Ragismic microtemperaments
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[[toc]] The ragisma is 4375/4374 with a monzo of |-1 -7 4 1>, the smallest 7-limit superparticular ratio. Since (10/9)^4 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal. =Ennealimmal= Ennealimmal temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the ennealimma comma, |1 -27 18>, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is <<18 27 18 1 -22 -34||. Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 equal, though its hardly likely anyone could tell the difference. If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example.) In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS. Commas: 2401/2400, 4375/4374 POTE generators: 36/35: 49.0205; 10/9: 182.354; 6/5: 315.687; 49/40: 350.980 Map: [<9 1 1 2|, <0 2 3 2|] Wedgie: <<18 27 18 1 -22 -34|| EDOs: [[27edo|27]], [[45edo|45]], [[72edo|72]], [[99edo|99]], [[171edo|171]], [[270edo|270]], [[441edo|441]], [[612edo|612]], [[3600edo|3600]] Badness: 0.00361 ==11 limit hemiennealimmal== Commas: 2401/2400, 4375/4374, 3025/3024 POTE generator: 99/98: 17.6219 or 6/5: 315.7114 Map: [<18 0 -1 22 48|, <0 2 3 2 1|] EDOs: 72, 198, 270, 342, 612, 954, 1566 Badness: 0.00628 ==13 limit hemiennealimmal== Commas: 676/675, 1001/1000, 1716/1715, 3025/3024 POTE generator ~99/98 = 17.7504 Map: [<18 0 -1 22 48 -19|, <0 2 3 2 1 6|] EDOs: 72, 198, 270 Badness: 0.0125 ==Semiennealimmal== Commas: 2401/2400, 4375/4374, 4000/3993 POTE generator: ~140/121 = 250.3367 Map: [<9 3 4 14 18|, <0 6 9 6 7|] EDOs: 72, 369, 441 Badness: 0.0342 ===13 limit semiennealimmal=== Commas: 1575/1573, 2080/2079, 2401/2400, 4375/4374 POTE generator: ~140/121 = 250.3375 Map: [<9 3 4 14 18 -8|, <0 6 9 6 7 22|] EDOs: 72, 441 Badness: 0.0261 ==Ennealimmic== Commas: 243/242, 441/440, 4375/4356 POTE generator: ~36/35 = 49.395 Map: [<9 1 1 12 -2|, <0 2 3 2 5|] EDOs: 72, 171, 243 Badness: 0.0203 ===13 limit ennealimmic=== Commas: 243/242, 364/363, 441/440, 625/624 POTE generator: ~36/35 = 49.341 Map: [<9 1 1 12 -2 -33|, <0 2 3 2 5 10|] EDOs: 72, 171, 243 Badness: 0.0233 ==Ennealimnic= Commas: 169/168, 243/242, 325/324, 441/440 POTE generator: ~36/35 = 49.708 Map: [<9 1 1 12 -2 20|, <0 2 3 2 5 2|] EDOs: 27e, 45f, 72, 315ff, 387cff, 459cdfff Badness: 0.0207 ==Semihemiennealimmal== Commas: 2401/2400, 4375/4374, 3025/3024, 4225/4224 POTE generator: Map: [<18 0 -1 22 48 88|, <0 4 6 4 2 -3|] EDOs: 126, 144, 270, 684, 954 Badness: 0.0131 ===17 limit ennealimmic=== Commas: 243/242, 364/363, 375/374, 441/440, 595/594 POTE generator: ~36/35 = 49.335 Map: [<9 1 1 12 -2 -33 -3|, <0 2 3 2 5 10 6|] EDOs: 72, 171, 243 Badness: 0.0146 =Gamera= Commas: 4375/4374, 589824/588245 POTE generator ~8/7 = 230.336 Map: [<1 6 10 3|, <0 -23 -40 -1|] EDOs: 26, 73, 99, 224, 323, 422, 735 Badness: 0.0376 =Supermajor= The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of <<37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS. Commas: 4375/4374, 52734375/52706752 POTE generator: ~9/7 = 435.082 Map: [<1 15 19 30|, <0 -37 -46 -75|] EDOs: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 Badness: 0.0108 =Enneadecal= Enndedecal temperament tempers out the enneadeca, |-14 -19 19>, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5 or 7 limits, and [[494edo]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning. Commas: 4375/4374, 703125/702464 POTE generator: ~3/2 = 701.880 Map: [<19 0 14 -37|, <0 1 1 3|] Generators: 28/27, 3 EDOs: 19, 152, 171, 665, 836, 1007, 2185 Badness: 0.0110 =Deca= Commas: 4375/4374, 165288374272/164794921875 POTE generator: ~460992/390625 = 284.423 Map: [<10 4 2 9|, <0 5 6 11|] EDOs: 80, 190, 270, 1270, 1540, 1810, 2080 Badness: 0.0806 ==11-limit== Commas: 3025/3024, 4375/4374, 422576/421875 