Schismatic family
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author xenwolf and made on 2016-09-03 07:24:39 UTC.
- The original revision id was 590996938.
- The revision comment was: Reverted to Aug 31, 2016 7:12 am: removal of DE link seems to be mistakenly
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
[[toc]] <span style="display: block; text-align: right;">Other languages: [[xenharmonie/Schismatische Temperaturen|Deutsch]] </span> =Five limit= The 5-limit parent comma for the schismatic family is the [[schisma]] of 32805/32768, which is the amount by which the Pythagorean comma exceeds the [[Didymus comma]] ([[81_80|81/80]]), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is |-15 8 1>, and flipping that yields <<1 -8 -15|| for the [[Wedgies and Multivals|wedgie]]. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768. The 5-limit version of the temperament is a [[Microtempering|microtemperament]], sometimes called **Helmholtz** or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. [[POTE tuning|POTE generator]]: ~3/2 = 701.736 Mapping generator: ~3 Map: [<1 0 15|, <0 1 -8|] EDOs: [[12edo|12]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[118edo|118]], [[171edo|171]], [[289edo|289]], 460, 749, 3456bc, 4205bc, 4954bc, 5703bc =Seven limit children= The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding |25 -14 0 -1> gives garibaldi, |-44 26 0 1> grackle, |6 -2 0 -1> schism and |-59 39 0 -1> pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2> and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3>, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3> with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4> and slices the fifth in four. =Garibaldi= Commas: 225/224, 3125/3087 7-limit minimax tuning: 7-limit: [|1 0 0 0>, |5/3 1/15 0 -1/15>, |5/3 -8/15 0 8/15>, |5/3 -14/15 0 14/15>] [[Eigenmonzo]]s: 2, 7/6 9-limit: [|1 0 0 0>, |25/16 1/8 0 -1/16>, |5/2 -1 0 1/2>, |25/8 -7/4 0 7/8>] Eigenmonzos: 2, 9/7 [[POTE tuning|POTE generator]]: ~3/2 = 702.085 Mapping generator: ~3 Map: [<1 0 15 25|, <0 1 -8 -14|] Wedgie: <<1 -8 -14 -15 -25 -10|| EDOs: 12, 29, 41, 53, [[94edo|94]], 241c, 335cd, 576cd Badness: 0.0216 ==11-limit== Commas: 225/224, 385/384, 2200/2187 Minimax tuning: [|1 0 0 0 0>, |25/16 1/8 0 -1/16 0>, |5/2 -1 0 1/2 0>, |25/8 -7/4 0 7/8 0>, |47/16 23/8 0 -23/16 0>] Eigenmonzos: 2, 9/7 [[POTE tuning|POTE generator]]: ~3/2 = 702.157 Mapping generator: ~3 Map: [<1 0 15 25 -33|, <0 1 -8 -14 23|] [[edo|Edos]]: [[41edo|41]], [[53edo|53]], [[94edo|94]], [[135edo|135]], 229c, 323c, 417ce Badness: 0.0274 ===13-limit=== Commas: 225/224, 275/273, 325/324, 385/384 [[POTE tuning|POTE generator]]: ~3/2 = 702.113 Mapping generator: ~3 Map: [<1 0 15 25 -33 -28|, <0 1 -8 -14 23 20|] EDOs: 12, 41, 53, 94, 429cdef, 523cdef Badness: 0.0207 ==Cassandra== Commas: 100/99, 225/224, 245/242 [[POTE tuning|POTE generator]]: ~3/2 = 702.321 Mapping generator: ~3 Map: [<1 0 15 25 32|, <0 1 -8 -14 -18|] EDOs: 12, 29, 41, 123c, 217ce, 258ce Badness: 0.0236 ===13-limit Cassandra=== Commas: 100/99, 105/104, 196/195, 245/242 [[POTE tuning|POTE generator]]: ~3/2 = 702.559 Mapping generator: ~3 Map: [<1 0 15 25 32 37|, <0 1 -8 -14 -18 -21|] EDOs: 12, 29, 41, 152cdf, 193cdf, 234cdf Badness: 0.0207 ==Helenus== Commas: 99/98 176/175 3125/3087 POTE generator: ~3/2 = 701.725 Mapping generator: ~3 Map: [<1 0 15 25 51|, <0 1 -8 -14 -30|] EDOs: 12, 53, 118d, 171de Badness: 0.0356 ===13-limit Helenus=== Commas: 99/98, 176/175, 275/273, 847/845 POTE generator: ~3/2 = 701.747 Mapping generator: ~3 Map: [<1 0 15 25 51 56|, <0 1 -8 -14 -30 -33|] EDOs: 53, 118d, 171de Badness: 0.0263 =Hemigari= Commas: 121/120, 225/224, 3125/3087 POTE generator: ~63/55 = 248.918 Mapping generator: ~110/63 Map: [<1 0 15 25 9|, <0 2 -16 -28 -7|] EDOs: 29, 53, 82e, 135e, 188ce Badness: 0.0507 ==13-limit== Commas: 121/120, 169/168, 225/224, 275/273 POTE generator: ~15/13 = 248.918 Mapping generator: ~26/15 Map: [<1 0 15 25 9 14|, <0 2 -16 -28 -7 -13|] EDOs: 29, 53, 82e, 135ef, 188cef Badness: 27.464 =Guiron= [[Comma|Commas]]: 1029/1024, 10976/10935 [[Minimax tuning]]: 7+9 limit: [|1 0 0 0>, |15/8 0 -1/8 0>, |0 0 1 0>, |65/24 0 1/24 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 5/4 [[POTE tuning|POTE generator]]: ~8/7 = 233.930 Mapping generator: ~8/7 Map: [<1 1 7 3|, <0 3 -24 -1|] Wedgie: <<3 -24 -1 -45 -10 65|| EDOs: 36, 41, [[77edo|77]], 118, 277d Badness: 0.0475 ==11-limit== [[Comma|Commas]]: 385/384, 441/440, 10976/10935 [[Minimax tuning]]: [|1 0 0 0 0>, |15/8 0 -1/8 0 0>, |0 0 1 0 0>, |65/24 0 1/24 0 0>, |37/6 0 -7/6 0 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 5/4 [[POTE tuning|POTE generator]]: ~8/7 = 233.931 Mapping generator: ~8/7 Map: [<1 1 7 3 -2|, <0 3 -24 -1 28|] [[edo|Edos]]: [[41edo|41]], [[77edo|77]], [[118edo|118]], [[159edo|159]], [[200edo|200]], [[277|277d]] Badness: 0.0266 ==13-limit== Commas: 196/195, 352/351, 385/384, 729/728 POTE generator: ~8/7 = 233.890 Mapping generator: ~8/7 Map: [<1 1 7 3 -2 0|, <0 3 -24 -1 28 19|] EDOs: 41, 77, 118 Badness: 0.0284 =Pogo= Commas: 32805/32768, 118098/117649 