line 457: |
Line 553: |
|
| |
|
| Map: [<2 1 9 11 8 15|, <0 2 -4 -5 -1 -7|] | | Map: [<2 1 9 11 8 15|, <0 2 -4 -5 -1 -7|] |
| | |
| EDOs: 24 | | EDOs: 24 |
| | |
| Badness: 0.0457 | | Badness: 0.0457 |
| | | [[Category:diaschismic]] |
| | | [[Category:family]] |
| </pre></div>
| | [[Category:theory]] |
| <h4>Original HTML content:</h4>
| | [[Category:todo:add_definition]] |
| <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>Diaschismic family</title></head><body><!-- ws:start:WikiTextTocRule:86:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><div style="margin-left: 2em;"><a href="#x-Seven limit children">Seven limit children</a></div>
| | [[Category:todo:intro]] |
| <!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><div style="margin-left: 1em;"><a href="#Srutal">Srutal</a></div>
| |
| <!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><div style="margin-left: 2em;"><a href="#Srutal-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><div style="margin-left: 2em;"><a href="#Srutal-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><div style="margin-left: 1em;"><a href="#Pajara">Pajara</a></div>
| |
| <!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><div style="margin-left: 2em;"><a href="#Pajara-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><div style="margin-left: 2em;"><a href="#Pajara-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><div style="margin-left: 2em;"><a href="#Pajara-Pajarous">Pajarous</a></div>
| |
| <!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 3em;"><a href="#Pajara-Pajarous-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 2em;"><a href="#Pajara-Pajaric">Pajaric</a></div>
| |
| <!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><div style="margin-left: 3em;"><a href="#Pajara-Pajaric-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><div style="margin-left: 2em;"><a href="#Pajara-Pajaro">Pajaro</a></div>
| |
| <!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><div style="margin-left: 2em;"><a href="#Pajara-Hemipaj">Hemipaj</a></div>
| |
| <!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><div style="margin-left: 1em;"><a href="#Diaschismic">Diaschismic</a></div>
| |
| <!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><div style="margin-left: 2em;"><a href="#Diaschismic-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><div style="margin-left: 2em;"><a href="#Diaschismic-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><div style="margin-left: 2em;"><a href="#Diaschismic-17-limit">17-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><div style="margin-left: 1em;"><a href="#Keen">Keen</a></div>
| |
| <!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 2em;"><a href="#Keen-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><div style="margin-left: 1em;"><a href="#Bidia">Bidia</a></div>
| |
| <!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><div style="margin-left: 2em;"><a href="#Bidia-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><div style="margin-left: 2em;"><a href="#Bidia-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><div style="margin-left: 1em;"><a href="#Echidna">Echidna</a></div>
| |
| <!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><div style="margin-left: 2em;"><a href="#Echidna-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><div style="margin-left: 2em;"><a href="#Echidna-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><div style="margin-left: 2em;"><a href="#Echidna-17-limit">17-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --><div style="margin-left: 1em;"><a href="#Echidnic">Echidnic</a></div>
| |
| <!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --><div style="margin-left: 2em;"><a href="#Echidnic-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --><div style="margin-left: 2em;"><a href="#Echidnic-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --><div style="margin-left: 1em;"><a href="#Shrutar">Shrutar</a></div>
| |
| <!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --><div style="margin-left: 2em;"><a href="#Shrutar-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><div style="margin-left: 2em;"><a href="#Shrutar-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><div style="margin-left: 2em;"><a href="#Shrutar-17-limit">17-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><div style="margin-left: 2em;"><a href="#Shrutar-19-limit">19-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><div style="margin-left: 1em;"><a href="#Sruti">Sruti</a></div>
| |