POTE generator: ~33/28 = 284.418 Map: [<10 4 2 9 18|, <0 5 6 11 7|] EDOs: 80, 190, 270, 1000, 1270 Badness: 0.0243 ==13-limit== Commas: 1001/1000, 3025/3024, 4225/4224, 4375/4374 POTE generator: ~33/28 = 284.398 Map: [<10 4 2 9 18 37|, <0 5 6 11 7 0|] EDOs: 80, 190, 270, 730, 1000 Badness: 0.0168 =Mitonic= Commas: 4375/4374, 2100875/2097152 POTE generator: ~10/9 = 182.458 Map: [<1 16 32 -15|, <0 -17 -35 21|] EDOs: 46, 125, 171 Badness: 0.0252 =Abigail= Commas: 4375/4374, 2147483648/2144153025 [[POTE tuning|POTE generator]]: 208.899 Map: [<2 7 13 -1|, <0 -11 -24 19|] Wedgie: <<22 48 -38 25 -122 -223|| EDOs: 46, 132, 178, 224, 270, 494, 764, 1034, 1798 Badness: 0.0370 ==11-limit== Comma: 3025/3024, 4375/4374, 20614528/20588575 [[POTE tuning|POTE generator]]: 208.901 Map: [<2 7 13 -1 1|, <0 -11 -24 19 17|] EDOs: 46, 132, 178, 224, 270, 494, 764 Badness: 0.0129 ==13-limit== Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095 [[POTE tuning|POTE generator]]: 208.903 Map: [<2 7 13 -1 1 -2|, <0 -11 -24 19 17 27|] EDOs: 46, 178, 224, 270, 494, 764, 1258 Badness: 0.00886 =Nearly Micro= =Octoid= Commas: 4375/4374, 16875/16807 POTE generator: ~7/5 = 583.940 Map: [<8 1 3 3|, <0 3 4 5|] Generators: 49/45, 7/5 EDOs: 72, 152, 224 Badness: 0.0427 ==11-limit== Commas: 540/539, 1375/1372, 4000/3993 POTE generator: ~7/5 = 583.692 Map: Map: [<8 1 3 3 16|, <0 3 4 5 3|] EDOs: 72, 152, 224 Badness: 0.0141 ==13-limit== Commas: 540/539, 1375/1372, 4000/3993, 625/624 POTE generator: ~7/5 = 583.905 Map: Map: [<8 1 3 3 16 -21|, <0 3 4 5 3 13|] EDOs: 72, 224 Badness: 0.0153 ==Music== http://www.archive.org/details/Dreyfus [[http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3|play]] =Amity= The generator for amity temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&53 temperament, or by its wedgie, <<5 13 -17 9 -41 -76||. [[99edo]] is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you. In the 5-limit amity is a genuine microtemperament, with 58/205 being a possible tuning. Another good choice is (64/5)^(1/13), which gives pure major thirds. ==5-limit== Comma: 1600000/1594323 POTE generator: ~243/200 = 339.519 Map: [<1 3 6|, <0 -5 -13|] EDOs: 7, 39, 46, 53, 152, 205, 463, 668, 873 Badness: 0.0220 ==7-limit== Commas: 4375/4374, 5120/5103 POTE generator: ~243/200 = 339.432 Map: [<1 3 6 -2|, <0 -5 -13 17|] EDOs: 7, 39, 46, 53, 99, 251, 350 Badness: 0.0236 ==Hitchcock== Commas: 121/120, 176/175, 2200/2187 POTE generator: ~11/9 = 339.340 Map: [<1 3 6 -2 6|, <0 -5 -13 17 -9|] EDOs: 7, 39, 46, 53, 99 Badness: 0.0352 ==Hemiamity== Commas: 4375/4374, 5120/5103, 3025/3024 POTE generator: ~ 243/200 = 339.493 Map: [<2 1 -1 13 13|, <0 5 13 -17 -14|] EDOs: 14, 46, 106, 152, 350 =Parakleismic= In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13>, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being <<13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie <<13 14 35 -36 ...|| adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118. Comma: 124440064/1220703125 POTE generator: ~6/5 = 315.240 Map: [<1 5 6|, <0 -13 -14|] EDOs: 19, 61, 80, 99, 118, 453, 571, 689, 1496 Badness: 0.0433 ==7-limit== Commas: 3136/3125, 4375/4374 POTE generator: ~6/5 = 315.181 Map: [<1 5 6 12|, <0 -13 -14 -35|] EDOs: 19, 80, 99, 217, 316, 415 Badness: 0.0274 ==11-limit== Commas: 385/384, 3136/3125, 4375/4374 POTE generator: ~6/5 = 315.251 Map: [<1 5 6 12 -6|, <0 -13 -14 -35 36|] EDOs: 19, 99, 118 Badness: 0.0497 ==Parkleismic== Commas: 176/175, 1375/1372, 2200/2187 POTE generator: ~6/5 = 315.060 Map: [<1 5 6 12 20|, <0 -13 -14 -35 -63|] EDOs: 15, 19, 80, 179 Badness: 0.0559 ===13-limit=== Commas: 169/168, 176/175, 325/324, 1375/1372 POTE generator: ~6/5 = 315.075 Map: [<1 5 6 12 20 10|, <0 -13 -14 -35 -63 -24|] EDOs: 15, 19, 80, 