POTE generator: ~9/7 = 433.901 Mapping generator: ~9/7 Map: [<2 1 22 2|, <0 3 -24 5|] EDOs: 36, 94, 130, 171, 183, 224, 354 Badness: 0.0796 ==11-limit== Commas: 540/539, 4000/3993, 32805/32768 POTE generator: ~9/7 = 433.911 Mapping generator: ~9/7 Map: [<2 1 22 2 25|, <0 3 -24 5 -25|] EDOs: 36, 94, 130, 224, 354, 578 Badness: 0.0319 ==13-limit== Commas: 540/539, 729/728, 4000/3993, 4225/4224 POTE generator: ~9/7 = 433.911 Map: [<2 1 22 2 25 -2|, <0 3 -24 5 -25 13|] EDOs: 36, 94, 130, 224, 354, 578 Badness: 0.0175 =Sanjaab= Commas: 225/224, 3125/3087, 1331/1323 POTE generator: ~11/10 = 165.974 Map: [<1 2 -1 -3 0|, <0 -3 24 42 25|] EDOs: 29, 65d, 94, 441cde, 535cde, 629cde Badness: 0.0580 ==13-limit== Commas: 225/224, 275/273, 847/845, 1331/1323 POTE generator: ~11/10 = 165.963 Map: [<1 2 -1 -3 0 -1|, <0 -3 24 42 25 34|] EDOs: 29, 65d, 94 Badness: 0.0338 =Squirrel= Commmas: 686/675, 32805/32768 POTE generator: ~160/147 = 166.140 Map: [<1 2 -1 1|, <0 -3 24 13|] EDOs: 29, 36, 65 Badness: 0.1747 ==11-limit== Commas: 245/242, 686/675, 896/891 POTE generator: ~11/10 = 166.097 Map: [<1 2 -1 1 0|, <0 -3 24 13 25|] EDOs: 29, 36, 65 Badness: 0.0683 ==13-limit== Commas: 91/90, 169/168, 245/242, 896/891 POTE generator: ~11/10 = 166.054 Map: [<1 2 -1 1 0 3|, <0 -3 24 13 25 5|] EDOs: 29, 36, 65f, 94df, 159df Badness: 0.0437 =Schism= Commas: 64/63, 360/343 POTE generator: ~3/2 = 701.556 Mapping generator: ~3 Map: [<1 0 15 6|, <0 1 -8 -2|] Wedgie: <<1 -8 -2 -15 -6 18|| EDOs: 12, 41d, 53d Badness: 0.0566 ==11-limit== Commas: 45/44, 64/63, 99/98 POTE generator ~3/2 = 702.136 Mapping generator: ~3 Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|] EDOs: 12, 29de, 41de Badness: 0.0375 =Pontiac= Commas: 32805/32768, 4375/4374 7-limit minimax: [|1 0 0 0>, |74/47 0 -1/47 1/47>, |113/47 0 8/47 -8/47>, |113/47 0 -39/47 39/47>] Eigenmonzos: 2, 7/5 9-limit minimax: [|1 0 0 0>, |3/2 1/5 -1/10 0>, |3 -8/5 4/5 0>, |-1/2 39/5 -39/10 0>] Eigenmonzos: 2, 10/9 [[POTE tuning|POTE generator]]: 701.757 Mapping generator: ~3 Map: [<1 0 15 -59|, <0 1 -8 39|] Wedgie: <<1 -8 39 -15 59 113|| [[edo|Edos]]: [[53edo|53]], [[118edo|118]], [[171edo|171]], [[1079edo|1079]], [[1250edo|1250]], [[1421edo|1421]] Badness: 0.0141 =Grackle= Commas: 126/125, 32805/32768 7-limit minimax Eigenmonzos: 2, 7/6 9-limit minimax Eigenmonzos: 2, 9/7 [[POTE tuning|POTE generator]]: 701.239 Mapping generator: ~3 Map: [<1 0 15 -44|, <0 1 -8 -26|] Wedgie: <<1 -8 -26 -15 -44 -38|| [[edo|Edos]]: [[77edo|77]], [[89edo|89]], [[101edo|101]], [[166edo|166c]], [[243edo|243c]] Badness: 0.070 =Bischismic= Commas: 3136/3125, 32805/32768 7-limit minimax Eigenmonzos: 2, 7/6 9-limit minimax Eigenmonzos: 2, 9/7 [[POTE tuning|POTE generator]]: 701.592 Mapping generator: ~3 Map: [<2 0 30 69|, <0 1 -8 -20|] Wedgie: <<2 -16 -40 -30 -69 -48|| [[edo|Edos]]: [[118edo|118]], [[130edo|130]], [[248edo|248]], [[378edo|378]], [[508edo|508]] Badness: 0.0547 ==11-limit== Commas: 441/440, 3136/3125, 8019/8000 POTE generator: ~3/2 = 701.612 Mapping generator: ~3 Map: [<2 0 30 69 102|, <0 1 -8 -20 -30|] EDOs: 12, 118, 130, 248 Badness: 0.0282 ==13-limit== Commas: 441/440, 729/728, 1001/1000, 3136/3125 POTE generator: ~3/2 = 701.590 Mapping generator: ~3 Map: [<2 0 30 69 102 -75|, <0 1 -8 -20 -30 26|] EDOs: 12, 118, 130, 248, 378 Badness: 0.0287 =Kleischismic= Commas: 32805/32768, 1500625/1492992 POTE generator: ~36/35 = 50.920 Mapping generator: ~35/24 Map: [<2 1 22 -15|, <0 2 -16 19|] Wedgie: <<4 -32 38 -60 49 178|| EDOs: 24, 94, 118, 212, 330, 542d, 872cd Badness: 0.1106 ==11-limit== Commas: 385/384, 9801/9800, 14641/14580 POTE generator: ~36/35 = 50.918 Mapping generator: ~16/11 Map: [<2 1 22 -15 8|, <0 2 -16 19 -1|] EDOs: 24, 94, 118, 212, 330e, 542de Badness: 0.0367 ==13-limit== Commas: 352/351, 385/384, 729/728, 1575/1573 POTE generator: ~36/35 = 50.938 Mapping generator: ~16/11 Map: [<2 1 22 -15 8 15|, <0 2 -16 19 -1 -7|] EDOs: 24, 94, 118, 212f Badness: 0.0376 =Term= Commas: 32805/32768, 250047/250000 7-limit minimax Eigenmonzos: 2, 6/5 9-limit minimax Eigenmonzos: 2, 9/7 [[POTE tuning|POTE generator]]: ~3/2 = 701.742 Mapping generator: ~3 Map: [<3 0 45 94|, <0 1 -8 -18|] Wedgie: <<3 -24 -54 -45 -94 -58|| [[edo|Edos]]: [[171edo|171]], [[1038edo|1038]], [[1209edo|1209]] Badness: 0.0200 =Sesquiquartififths= Commas: 2401/2400, 32805/32768 7-limit minimax Eigenmonzos: 2, 7/6 9-limit minimax Eigenmonzos: 2, 9/7 [[POTE tuning|POTE generator]]: ~448/405 = 175.434 Mapping generator: ~448/405 Map: [<1 1 7 5|, <0 4 -32 -15|] Wedgie: <<4 -32 -15 -60 -35 55|| [[edo|Edos]]: [[171edo|171]], [[643edo|643]], [[2100edo]] Badness: 0.0112 ==Sesquart== Commas: 243/242, 441/440, 16384/16335 POTE generator: ~256/231 = 175.406 Mapping generator: ~256/231 Map: [<1 1 7 5 2|, <0 4 -32 -15 10|] EDOs: 41, 89, 130 Badness: 0.0293 ===13-limit=== Commas: 243/242, 364/363, 441/440, 105644/105625 POTE generator: ~256/231 = 175.409 Map: [<1 1 7 5 2 -2|, <0 4 -32 -15 10 39|] EDOs: 41, 89, 130, 301e, 431e Badness: 0.0224 ==Bisesqui== Commas: 2401/2400, 9801/9800, 32805/32768 [[POTE tuning|POTE generator]]: ~448/405 = 175.435 Map: [<2 2 14 