| <!-- ws:end:WikiTextTocRule:121 --><!-- ws:start:WikiTextTocRule:122: --><div style="margin-left: 2em;"><a href="#Sruti-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:122 --><!-- ws:start:WikiTextTocRule:123: --><div style="margin-left: 2em;"><a href="#Sruti-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:123 --><!-- ws:start:WikiTextTocRule:124: --><div style="margin-left: 1em;"><a href="#Anguirus">Anguirus</a></div>
| |
| <!-- ws:end:WikiTextTocRule:124 --><!-- ws:start:WikiTextTocRule:125: --><div style="margin-left: 2em;"><a href="#Anguirus-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:125 --><!-- ws:start:WikiTextTocRule:126: --><div style="margin-left: 2em;"><a href="#Anguirus-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:126 --><!-- ws:start:WikiTextTocRule:127: --><div style="margin-left: 1em;"><a href="#Shru">Shru</a></div>
| |
| <!-- ws:end:WikiTextTocRule:127 --><!-- ws:start:WikiTextTocRule:128: --><div style="margin-left: 2em;"><a href="#Shru-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:128 --><!-- ws:start:WikiTextTocRule:129: --><div style="margin-left: 2em;"><a href="#Shru-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:129 --><!-- ws:start:WikiTextTocRule:130: --></div>
| |
| <!-- ws:end:WikiTextTocRule:130 -->The 5-limit parent comma for the <strong>diaschismic family</strong> is 2048/2025, the <a class="wiki_link" href="/diaschisma">diaschisma</a>. Its monzo is |11 -4 -2&gt;, and flipping that yields &lt;&lt;2 -4 -11|| for the wedgie for 5-limit diaschismic, or <strong>srutal</strong>, temperament. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. <a class="wiki_link" href="/34edo">34edo</a> is a good tuning choice, with <a class="wiki_link" href="/46edo">46edo</a>, <a class="wiki_link" href="/56edo">56edo</a>, <a class="wiki_link" href="/58edo">58edo</a> or <a class="wiki_link" href="/80edo">80edo</a> being other possibilities. Both <a class="wiki_link" href="/12edo">12edo</a> and <a class="wiki_link" href="/22edo">22edo</a> support it, and retuning them to a MOS of diaschismic gives two scale possibilities.<br />
| |
| <br />
| |
| <a class="wiki_link" href="/Tuning%20Ranges%20of%20Regular%20Temperaments">valid range</a>: [600.000 to 720.000] (2 to 5)<br />
| |
| nice range: [701.955, 706.843]<br />
| |
| strict range: [701.955, 706.843]<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 704.898<br />
| |
| <br />
| |
| Map: [&lt;2 0 11|, &lt;0 1 -2|]<br />
| |
| EDOs: 34, 46, 80, 206c, 286bc<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2>
| |
| The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> defines which 7-limit family member we are looking at. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as <a class="wiki_link" href="/36_35">36/35</a>, the septimal quarter-tone) and echidna has a generator of 9/7. Adding 4375/4374 does no significant tuning damage, so for that we keep the 5-limit label srutal.<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Srutal"></a><!-- ws:end:WikiTextHeadingRule:2 -->Srutal</h1>
| |
| Commas: 2048/2025, 4375/4374<br />
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| <br />
| |
| valid range: [703.448, 705.882] (58 to 34d)<br />
| |
| nice range: [701.955, 706.843]<br />
| |
| strict range: [703.448, 705.882]<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 704.814<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 -42|, &lt;0 1 -2 15|]<br />
| |
| Wedgie: &lt;&lt;2 -4 30 -11 42 81||<br />
| |
| EDOs: 46, 80, 126, 206cd, 332bcd<br />
| |
| Badness: 0.0915<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Srutal-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h2>
| |
| Commas: 176/175, 896/891, 1331/1323<br />
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| <br />
| |
| valid range: [704.348, 705.882] (46 to 34d)<br />
| |
| nice range: [701.955, 706.843]<br />
| |
| strict range: [704.348, 705.882]<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 704.856<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 -42 -28|, &lt;0 1 -2 15 11|]<br />
| |
| EDOs: 46, 80, 126, 206cd<br />
| |
| Badness: 0.0353<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Srutal-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h2>
| |
| Commas: 169/168, 176/175, 325/324, 364/363<br />
| |
| <br />
| |
| valid range: [704.348, 705.882] (46 to 34d)<br />
| |
| nice range: [701.955, 706.843]<br />
| |
| strict range: [704.348, 705.882]<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 704.881<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 -42 -28 -18|, &lt;0 1 -2 15 11 8|]<br />
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| EDOs: 34d, 46, 80, 206cd, 286bcde<br />
| |
| Badness: 0.0253<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Pajara"></a><!-- ws:end:WikiTextHeadingRule:8 -->Pajara</h1>