179 Badness: 0.0366 =Quincy= Commas: 4375/4374, 823543/819200 POTE generator: ~1728/1715 = 16.613 Map: [<1 2 2 3|, <0 -30 -49 -14|] EDOs: 72, 217, 289 Badness: 0.0797 ==11-limit== Commas: 441/440, 4000/3993, 41503/41472 POTE generator: ~100/99 = 16.613 Map: [<1 2 2 3 4|, <0 -30 -49 -14 -39|] EDOs: 72, 217, 289 Badness: 0.0309 ==13-limit== Commas: 364/363, 441/440, 676/675, 4375/4374 POTE generator: ~100/99 = 16.602 Map: [<1 2 2 3 4 5|, <0 -30 -49 -14 -39 -94|] EDOs: 72, 145, 217, 289 Badness: 0.0239 ==17-limit== Commas: 364/363, 441/440, 595/594, 1001/1000, 1156/1155 POTE generator: ~100/99 = 16.602 Map: [<1 2 2 3 4 5 5|, <0 -30 -49 -14 -39 -94 -66|] EDOs: 72, 145, 217, 289 Badness: 0.0147 ==19-limit== Commas: 343/342, 364/363, 441/440, 595/594, 676/675, 2601/2600 POTE generator: ~100/99 = 16.594 Map: [<1 2 2 3 4 5 5 4|, <0 -30 -49 -14 -39 -94 -66 18|] EDOs: 72, 145, 217 Badness: 0.0152
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<html><head><title>Ragismic microtemperaments</title></head><body><!-- ws:start:WikiTextTocRule:80:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 1em;"><a href="#Ennealimmal">Ennealimmal</a></div> <!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 2em;"><a href="#Ennealimmal-11 limit hemiennealimmal">11 limit hemiennealimmal</a></div> <!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><div style="margin-left: 2em;"><a href="#Ennealimmal-13 limit hemiennealimmal">13 limit hemiennealimmal</a></div> <!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><div style="margin-left: 2em;"><a href="#Ennealimmal-Semiennealimmal">Semiennealimmal</a></div> <!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><div style="margin-left: 3em;"><a href="#Ennealimmal-Semiennealimmal-13 limit semiennealimmal">13 limit semiennealimmal</a></div> <!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><div style="margin-left: 2em;"><a href="#Ennealimmal-Ennealimmic">Ennealimmic</a></div> <!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><div style="margin-left: 3em;"><a href="#Ennealimmal-Ennealimmic-13 limit ennealimmic">13 limit ennealimmic</a></div> <!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><div style="margin-left: 1em;"><a href="#x=Ennealimnic">=Ennealimnic</a></div> <!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><div style="margin-left: 2em;"><a href="#x=Ennealimnic-Semihemiennealimmal">Semihemiennealimmal</a></div> <!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><div style="margin-left: 3em;"><a href="#x=Ennealimnic-Semihemiennealimmal-17 limit ennealimmic">17 limit ennealimmic</a></div> <!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><div style="margin-left: 1em;"><a href="#Gamera">Gamera</a></div> <!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><div style="margin-left: 1em;"><a href="#Supermajor">Supermajor</a></div> <!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><div style="margin-left: 1em;"><a href="#Enneadecal">Enneadecal</a></div> <!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><div style="margin-left: 1em;"><a href="#Deca">Deca</a></div> <!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 2em;"><a href="#Deca-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 2em;"><a href="#Deca-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><div style="margin-left: 1em;"><a href="#Mitonic">Mitonic</a></div> <!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><div style="margin-left: 1em;"><a href="#Abigail">Abigail</a></div> <!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><div style="margin-left: 2em;"><a href="#Abigail-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><div style="margin-left: 2em;"><a href="#Abigail-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><div style="margin-left: 1em;"><a href="#Nearly Micro">Nearly Micro</a></div> <!