Schismatic family: Difference between revisions

Revision as of 07:24, 3 September 2016

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[[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:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule: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:&lt;h1&gt; --><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&gt;, and flipping that yields &lt;&lt;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: [&lt;1 0 15|, &lt;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:&lt;h1&gt; --><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&gt; gives garibaldi, |-44 26 0 1&gt; grackle, |6 -2 0 -1&gt; schism and |-59 39 0 -1&gt; pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2&gt; and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3&gt;, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3&gt; with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4&gt; and slices the fifth in four.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><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&gt;, |5/3 1/15 0 -1/15&gt;,<br />
|5/3 -8/15 0 8/15&gt;, |5/3 -14/15 0 14/15&gt;]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 7/6<br />
<br />
9-limit: [|1 0 0 0&gt;, |25/16 1/8 0 -1/16&gt;,<br />
|5/2 -1 0 1/2&gt;, |25/8 -7/4 0 7/8&gt;]<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: [&lt;1 0 15 25|, &lt;0 1 -8 -14|]<br />
Wedgie: &lt;&lt;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:&lt;h2&gt; --><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&gt;, |25/16 1/8 0 -1/16 0&gt;, |5/2 -1 0 1/2 0&gt;,<br />
|25/8 -7/4 0 7/8 0&gt;, |47/16 23/8 0 -23/16 0&gt;]<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: [&lt;1 0 15 25 -33|, &lt;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:&lt;h3&gt; --><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: [&lt;1 0 15 25 -33 -28|, &lt;0 1 -8 -14 23 20|]<br />
EDOs: 12, 41, 53, 94, 429cdef, 523cdef<br />
Badness: 0.0207<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><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: [&lt;1 0 15 25 32|, &lt;0 1 -8 -14 -18|]<br />
EDOs: 12, 29, 41, 123c, 217ce, 258ce<br />
Badness: 0.0236<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><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: [&lt;1 0 15 25 32 37|, &lt;0 1 -8 -14 -18 -21|]<br />
EDOs: 12, 29, 41, 152cdf, 193cdf, 234cdf<br />
Badness: 0.0207<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><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: [&lt;1 0 15 25 51|, &lt;0 1 -8 -14 -30|]<br />
EDOs: 12, 53, 118d, 171de<br />
Badness: 0.0356<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><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: [&lt;1 0 15 25 51 56|, &lt;0 1 -8 -14 -30 -33|]<br />
EDOs: 53, 118d, 171de<br />
Badness: 0.0263<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><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: [&lt;1 0 15 25 9|, &lt;0 2 -16 -28 -7|]<br />
EDOs: 29, 53, 82e, 135e, 188ce<br />
Badness: 0.0507<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><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: [&lt;1 0 15 25 9 14|, &lt;0 2 -16 -28 -7 -13|]<br />
EDOs: 29, 53, 82e, 135ef, 188cef<br />
Badness: 27.464<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><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&gt;, |15/8 0 -1/8 0&gt;,<br />
|0 0 1 0&gt;, |65/24 0 1/24 0&gt;]<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: [&lt;1 1 7 3|, &lt;0 3 -24 -1|]<br />
Wedgie: &lt;&lt;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:&lt;h2&gt; --><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&gt;, |15/8 0 -1/8 0 0&gt;,<br />
|0 0 1 0 0&gt;, |65/24 0 1/24 0 0&gt;,<br />
|37/6 0 -7/6 0 0&gt;]<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: [&lt;1 1 7 3 -2|, &lt;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:&lt;h2&gt; --><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: [&lt;1 1 7 3 -2 0|, &lt;0 3 -24 -1 28 19|]<br />
EDOs: 41, 77, 118<br />
Badness: 0.0284<br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><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: [&lt;2 1 22 2|, &lt;0 3 -24 5|]<br />
EDOs: 36, 94, 130, 171, 183, 224, 354<br />
Badness: 0.0796<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><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: [&lt;2 1 22 2 25|, &lt;0 3 -24 5 -25|]<br />
EDOs: 36, 94, 130, 224, 354, 578<br />
Badness: 0.0319<br />
<br />
<!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><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: [&lt;2 1 22 2 25 -2|, &lt;0 3 -24 5 -25 13|]<br />
EDOs: 36, 94, 130, 224, 354, 578<br />
Badness: 0.0175<br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><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: [&lt;1 2 -1 -3 0|, &lt;0 -3 24 42 25|]<br />
EDOs: 29, 65d, 94, 441cde, 535cde, 629cde<br />
Badness: 0.0580<br />
<br />
<!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><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: [&lt;1 2 -1 -3 0 -1|, &lt;0 -3 24 42 25 34|]<br />
EDOs: 29, 65d, 94<br />
Badness: 0.0338<br />
<br />
<!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><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: [&lt;1 2 -1 1|, &lt;0 -3 24 13|]<br />
EDOs: 29, 36, 65<br />
Badness: 0.1747<br />
<br />
<!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><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: [&lt;1 2 -1 1 0|, &lt;0 -3 24 13 25|]<br />
EDOs: 29, 36, 65<br />
Badness: 0.0683<br />
<br />
<!-- ws:start:WikiTextHeadingRule:42:&lt;h2&gt; --><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: [&lt;1 2 -1 1 0 3|, &lt;0 -3 24 13 25 5|]<br />
EDOs: 29, 36, 65f, 94df, 159df<br />
Badness: 0.0437<br />
<br />
<!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><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: [&lt;1 0 15 6|, &lt;0 1 -8 -2|]<br />
Wedgie: &lt;&lt;1 -8 -2 -15 -6 18||<br />
EDOs: 12, 41d, 53d<br />
Badness: 0.0566<br />
<br />
<!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><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: [&lt;1 0 15 6 13|, &lt;0 1 -8 -2 -6|]<br />
EDOs: 12, 29de, 41de<br />
Badness: 0.0375<br />
<br />
<!-- ws:start:WikiTextHeadingRule:48:&lt;h1&gt; --><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&gt;, |74/47 0 -1/47 1/47&gt;, |113/47 0 8/47 -8/47&gt;,<br />