| |
| Main article: <a class="wiki_link" href="/Pajara">Pajara</a><br />
| |
| Pajara, with wedgie &lt;&lt;2 -4 -4 -11 -12 2|| is closely associated with 22et (not to mention <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a>) but other tunings are possible. The 1/2 octave period serves as both a <a class="wiki_link" href="/10_7">10/7</a> and a <a class="wiki_link" href="/7_5">7/5</a>. Aside from 22et, 34 with the val &lt;34 54 79 96| and 56 with the val &lt;56 89 130 158| are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12et and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.<br />
| |
| <br />
| |
| Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.<br />
| |
| <br />
| |
| Commas: 50/49, 64/63<br />
| |
| <br />
| |
| valid range: [700.000, 720.000] (12 to 10)<br />
| |
| nice range: [701.955, 715.587]<br />
| |
| strict range: [701.955, 715.587]<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 707.048<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 12|, &lt;0 1 -2 -2|]<br />
| |
| EDOs: 22, 34, 56<br />
| |
| Badness: 0.0200<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Pajara-11-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->11-limit</h2>
| |
| Commas: 50/49, 64/63, 99/98<br />
| |
| <br />
| |
| valid range: [700.000, 709.091] (12 to 22)<br />
| |
| nice range: [701.955, 715.587]<br />
| |
| strict range: [701.955, 709.091]<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 706.885<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 12 26|, &lt;0 1 -2 -2 -6|]<br />
| |
| EDOs: 22, 34, 56, 146<br />
| |
| Badness: 0.0203<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Pajara-13-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit</h2>
| |
| Commas: 50/49, 64/63, 65/63, 99/98<br />
| |
| <br />
| |
| valid range: []<br />
| |
| nice range: [701.955, 738.573]<br />
| |
| strict range: []<br />
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| <br />
| |
| POTE generator: ~3/2 = 708.919<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 12 26 1|, &lt;0 1 -2 -2 -6 2|]<br />
| |
| EDOs: 12, 22<br />
| |
| Badness: 0.0276<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Pajara-Pajarous"></a><!-- ws:end:WikiTextHeadingRule:14 -->Pajarous</h2>
| |
| Commas: 50/49, 55/54, 64/63<br />
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| <br />
| |
| valid range: 709.091 (22)<br />
| |
| nice range: [701.955, 715.803]<br />
| |
| strict range: 709.091<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 709.578<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 12 -9|, &lt;0 1 -2 -2 5|]<br />
| |
| EDOs: 10, 12e, 22, 120bce, 142bce<br />
| |
| Badness: 0.0283<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Pajara-Pajarous-13-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->13-limit</h3>
| |
| Commas: 50/49, 55/54, 64/63, 65/63<br />
| |
| <br />
| |
| valid range: []<br />
| |
| nice range: [701.955, 738.573]<br />
| |
| strict range: []<br />
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| <br />
| |
| POTE generator: ~3/2 = 710.240<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 12 -9 1|, &lt;0 1 -2 -2 5 2|]<br />
| |
| EDOs: 10, 22, 54f, 76bdf<br />
| |
| Badness: 0.0252<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Pajara-Pajaric"></a><!-- ws:end:WikiTextHeadingRule:18 -->Pajaric</h2>
| |
| Commas: 45/44, 50/49, 56/55<br />
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| <br />
| |
| POTE generator: ~3/2 = 705.524<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 12 7|, &lt;0 1 -2 -2 0|]<br />
| |
| EDOs: 10, 12, 22e, 34de<br />
| |
| Badness: 0.0238<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Pajara-Pajaric-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h3>
| |
| Commas: 40/39, 45/44, 50/49, 56/55<br />
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| <br />
| |
| POTE generator: ~3/2 = 707.442<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 12 7 17|, &lt;0 1 -2 -2 0 -3|]<br />
| |
| EDOs: 10, 12f, 22ef, 34def<br />
| |
| Badness: 0.0205<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Pajara-Pajaro"></a><!-- ws:end:WikiTextHeadingRule:22 -->Pajaro</h2>
| |
| Commas: 40/39, 50/49, 55/54, 64/63<br />
| |
| <br />
| |
| POTE generator ~3/2 = 710.818<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 12 -9 17|, &lt;0 1 -2 -2 5 -3|]<br />
| |
| EDOs: 10, 22f, 32f, 54f<br />
| |
| Badness: 0.0274<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Pajara-Hemipaj"></a><!-- ws:end:WikiTextHeadingRule:24 -->Hemipaj</h2>
| |
| Commas: 50/49, 64/63, 121/120 <br />
| |
| <br />
| |
| POTE generator: ~11/8 = 546.383<br />
| |
| <br />
| |
| Map: [&lt;2 1 9 10 8|, &lt;0 2 -4 -4 -1|]<br />
