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 genewardsmith and made on 2012-05-21 17:10:23 UTC.
- The original revision id was 338060268.
- The revision comment was:
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]] =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 just 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 schisma 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]]: 701.736 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]]: 702.085 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]]: 702.157 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]]: 702.113 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]]: 702.321 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]]: 702.559 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 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 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: ~220/189 = 248.918 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: ~52/45 = 248.918 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]]: 233.930 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]]: 233.931 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 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 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 Map: [<2 1 22 2 25|, <0 3 -24 5 -25|] EDOs: 36, 94, 130, 224, 354, 578 Badness: 0.0319 =Schism= Commas: 64/63, 360/343 POTE generator: ~3/2 = 701.556 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 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 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 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 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 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 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 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 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 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]]: 701.742 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]]: 175.434 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 Map: [<1 1 7 5 2|, <0 4 -32 -15 10|] EDOs: 41, 89, 130 Badness: 0.0293 =Sextilififths= Commas: 32768/32805, 235298/234375 POTE generator: ~21/20 = 83.053 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 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 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 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 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 Map: [<1 0 15 12 -7|, <0 5 -40 -29 33|] EDOs: 41, 142, 183, 224, 631d, 855d, 1079d Badness: 0.0315 =Octant= Commas: 32805/32768, 2259436291848/2251875390625 POTE generator: ~3/2 = 701.713 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 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 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:76:<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:76 --><!-- ws:start:WikiTextTocRule:77: --><div style="margin-left: 1em;"><a href="#Five limit">Five limit</a></div> <!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><div style="margin-left: 1em;"><a href="#Seven limit children">Seven limit children</a></div> <!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><div style="margin-left: 1em;"><a href="#Garibaldi">Garibaldi</a></div> <!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><div style="margin-left: 2em;"><a href="#Garibaldi-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 3em;"><a href="#Garibaldi-11-limit-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 2em;"><a href="#Garibaldi-Cassandra">Cassandra</a></div> <!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><div style="margin-left: 3em;"><a href="#Garibaldi-Cassandra-13-limit Cassandra">13-limit Cassandra</a></div> <!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><div style="margin-left: 2em;"><a href="#Garibaldi-Helenus">Helenus</a></div> <!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><div style="margin-left: 3em;"><a href="#Garibaldi-Helenus-13-limit Helenus">13-limit Helenus</a></div> <!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><div style="margin-left: 1em;"><a href="#Hemigari">Hemigari</a></div> <!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><div style="margin-left: 2em;"><a href="#Hemigari-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><div style="margin-left: 1em;"><a href="#Guiron">Guiron</a></div> <!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><div style="margin-left: 2em;"><a href="#Guiron-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><div style="margin-left: 2em;"><a href="#Guiron-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><div style="margin-left: 1em;"><a href="#Pogo">Pogo</a></div> <!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><div style="margin-left: 2em;"><a href="#Pogo-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><div style="margin-left: 1em;"><a href="#Schism">Schism</a></div> <!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><div style="margin-left: 2em;"><a href="#Schism-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 1em;"><a href="#Pontiac">Pontiac</a></div> <!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 1em;"><a href="#Grackle">Grackle</a></div> <!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><div style="margin-left: 1em;"><a href="#Bischismic">Bischismic</a></div> <!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><div style="margin-left: 2em;"><a href="#Bischismic-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><div style="margin-left: 2em;"><a href="#Bischismic-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><div style="margin-left: 1em;"><a href="#Kleischismic">Kleischismic</a></div> <!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><div style="margin-left: 2em;"><a href="#Kleischismic-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><div style="margin-left: 2em;"><a href="#Kleischismic-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><div style="margin-left: 1em;"><a href="#Term">Term</a></div> <!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><div style="margin-left: 1em;"><a href="#Sesquiquartififths">Sesquiquartififths</a></div> <!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 2em;"><a href="#Sesquiquartififths-Sesquart">Sesquart</a></div> <!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><div style="margin-left: 1em;"><a href="#Sextilififths">Sextilififths</a></div> <!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><div style="margin-left: 2em;"><a href="#Sextilififths-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><div style="margin-left: 2em;"><a href="#Sextilififths-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><div style="margin-left: 1em;"><a href="#Tsaharuk">Tsaharuk</a></div> <!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><div style="margin-left: 1em;"><a href="#Quanharuk">Quanharuk</a></div> <!