|113/47 0 -39/47 39/47&gt;]<br />
Eigenmonzos: 2, 7/5<br />
<br />
9-limit minimax:<br />
[|1 0 0 0&gt;, |3/2 1/5 -1/10 0&gt;,<br />
|3 -8/5 4/5 0&gt;, |-1/2 39/5 -39/10 0&gt;]<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: [&lt;1 0 15 -59|, &lt;0 1 -8 39|]<br />
Wedgie: &lt;&lt;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:&lt;h1&gt; --><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: [&lt;1 0 15 -44|, &lt;0 1 -8 -26|]<br />
Wedgie: &lt;&lt;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:&lt;h1&gt; --><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: [&lt;2 0 30 69|, &lt;0 1 -8 -20|]<br />
Wedgie: &lt;&lt;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:&lt;h2&gt; --><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: [&lt;2 0 30 69 102|, &lt;0 1 -8 -20 -30|]<br />
EDOs: 12, 118, 130, 248<br />
Badness: 0.0282<br />
<br />
<!-- ws:start:WikiTextHeadingRule:56:&lt;h2&gt; --><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: [&lt;2 0 30 69 102 -75|, &lt;0 1 -8 -20 -30 26|]<br />
EDOs: 12, 118, 130, 248, 378<br />
Badness: 0.0287<br />
<br />
<!-- ws:start:WikiTextHeadingRule:58:&lt;h1&gt; --><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: [&lt;2 1 22 -15|, &lt;0 2 -16 19|]<br />
Wedgie: &lt;&lt;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:&lt;h2&gt; --><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: [&lt;2 1 22 -15 8|, &lt;0 2 -16 19 -1|]<br />
EDOs: 24, 94, 118, 212, 330e, 542de<br />
Badness: 0.0367<br />
<br />
<!-- ws:start:WikiTextHeadingRule:62:&lt;h2&gt; --><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: [&lt;2 1 22 -15 8 15|, &lt;0 2 -16 19 -1 -7|]<br />
EDOs: 24, 94, 118, 212f<br />
Badness: 0.0376<br />
<br />
<!-- ws:start:WikiTextHeadingRule:64:&lt;h1&gt; --><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: [&lt;3 0 45 94|, &lt;0 1 -8 -18|]<br />
Wedgie: &lt;&lt;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:&lt;h1&gt; --><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: [&lt;1 1 7 5|, &lt;0 4 -32 -15|]<br />
Wedgie: &lt;&lt;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:&lt;h2&gt; --><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: [&lt;1 1 7 5 2|, &lt;0 4 -32 -15 10|]<br />
EDOs: 41, 89, 130<br />
Badness: 0.0293<br />
<br />
<!-- ws:start:WikiTextHeadingRule:70:&lt;h3&gt; --><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: [&lt;1 1 7 5 2 -2|, &lt;0 4 -32 -15 10 39|]<br />
EDOs: 41, 89, 130, 301e, 431e<br />
Badness: 0.0224<br />
<br />
<!-- ws:start:WikiTextHeadingRule:72:&lt;h2&gt; --><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: [&lt;2 2 14 10 23|, &lt;0 4 -32 -15 -55|]<br />
EDOs: 130, 212, 342, 1156, 1498<br />
Badness: 0.0170<br />
<br />
<!-- ws:start:WikiTextHeadingRule:74:&lt;h1&gt; --><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: [&lt;1 2 -1 -1|, &lt;0 -6 48 55|]<br />
EDOs: 29, 101, 130, 289, 419<br />
Badness: 0.1088<br />
<br />
<!-- ws:start:WikiTextHeadingRule:76:&lt;h2&gt; --><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: [&lt;1 2 -1 -1 0|, &lt;0 -6 48 55 50|]<br />
EDOs: 29, 130, 289<br />
Badness: 0.0455<br />
<br />
<!-- ws:start:WikiTextHeadingRule:78:&lt;h2&gt; --><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: [&lt;1 2 -1 -1 0 1|, &lt;0 -6 48 55 50 39|]<br />
EDOs: 29, 130, 289<br />
Badness: 0.0253<br />
<br />
<!-- ws:start:WikiTextHeadingRule:80:&lt;h1&gt; --><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: [&lt;1 1 7 0|, &lt;0 5 -40 24|]<br />
Wedgie: &lt;&lt;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:&lt;h1&gt; --><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: [&lt;1 0 15 12|, &lt;0 5 -40 -29|]<br />
Wedgie: &lt;&lt;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:&lt;h2&gt; --><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: [&lt;1 0 15 12 -7|, &lt;0 5 -40 -29 33|]<br />
EDOs: 41, 142, 183, 224, 631d, 855d, 1079d<br />
Badness: 0.0315<br />
<br />
<!-- ws:start:WikiTextHeadingRule:86:&lt;h2&gt; --><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: [&lt;1 0 15 12 -7 -15|, &lt;0 5 -40 -29 33 59|]<br />
EDOs: 41, 142, 183, 224, 631d, 855d<br />
Badness: 0.0214<br />
<br />
<!-- ws:start:WikiTextHeadingRule:88:&lt;h1&gt; --><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: [&lt;8 0 120 -117|, &lt;0 1 -8 11|]<br />
Wedgie: &lt;&lt;8 -64 88 -120 117 384||<br />
EDOs: 24, 224, 472, 696, 1168<br />
Badness: 0.1572<br />
<br />
<!-- ws:start:WikiTextHeadingRule:90:&lt;h2&gt; --><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: [&lt;8 0 120 -117 15|, &lt;0 1 -8 11 1|]<br />
EDOs: 24, 224, 472, 696, 1168<br />
Badness: 0.0448<br />
<br />
<!-- ws:start:WikiTextHeadingRule:92:&lt;h2&gt; --><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: [&lt;8 0 120 -117 15 93|, &lt;0 1 -8 11 1 -5|]<br />
EDOs: 24, 224, 472, 696<br />
Badness: 0.0304</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-08-31 01:13:15 UTC</tt>.<br>
+: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-09-03 07:24:39 UTC</tt>.<br>
-: The original revision id was <tt>590581204</tt>.<br>
+: The original revision id was <tt>590996938</tt>.<br>
-: The revision comment was: <tt></tt><br>
+: The revision comment was: <tt>Reverted to Aug 31, 2016 7:12 am: removal of DE link seems to be mistakenly</tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc]]
-**&lt;span style="font-size: 140%;"&gt;Five limit&lt;/span&gt;**
+&lt;span style="display: block; text-align: right;"&gt;Other languages: [[xenharmonie/Schismatische Temperaturen|Deutsch]]
+&lt;/span&gt;
+=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&gt;, and flipping that yields &lt;&lt;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.
@@ Line 533: / Line 535: @@
 Badness: 0.0304</pre></div>
 <h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Schismatic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:92:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:92 --&gt;&lt;!-- ws:start:WikiTextTocRule:93: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Seven limit children"&gt;Seven limit children&lt;/a&gt;&lt;/div&gt;