| |
| EDOs: 20, 22, 68d, 90d<br />
| |
| Badness: 0.0389<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Diaschismic"></a><!-- ws:end:WikiTextHeadingRule:26 -->Diaschismic</h1>
| |
| A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46&amp;58. However described, diaschismic has wedgie &lt;&lt;2 -4 -16 -11 -31 -26||, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. <a class="wiki_link" href="/58edo">58et</a> provides an excellent tuning, but an alternative is to make <a class="wiki_link" href="/7_4">7/4</a> just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et.<br />
| |
| <br />
| |
| Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363. The 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.<br />
| |
| <br />
| |
| Commas: 126/125, 2048/2025<br />
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| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 703.681<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 31|, &lt;0 1 -2 -8|]<br />
| |
| EDOs: 46, 58, 104c, 162c<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Diaschismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->11-limit</h2>
| |
| Commas: 126/125, 176/175, 896/891<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 703.714<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 31 45|, &lt;0 1 -2 -8 -12|]<br />
| |
| EDOs: 46, 58, 104c, 162ce<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Diaschismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->13-limit</h2>
| |
| Commas: 126/125, 196/195, 364/363, 2048/2025<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 703.704<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 31 45 55|, &lt;0 1 -2 -8 -12 -15|]<br />
| |
| EDOs: <a class="wiki_link" href="/46edo">46</a>, <a class="wiki_link" href="/58edo">58</a>, <a class="wiki_link" href="/104edo">104c</a>, <a class="wiki_link" href="/162edo">162cef</a><br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Diaschismic-17-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->17-limit</h2>
| |
| Commas: 126/125, 136/135, 176/175, 196/195, 256/255<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 703.812<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 31 45 55 5|, &lt;0 1 -2 -8 -12 -15 1|]<br />
| |
| EDOs: 46, 58, 104c<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Keen"></a><!-- ws:end:WikiTextHeadingRule:34 -->Keen</h1>
| |
| Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie &lt;&lt;2 -4 18 -11 23 53||. It may also be described as the 22&amp;56 temperament. <a class="wiki_link" href="/78edo">78et</a> is a good tuning choice, and remains a good one in the 11-limit, where keen, &lt;&lt;2 -4 18 -12 ...||, is really more interesting, adding 100/99 and 385/384 to the commas.<br />
| |
| <br />
| |
| Commas: 2048/2025, 875/864<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 707.571<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 -23|, &lt;0 1 -2 9|]<br />
| |
| EDOs: 22, 56, 78, 134b, 212b, 290b<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Keen-11-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->11-limit</h2>
| |
| Commas: 100/99, 385/384, 1232/1215<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 707.609<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 -23 26|, &lt;0 1 -2 9 -6|]<br />
| |
| EDOs: 22, 56, 78, 212bf, 290bf<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="Bidia"></a><!-- ws:end:WikiTextHeadingRule:38 -->Bidia</h1>
| |
| Bidia adds 3136/3125 to the commas, splitting the period into 1/4 octave. It may be called the 12&amp;56 temperament.<br />
| |
| <br />
| |
| Commas: 2048/2025, 3136/3125<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 705.364<br />
| |
| <br />
| |
| Map: [&lt;4 0 22 43|,&lt;0 1 -2 -5|]<br />
| |
| Wedgie: &lt;&lt;4 -8 -20 -22 -43 -24||<br />
| |
| EDOs: 12, 56, 68, 80, 148d<br />
| |
| Badness: 0.0565<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Bidia-11-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->11-limit</h2>
| |
| Commas: 176/175, 896/891, 1375/1372<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 705.087<br />
| |
| <br />
| |
| Map: [&lt;4 0 22 43 71|,&lt;0 1 -2 -5 -9|]<br />
| |
| EDOs: 12, 68, 80<br />
| |
| Badness: 0.0402<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:42:&lt;h2&gt; --><h2 id="toc21"><a name="Bidia-13-limit"></a><!-- ws:end:WikiTextHeadingRule:42 -->13-limit</h2>
| |
| Commas: 176/175, 325/324, 640/637, 896/891<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 705.301<br />
| |
| <br />
| |
| Map: [&lt;4 0 22 43 71|,&lt;0 1 -2 -5 -9|]<br />
| |
| EDOs: 12, 68, 80, 148d, 228bcd, 376bcdf<br />
| |
