Schismatic family: Difference between revisions
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{{Vals|legend=1| 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc }} | {{Vals|legend=1| 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc }} | ||
[[Badness]]: 0. | [[Badness]]: 0.004259 | ||
== Seven-limit extensions == | == Seven-limit extensions == | ||
| Line 74: | Line 74: | ||
{{Val list|legend=1| 12, 29, 41, 53, 94, 241c, 335cd, 576ccd }} | {{Val list|legend=1| 12, 29, 41, 53, 94, 241c, 335cd, 576ccd }} | ||
[[Badness]]: 0. | [[Badness]]: 0.021644 | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 95: | Line 94: | ||
{{Val list|legend=1| 41, 53, 94, 229c, 323c, 417ce }} | {{Val list|legend=1| 41, 53, 94, 229c, 323c, 417ce }} | ||
Badness: 0. | Badness: 0.027396 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 111: | Line 109: | ||
{{Val list|legend=1| 41, 53, 94, 429cdef, 523cdef }} | {{Val list|legend=1| 41, 53, 94, 429cdef, 523cdef }} | ||
Badness: 0. | Badness: 0.020676 | ||
== Andromeda == | == Andromeda == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 127: | Line 124: | ||
{{Val list|legend=1| 12, 29, 41, 217ce, 258ce }} | {{Val list|legend=1| 12, 29, 41, 217ce, 258ce }} | ||
Badness: 0. | Badness: 0.023556 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 143: | Line 139: | ||
{{Val list|legend=1| 12f, 29, 41, 152cdf, 193cdf, 234cdf }} | {{Val list|legend=1| 12f, 29, 41, 152cdf, 193cdf, 234cdf }} | ||
Badness: 0. | Badness: 0.020749 | ||
== Helenus == | == Helenus == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 157: | Line 152: | ||
POTE generator: ~3/2 = 701.725 | POTE generator: ~3/2 = 701.725 | ||
{{Val list|legend=1| 53, 118d, 171de }} | {{Val list|legend=1| 12, 41e, 53, 118d, 171de }} | ||
Badness: 0. | Badness: 0.035637 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 173: | Line 167: | ||
POTE generator: ~3/2 = 701.747 | POTE generator: ~3/2 = 701.747 | ||
{{Val list|legend=1| 53, 118d, 171de }} | {{Val list|legend=1| 12f, 41ef, 53, 118d, 171de }} | ||
Badness: 0. | Badness: 0.026284 | ||
== Hemigari == | == Hemigari == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 191: | Line 184: | ||
{{Val list|legend=1| 29, 53, 82e, 135e, 188ce }} | {{Val list|legend=1| 29, 53, 82e, 135e, 188ce }} | ||
Badness: 0. | Badness: 0.050681 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 210: | Line 202: | ||
== Sanjaab == | == Sanjaab == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 223: | Line 214: | ||
{{Val list|legend=1| 29, 65d, 94, 441cde, 535cde, 629cde }} | {{Val list|legend=1| 29, 65d, 94, 441cde, 535cde, 629cde }} | ||
Badness: 0. | Badness: 0.058040 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 239: | Line 229: | ||
{{Val list|legend=1| 29, 65d, 94 }} | {{Val list|legend=1| 29, 65d, 94 }} | ||
Badness: 0. | Badness: 0.033849 | ||
= Schism = | = Schism = | ||
| Line 261: | Line 251: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 277: | Line 266: | ||
= Pontiac = | = Pontiac = | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 305: | Line 293: | ||
{{Val list|legend=1| 53, 118, 171, 1592c, 1763c, …, 2960cd, 3131bcd }} | {{Val list|legend=1| 53, 118, 171, 1592c, 1763c, …, 2960cd, 3131bcd }} | ||
[[Badness]]: 0. | [[Badness]]: 0.014133 | ||
= Grackle = | = Grackle = | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 325: | Line 312: | ||
* [[9-odd-limit]] eigenmonzos: 2, 9/7 | * [[9-odd-limit]] eigenmonzos: 2, 9/7 | ||
{{Val list|legend=1| 77, 166c, 243c }} | {{Val list|legend=1| 12, 65, 77, 166c, 243c }} | ||
[[Badness]]: 0. | [[Badness]]: 0.070407 | ||
= Bischismic = | = Bischismic = | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 347: | Line 333: | ||
* [[9-odd-limit]] eigenmonzos: 2, 9/7 | * [[9-odd-limit]] eigenmonzos: 2, 9/7 | ||
{{Val list|legend=1| 118, 130, 248, 378, 508 }} | {{Val list|legend=1| 12, 118, 130, 248, 378, 508 }} | ||
[[Badness]]: 0.0547 | [[Badness]]: 0.0547 | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 363: | Line 348: | ||
POTE generator: ~3/2 = 701.612 | POTE generator: ~3/2 = 701.612 | ||
{{Val list|legend=1| 118, 130, 248 }} | {{Val list|legend=1| 12, 118, 130, 248 }} | ||
Badness: 0.0282 | Badness: 0.0282 | ||