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Schismatic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:94:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:94 --&gt;&lt;!-- ws:start:WikiTextTocRule:95: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Five limit"&gt;Five limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:93 --&gt;&lt;!-- ws:start:WikiTextTocRule:94: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Garibaldi"&gt;Garibaldi&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:95 --&gt;&lt;!-- ws:start:WikiTextTocRule:96: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Seven limit children"&gt;Seven limit children&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:94 --&gt;&lt;!-- ws:start:WikiTextTocRule:95: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Garibaldi-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:96 --&gt;&lt;!-- ws:start:WikiTextTocRule:97: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Garibaldi"&gt;Garibaldi&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:95 --&gt;&lt;!-- ws:start:WikiTextTocRule:96: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Garibaldi-11-limit-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:97 --&gt;&lt;!-- ws:start:WikiTextTocRule:98: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Garibaldi-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:96 --&gt;&lt;!-- ws:start:WikiTextTocRule:97: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Garibaldi-Cassandra"&gt;Cassandra&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:98 --&gt;&lt;!-- ws:start:WikiTextTocRule:99: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Garibaldi-11-limit-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:97 --&gt;&lt;!-- ws:start:WikiTextTocRule:98: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Garibaldi-Cassandra-13-limit Cassandra"&gt;13-limit Cassandra&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:99 --&gt;&lt;!-- ws:start:WikiTextTocRule:100: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Garibaldi-Cassandra"&gt;Cassandra&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:98 --&gt;&lt;!-- ws:start:WikiTextTocRule:99: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Garibaldi-Helenus"&gt;Helenus&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:100 --&gt;&lt;!-- ws:start:WikiTextTocRule:101: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Garibaldi-Cassandra-13-limit Cassandra"&gt;13-limit Cassandra&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:99 --&gt;&lt;!-- ws:start:WikiTextTocRule:100: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Garibaldi-Helenus-13-limit Helenus"&gt;13-limit Helenus&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:101 --&gt;&lt;!-- ws:start:WikiTextTocRule:102: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Garibaldi-Helenus"&gt;Helenus&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:100 --&gt;&lt;!-- ws:start:WikiTextTocRule:101: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Hemigari"&gt;Hemigari&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:102 --&gt;&lt;!-- ws:start:WikiTextTocRule:103: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Garibaldi-Helenus-13-limit Helenus"&gt;13-limit Helenus&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:101 --&gt;&lt;!-- ws:start:WikiTextTocRule:102: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Hemigari-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:103 --&gt;&lt;!-- ws:start:WikiTextTocRule:104: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Hemigari"&gt;Hemigari&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:102 --&gt;&lt;!-- ws:start:WikiTextTocRule:103: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Guiron"&gt;Guiron&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:104 --&gt;&lt;!-- ws:start:WikiTextTocRule:105: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Hemigari-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:103 --&gt;&lt;!-- ws:start:WikiTextTocRule:104: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Guiron-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:105 --&gt;&lt;!-- ws:start:WikiTextTocRule:106: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Guiron"&gt;Guiron&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:104 --&gt;&lt;!-- ws:start:WikiTextTocRule:105: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Guiron-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:106 --&gt;&lt;!-- ws:start:WikiTextTocRule:107: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Guiron-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:105 --&gt;&lt;!-- ws:start:WikiTextTocRule:106: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Pogo"&gt;Pogo&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:107 --&gt;&lt;!-- ws:start:WikiTextTocRule:108: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Guiron-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:106 --&gt;&lt;!-- ws:start:WikiTextTocRule:107: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Pogo-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&lt;!-- ws:start:WikiTextTocRule:109: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Pogo"&gt;Pogo&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:107 --&gt;&lt;!-- ws:start:WikiTextTocRule:108: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Pogo-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:109 --&gt;&lt;!-- ws:start:WikiTextTocRule:110: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Pogo-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&lt;!-- ws:start:WikiTextTocRule:109: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Sanjaab"&gt;Sanjaab&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:110 --&gt;&lt;!-- ws:start:WikiTextTocRule:111: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Pogo-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:109 --&gt;&lt;!-- ws:start:WikiTextTocRule:110: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Sanjaab-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:111 --&gt;&lt;!-- ws:start:WikiTextTocRule:112: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Sanjaab"&gt;Sanjaab&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:110 --&gt;&lt;!-- ws:start:WikiTextTocRule:111: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Squirrel"&gt;Squirrel&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:112 --&gt;&lt;!-- ws:start:WikiTextTocRule:113: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Sanjaab-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:111 --&gt;&lt;!-- ws:start:WikiTextTocRule:112: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Squirrel-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:113 --&gt;&lt;!