| Badness: 0.0411<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><h1 id="toc22"><a name="Echidna"></a><!-- ws:end:WikiTextHeadingRule:44 -->Echidna</h1>
| |
| Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie &lt;&lt;6 -12 10 -33 -1 57|| and may be called the 22&amp;58 temperament. <a class="wiki_link" href="/58edo">58et</a> or <a class="wiki_link" href="/80edo">80et</a> make for good tunings, or their vals can be add to &lt;138 219 321 388|.<br />
| |
| <br />
| |
| Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-limit diamond to within about six cents of error, within a compass of 24 notes. The 28 note 2MOS gives scope for this, and the 36 note MOS much more.<br />
| |
| <br />
| |
| Commas: 2048/2025, 1728/1715<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 434.856<br />
| |
| <br />
| |
| Map: [&lt;2 1 9 2|, &lt;0 3 -6 5|]<br />
| |
| EDOs: 22, 58, 80, 138cd, 218cd<br />
| |
| Badness: 0.0580<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc23"><a name="Echidna-11-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->11-limit</h2>
| |
| Commas: 176/175, 896/891, 540/539<br />
| |
| <br />
| |
| 11-limit minimax<br />
| |
| [|1 0 0 0 0&gt;, |7/4 0 0 1/4 -1/4&gt;, |2 0 0 -1/2 1/2&gt;,<br />
| |
| |37/12 0 0 5/12 -5/12&gt;, |37/12 0 0 -7/12 7/12&gt;]<br />
| |
| Eigenmonzos: 2, 11/7<br />
| |
| <br />
| |
| Minimax generator: (224/11)^(1/12) = 434.792<br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 434.852<br />
| |
| <br />
| |
| Map: [&lt;2 1 9 2 12|, &lt;0 3 -6 5 -7|]<br />
| |
| EDOs: 22, 58, 80, 138cde, 218cde<br />
| |
| Badness: 0.0260<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:48:&lt;h2&gt; --><h2 id="toc24"><a name="Echidna-13-limit"></a><!-- ws:end:WikiTextHeadingRule:48 -->13-limit</h2>
| |
| Commas: 176/175, 351/350, 364/363, 540/539<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 434.756<br />
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| <br />
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| Map: [&lt;2 1 9 2 12 19|, &lt;0 3 -6 5 -7 -16|]<br />
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| EDOs: 22, 58, 80, 138cde<br />
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| Badness: 0.0237<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:50:&lt;h2&gt; --><h2 id="toc25"><a name="Echidna-17-limit"></a><!-- ws:end:WikiTextHeadingRule:50 -->17-limit</h2>
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| Commas: 136/135, 176/175, 221/220, 256/255, 540/539<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 434.816<br />
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| <br />
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| Map: [&lt;2 1 9 2 12 19 6|, &lt;0 3 -6 5 -7 -16 3|]<br />
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| EDOs: 22, 58, 80, 138cde<br />
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| Badness: 0.0203<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:52:&lt;h1&gt; --><h1 id="toc26"><a name="Echidnic"></a><!-- ws:end:WikiTextHeadingRule:52 -->Echidnic</h1>
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| Commas: 686/675, 1029/1024<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 234.492<br />
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| <br />
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| Map: [&lt;2 2 7 6|, &lt;0 3 -6 -1|]<br />
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| EDOs: 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bcd<br />
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| Badness: 0.0722<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:54:&lt;h2&gt; --><h2 id="toc27"><a name="Echidnic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:54 -->11-limit</h2>
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| Commas: 385/384, 441/440, 686/675<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 235.096<br />
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| <br />
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| Map: [&lt;2 2 7 6 3|, &lt;0 3 -6 -1 10|]<br />
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| EDOs: 10, 46, 102, 148, 342bcd<br />
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| Badness: 0.0451<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:56:&lt;h2&gt; --><h2 id="toc28"><a name="Echidnic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:56 -->13-limit</h2>
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| Commas: 91/90, 169/168, 385/384, 441/440<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 235.088<br />
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| <br />
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| Map: [&lt;2 2 7 6 3 7|, &lt;0 3 -6 -1 10 1|]<br />