10 23|, <0 4 -32 -15 -55|] EDOs: 130, 212, 342, 1156, 1498 Badness: 0.0170 =Sextilififths= Commas: 32768/32805, 235298/234375 POTE generator: ~21/20 = 83.053 Mapping generator: ~21/20 Map: [<1 2 -1 -1|, <0 -6 48 55|] EDOs: 29, 101, 130, 289, 419 Badness: 0.1088 ==11-limit== Commas: 441/440, 4000/3993, 235298/234375 POTE generator: ~21/20 = 83.049 Mapping generator: ~21/20 Map: [<1 2 -1 -1 0|, <0 -6 48 55 50|] EDOs: 29, 130, 289 Badness: 0.0455 ==13-limit== Commas: 363/363, 441/440, 676/675, 10985/10976 POTE generator: ~21/20 = 83.049 Mapping generator: ~21/20 Map: [<1 2 -1 -1 0 1|, <0 -6 48 55 50 39|] EDOs: 29, 130, 289 Badness: 0.0253 =Tsaharuk= Commas: 32805/32768, 420175/419904 POTE generator: ~243/224 = 140.350 Mapping generator: ~243/224 Map: [<1 1 7 0|, <0 5 -40 24|] Wedgie: <<5 -40 24 -75 24 168|| EDOs: 17, 26, 43, 60, 77, 94, 171 Badness: 0.0307 =Quanharuk= Commas: 16875/16807, 32805/32768 POTE generator: ~56/45 = 380.355 Mapping generator: ~56/45 Map: [<1 0 15 12|, <0 5 -40 -29|] Wedgie: <<5 -40 -29 -75 -60 45|| EDOs: 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd Badness: 0.0720 ==11-limit== Commas: 540/539, 1375/1372, 32805/32768 POTE generator: ~56/45 = 380.352 Mapping generator: ~56/45 Map: [<1 0 15 12 -7|, <0 5 -40 -29 33|] EDOs: 41, 142, 183, 224, 631d, 855d, 1079d Badness: 0.0315 ==13-limit== Commas: 540/539, 1375/1372, 4096/4095, 6656/6655 POTE generator: ~56/45 = 380.351 Map: [<1 0 15 12 -7 -15|, <0 5 -40 -29 33 59|] EDOs: 41, 142, 183, 224, 631d, 855d Badness: 0.0214 =Octant= Commas: 32805/32768, 2259436291848/2251875390625 POTE generator: ~3/2 = 701.713 Mapping generator: ~3 Map: [<8 0 120 -117|, <0 1 -8 11|] Wedgie: <<8 -64 88 -120 117 384|| EDOs: 24, 224, 472, 696, 1168 Badness: 0.1572 ==11-limit== Commas: 9801/9800, 32805/32768, 46656/46585 POTE generator: ~3/2 = 701.713 Mapping generator: ~3 Map: [<8 0 120 -117 15|, <0 1 -8 11 1|] EDOs: 24, 224, 472, 696, 1168 Badness: 0.0448 ==13-limit== Commas: 729/728, 1575/1573, 2200/2197, 6656/6655 POTE generator: ~3/2 = 701.725 Mapping generator: ~3 Map: [<8 0 120 -117 15 93|, <0 1 -8 11 1 -5|] EDOs: 24, 224, 472, 696 Badness: 0.0304
Original HTML content:
<html><head><title>Schismatic family</title></head><body><!-- ws:start:WikiTextTocRule:94:<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:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 1em;"><a href="#Five limit">Five limit</a></div> <!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 1em;"><a href="#Seven limit children">Seven limit children</a></div> <!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><div style="margin-left: 1em;"><a href="#Garibaldi">Garibaldi</a></div> <!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><div style="margin-left: 2em;"><a href="#Garibaldi-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><div style="margin-left: 3em;"><a href="#Garibaldi-11-limit-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><div style="margin-left: 2em;"><a href="#Garibaldi-Cassandra">Cassandra</a></div> <!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><div style="margin-left: 3em;"><a href="#Garibaldi-Cassandra-13-limit Cassandra">13-limit Cassandra</a></div> <!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><div style="margin-left: 2em;"><a href="#Garibaldi-Helenus">Helenus</a></div> <!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><div style="margin-left: 3em;"><a href="#Garibaldi-Helenus-13-limit Helenus">13-limit Helenus</a></div> <!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><div style="margin-left: 1em;"><a href="#Hemigari">Hemigari</a></div> <!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 2em;"><a href="#Hemigari-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><div style="margin-left: 1em;"><a href="#Guiron">Guiron</a></div> <!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><div style="margin-left: 2em;"><a href="#Guiron-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><div style="margin-left: 2em;"><a href="#Guiron-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><div style="margin-left: 1em;"><a href="#Pogo">Pogo</a></div> <!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><div style="margin-left: 2em;"><a href="#Pogo-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><div style="margin-left: 2em;"><a href="#Pogo-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><div style="margin-left: 1em;"><a href="#Sanjaab">Sanjaab</a></div> <!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --><div style="margin-left: 2em;"><a href="#Sanjaab-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --><div style="margin-left: 1em;"><a href="#Squirrel">Squirrel</a></div> <!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --><div style="margin-left: 2em;"><a href="#Squirrel-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --><div style="margin-left: 2em;"><a href="#Squirrel-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --><div style="margin-left: 1em;"><a href="#Schism">Schism</a></div> <!