-- ws:start:WikiTextTocRule:114: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Squirrel"&gt;Squirrel&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:112 --&gt;&lt;!-- ws:start:WikiTextTocRule:113: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Squirrel-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Squirrel-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:113 --&gt;&lt;!-- ws:start:WikiTextTocRule:114: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Schism"&gt;Schism&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:115 --&gt;&lt;!-- ws:start:WikiTextTocRule:116: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Squirrel-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Schism-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:116 --&gt;&lt;!-- ws:start:WikiTextTocRule:117: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Schism"&gt;Schism&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:115 --&gt;&lt;!-- ws:start:WikiTextTocRule:116: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Pontiac"&gt;Pontiac&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:117 --&gt;&lt;!-- ws:start:WikiTextTocRule:118: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Schism-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:116 --&gt;&lt;!-- ws:start:WikiTextTocRule:117: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Grackle"&gt;Grackle&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:118 --&gt;&lt;!-- ws:start:WikiTextTocRule:119: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Pontiac"&gt;Pontiac&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:117 --&gt;&lt;!-- ws:start:WikiTextTocRule:118: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Bischismic"&gt;Bischismic&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:119 --&gt;&lt;!-- ws:start:WikiTextTocRule:120: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Grackle"&gt;Grackle&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:118 --&gt;&lt;!-- ws:start:WikiTextTocRule:119: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Bischismic-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:120 --&gt;&lt;!-- ws:start:WikiTextTocRule:121: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Bischismic"&gt;Bischismic&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:119 --&gt;&lt;!-- ws:start:WikiTextTocRule:120: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Bischismic-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:121 --&gt;&lt;!-- ws:start:WikiTextTocRule:122: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Bischismic-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:120 --&gt;&lt;!-- ws:start:WikiTextTocRule:121: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Kleischismic"&gt;Kleischismic&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:122 --&gt;&lt;!-- ws:start:WikiTextTocRule:123: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Bischismic-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:121 --&gt;&lt;!-- ws:start:WikiTextTocRule:122: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Kleischismic-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:123 --&gt;&lt;!-- ws:start:WikiTextTocRule:124: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Kleischismic"&gt;Kleischismic&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:122 --&gt;&lt;!-- ws:start:WikiTextTocRule:123: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Kleischismic-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:124 --&gt;&lt;!-- ws:start:WikiTextTocRule:125: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Kleischismic-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:123 --&gt;&lt;!-- ws:start:WikiTextTocRule:124: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Term"&gt;Term&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:125 --&gt;&lt;!-- ws:start:WikiTextTocRule:126: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Kleischismic-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:124 --&gt;&lt;!-- ws:start:WikiTextTocRule:125: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Sesquiquartififths"&gt;Sesquiquartififths&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:126 --&gt;&lt;!-- ws:start:WikiTextTocRule:127: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Term"&gt;Term&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:125 --&gt;&lt;!-- ws:start:WikiTextTocRule:126: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Sesquiquartififths-Sesquart"&gt;Sesquart&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:127 --&gt;&lt;!-- ws:start:WikiTextTocRule:128: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Sesquiquartififths"&gt;Sesquiquartififths&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:126 --&gt;&lt;!-- ws:start:WikiTextTocRule:127: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Sesquiquartififths-Sesquart-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:128 --&gt;&lt;!-- ws:start:WikiTextTocRule:129: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Sesquiquartififths-Sesquart"&gt;Sesquart&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:127 --&gt;&lt;!-- ws:start:WikiTextTocRule:128: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Sesquiquartififths-Bisesqui"&gt;Bisesqui&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:129 --&gt;&lt;!-- ws:start:WikiTextTocRule:130: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Sesquiquartififths-Sesquart-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:128 --&gt;&lt;!-- ws:start:WikiTextTocRule:129: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Sextilififths"&gt;Sextilififths&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:130 --&gt;&lt;!-- ws:start:WikiTextTocRule:131: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Sesquiquartififths-Bisesqui"&gt;Bisesqui&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:129 --&gt;&lt;!-- ws:start:WikiTextTocRule:130: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Sextilififths-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:131 --&gt;&lt;!-- ws:start:WikiTextTocRule:132: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Sextilififths"&gt;Sextilififths&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:130 --&gt;&lt;!-- ws:start:WikiTextTocRule:131: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Sextilififths-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:132 --&gt;&lt;!