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><div style="margin-left: 1em;"><a href="#Octoid">Octoid</a></div> <!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><div style="margin-left: 2em;"><a href="#Octoid-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><div style="margin-left: 2em;"><a href="#Octoid-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 2em;"><a href="#Octoid-Music">Music</a></div> <!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><div style="margin-left: 1em;"><a href="#Amity">Amity</a></div> <!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><div style="margin-left: 2em;"><a href="#Amity-5-limit">5-limit</a></div> <!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><div style="margin-left: 2em;"><a href="#Amity-7-limit">7-limit</a></div> <!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><div style="margin-left: 2em;"><a href="#Amity-Hitchcock">Hitchcock</a></div> <!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><div style="margin-left: 2em;"><a href="#Amity-Hemiamity">Hemiamity</a></div> <!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><div style="margin-left: 1em;"><a href="#Parakleismic">Parakleismic</a></div> <!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><div style="margin-left: 2em;"><a href="#Parakleismic-7-limit">7-limit</a></div> <!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --><div style="margin-left: 2em;"><a href="#Parakleismic-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --><div style="margin-left: 2em;"><a href="#Parakleismic-Parkleismic">Parkleismic</a></div> <!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --><div style="margin-left: 3em;"><a href="#Parakleismic-Parkleismic-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --><div style="margin-left: 1em;"><a href="#Quincy">Quincy</a></div> <!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --><div style="margin-left: 2em;"><a href="#Quincy-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><div style="margin-left: 2em;"><a href="#Quincy-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><div style="margin-left: 2em;"><a href="#Quincy-17-limit">17-limit</a></div> <!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><div style="margin-left: 2em;"><a href="#Quincy-19-limit">19-limit</a></div> <!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --></div> <!-- ws:end:WikiTextTocRule:121 -->The ragisma is 4375/4374 with a monzo of |-1 -7 4 1>, the smallest 7-limit superparticular ratio. Since (10/9)^4 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Ennealimmal"></a><!-- ws:end:WikiTextHeadingRule:0 -->Ennealimmal</h1> Ennealimmal temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the ennealimma comma, |1 -27 18>, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is <<18 27 18 1 -22 -34||.<br /> <br /> Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 equal, though its hardly likely anyone could tell the difference.<br /> <br /> If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example.) In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS.<br /> <br /> Commas: 2401/2400, 4375/4374<br /> <br /> POTE generators: 36/35: 49.0205; 10/9: 182.354; 6/5: 315.687; 49/40: 350.980<br /> <br /> Map: [<9 1 1 2|, <0 2 3 2|]<br /> Wedgie: <<18 27 18 1 -22 -34||<br /> EDOs: <a class="wiki_link" href="/27edo">27</a>, <a class="wiki_link" href="/45edo">45</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/99edo">99</a>, <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/270edo">270</a>, <a class="wiki_link" href="/441edo">441</a>, <a class="wiki_link" href="/612edo">612</a>, <a class="wiki_link" href="/3600edo">3600</a><br /> Badness: 0.00361<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Ennealimmal-11 limit hemiennealimmal"></a><!