== 13-limit == | == 13-limit == | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 379: | Line 363: | ||
POTE generator: ~3/2 = 701.590 | POTE generator: ~3/2 = 701.590 | ||
{{Val list|legend=1| 118, 130, 248, 378 }} | {{Val list|legend=1| 12, 118, 130, 248, 378 }} | ||
Badness: 0.0287 | Badness: 0.0287 | ||
= Kleischismic = | = Kleischismic = | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 402: | Line 385: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 418: | Line 400: | ||
== 13-limit == | == 13-limit == | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 434: | Line 415: | ||
= Pogo = | = Pogo = | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 450: | Line 430: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 466: | Line 445: | ||
== 13-limit == | == 13-limit == | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 482: | Line 460: | ||
= Squirrel = | = Squirrel = | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 496: | Line 473: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 510: | Line 486: | ||
== 13-limit == | == 13-limit == | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 524: | Line 499: | ||
= Term = | = Term = | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 541: | Line 515: | ||
* [[9-odd-limit]] eigenmonzos: 2, 9/7 | * [[9-odd-limit]] eigenmonzos: 2, 9/7 | ||
{{Val list|legend=1| 171, 867, 1038, 1209, 1380, 1551, 1722 }} | {{Val list|legend=1| 12, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }} | ||
[[Badness]]: 0. | [[Badness]]: 0.019950 | ||
= Sesquiquartififths = | = Sesquiquartififths = | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 568: | Line 541: | ||
== Sesquart == | == Sesquart == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 584: | Line 556: | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 598: | Line 569: | ||
== Bisesqui == | == Bisesqui == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 612: | Line 582: | ||
= Sextilififths = | = Sextilififths = | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 628: | Line 597: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 644: | Line 612: | ||
== 13-limit == | == 13-limit == | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 679: | Line 646: | ||
= Quanharuk = | = Quanharuk = | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 697: | Line 663: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 713: | Line 678: | ||
== 13-limit == | == 13-limit == | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 729: | Line 693: | ||
= Octant = | = Octant = | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 747: | Line 710: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 763: | Line 725: | ||
== 13-limit == | == 13-limit == | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Revision as of 22:42, 12 April 2021
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), 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 wedgie. This tells us the generator is a fifth and 5/4 is represented by a diminished fourth. In fact, 10 = (4/3)8 × 32805/32768.
Schismatic aka Helmholtz
The 5-limit version of the temperament is a microtemperament, sometimes called Helmholtz, schismic 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. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.
Subgroup: 2.3.5
Comma list: 32805/32768
Mapping: [⟨1 0 15], ⟨0 1 -8]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 701.736
- valid range: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
- nice range: ~3/2 = [701.711, 701.955]
- strict range: ~3/2 = [701.711, 701.955]
Optimal ET sequence: 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc
Badness: 0.004259
Seven-limit extensions
The second comma of the normal comma list defines which 7-limit family member we are looking at.
- Garibaldi adds [25 -14 0 -1⟩,
- Grackle adds [-44 26 0 1⟩,
- Schism adds [6 -2 0 -1⟩,
- Pontiac adds [-59 39 0 -1⟩.
Those all have a fifth as generator.
- Bischismic adds [-69 40 0 2⟩ and has a fifth generator with a half-octave period.
- Guiron adds [-10 1 0 3⟩, with an 8/7 generator, three of which give the fifth.
- Term adds [-94 54 0 3⟩ with a 1/3 octave period.
- Sesquiquartififths adds [-35 15 0 4⟩ and slices the fifth in four.
Temperaments not discussed here include salsa, guiron, hemischis and karadeniz.