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><div style="margin-left: 2em;"><a href="#Schism-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><div style="margin-left: 1em;"><a href="#Pontiac">Pontiac</a></div> <!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><div style="margin-left: 1em;"><a href="#Grackle">Grackle</a></div> <!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><div style="margin-left: 1em;"><a href="#Bischismic">Bischismic</a></div> <!-- ws:end:WikiTextTocRule:121 --><!-- ws:start:WikiTextTocRule:122: --><div style="margin-left: 2em;"><a href="#Bischismic-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:122 --><!-- ws:start:WikiTextTocRule:123: --><div style="margin-left: 2em;"><a href="#Bischismic-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:123 --><!-- ws:start:WikiTextTocRule:124: --><div style="margin-left: 1em;"><a href="#Kleischismic">Kleischismic</a></div> <!-- ws:end:WikiTextTocRule:124 --><!-- ws:start:WikiTextTocRule:125: --><div style="margin-left: 2em;"><a href="#Kleischismic-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:125 --><!-- ws:start:WikiTextTocRule:126: --><div style="margin-left: 2em;"><a href="#Kleischismic-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:126 --><!-- ws:start:WikiTextTocRule:127: --><div style="margin-left: 1em;"><a href="#Term">Term</a></div> <!-- ws:end:WikiTextTocRule:127 --><!-- ws:start:WikiTextTocRule:128: --><div style="margin-left: 1em;"><a href="#Sesquiquartififths">Sesquiquartififths</a></div> <!-- ws:end:WikiTextTocRule:128 --><!-- ws:start:WikiTextTocRule:129: --><div style="margin-left: 2em;"><a href="#Sesquiquartififths-Sesquart">Sesquart</a></div> <!-- ws:end:WikiTextTocRule:129 --><!-- ws:start:WikiTextTocRule:130: --><div style="margin-left: 3em;"><a href="#Sesquiquartififths-Sesquart-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:130 --><!-- ws:start:WikiTextTocRule:131: --><div style="margin-left: 2em;"><a href="#Sesquiquartififths-Bisesqui">Bisesqui</a></div> <!-- ws:end:WikiTextTocRule:131 --><!-- ws:start:WikiTextTocRule:132: --><div style="margin-left: 1em;"><a href="#Sextilififths">Sextilififths</a></div> <!-- ws:end:WikiTextTocRule:132 --><!-- ws:start:WikiTextTocRule:133: --><div style="margin-left: 2em;"><a href="#Sextilififths-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:133 --><!-- ws:start:WikiTextTocRule:134: --><div style="margin-left: 2em;"><a href="#Sextilififths-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:134 --><!-- ws:start:WikiTextTocRule:135: --><div style="margin-left: 1em;"><a href="#Tsaharuk">Tsaharuk</a></div> <!-- ws:end:WikiTextTocRule:135 --><!-- ws:start:WikiTextTocRule:136: --><div style="margin-left: 1em;"><a href="#Quanharuk">Quanharuk</a></div> <!-- ws:end:WikiTextTocRule:136 --><!-- ws:start:WikiTextTocRule:137: --><div style="margin-left: 2em;"><a href="#Quanharuk-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:137 --><!-- ws:start:WikiTextTocRule:138: --><div style="margin-left: 2em;"><a href="#Quanharuk-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:138 --><!-- ws:start:WikiTextTocRule:139: --><div style="margin-left: 1em;"><a href="#Octant">Octant</a></div> <!-- ws:end:WikiTextTocRule:139 --><!-- ws:start:WikiTextTocRule:140: --><div style="margin-left: 2em;"><a href="#Octant-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:140 --><!-- ws:start:WikiTextTocRule:141: --><div style="margin-left: 2em;"><a href="#Octant-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:141 --><!-- ws:start:WikiTextTocRule:142: --></div> <!-- ws:end:WikiTextTocRule:142 --><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/Schismatische%20Temperaturen">Deutsch</a><br /> </span><br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Five limit"></a><!-- ws:end:WikiTextHeadingRule:0 -->Five limit</h1> The 5-limit parent comma for the schismatic family is the <a class="wiki_link" href="/schisma">schisma</a> of 32805/32768, which is the amount by which the Pythagorean comma exceeds the <a class="wiki_link" href="/Didymus%20comma">Didymus comma</a> (<a class="wiki_link" href="/81_80">81/80</a>), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its <a class="wiki_link" href="/monzo">monzo</a> is |-15 8 1>, and flipping that yields <<1 -8 -15|| for the <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768.<br /> <br /> The 5-limit version of the temperament is a <a class="wiki_link" href="/Microtempering">microtemperament</a>, sometimes called <strong>Helmholtz</strong> or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. <a class="wiki_link" href="/53edo">53edo</a> is a possible tuning for schismatic, but you need <a class="wiki_link" href="/118edo">118edo</a> if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 701.736<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 15|, <0 1 -8|]<br /> EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/29edo">29</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/289edo">289</a>, 460, 749, 3456bc, 4205bc, 4954bc, 5703bc<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:2 -->Seven limit children</h1> 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. Adding |25 -14 0 -1> gives garibaldi, |-44 26 0 1> grackle, |6 -2 0 -1> schism and |-59 39 0 -1> pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2> and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3>, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3> with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4> and slices the fifth in four.