-- ws:end:WikiTextHeadingRule:2 -->11 limit hemiennealimmal</h2> Commas: 2401/2400, 4375/4374, 3025/3024<br /> <br /> POTE generator: 99/98: 17.6219 or 6/5: 315.7114<br /> <br /> Map: [<18 0 -1 22 48|, <0 2 3 2 1|]<br /> EDOs: 72, 198, 270, 342, 612, 954, 1566<br /> Badness: 0.00628<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Ennealimmal-13 limit hemiennealimmal"></a><!-- ws:end:WikiTextHeadingRule:4 -->13 limit hemiennealimmal</h2> Commas: 676/675, 1001/1000, 1716/1715, 3025/3024<br /> <br /> POTE generator ~99/98 = 17.7504<br /> <br /> Map: [<18 0 -1 22 48 -19|, <0 2 3 2 1 6|]<br /> EDOs: 72, 198, 270<br /> Badness: 0.0125<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Ennealimmal-Semiennealimmal"></a><!-- ws:end:WikiTextHeadingRule:6 -->Semiennealimmal</h2> Commas: 2401/2400, 4375/4374, 4000/3993<br /> <br /> POTE generator: ~140/121 = 250.3367<br /> <br /> Map: [<9 3 4 14 18|, <0 6 9 6 7|]<br /> EDOs: 72, 369, 441<br /> Badness: 0.0342<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="Ennealimmal-Semiennealimmal-13 limit semiennealimmal"></a><!-- ws:end:WikiTextHeadingRule:8 -->13 limit semiennealimmal</h3> Commas: 1575/1573, 2080/2079, 2401/2400, 4375/4374<br /> <br /> POTE generator: ~140/121 = 250.3375<br /> <br /> Map: [<9 3 4 14 18 -8|, <0 6 9 6 7 22|]<br /> EDOs: 72, 441<br /> Badness: 0.0261<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Ennealimmal-Ennealimmic"></a><!-- ws:end:WikiTextHeadingRule:10 -->Ennealimmic</h2> Commas: 243/242, 441/440, 4375/4356<br /> <br /> POTE generator: ~36/35 = 49.395<br /> <br /> Map: [<9 1 1 12 -2|, <0 2 3 2 5|]<br /> EDOs: 72, 171, 243<br /> Badness: 0.0203<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="Ennealimmal-Ennealimmic-13 limit ennealimmic"></a><!-- ws:end:WikiTextHeadingRule:12 -->13 limit ennealimmic</h3> Commas: 243/242, 364/363, 441/440, 625/624<br /> <br /> POTE generator: ~36/35 = 49.341<br /> <br /> Map: [<9 1 1 12 -2 -33|, <0 2 3 2 5 10|]<br /> EDOs: 72, 171, 243<br /> Badness: 0.0233<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="x=Ennealimnic"></a><!-- ws:end:WikiTextHeadingRule:14 -->=Ennealimnic</h1> Commas: 169/168, 243/242, 325/324, 441/440<br /> <br /> POTE generator: ~36/35 = 49.708<br /> <br /> Map: [<9 1 1 12 -2 20|, <0 2 3 2 5 2|]<br /> EDOs: 27e, 45f, 72, 315ff, 387cff, 459cdfff<br /> Badness: 0.0207<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="x=Ennealimnic-Semihemiennealimmal"></a><!-- ws:end:WikiTextHeadingRule:16 -->Semihemiennealimmal</h2> Commas: 2401/2400, 4375/4374, 3025/3024, 4225/4224<br /> <br /> POTE generator:<br /> <br /> Map: [<18 0 -1 22 48 88|, <0 4 6 4 2 -3|]<br /> EDOs: 126, 144, 270, 684, 954<br /> Badness: 0.0131<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h3> --><h3 id="toc9"><a name="x=Ennealimnic-Semihemiennealimmal-17 limit ennealimmic"></a><!-- ws:end:WikiTextHeadingRule:18 -->17 limit ennealimmic</h3> Commas: 243/242, 364/363, 375/374, 441/440, 595/594<br /> <br /> POTE generator: ~36/35 = 49.335<br /> <br /> Map: [<9 1 1 12 -2 -33 -3|, <0 2 3 2 5 10 6|]<br /> EDOs: 72, 171, 243<br /> Badness: 0.0146<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h1> --><h1 id="toc10"><a name="Gamera"></a><!-- ws:end:WikiTextHeadingRule:20 -->Gamera</h1> Commas: 4375/4374, 589824/588245<br /> <br /> POTE generator ~8/7 = 230.336<br /> <br /> Map: [<1 6 10 3|, <0 -23 -40 -1|]<br /> EDOs: 26, 73, 99, 224, 323, 422, 735<br /> Badness: 0.0376<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h1> --><h1 id="toc11"><a name="Supermajor"></a><!