-- ws:start:WikiTextTocRule:133: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Sextilififths-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:131 --&gt;&lt;!-- ws:start:WikiTextTocRule:132: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Tsaharuk"&gt;Tsaharuk&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:133 --&gt;&lt;!-- ws:start:WikiTextTocRule:134: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Sextilififths-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:132 --&gt;&lt;!-- ws:start:WikiTextTocRule:133: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Quanharuk"&gt;Quanharuk&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:134 --&gt;&lt;!-- ws:start:WikiTextTocRule:135: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Tsaharuk"&gt;Tsaharuk&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:133 --&gt;&lt;!-- ws:start:WikiTextTocRule:134: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Quanharuk-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:135 --&gt;&lt;!-- ws:start:WikiTextTocRule:136: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Quanharuk"&gt;Quanharuk&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:134 --&gt;&lt;!-- ws:start:WikiTextTocRule:135: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Quanharuk-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:136 --&gt;&lt;!-- ws:start:WikiTextTocRule:137: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Quanharuk-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:135 --&gt;&lt;!-- ws:start:WikiTextTocRule:136: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Octant"&gt;Octant&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:137 --&gt;&lt;!-- ws:start:WikiTextTocRule:138: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Quanharuk-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:136 --&gt;&lt;!-- ws:start:WikiTextTocRule:137: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Octant-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:138 --&gt;&lt;!-- ws:start:WikiTextTocRule:139: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Octant"&gt;Octant&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:137 --&gt;&lt;!-- ws:start:WikiTextTocRule:138: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Octant-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:139 --&gt;&lt;!-- ws:start:WikiTextTocRule:140: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Octant-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:138 --&gt;&lt;!-- ws:start:WikiTextTocRule:139: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:140 --&gt;&lt;!-- ws:start:WikiTextTocRule:141: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Octant-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:139 --&gt;&lt;strong&gt;&lt;span style="font-size: 140%;"&gt;Five limit&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:141 --&gt;&lt;!-- ws:start:WikiTextTocRule:142: --&gt;&lt;/div&gt;
-The 5-limit parent comma for the schismatic family is the &lt;a class="wiki_link" href="/schisma"&gt;schisma&lt;/a&gt; of 32805/32768, which is the amount by which the Pythagorean comma exceeds the &lt;a class="wiki_link" href="/Didymus%20comma"&gt;Didymus comma&lt;/a&gt; (&lt;a class="wiki_link" href="/81_80"&gt;81/80&lt;/a&gt;), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt; is |-15 8 1&amp;gt;, and flipping that yields &amp;lt;&amp;lt;1 -8 -15|| for the &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt;. 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.&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:142 --&gt;&lt;span style="display: block; text-align: right;"&gt;Other languages: &lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/Schismatische%20Temperaturen"&gt;Deutsch&lt;/a&gt;&lt;br /&gt;
+&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Five limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Five limit&lt;/h1&gt;
+ The 5-limit parent comma for the schismatic family is the &lt;a class="wiki_link" href="/schisma"&gt;schisma&lt;/a&gt; of 32805/32768, which is the amount by which the Pythagorean comma exceeds the &lt;a class="wiki_link" href="/Didymus%20comma"&gt;Didymus comma&lt;/a&gt; (&lt;a class="wiki_link" href="/81_80"&gt;81/80&lt;/a&gt;), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt; is |-15 8 1&amp;gt;, and flipping that yields &amp;lt;&amp;lt;1 -8 -15|| for the &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt;. 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.&lt;br /&gt;
 &lt;br /&gt;
 The 5-limit version of the temperament is a &lt;a class="wiki_link" href="/Microtempering"&gt;microtemperament&lt;/a&gt;, sometimes called &lt;strong&gt;Helmholtz&lt;/strong&gt; 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. &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt; is a possible tuning for schismatic, but you need &lt;a class="wiki_link" href="/118edo"&gt;118edo&lt;/a&gt; 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.&lt;br /&gt;
@@ Line 591: / Line 596: @@
 EDOs: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/29edo"&gt;29&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/118edo"&gt;118&lt;/a&gt;, &lt;a class="wiki_link" href="/171edo"&gt;171&lt;/a&gt;, &lt;a class="wiki_link" href="/289edo"&gt;289&lt;/a&gt;, 460, 749, 3456bc, 4205bc, 4954bc, 5703bc&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Seven limit children&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Seven limit children&lt;/h1&gt;
   The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. Adding |25 -14 0 -1&amp;gt; gives garibaldi, |-44 26 0 1&amp;gt; grackle, |6 -2 0 -1&amp;gt; schism and |-59 39 0 -1&amp;gt; pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2&amp;gt; and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3&amp;gt;, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3&amp;gt; with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4&amp;gt; and slices the fifth in four.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Garibaldi"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Garibaldi&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Garibaldi"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Garibaldi&lt;/h1&gt;
   Commas: 225/224, 3125/3087&lt;br /&gt;
 &lt;br /&gt;
@@ Line 614: / Line 619: @@