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Garibaldi"></a><!-- ws:end:WikiTextHeadingRule:4 -->Garibaldi</h1> Commas: 225/224, 3125/3087<br /> <br /> 7-limit minimax tuning:<br /> 7-limit: [|1 0 0 0>, |5/3 1/15 0 -1/15>,<br /> |5/3 -8/15 0 8/15>, |5/3 -14/15 0 14/15>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 7/6<br /> <br /> 9-limit: [|1 0 0 0>, |25/16 1/8 0 -1/16>,<br /> |5/2 -1 0 1/2>, |25/8 -7/4 0 7/8>]<br /> Eigenmonzos: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 702.085<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 15 25|, <0 1 -8 -14|]<br /> Wedgie: <<1 -8 -14 -15 -25 -10||<br /> EDOs: 12, 29, 41, 53, <a class="wiki_link" href="/94edo">94</a>, 241c, 335cd, 576cd<br /> Badness: 0.0216<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Garibaldi-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h2> Commas: 225/224, 385/384, 2200/2187<br /> <br /> Minimax tuning:<br /> [|1 0 0 0 0>, |25/16 1/8 0 -1/16 0>, |5/2 -1 0 1/2 0>,<br /> |25/8 -7/4 0 7/8 0>, |47/16 23/8 0 -23/16 0>]<br /> Eigenmonzos: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 702.157<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 15 25 -33|, <0 1 -8 -14 23|]<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/94edo">94</a>, <a class="wiki_link" href="/135edo">135</a>, 229c, 323c, 417ce<br /> Badness: 0.0274<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="Garibaldi-11-limit-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h3> Commas: 225/224, 275/273, 325/324, 385/384<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 702.113<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 15 25 -33 -28|, <0 1 -8 -14 23 20|]<br /> EDOs: 12, 41, 53, 94, 429cdef, 523cdef<br /> Badness: 0.0207<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Garibaldi-Cassandra"></a><!-- ws:end:WikiTextHeadingRule:10 -->Cassandra</h2> Commas: 100/99, 225/224, 245/242<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 702.321<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 15 25 32|, <0 1 -8 -14 -18|]<br /> EDOs: 12, 29, 41, 123c, 217ce, 258ce<br /> Badness: 0.0236<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="Garibaldi-Cassandra-13-limit Cassandra"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit Cassandra</h3> Commas: 100/99, 105/104, 196/195, 245/242<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 702.559<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 15 25 32 37|, <0 1 -8 -14 -18 -21|]<br /> EDOs: 12, 29, 41, 152cdf, 193cdf, 234cdf<br /> Badness: 0.0207<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="Garibaldi-Helenus"></a><!-- ws:end:WikiTextHeadingRule:14 -->Helenus</h2> Commas: 99/98 176/175 3125/3087<br /> <br /> POTE generator: ~3/2 = 701.725<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 15 25 51|, <0 1 -8 -14 -30|]<br /> EDOs: 12, 53, 118d, 171de<br /> Badness: 0.0356<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h3> --><h3 id="toc8"><a name="Garibaldi-Helenus-13-limit Helenus"></a><!-- ws:end:WikiTextHeadingRule:16 -->13-limit Helenus</h3> Commas: 99/98, 176/175, 275/273, 847/845<br /> <br /> POTE generator: ~3/2 = 701.747<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 15 25 51 56|, <0 1 -8 -14 -30 -33|]<br /> EDOs: 53, 118d, 171de<br /> Badness: 0.0263<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h1> --><h1 id="toc9"><a name="Hemigari"></a><!-- ws:end:WikiTextHeadingRule:18 -->Hemigari</h1> Commas: 121/120, 225/224, 3125/3087<br /> <br /> POTE generator: ~63/55 = 248.918<br /> Mapping generator: ~110/63<br /> <br /> Map: [<1 0 15 25 9|, <0 2 -16 -28 -7|]<br /> EDOs: 29, 53, 82e, 135e, 188ce<br /> Badness: 0.0507<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h2> --><h2 id="toc10"><a name="Hemigari-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h2> Commas: 121/120, 169/168, 225/224, 275/273<br /> <br /> POTE generator: ~15/13 = 248.918<br /> Mapping generator: ~26/15<br /> <br /> Map: [<1 0 15 25 9 14|, <0 2 -16 -28 -7 -13|]<br /> EDOs: 29, 53, 82e, 135ef, 188cef<br /> Badness: 27.464<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h1> --><h1 id="toc11"><a name="Guiron"></a><!-- ws:end:WikiTextHeadingRule:22 -->Guiron</h1> <a class="wiki_link" href="/Comma">Commas</a>: 1029/1024, 10976/10935<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br /> 7+9 limit: [|1 0 0 0>, |15/8 0 -1/8 0>,<br /> |0 0 1 0>, |65/24 0 1/24 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 5/4<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~8/7 = 233.930<br /> Mapping generator: ~8/7<br /> <br /> Map: [<1 1 7 3|, <0 3 -24 -1|]<br /> Wedgie: <<3 -24 -1 -45 -10 65||<br /> EDOs: 36, 41, <a class="wiki_link" href="/77edo">77</a>, 118, 277d<br /> Badness: 0.0475<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h2> --><h2 id="toc12"><a name="Guiron-11-limit"></a><!