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><div style="margin-left: 2em;"><a href="#Quanharuk-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><div style="margin-left: 1em;"><a href="#Octant">Octant</a></div> <!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --><div style="margin-left: 2em;"><a href="#Octant-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --><div style="margin-left: 2em;"><a href="#Octant-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --></div> <!-- ws:end:WikiTextTocRule:115 --><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 just 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 schisma 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>: 701.736<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>: 702.085<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>: 702.157<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>: 702.113<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>: 702.321<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>: 702.559<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 /> <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 /> <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: ~220/189 = 248.918<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: ~52/45 = 248.918<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>: 233.930<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>: 233.931<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>, [277|277d]]<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 /> <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 /> <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 /> <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:<h1> --><h1 id="toc16"><a name="Schism"></a><!-- ws:end:WikiTextHeadingRule:32 -->Schism</h1> Commas: 64/63, 360/343<br /> <br /> POTE generator: ~3/2 = 701.556<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:34:<h2> --><h2 id="toc17"><a name="Schism-11-limit"></a><!-- ws:end:WikiTextHeadingRule:34 -->11-limit</h2> Commas: 45/44, 64/63, 99/98<br /> <br /> POTE generator ~3/2 = 702.136<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:36:<h1> --><h1 id="toc18"><a name="Pontiac"></a><!-- ws:end:WikiTextHeadingRule:36 -->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 /> <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:38:<h1> --><h1 id="toc19"><a name="Grackle"></a><!-- ws:end:WikiTextHeadingRule:38 -->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 /> <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:40:<h1> --><h1 id="toc20"><a name="Bischismic"></a><!-- ws:end:WikiTextHeadingRule:40 -->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 /> <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:42:<h2> --><h2 id="toc21"><a name="Bischismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:42 -->11-limit</h2> Commas: 441/440, 3136/3125, 8019/8000<br /> <br /> POTE generator: ~3/2 = 701.612<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:44:<h2> --><h2 id="toc22"><a name="Bischismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:44 -->13-limit</h2> Commas: 441/440, 729/728, 1001/1000, 3136/3125<br /> <br /> POTE generator: ~3/2 = 701.590<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:46:<h1> --><h1 id="toc23"><a name="Kleischismic"></a><!-- ws:end:WikiTextHeadingRule:46 -->Kleischismic</h1> Commas: 32805/32768, 1500625/1492992<br /> <br /> POTE generator: ~36/35 = 50.920<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:48:<h2> --><h2 id="toc24"><a name="Kleischismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:48 -->11-limit</h2> Commas: 385/384, 9801/9800, 14641/14580<br /> <br /> POTE generator: ~36/35 = 50.918<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:50:<h2> --><h2 id="toc25"><a name="Kleischismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:50 -->13-limit</h2> Commas: 352/351, 385/384, 729/728, 1575/1573<br /> <br /> POTE generator: ~36/35 = 50.938<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:52:<h1> --><h1 id="toc26"><a name="Term"></a><!-- ws:end:WikiTextHeadingRule:52 -->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>: 701.742<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:54:<h1> --><h1 id="toc27"><a name="Sesquiquartififths"></a><!-- ws:end:WikiTextHeadingRule:54 -->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>: 175.434<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:56:<h2> --><h2 id="toc28"><a name="Sesquiquartififths-Sesquart"></a><!-- ws:end:WikiTextHeadingRule:56 -->Sesquart</h2> Commas: 243/242, 441/440, 16384/16335<br /> <br /> POTE generator: ~256/231 = 175.406<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:58:<h1> --><h1 id="toc29"><a name="Sextilififths"></a><!-- ws:end:WikiTextHeadingRule:58 -->Sextilififths</h1> Commas: 32768/32805, 235298/234375<br /> <br /> POTE generator: ~21/20 = 83.053<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:60:<h2> --><h2 id="toc30"><a name="Sextilififths-11-limit"></a><!-- ws:end:WikiTextHeadingRule:60 -->11-limit</h2> Commas: 441/440, 4000/3993, 235298/234375<br /> <br /> POTE generator: ~21/20 = 83.049<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:62:<h2> --><h2 id="toc31"><a name="Sextilififths-13-limit"></a><!-- ws:end:WikiTextHeadingRule:62 -->13-limit</h2> Commas: 363/363, 441/440, 676/675, 10985/10976<br /> <br /> POTE generator: ~21/20 = 83.049<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:64:<h1> --><h1 id="toc32"><a name="Tsaharuk"></a><!-- ws:end:WikiTextHeadingRule:64 -->Tsaharuk</h1> Commas: 32805/32768, 420175/419904<br /> <br /> POTE generator: ~243/224 = 140.350<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:66:<h1> --><h1 id="toc33"><a name="Quanharuk"></a><!-- ws:end:WikiTextHeadingRule:66 -->Quanharuk</h1> Commas: 16875/16807, 32805/32768<br /> <br /> POTE generator: ~56/45 = 380.355<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:68:<h2> --><h2 id="toc34"><a name="Quanharuk-11-limit"></a><!-- ws:end:WikiTextHeadingRule:68 -->11-limit</h2> Commas: 540/539, 1375/1372, 32805/32768<br /> <br /> POTE generator: ~56/45 = 380.352<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:70:<h1> --><h1 id="toc35"><a name="Octant"></a><!-- ws:end:WikiTextHeadingRule:70 -->Octant</h1> Commas: 32805/32768, 2259436291848/2251875390625<br /> <br /> POTE generator: ~3/2 = 701.713<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:72:<h2> --><h2 id="toc36"><a name="Octant-11-limit"></a><!-- ws:end:WikiTextHeadingRule:72 -->11-limit</h2> Commas: 9801/9800, 32805/32768, 46656/46585<br /> <br /> POTE generator: ~3/2 = 701.713<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:74:<h2> --><h2 id="toc37"><a name="Octant-13-limit"></a><!-- ws:end:WikiTextHeadingRule:74 -->13-limit</h2> Commas: 729/728, 1575/1573, 2200/2197, 6656/6655<br /> <br /> POTE generator: ~3/2 = 701.725<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>