 Badness: 0.0216&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Garibaldi-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Garibaldi-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;11-limit&lt;/h2&gt;
   Commas: 225/224, 385/384, 2200/2187&lt;br /&gt;
 &lt;br /&gt;
@@ Line 629: / Line 634: @@
 Badness: 0.0274&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="Garibaldi-11-limit-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="Garibaldi-11-limit-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;13-limit&lt;/h3&gt;
   Commas: 225/224, 275/273, 325/324, 385/384&lt;br /&gt;
 &lt;br /&gt;
@@ Line 639: / Line 644: @@
 Badness: 0.0207&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Garibaldi-Cassandra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Cassandra&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Garibaldi-Cassandra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Cassandra&lt;/h2&gt;
   Commas: 100/99, 225/224, 245/242&lt;br /&gt;
 &lt;br /&gt;
@@ Line 649: / Line 654: @@
 Badness: 0.0236&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Garibaldi-Cassandra-13-limit Cassandra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit Cassandra&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="Garibaldi-Cassandra-13-limit Cassandra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit Cassandra&lt;/h3&gt;
   Commas: 100/99, 105/104, 196/195, 245/242&lt;br /&gt;
 &lt;br /&gt;
@@ Line 659: / Line 664: @@
 Badness: 0.0207&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Garibaldi-Helenus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Helenus&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Garibaldi-Helenus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Helenus&lt;/h2&gt;
   Commas: 99/98 176/175 3125/3087&lt;br /&gt;
 &lt;br /&gt;
@@ Line 669: / Line 674: @@
 Badness: 0.0356&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="Garibaldi-Helenus-13-limit Helenus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit Helenus&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="Garibaldi-Helenus-13-limit Helenus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;13-limit Helenus&lt;/h3&gt;
   Commas: 99/98, 176/175, 275/273, 847/845&lt;br /&gt;
 &lt;br /&gt;
@@ Line 679: / Line 684: @@
 Badness: 0.0263&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Hemigari"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Hemigari&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Hemigari"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Hemigari&lt;/h1&gt;
   Commas: 121/120, 225/224, 3125/3087&lt;br /&gt;
 &lt;br /&gt;
@@ Line 689: / Line 694: @@
 Badness: 0.0507&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Hemigari-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;13-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Hemigari-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;13-limit&lt;/h2&gt;
   Commas: 121/120, 169/168, 225/224, 275/273&lt;br /&gt;
 &lt;br /&gt;
@@ Line 699: / Line 704: @@
 Badness: 27.464&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Guiron"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Guiron&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Guiron"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Guiron&lt;/h1&gt;
   &lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 1029/1024, 10976/10935&lt;br /&gt;
 &lt;br /&gt;
@@ Line 715: / Line 720: @@
 Badness: 0.0475&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Guiron-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;11-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Guiron-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;11-limit&lt;/h2&gt;
   &lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 385/384, 441/440, 10976/10935&lt;br /&gt;
 &lt;br /&gt;
@@ Line 731: / Line 736: @@
 Badness: 0.0266&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Guiron-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;13-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Guiron-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;13-limit&lt;/h2&gt;
   Commas: 196/195, 352/351, 385/384, 729/728&lt;br /&gt;
 &lt;br /&gt;
@@ Line 741: / Line 746: @@
 Badness: 0.0284&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc13"&gt;&lt;a name="Pogo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Pogo&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Pogo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Pogo&lt;/h1&gt;
   Commas: 32805/32768, 118098/117649&lt;br /&gt;
 &lt;br /&gt;
@@ Line 751: / Line 756: @@
 Badness: 0.0796&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Pogo-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;11-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Pogo-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;11-limit&lt;/h2&gt;
   Commas: 540/539, 4000/3993, 32805/32768&lt;br /&gt;
 &lt;br /&gt;
@@ Line 761: / Line 766: @@
 Badness: 0.0319&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Pogo-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;13-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Pogo-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;13-limit&lt;/h2&gt;
   Commas: 540/539, 729/728, 4000/3993, 4225/4224&lt;br /&gt;
 &lt;br /&gt;
@@ Line 770: / Line 775: @@
 Badness: 0.0175&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc16"&gt;&lt;a name="Sanjaab"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;Sanjaab&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc17"&gt;&lt;a name="Sanjaab"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Sanjaab&lt;/h1&gt;
   Commas: 225/224, 3125/3087, 1331/1323&lt;br /&gt;
 &lt;br /&gt;
@@ Line 779: / Line 784: @@
 Badness: 0.0580&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Sanjaab-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;13-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Sanjaab-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;13-limit&lt;/h2&gt;
   Commas: 225/224, 275/273, 847/845, 1331/1323&lt;br /&gt;
 &lt;br /&gt;
@@ Line 788: / Line 793: @@
 Badness: 0.0338&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc18"&gt;&lt;a name="Squirrel"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Squirrel&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc19"&gt;&lt;a name="Squirrel"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;Squirrel&lt;/h1&gt;
   Commmas: 686/675, 32805/32768&lt;br /&gt;
 &lt;br /&gt;
@@ Line 797: / Line 802: @@