-- ws:end:WikiTextHeadingRule:24 -->11-limit</h2> <a class="wiki_link" href="/Comma">Commas</a>: 385/384, 441/440, 10976/10935<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br /> [|1 0 0 0 0>, |15/8 0 -1/8 0 0>,<br /> |0 0 1 0 0>, |65/24 0 1/24 0 0>,<br /> |37/6 0 -7/6 0 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 5/4<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~8/7 = 233.931<br /> Mapping generator: ~8/7<br /> <br /> Map: [<1 1 7 3 -2|, <0 3 -24 -1 28|]<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/77edo">77</a>, <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/159edo">159</a>, <a class="wiki_link" href="/200edo">200</a>, <a class="wiki_link" href="/277">277d</a><br /> Badness: 0.0266<br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h2> --><h2 id="toc13"><a name="Guiron-13-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->13-limit</h2> Commas: 196/195, 352/351, 385/384, 729/728<br /> <br /> POTE generator: ~8/7 = 233.890<br /> Mapping generator: ~8/7<br /> <br /> Map: [<1 1 7 3 -2 0|, <0 3 -24 -1 28 19|]<br /> EDOs: 41, 77, 118<br /> Badness: 0.0284<br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h1> --><h1 id="toc14"><a name="Pogo"></a><!-- ws:end:WikiTextHeadingRule:28 -->Pogo</h1> Commas: 32805/32768, 118098/117649<br /> <br /> POTE generator: ~9/7 = 433.901<br /> Mapping generator: ~9/7<br /> <br /> Map: [<2 1 22 2|, <0 3 -24 5|]<br /> EDOs: 36, 94, 130, 171, 183, 224, 354<br /> Badness: 0.0796<br /> <br /> <!-- ws:start:WikiTextHeadingRule:30:<h2> --><h2 id="toc15"><a name="Pogo-11-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->11-limit</h2> Commas: 540/539, 4000/3993, 32805/32768<br /> <br /> POTE generator: ~9/7 = 433.911<br /> Mapping generator: ~9/7<br /> <br /> Map: [<2 1 22 2 25|, <0 3 -24 5 -25|]<br /> EDOs: 36, 94, 130, 224, 354, 578<br /> Badness: 0.0319<br /> <br /> <!-- ws:start:WikiTextHeadingRule:32:<h2> --><h2 id="toc16"><a name="Pogo-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->13-limit</h2> Commas: 540/539, 729/728, 4000/3993, 4225/4224<br /> <br /> POTE generator: ~9/7 = 433.911<br /> <br /> Map: [<2 1 22 2 25 -2|, <0 3 -24 5 -25 13|]<br /> EDOs: 36, 94, 130, 224, 354, 578<br /> Badness: 0.0175<br /> <br /> <!-- ws:start:WikiTextHeadingRule:34:<h1> --><h1 id="toc17"><a name="Sanjaab"></a><!-- ws:end:WikiTextHeadingRule:34 -->Sanjaab</h1> Commas: 225/224, 3125/3087, 1331/1323<br /> <br /> POTE generator: ~11/10 = 165.974<br /> <br /> Map: [<1 2 -1 -3 0|, <0 -3 24 42 25|]<br /> EDOs: 29, 65d, 94, 441cde, 535cde, 629cde<br /> Badness: 0.0580<br /> <br /> <!-- ws:start:WikiTextHeadingRule:36:<h2> --><h2 id="toc18"><a name="Sanjaab-13-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->13-limit</h2> Commas: 225/224, 275/273, 847/845, 1331/1323<br /> <br /> POTE generator: ~11/10 = 165.963<br /> <br /> Map: [<1 2 -1 -3 0 -1|, <0 -3 24 42 25 34|]<br /> EDOs: 29, 65d, 94<br /> Badness: 0.0338<br /> <br /> <!-- ws:start:WikiTextHeadingRule:38:<h1> --><h1 id="toc19"><a name="Squirrel"></a><!-- ws:end:WikiTextHeadingRule:38 -->Squirrel</h1> Commmas: 686/675, 32805/32768<br /> <br /> POTE generator: ~160/147 = 166.140<br /> <br /> Map: [<1 2 -1 1|, <0 -3 24 13|]<br /> EDOs: 29, 36, 65<br /> Badness: 0.1747<br /> <br /> <!-- ws:start:WikiTextHeadingRule:40:<h2> --><h2 id="toc20"><a name="Squirrel-11-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->11-limit</h2> Commas: 245/242, 686/675, 896/891<br /> <br /> POTE generator: ~11/10 = 166.097<br /> <br /> Map: [<1 2 -1 1 0|, <0 -3 24 13 25|]<br /> EDOs: 29, 36, 65<br /> Badness: 0.0683<br /> <br /> <!-- ws:start:WikiTextHeadingRule:42:<h2> --><h2 id="toc21"><a name="Squirrel-13-limit"></a><!-- ws:end:WikiTextHeadingRule:42 -->13-limit</h2> Commas: 91/90, 169/168, 245/242, 896/891<br /> <br /> POTE generator: ~11/10 = 166.054<br /> <br /> Map: [<1 2 -1 1 0 3|, <0 -3 24 13 25 5|]<br /> EDOs: 29, 36, 65f, 94df, 159df<br /> Badness: 0.0437<br /> <br /> <!-- ws:start:WikiTextHeadingRule:44:<h1> --><h1 id="toc22"><a name="Schism"></a><!-- ws:end:WikiTextHeadingRule:44 -->Schism</h1> Commas: 64/63, 360/343<br /> <br /> POTE generator: ~3/2 = 701.556<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 15 6|, <0 1 -8 -2|]<br /> Wedgie: <<1 -8 -2 -15 -6 18||<br /> EDOs: 12, 41d, 53d<br /> Badness: 0.0566<br /> <br /> <!-- ws:start:WikiTextHeadingRule:46:<h2> --><h2 id="toc23"><a name="Schism-11-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->11-limit</h2> Commas: 45/44, 64/63, 99/98<br /> <br /> POTE generator ~3/2 = 702.136<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|]<br /> EDOs: 12, 29de, 41de<br /> Badness: 0.0375<br /> <br /> <!-- ws:start:WikiTextHeadingRule:48:<h1> --><h1 id="toc24"><a name="Pontiac"></a><!