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| EDOs: 10, 46, 102, 148f, 194bcdf<br />
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| Badness: 0.0289<br />
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| <br />
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| Compositions:<br />
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| <br />
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| <!-- ws:start:WikiTextUrlRule:582:http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 --><a class="wiki_link_ext" href="http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3" rel="nofollow">http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3</a><!-- ws:end:WikiTextUrlRule:582 --><br />
| |
| (the description says &quot;lemba&quot; which has a similar sale structure but different mapping for 5)<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:58:&lt;h1&gt; --><h1 id="toc29"><a name="Shrutar"></a><!-- ws:end:WikiTextHeadingRule:58 -->Shrutar</h1>
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| Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. With wedgie &lt;&lt;4 -8 14 -22 11 55||, it can also be described as 22&amp;46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. <a class="wiki_link" href="/68edo">68edo</a> makes for a good tuning, but another and excellent choice is a generator of 14^(1/7), making 7s just.<br />
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| <br />
| |
| By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14^(1/7) generator can again be used as tunings.<br />
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| <br />
| |
| Commas: 2048/2025, 245/243<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 52.811<br />
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| <br />
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| Map: [&lt;2 1 9 -2|, &lt;0 2 -4 7|]<br />
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| EDOs: 22, 46, 68, 182b, 250bc<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:60:&lt;h2&gt; --><h2 id="toc30"><a name="Shrutar-11-limit"></a><!-- ws:end:WikiTextHeadingRule:60 -->11-limit</h2>
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| Commas: 2048/2025, 245/243, 121/120<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 52.680<br />
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| <br />
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| Map: [&lt;2 1 9 -2 8|, &lt;0 2 -4 7 -1|]<br />
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| EDOs: 22, 46, 68, 114, 296bce, 410bce<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:62:&lt;h2&gt; --><h2 id="toc31"><a name="Shrutar-13-limit"></a><!-- ws:end:WikiTextHeadingRule:62 -->13-limit</h2>
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| Commas: 121/120, 176/175, 196/195, 245/243<br />
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| <br />
| |
| POTE generator: ~28/27 = 52.654<br />
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| <br />
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| Map: [&lt;2 1 9 -2 8 -10|, &lt;0 2 -4 7 -1 16|]<br />
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| EDOs: 22, 24, 46, 68, 114<br />
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| Badness: 0.0281<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:64:&lt;h2&gt; --><h2 id="toc32"><a name="Shrutar-17-limit"></a><!-- ws:end:WikiTextHeadingRule:64 -->17-limit</h2>
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| Commas: 121/120, 136/135, 154/153, 176/175, 196/195<br />
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| <br />
| |
| POTE generator: ~28/27 = 52.647<br />
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| <br />
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| Map: [&lt;2 1 9 -2 8 -10 6|, &lt;0 2 -4 7 -1 16 2|]<br />
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| EDOs: 22, 24, 46, 68, 114<br />
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| Badness: 0.0187<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:66:&lt;h2&gt; --><h2 id="toc33"><a name="Shrutar-19-limit"></a><!-- ws:end:WikiTextHeadingRule:66 -->19-limit</h2>
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| Commas: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342<br />
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| <br />
| |
| POTE generator: ~28/27 = 52.730<br />
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| <br />
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| Map: [&lt;2 1 9 -2 8 -10 6 -10|, &lt;0 2 -4 7 -1 16 2 17|]<br />
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| EDOs: 22, 24, 46, 68, 114, 182bef<br />