 Badness: 0.1747&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Squirrel-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;11-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Squirrel-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;11-limit&lt;/h2&gt;
   Commas: 245/242, 686/675, 896/891&lt;br /&gt;
 &lt;br /&gt;
@@ Line 806: / Line 811: @@
 Badness: 0.0683&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Squirrel-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;13-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Squirrel-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;13-limit&lt;/h2&gt;
   Commas: 91/90, 169/168, 245/242, 896/891&lt;br /&gt;
 &lt;br /&gt;
@@ Line 815: / Line 820: @@
 Badness: 0.0437&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc21"&gt;&lt;a name="Schism"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;Schism&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc22"&gt;&lt;a name="Schism"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;Schism&lt;/h1&gt;
   Commas: 64/63, 360/343&lt;br /&gt;
 &lt;br /&gt;
@@ Line 826: / Line 831: @@
 Badness: 0.0566&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Schism-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;11-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="Schism-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;11-limit&lt;/h2&gt;
   Commas: 45/44, 64/63, 99/98&lt;br /&gt;
 &lt;br /&gt;
@@ Line 836: / Line 841: @@
 Badness: 0.0375&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc23"&gt;&lt;a name="Pontiac"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;Pontiac&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc24"&gt;&lt;a name="Pontiac"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;Pontiac&lt;/h1&gt;
   Commas: 32805/32768, 4375/4374&lt;br /&gt;
 &lt;br /&gt;
@@ Line 857: / Line 862: @@
 Badness: 0.0141&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc24"&gt;&lt;a name="Grackle"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;Grackle&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc25"&gt;&lt;a name="Grackle"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;Grackle&lt;/h1&gt;
   Commas: 126/125, 32805/32768&lt;br /&gt;
 &lt;br /&gt;
@@ Line 874: / Line 879: @@
 Badness: 0.070&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc25"&gt;&lt;a name="Bischismic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;Bischismic&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc26"&gt;&lt;a name="Bischismic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;Bischismic&lt;/h1&gt;
   Commas: 3136/3125, 32805/32768&lt;br /&gt;
 &lt;br /&gt;
@@ Line 891: / Line 896: @@
 Badness: 0.0547&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc26"&gt;&lt;a name="Bischismic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;11-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc27"&gt;&lt;a name="Bischismic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;11-limit&lt;/h2&gt;
   Commas: 441/440, 3136/3125, 8019/8000&lt;br /&gt;
 &lt;br /&gt;
@@ Line 901: / Line 906: @@
 Badness: 0.0282&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc27"&gt;&lt;a name="Bischismic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;13-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc28"&gt;&lt;a name="Bischismic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;13-limit&lt;/h2&gt;
   Commas: 441/440, 729/728, 1001/1000, 3136/3125&lt;br /&gt;
 &lt;br /&gt;
@@ Line 911: / Line 916: @@
 Badness: 0.0287&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc28"&gt;&lt;a name="Kleischismic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;Kleischismic&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc29"&gt;&lt;a name="Kleischismic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;Kleischismic&lt;/h1&gt;
   Commas: 32805/32768, 1500625/1492992&lt;br /&gt;
 &lt;br /&gt;
@@ Line 922: / Line 927: @@
 Badness: 0.1106&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc29"&gt;&lt;a name="Kleischismic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;11-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc30"&gt;&lt;a name="Kleischismic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;11-limit&lt;/h2&gt;
   Commas: 385/384, 9801/9800, 14641/14580&lt;br /&gt;
 &lt;br /&gt;
@@ Line 932: / Line 937: @@
 Badness: 0.0367&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc30"&gt;&lt;a name="Kleischismic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;13-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc31"&gt;&lt;a name="Kleischismic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;13-limit&lt;/h2&gt;
   Commas: 352/351, 385/384, 729/728, 1575/1573&lt;br /&gt;
 &lt;br /&gt;
@@ Line 942: / Line 947: @@
 Badness: 0.0376&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc31"&gt;&lt;a name="Term"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;Term&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc32"&gt;&lt;a name="Term"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt;Term&lt;/h1&gt;
   Commas: 32805/32768, 250047/250000&lt;br /&gt;
 &lt;br /&gt;
@@ Line 959: / Line 964: @@
 Badness: 0.0200&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc32"&gt;&lt;a name="Sesquiquartififths"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt;Sesquiquartififths&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:66:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc33"&gt;&lt;a name="Sesquiquartififths"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:66 --&gt;Sesquiquartififths&lt;/h1&gt;
   Commas: 2401/2400, 32805/32768&lt;br /&gt;
 &lt;br /&gt;
@@ Line 976: / Line 981: @@
 Badness: 0.0112&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:66:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc33"&gt;&lt;a name="Sesquiquartififths-Sesquart"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:66 --&gt;Sesquart&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc34"&gt;&lt;a name="Sesquiquartififths-Sesquart"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;Sesquart&lt;/h2&gt;
   Commas: 243/242, 441/440, 16384/16335&lt;br /&gt;
 &lt;br /&gt;
@@ Line 986: / Line 991: @@
 Badness: 0.0293&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc34"&gt;&lt;a name="Sesquiquartififths-Sesquart-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;13-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:70:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc35"&gt;&lt;a name="Sesquiquartififths-Sesquart-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&gt;13-limit&lt;/h3&gt;