-- ws:end:WikiTextHeadingRule:48 -->Pontiac</h1> Commas: 32805/32768, 4375/4374<br /> <br /> 7-limit minimax:<br /> [|1 0 0 0>, |74/47 0 -1/47 1/47>, |113/47 0 8/47 -8/47>,<br /> |113/47 0 -39/47 39/47>]<br /> Eigenmonzos: 2, 7/5<br /> <br /> 9-limit minimax:<br /> [|1 0 0 0>, |3/2 1/5 -1/10 0>,<br /> |3 -8/5 4/5 0>, |-1/2 39/5 -39/10 0>]<br /> Eigenmonzos: 2, 10/9<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.757<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 15 -59|, <0 1 -8 39|]<br /> Wedgie: <<1 -8 39 -15 59 113||<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/1079edo">1079</a>, <a class="wiki_link" href="/1250edo">1250</a>, <a class="wiki_link" href="/1421edo">1421</a><br /> Badness: 0.0141<br /> <br /> <!-- ws:start:WikiTextHeadingRule:50:<h1> --><h1 id="toc25"><a name="Grackle"></a><!-- ws:end:WikiTextHeadingRule:50 -->Grackle</h1> Commas: 126/125, 32805/32768<br /> <br /> 7-limit minimax<br /> Eigenmonzos: 2, 7/6<br /> <br /> 9-limit minimax<br /> Eigenmonzos: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.239<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 15 -44|, <0 1 -8 -26|]<br /> Wedgie: <<1 -8 -26 -15 -44 -38||<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/77edo">77</a>, <a class="wiki_link" href="/89edo">89</a>, <a class="wiki_link" href="/101edo">101</a>, <a class="wiki_link" href="/166edo">166c</a>, <a class="wiki_link" href="/243edo">243c</a><br /> Badness: 0.070<br /> <br /> <!-- ws:start:WikiTextHeadingRule:52:<h1> --><h1 id="toc26"><a name="Bischismic"></a><!-- ws:end:WikiTextHeadingRule:52 -->Bischismic</h1> Commas: 3136/3125, 32805/32768<br /> <br /> 7-limit minimax<br /> Eigenmonzos: 2, 7/6<br /> <br /> 9-limit minimax<br /> Eigenmonzos: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.592<br /> Mapping generator: ~3<br /> <br /> Map: [<2 0 30 69|, <0 1 -8 -20|]<br /> Wedgie: <<2 -16 -40 -30 -69 -48||<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/130edo">130</a>, <a class="wiki_link" href="/248edo">248</a>, <a class="wiki_link" href="/378edo">378</a>, <a class="wiki_link" href="/508edo">508</a><br /> Badness: 0.0547<br /> <br /> <!-- ws:start:WikiTextHeadingRule:54:<h2> --><h2 id="toc27"><a name="Bischismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:54 -->11-limit</h2> Commas: 441/440, 3136/3125, 8019/8000<br /> <br /> POTE generator: ~3/2 = 701.612<br /> Mapping generator: ~3<br /> <br /> Map: [<2 0 30 69 102|, <0 1 -8 -20 -30|]<br /> EDOs: 12, 118, 130, 248<br /> Badness: 0.0282<br /> <br /> <!-- ws:start:WikiTextHeadingRule:56:<h2> --><h2 id="toc28"><a name="Bischismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:56 -->13-limit</h2> Commas: 441/440, 729/728, 1001/1000, 3136/3125<br /> <br /> POTE generator: ~3/2 = 701.590<br /> Mapping generator: ~3<br /> <br /> Map: [<2 0 30 69 102 -75|, <0 1 -8 -20 -30 26|]<br /> EDOs: 12, 118, 130, 248, 378<br /> Badness: 0.0287<br /> <br /> <!-- ws:start:WikiTextHeadingRule:58:<h1> --><h1 id="toc29"><a name="Kleischismic"></a><!-- ws:end:WikiTextHeadingRule:58 -->Kleischismic</h1> Commas: 32805/32768, 1500625/1492992<br /> <br /> POTE generator: ~36/35 = 50.920<br /> Mapping generator: ~35/24<br /> <br /> Map: [<2 1 22 -15|, <0 2 -16 19|]<br /> Wedgie: <<4 -32 38 -60 49 178||<br /> EDOs: 24, 94, 118, 212, 330, 542d, 872cd<br /> Badness: 0.1106<br /> <br /> <!-- ws:start:WikiTextHeadingRule:60:<h2> --><h2 id="toc30"><a name="Kleischismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:60 -->11-limit</h2> Commas: 385/384, 9801/9800, 14641/14580<br /> <br /> POTE generator: ~36/35 = 50.918<br /> Mapping generator: ~16/11<br /> <br /> Map: [<2 1 22 -15 8|, <0 2 -16 19 -1|]<br /> EDOs: 24, 94, 118, 212, 330e, 542de<br /> Badness: 0.0367<br /> <br /> <!-- ws:start:WikiTextHeadingRule:62:<h2> --><h2 id="toc31"><a name="Kleischismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:62 -->13-limit</h2> Commas: 352/351, 385/384, 729/728, 1575/1573<br /> <br /> POTE generator: ~36/35 = 50.938<br /> Mapping generator: ~16/11<br /> <br /> Map: [<2 1 22 -15 8 15|, <0 2 -16 19 -1 -7|]<br /> EDOs: 24, 94, 118, 212f<br /> Badness: 0.0376<br /> <br /> <!-- ws:start:WikiTextHeadingRule:64:<h1> --><h1 id="toc32"><a name="Term"></a><!-- ws:end:WikiTextHeadingRule:64 -->Term</h1> Commas: 32805/32768, 250047/250000<br /> <br /> 7-limit minimax<br /> Eigenmonzos: 2, 6/5<br /> <br /> 9-limit minimax<br /> Eigenmonzos: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 701.742<br /> Mapping generator: ~3<br /> <br /> Map: [<3 0 45 94|, <0 1 -8 -18|]<br /> Wedgie: <<3 -24 -54 -45 -94 -58||<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/1038edo">1038</a>, <a class="wiki_link" href="/1209edo">1209</a><br /> Badness: 0.0200<br /> <br /> <!-- ws:start:WikiTextHeadingRule:66:<h1> --><h1 id="toc33"><a name="Sesquiquartififths"></a><!-- ws:end:WikiTextHeadingRule:66 -->Sesquiquartififths</h1> Commas: 2401/2400, 32805/32768<br /> <br /> 7-limit minimax<br /> Eigenmonzos: 2, 7/6<br /> <br /> 9-limit minimax<br /> Eigenmonzos: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~448/405 = 175.434<br /> Mapping generator: ~448/405<br /> <br /> Map: [<1 1 7 5|, <0 4 -32 -15|]<br /> Wedgie: <<4 -32 -15 -60 -35 55||<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/643edo">643</a>, <a class="wiki_link" href="/2100edo">2100edo</a><br /> Badness: 0.0112<br /> <br /> <!