-- ws:end:WikiTextHeadingRule:22 -->Supermajor</h1> The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of <<37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.<br /> <br /> Commas: 4375/4374, 52734375/52706752<br /> <br /> POTE generator: ~9/7 = 435.082<br /> <br /> Map: [<1 15 19 30|, <0 -37 -46 -75|]<br /> EDOs: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214<br /> Badness: 0.0108<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h1> --><h1 id="toc12"><a name="Enneadecal"></a><!-- ws:end:WikiTextHeadingRule:24 -->Enneadecal</h1> Enndedecal temperament tempers out the enneadeca, |-14 -19 19>, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of <a class="wiki_link" href="/19edo">19edo</a> up to just ones. <a class="wiki_link" href="/171edo">171edo</a> is a good tuning for either the 5 or 7 limits, and <a class="wiki_link" href="/494edo">494edo</a> shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use <a class="wiki_link" href="/665edo">665edo</a> for a tuning.<br /> <br /> Commas: 4375/4374, 703125/702464<br /> <br /> POTE generator: ~3/2 = 701.880<br /> <br /> Map: [<19 0 14 -37|, <0 1 1 3|]<br /> Generators: 28/27, 3<br /> EDOs: 19, 152, 171, 665, 836, 1007, 2185<br /> Badness: 0.0110<br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h1> --><h1 id="toc13"><a name="Deca"></a><!-- ws:end:WikiTextHeadingRule:26 -->Deca</h1> Commas: 4375/4374, 165288374272/164794921875<br /> <br /> POTE generator: ~460992/390625 = 284.423<br /> <br /> Map: [<10 4 2 9|, <0 5 6 11|]<br /> EDOs: 80, 190, 270, 1270, 1540, 1810, 2080<br /> Badness: 0.0806<br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h2> --><h2 id="toc14"><a name="Deca-11-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->11-limit</h2> Commas: 3025/3024, 4375/4374, 422576/421875<br /> <br /> POTE generator: ~33/28 = 284.418<br /> <br /> Map: [<10 4 2 9 18|, <0 5 6 11 7|]<br /> EDOs: 80, 190, 270, 1000, 1270<br /> Badness: 0.0243<br /> <br /> <!-- ws:start:WikiTextHeadingRule:30:<h2> --><h2 id="toc15"><a name="Deca-13-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->13-limit</h2> Commas: 1001/1000, 3025/3024, 4225/4224, 4375/4374<br /> <br /> POTE generator: ~33/28 = 284.398<br /> <br /> Map: [<10 4 2 9 18 37|, <0 5 6 11 7 0|]<br /> EDOs: 80, 190, 270, 730, 1000<br /> Badness: 0.0168<br /> <br /> <!-- ws:start:WikiTextHeadingRule:32:<h1> --><h1 id="toc16"><a name="Mitonic"></a><!-- ws:end:WikiTextHeadingRule:32 -->Mitonic</h1> Commas: 4375/4374, 2100875/2097152<br /> <br /> POTE generator: ~10/9 = 182.458<br /> <br /> Map: [<1 16 32 -15|, <0 -17 -35 21|]<br /> EDOs: 46, 125, 171<br /> Badness: 0.0252<br /> <br /> <!-- ws:start:WikiTextHeadingRule:34:<h1> --><h1 id="toc17"><a name="Abigail"></a><!-- ws:end:WikiTextHeadingRule:34 -->Abigail</h1> Commas: 4375/4374, 2147483648/2144153025<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 208.899<br /> <br /> Map: [<2 7 13 -1|, <0 -11 -24 19|]<br /> Wedgie: <<22 48 -38 25 -122 -223||<br /> EDOs: 46, 132, 178, 224, 270, 494, 764, 1034, 1798<br /> Badness: 0.0370<br /> <br /> <!-- ws:start:WikiTextHeadingRule:36:<h2> --><h2 id="toc18"><a name="Abigail-11-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->11-limit</h2> Comma: 3025/3024, 4375/4374, 20614528/20588575<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 208.901<br /> <br /> Map: [<2 7 13 -1 1|, <0 -11 -24 19 17|]<br /> EDOs: 46, 132, 178, 224, 270, 494, 764<br /> Badness: 0.0129<br /> <br /> <!-- ws:start:WikiTextHeadingRule:38:<h2> --><h2 id="toc19"><a name="Abigail-13-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->13-limit</h2> Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 208.903<br /> <br /> Map: [<2 7 13 -1 1 -2|, <0 -11 -24 19 17 27|]<br /> EDOs: 46, 178, 224, 270, 494, 764, 1258<br /> Badness: 0.00886<br /> <br /> <!