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| Badness: 0.0175<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:68:&lt;h1&gt; --><h1 id="toc34"><a name="Sruti"></a><!-- ws:end:WikiTextHeadingRule:68 -->Sruti</h1>
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| Commas: 2048/2025, 19683/19600<br />
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| <br />
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| POTE generator: ~175/144 = 351.876<br />
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| <br />
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| Map: [&lt;2 0 11 -15|, &lt;0 2 -4 13|]<br />
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| Wedgie: &lt;&lt;4 -8 26 -22 30 83||<br />
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| EDOs: 24, 34d, 58, 150cd, 208cd, 266cd<br />
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| Badness: 0.1174<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:70:&lt;h2&gt; --><h2 id="toc35"><a name="Sruti-11-limit"></a><!-- ws:end:WikiTextHeadingRule:70 -->11-limit</h2>
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| Commas: 176/175, 243/242, 896/891<br />
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| <br />
| |
| POTE generator: ~11/9 = 351.863<br />
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| <br />
| |
| Map: [&lt;2 0 11 -15 -1|, &lt;0 2 -4 13 5|]<br />
| |
| EDOs: 24, 34d, 58, 150cde, 208cde<br />
| |
| Badness: 0.0415<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:72:&lt;h2&gt; --><h2 id="toc36"><a name="Sruti-13-limit"></a><!-- ws:end:WikiTextHeadingRule:72 -->13-limit</h2>
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| Commas: 144/143, 176/175, 351/350, 676/675<br />
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| <br />
| |
| POTE generator: ~11/9 = 351.886<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 -15 -1 9|, &lt;0 2 -4 13 5 -1|]<br />
| |
| EDOs: 24, 34d, 58, 150cdef, 208cdef<br />
| |
| Badness: 0.0238<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:74:&lt;h1&gt; --><h1 id="toc37"><a name="Anguirus"></a><!-- ws:end:WikiTextHeadingRule:74 -->Anguirus</h1>
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| Commas: 49/48, 2048/2025<br />
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| <br />
| |
| POTE generator: ~8/7 = 246.979<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 4|, &lt;0 2 -4 1|]<br />
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| Wedgie: &lt;&lt;4 -8 2 -22 -8 27||<br />
| |
| EDOs: 10, 24, 34<br />
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| Badness: 0.0780<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:76:&lt;h2&gt; --><h2 id="toc38"><a name="Anguirus-11-limit"></a><!-- ws:end:WikiTextHeadingRule:76 -->11-limit</h2>
| |
| Commas: 49/48, 56/55, 243/242<br />
| |
| <br />
| |
| POTE generator: ~8/7 = 247.816<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 4 -1|, &lt;0 2 -4 1 5|]<br />
| |
| EDOs: 10, 24, 34, 58d, 92de<br />
| |
| Badness: 0.0493<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:78:&lt;h2&gt; --><h2 id="toc39"><a name="Anguirus-13-limit"></a><!-- ws:end:WikiTextHeadingRule:78 -->13-limit</h2>
| |
| Commas: 49/48 56/55 91/90 352/351<br />
| |
| <br />
| |
| POTE generator: ~8/7 = 247.691<br />
| |
| <br />
| |
| Map: [&lt;2 0 11 4 -1 9|, &lt;0 2 -4 1 5 -1|]<br />
| |
| EDOs: 10, 24, 34, 58d, 92def<br />
| |
| Badness: 0.0308<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:80:&lt;h1&gt; --><h1 id="toc40"><a name="Shru"></a><!-- ws:end:WikiTextHeadingRule:80 -->Shru</h1>
| |
| Commas: 392/375, 1323/1280<br />
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| <br />
| |
| POTE generator: ~64/63 = 50.135<br />
| |
| <br />
| |
| Map: [&lt;2 1 9 11|, &lt;0 2 -4 -5|]<br />
| |
| Wedgie: &lt;&lt;4 -8 -10 -22 -27 -1||<br />
| |
| EDOs: 22d, 24<br />
| |
| Badness: 0.1576<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:82:&lt;h2&gt; --><h2 id="toc41"><a name="Shru-11-limit"></a><!-- ws:end:WikiTextHeadingRule:82 -->11-limit</h2>
| |
| Commas: 56/55, 77/75, 1323/1280<br />
| |
| <br />
| |
| POTE generator: ~64/63 = 50.130<br />
| |
| <br />
| |
| Map: [&lt;2 1 9 11 8|, &lt;0 2 -4 -5 -1|]<br />
| |
| EDOs: 22d, 24<br />
| |
| Badness: 0.0635<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:84:&lt;h2&gt; --><h2 id="toc42"><a name="Shru-13-limit"></a><!-- ws:end:WikiTextHeadingRule:84 -->13-limit</h2>
| |
| Commas: 56/55, 77/75, 105/104, 507/500<br />
| |
| <br />
| |
| POTE generator: ~64/63 = 50.535<br />
| |
| <br />
| |
| Map: [&lt;2 1 9 11 8 15|, &lt;0 2 -4 -5 -1 -7|]<br />
| |
| EDOs: 24<br />
| |
| Badness: 0.0457</body></html></pre></div>
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