-- ws:start:WikiTextHeadingRule:68:<h2> --><h2 id="toc34"><a name="Sesquiquartififths-Sesquart"></a><!-- ws:end:WikiTextHeadingRule:68 -->Sesquart</h2> Commas: 243/242, 441/440, 16384/16335<br /> <br /> POTE generator: ~256/231 = 175.406<br /> Mapping generator: ~256/231<br /> <br /> Map: [<1 1 7 5 2|, <0 4 -32 -15 10|]<br /> EDOs: 41, 89, 130<br /> Badness: 0.0293<br /> <br /> <!-- ws:start:WikiTextHeadingRule:70:<h3> --><h3 id="toc35"><a name="Sesquiquartififths-Sesquart-13-limit"></a><!-- ws:end:WikiTextHeadingRule:70 -->13-limit</h3> Commas: 243/242, 364/363, 441/440, 105644/105625<br /> <br /> POTE generator: ~256/231 = 175.409<br /> <br /> Map: [<1 1 7 5 2 -2|, <0 4 -32 -15 10 39|]<br /> EDOs: 41, 89, 130, 301e, 431e<br /> Badness: 0.0224<br /> <br /> <!-- ws:start:WikiTextHeadingRule:72:<h2> --><h2 id="toc36"><a name="Sesquiquartififths-Bisesqui"></a><!-- ws:end:WikiTextHeadingRule:72 -->Bisesqui</h2> Commas: 2401/2400, 9801/9800, 32805/32768<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~448/405 = 175.435<br /> <br /> Map: [<2 2 14 10 23|, <0 4 -32 -15 -55|]<br /> EDOs: 130, 212, 342, 1156, 1498<br /> Badness: 0.0170<br /> <br /> <!-- ws:start:WikiTextHeadingRule:74:<h1> --><h1 id="toc37"><a name="Sextilififths"></a><!-- ws:end:WikiTextHeadingRule:74 -->Sextilififths</h1> Commas: 32768/32805, 235298/234375<br /> <br /> POTE generator: ~21/20 = 83.053<br /> Mapping generator: ~21/20<br /> <br /> Map: [<1 2 -1 -1|, <0 -6 48 55|]<br /> EDOs: 29, 101, 130, 289, 419<br /> Badness: 0.1088<br /> <br /> <!-- ws:start:WikiTextHeadingRule:76:<h2> --><h2 id="toc38"><a name="Sextilififths-11-limit"></a><!-- ws:end:WikiTextHeadingRule:76 -->11-limit</h2> Commas: 441/440, 4000/3993, 235298/234375<br /> <br /> POTE generator: ~21/20 = 83.049<br /> Mapping generator: ~21/20<br /> <br /> Map: [<1 2 -1 -1 0|, <0 -6 48 55 50|]<br /> EDOs: 29, 130, 289<br /> Badness: 0.0455<br /> <br /> <!-- ws:start:WikiTextHeadingRule:78:<h2> --><h2 id="toc39"><a name="Sextilififths-13-limit"></a><!-- ws:end:WikiTextHeadingRule:78 -->13-limit</h2> Commas: 363/363, 441/440, 676/675, 10985/10976<br /> <br /> POTE generator: ~21/20 = 83.049<br /> Mapping generator: ~21/20<br /> <br /> Map: [<1 2 -1 -1 0 1|, <0 -6 48 55 50 39|]<br /> EDOs: 29, 130, 289<br /> Badness: 0.0253<br /> <br /> <!-- ws:start:WikiTextHeadingRule:80:<h1> --><h1 id="toc40"><a name="Tsaharuk"></a><!-- ws:end:WikiTextHeadingRule:80 -->Tsaharuk</h1> Commas: 32805/32768, 420175/419904<br /> <br /> POTE generator: ~243/224 = 140.350<br /> Mapping generator: ~243/224<br /> <br /> Map: [<1 1 7 0|, <0 5 -40 24|]<br /> Wedgie: <<5 -40 24 -75 24 168||<br /> EDOs: 17, 26, 43, 60, 77, 94, 171<br /> Badness: 0.0307<br /> <br /> <!-- ws:start:WikiTextHeadingRule:82:<h1> --><h1 id="toc41"><a name="Quanharuk"></a><!-- ws:end:WikiTextHeadingRule:82 -->Quanharuk</h1> Commas: 16875/16807, 32805/32768<br /> <br /> POTE generator: ~56/45 = 380.355<br /> Mapping generator: ~56/45<br /> <br /> Map: [<1 0 15 12|, <0 5 -40 -29|]<br /> Wedgie: <<5 -40 -29 -75 -60 45||<br /> EDOs: 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd<br /> Badness: 0.0720<br /> <br /> <!-- ws:start:WikiTextHeadingRule:84:<h2> --><h2 id="toc42"><a name="Quanharuk-11-limit"></a><!-- ws:end:WikiTextHeadingRule:84 -->11-limit</h2> Commas: 540/539, 1375/1372, 32805/32768<br /> <br /> POTE generator: ~56/45 = 380.352<br /> Mapping generator: ~56/45<br /> <br /> Map: [<1 0 15 12 -7|, <0 5 -40 -29 33|]<br /> EDOs: 41, 142, 183, 224, 631d, 855d, 1079d<br /> Badness: 0.0315<br /> <br /> <!-- ws:start:WikiTextHeadingRule:86:<h2> --><h2 id="toc43"><a name="Quanharuk-13-limit"></a><!-- ws:end:WikiTextHeadingRule:86 -->13-limit</h2> Commas: 540/539, 1375/1372, 4096/4095, 6656/6655<br /> <br /> POTE generator: ~56/45 = 380.351<br /> <br /> Map: [<1 0 15 12 -7 -15|, <0 5 -40 -29 33 59|]<br /> EDOs: 41, 142, 183, 224, 631d, 855d<br /> Badness: 0.0214<br /> <br /> <!-- ws:start:WikiTextHeadingRule:88:<h1> --><h1 id="toc44"><a name="Octant"></a><!-- ws:end:WikiTextHeadingRule:88 -->Octant</h1> Commas: 32805/32768, 2259436291848/2251875390625<br /> <br /> POTE generator: ~3/2 = 701.713<br /> Mapping generator: ~3<br /> <br /> Map: [<8 0 120 -117|, <0 1 -8 11|]<br /> Wedgie: <<8 -64 88 -120 117 384||<br /> EDOs: 24, 224, 472, 696, 1168<br /> Badness: 0.1572<br /> <br /> <!-- ws:start:WikiTextHeadingRule:90:<h2> --><h2 id="toc45"><a name="Octant-11-limit"></a><!-- ws:end:WikiTextHeadingRule:90 -->11-limit</h2> Commas: 9801/9800, 32805/32768, 46656/46585<br /> <br /> POTE generator: ~3/2 = 701.713<br /> Mapping generator: ~3<br /> <br /> Map: [<8 0 120 -117 15|, <0 1 -8 11 1|]<br /> EDOs: 24, 224, 472, 696, 1168<br /> Badness: 0.0448<br /> <br /> <!-- ws:start:WikiTextHeadingRule:92:<h2> --><h2 id="toc46"><a name="Octant-13-limit"></a><!-- ws:end:WikiTextHeadingRule:92 -->13-limit</h2> Commas: 729/728, 1575/1573, 2200/2197, 6656/6655<br /> <br /> POTE generator: ~3/2 = 701.725<br /> Mapping generator: ~3<br /> <br /> Map: [<8 0 120 -117 15 93|, <0 1 -8 11 1 -5|]<br /> EDOs: 24, 224, 472, 696<br /> Badness: 0.0304</body></html>