   Commas: 243/242, 364/363, 441/440, 105644/105625&lt;br /&gt;
 &lt;br /&gt;
@@ Line 995: / Line 1,000: @@
 Badness: 0.0224&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:70:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc35"&gt;&lt;a name="Sesquiquartififths-Bisesqui"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&gt;Bisesqui&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc36"&gt;&lt;a name="Sesquiquartififths-Bisesqui"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;Bisesqui&lt;/h2&gt;
   Commas: 2401/2400, 9801/9800, 32805/32768&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,004: / Line 1,009: @@
 Badness: 0.0170&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc36"&gt;&lt;a name="Sextilififths"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;Sextilififths&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:74:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc37"&gt;&lt;a name="Sextilififths"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:74 --&gt;Sextilififths&lt;/h1&gt;
   Commas: 32768/32805, 235298/234375&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,014: / Line 1,019: @@
 Badness: 0.1088&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:74:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc37"&gt;&lt;a name="Sextilififths-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:74 --&gt;11-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:76:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc38"&gt;&lt;a name="Sextilififths-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:76 --&gt;11-limit&lt;/h2&gt;
   Commas: 441/440, 4000/3993, 235298/234375&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,024: / Line 1,029: @@
 Badness: 0.0455&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:76:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc38"&gt;&lt;a name="Sextilififths-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:76 --&gt;13-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:78:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc39"&gt;&lt;a name="Sextilififths-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:78 --&gt;13-limit&lt;/h2&gt;
   Commas: 363/363, 441/440, 676/675, 10985/10976&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,034: / Line 1,039: @@
 Badness: 0.0253&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:78:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc39"&gt;&lt;a name="Tsaharuk"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:78 --&gt;Tsaharuk&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:80:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc40"&gt;&lt;a name="Tsaharuk"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:80 --&gt;Tsaharuk&lt;/h1&gt;
   Commas: 32805/32768, 420175/419904&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,045: / Line 1,050: @@
 Badness: 0.0307&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:80:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc40"&gt;&lt;a name="Quanharuk"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:80 --&gt;Quanharuk&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:82:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc41"&gt;&lt;a name="Quanharuk"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:82 --&gt;Quanharuk&lt;/h1&gt;
   Commas: 16875/16807, 32805/32768&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,056: / Line 1,061: @@
 Badness: 0.0720&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:82:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc41"&gt;&lt;a name="Quanharuk-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:82 --&gt;11-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:84:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc42"&gt;&lt;a name="Quanharuk-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:84 --&gt;11-limit&lt;/h2&gt;
   Commas: 540/539, 1375/1372, 32805/32768&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,066: / Line 1,071: @@
 Badness: 0.0315&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:84:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc42"&gt;&lt;a name="Quanharuk-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:84 --&gt;13-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:86:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc43"&gt;&lt;a name="Quanharuk-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:86 --&gt;13-limit&lt;/h2&gt;
   Commas: 540/539, 1375/1372, 4096/4095, 6656/6655&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,075: / Line 1,080: @@
 Badness: 0.0214&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:86:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc43"&gt;&lt;a name="Octant"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:86 --&gt;Octant&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:88:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc44"&gt;&lt;a name="Octant"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:88 --&gt;Octant&lt;/h1&gt;
   Commas: 32805/32768, 2259436291848/2251875390625&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,086: / Line 1,091: @@
 Badness: 0.1572&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:88:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc44"&gt;&lt;a name="Octant-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:88 --&gt;11-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:90:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc45"&gt;&lt;a name="Octant-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:90 --&gt;11-limit&lt;/h2&gt;
   Commas: 9801/9800, 32805/32768, 46656/46585&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,096: / Line 1,101: @@
 Badness: 0.0448&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:90:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc45"&gt;&lt;a name="Octant-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:90 --&gt;13-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:92:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc46"&gt;&lt;a name="Octant-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:92 --&gt;13-limit&lt;/h2&gt;
   Commas: 729/728, 1575/1573, 2200/2197, 6656/6655&lt;br /&gt;
 &lt;br /&gt;

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