-- ws:start:WikiTextHeadingRule:40:<h1> --><h1 id="toc20"><a name="Nearly Micro"></a><!-- ws:end:WikiTextHeadingRule:40 -->Nearly Micro</h1> <br /> <!-- ws:start:WikiTextHeadingRule:42:<h1> --><h1 id="toc21"><a name="Octoid"></a><!-- ws:end:WikiTextHeadingRule:42 -->Octoid</h1> Commas: 4375/4374, 16875/16807<br /> <br /> POTE generator: ~7/5 = 583.940<br /> <br /> Map: [<8 1 3 3|, <0 3 4 5|]<br /> Generators: 49/45, 7/5<br /> EDOs: 72, 152, 224<br /> Badness: 0.0427<br /> <br /> <!-- ws:start:WikiTextHeadingRule:44:<h2> --><h2 id="toc22"><a name="Octoid-11-limit"></a><!-- ws:end:WikiTextHeadingRule:44 -->11-limit</h2> Commas: 540/539, 1375/1372, 4000/3993<br /> <br /> POTE generator: ~7/5 = 583.692<br /> <br /> Map: Map: [<8 1 3 3 16|, <0 3 4 5 3|]<br /> EDOs: 72, 152, 224<br /> Badness: 0.0141<br /> <br /> <!-- ws:start:WikiTextHeadingRule:46:<h2> --><h2 id="toc23"><a name="Octoid-13-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->13-limit</h2> Commas: 540/539, 1375/1372, 4000/3993, 625/624<br /> <br /> POTE generator: ~7/5 = 583.905<br /> <br /> Map: Map: [<8 1 3 3 16 -21|, <0 3 4 5 3 13|]<br /> EDOs: 72, 224<br /> Badness: 0.0153<br /> <br /> <!-- ws:start:WikiTextHeadingRule:48:<h2> --><h2 id="toc24"><a name="Octoid-Music"></a><!-- ws:end:WikiTextHeadingRule:48 -->Music</h2> <!-- ws:start:WikiTextUrlRule:461:http://www.archive.org/details/Dreyfus --><a class="wiki_link_ext" href="http://www.archive.org/details/Dreyfus" rel="nofollow">http://www.archive.org/details/Dreyfus</a><!-- ws:end:WikiTextUrlRule:461 --><br /> <a class="wiki_link_ext" href="http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3" rel="nofollow">play</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:50:<h1> --><h1 id="toc25"><a name="Amity"></a><!-- ws:end:WikiTextHeadingRule:50 -->Amity</h1> The generator for amity temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&53 temperament, or by its wedgie, <<5 13 -17 9 -41 -76||. <a class="wiki_link" href="/99edo">99edo</a> is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.<br /> <br /> In the 5-limit amity is a genuine microtemperament, with 58/205 being a possible tuning. Another good choice is (64/5)^(1/13), which gives pure major thirds.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:52:<h2> --><h2 id="toc26"><a name="Amity-5-limit"></a><!-- ws:end:WikiTextHeadingRule:52 -->5-limit</h2> Comma: 1600000/1594323<br /> <br /> POTE generator: ~243/200 = 339.519<br /> <br /> Map: [<1 3 6|, <0 -5 -13|]<br /> EDOs: 7, 39, 46, 53, 152, 205, 463, 668, 873<br /> Badness: 0.0220<br /> <br /> <!-- ws:start:WikiTextHeadingRule:54:<h2> --><h2 id="toc27"><a name="Amity-7-limit"></a><!-- ws:end:WikiTextHeadingRule:54 -->7-limit</h2> Commas: 4375/4374, 5120/5103<br /> <br /> POTE generator: ~243/200 = 339.432<br /> <br /> Map: [<1 3 6 -2|, <0 -5 -13 17|]<br /> EDOs: 7, 39, 46, 53, 99, 251, 350<br /> Badness: 0.0236<br /> <br /> <!-- ws:start:WikiTextHeadingRule:56:<h2> --><h2 id="toc28"><a name="Amity-Hitchcock"></a><!-- ws:end:WikiTextHeadingRule:56 -->Hitchcock</h2> Commas: 121/120, 176/175, 2200/2187<br /> <br /> POTE generator: ~11/9 = 339.340<br /> <br /> Map: [<1 3 6 -2 6|, <0 -5 -13 17 -9|]<br /> EDOs: 7, 39, 46, 53, 99<br /> Badness: 0.0352<br /> <br /> <!-- ws:start:WikiTextHeadingRule:58:<h2> --><h2 id="toc29"><a name="Amity-Hemiamity"></a><!-- ws:end:WikiTextHeadingRule:58 -->Hemiamity</h2> Commas: 4375/4374, 5120/5103, 3025/3024<br /> <br /> POTE generator: ~ 243/200 = 339.493<br /> <br /> Map: [<2 1 -1 13 13|, <0 5 13 -17 -14|]<br /> EDOs: 14, 46, 106, 152, 350<br /> <br /> <!-- ws:start:WikiTextHeadingRule:60:<h1> --><h1 id="toc30"><a name="Parakleismic"></a><!-- ws:end:WikiTextHeadingRule:60 -->Parakleismic</h1> In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13>, with the <a class="wiki_link" href="/118edo">118edo</a> tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being <<13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie <<13 14 35 -36 ...|| adding 385/384. For the 7-limit <a class="wiki_link" href="/99edo">99edo</a> may be preferred, but in the 11-limit it is best to stick with 118.<br /> <br /> Comma: 124440064/1220703125<br /> <br /> POTE generator: ~6/5 = 315.240<br /> <br /> Map: [<1 5 6|, <0 -13 -14|]<br /> EDOs: 19, 61, 80, 99, 118, 453, 571, 689, 1496<br /> Badness: 0.0433<br /> <br /> <!-- ws:start:WikiTextHeadingRule:62:<h2> --><h2 id="toc31"><a name="Parakleismic-7-limit"></a><!-- ws:end:WikiTextHeadingRule:62 -->7-limit</h2> Commas: 3136/3125, 4375/4374<br /> <br /> POTE generator: ~6/5 = 315.181<br /> <br /> Map: [<1 5 6 12|, <0 -13 -14 -35|]<br /> EDOs: 19, 80, 99, 217, 316, 415<br /> Badness: 0.0274<br /> <br /> <!-- ws:start:WikiTextHeadingRule:64:<h2> --><h2 id="toc32"><a name="Parakleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:64 -->11-limit</h2> Commas: 385/384, 3136/3125, 4375/4374<br /> <br /> POTE generator: ~6/5 = 315.251<br /> <br /> Map: [<1 5 6 12 -6|, <0 -13 -14 -35 36|]<br /> EDOs: 19, 99, 118<br /> Badness: 0.0497<br /> <br /> <!-- ws:start:WikiTextHeadingRule:66:<h2> --><h2 id="toc33"><a name="Parakleismic-Parkleismic"></a><!-- ws:end:WikiTextHeadingRule:66 -->Parkleismic</h2> Commas: 176/175, 1375/1372, 2200/2187<br /> <br /> POTE generator: ~6/5 = 315.060<br /> <br /> Map: [<1 5 6 12 20|, <0 -13 -14 -35 -63|]<br /> EDOs: 15, 19, 80, 179<br /> Badness: 0.0559<br /> <br /> <!-- ws:start:WikiTextHeadingRule:68:<h3> --><h3 id="toc34"><a name="Parakleismic-Parkleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:68 -->13-limit</h3> Commas: 169/168, 176/175, 325/324, 1375/1372<br /> <br /> POTE generator: ~6/5 = 315.075<br /> <br /> Map: [<1 5 6 12 20 10|, <0 -13 -14 -35 -63 -24|]<br /> EDOs: 15, 19, 80, 179<br /> Badness: 0.0366<br /> <br /> <!-- ws:start:WikiTextHeadingRule:70:<h1> --><h1 id="toc35"><a name="Quincy"></a><!-- ws:end:WikiTextHeadingRule:70 -->Quincy</h1> Commas: 4375/4374, 823543/819200<br /> <br /> POTE generator: ~1728/1715 = 16.613<br /> <br /> Map: [<1 2 2 3|, <0 -30 -49 -14|]<br /> EDOs: 72, 217, 289<br /> Badness: 0.0797<br /> <br /> <!-- ws:start:WikiTextHeadingRule:72:<h2> --><h2 id="toc36"><a name="Quincy-11-limit"></a><!-- ws:end:WikiTextHeadingRule:72 -->11-limit</h2> Commas: 441/440, 4000/3993, 41503/41472<br /> <br /> POTE generator: ~100/99 = 16.613<br /> <br /> Map: [<1 2 2 3 4|, <0 -30 -49 -14 -39|]<br /> EDOs: 72, 217, 289<br /> Badness: 0.0309<br /> <br /> <!-- ws:start:WikiTextHeadingRule:74:<h2> --><h2 id="toc37"><a name="Quincy-13-limit"></a><!-- ws:end:WikiTextHeadingRule:74 -->13-limit</h2> Commas: 364/363, 441/440, 676/675, 4375/4374<br /> <br /> POTE generator: ~100/99 = 16.602<br /> <br /> Map: [<1 2 2 3 4 5|, <0 -30 -49 -14 -39 -94|]<br /> EDOs: 72, 145, 217, 289<br /> Badness: 0.0239<br /> <br /> <!-- ws:start:WikiTextHeadingRule:76:<h2> --><h2 id="toc38"><a name="Quincy-17-limit"></a><!-- ws:end:WikiTextHeadingRule:76 -->17-limit</h2> Commas: 364/363, 441/440, 595/594, 1001/1000, 1156/1155<br /> <br /> POTE generator: ~100/99 = 16.602<br /> <br /> Map: [<1 2 2 3 4 5 5|, <0 -30 -49 -14 -39 -94 -66|]<br /> EDOs: 72, 145, 217, 289<br /> Badness: 0.0147<br /> <br /> <!-- ws:start:WikiTextHeadingRule:78:<h2> --><h2 id="toc39"><a name="Quincy-19-limit"></a><!-- ws:end:WikiTextHeadingRule:78 -->19-limit</h2> Commas: 343/342, 364/363, 441/440, 595/594, 676/675, 2601/2600<br /> <br /> POTE generator: ~100/99 = 16.594<br /> <br /> Map: [<1 2 2 3 4 5 5 4|, <0 -30 -49 -14 -39 -94 -66 18|]<br /> EDOs: 72, 145, 217<br /> Badness: 0.0152</body></html>