Garibaldi
Subgroup: 2.3.5.7
Comma list: 225/224, 3125/3087
Mapping: [⟨1 0 15 25], ⟨0 1 -8 -14]]
Mapping generators: ~2, ~3
Wedgie: ⟨⟨ 1 -8 -14 -15 -25 -10 ]]
POTE generator: ~3/2 = 702.085
- [[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⟩]
- Eigenmonzos: 2, 7/6
- [[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
- valid range: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
- nice range: ~3/2 = [701.711, 702.915]
- strict range: ~3/2 = [701.711, 702.915]
Badness: 0.021644
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 2200/2187
Mapping: [⟨1 0 15 25 -33], ⟨0 1 -8 -14 23]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 702.157
Minimax tuning:
- 11-odd-limit
- [[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
Badness: 0.027396
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 275/273, 325/324, 385/384
Mapping: [⟨1 0 15 25 -33 -28], ⟨0 1 -8 -14 23 20]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 702.113
Badness: 0.020676
Andromeda
Subgroup: 2.3.5.7.11
Comma list: 100/99, 225/224, 245/242
Mapping: [⟨1 0 15 25 32], ⟨0 1 -8 -14 -18]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 702.321
Badness: 0.023556
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 105/104, 196/195, 245/242
Mapping: [⟨1 0 15 25 32 37], ⟨0 1 -8 -14 -18 -21]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 702.559
Badness: 0.020749
Helenus
Subgroup: 2.3.5.7.11
Comma list: 99/98, 176/175, 3125/3087
Mapping: [⟨1 0 15 25 51], ⟨0 1 -8 -14 -30]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 701.725
Badness: 0.035637
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 176/175, 275/273, 847/845
Mapping: [⟨1 0 15 25 51 56], ⟨0 1 -8 -14 -30 -33]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 701.747
Badness: 0.026284
Hemigari
Subgroup: 2.3.5.7.11
Comma list: 121/120, 225/224, 3125/3087
Mapping: [⟨1 0 15 25 9], ⟨0 2 -16 -28 -7]]
Mapping generators: ~2, ~110/63
POTE generator: ~63/55 = 248.918
Badness: 0.050681
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 169/168, 225/224, 275/273
Mapping: [⟨1 0 15 25 9 14], ⟨0 2 -16 -28 -7 -13]]
Mapping generators: ~2, ~26/15
POTE generator: ~15/13 = 248.918
Badness: 0.027464
Sanjaab
Subgroup: 2.3.5.7.11
Comma list: 225/224, 1331/1323, 3125/3087
Mapping: [⟨1 2 -1 -3 0], ⟨0 -3 24 42 25]]
Mapping generators: ~2, ~11/10
POTE generator: ~11/10 = 165.974
Badness: 0.058040
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 275/273, 847/845, 1331/1323
Mapping: [⟨1 2 -1 -3 0 -1], ⟨0 -3 24 42 25 34]]
Mapping generators: ~2, ~11/10
POTE generator: ~11/10 = 165.963
Badness: 0.033849
Schism
Subgroup: 2.3.5.7
Comma list: 64/63, 360/343
Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 701.556
Wedgie: ⟨⟨ 1 -8 -2 -15 -6 18 ]]
Badness: 0.0566
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 64/63, 99/98
Mapping: [⟨1 0 15 6 13], ⟨0 1 -8 -2 -6]]
Mapping generators: ~2, ~3
POTE generator ~3/2 = 702.136
Badness: 0.0375
Pontiac
Subgroup: 2.3.5.7
Comma list: 4375/4374, 32805/32768
Mapping: [⟨1 0 15 -59], ⟨0 1 -8 39]]
Mapping generators: ~2, ~3
Wedgie: ⟨⟨ 1 -8 39 -15 59 113 ]]
POTE generator: ~3/2 = 701.757
- [[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
- [[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
- valid range: ~3/2 = [701.538, 701.886] (38\65 to 31\53)
- nice range: ~3/2 = [701.711, 701.955]
- strict range: ~3/2 = [701.711, 701.886]
Badness: 0.014133
Grackle
Subgroup: 2.3.5.7
Comma list: 126/125, 32805/32768
Mapping: [⟨1 0 15 -44], ⟨0 1 -8 -26]]
Mapping generators: ~2, ~3
Wedgie: ⟨⟨ 1 -8 -26 -15 -44 -38 ]]
POTE generator: ~3/2 = 701.239
- 7-odd-limit eigenmonzos: 2, 7/6
- 9-odd-limit eigenmonzos: 2, 9/7
Badness: 0.070407
Bischismic
Subgroup: 2.3.5.7
Comma list: 3136/3125, 32805/32768
Mapping: [⟨2 0 30 69], ⟨0 1 -8 -20]]
Mapping generators: ~567/400, ~3
Wedgie: ⟨⟨ 2 -16 -40 -30 -69 -48 ]]
POTE generator: ~3/2 = 701.592
- 7-odd-limit eigenmonzos: 2, 7/6
- 9-odd-limit eigenmonzos: 2, 9/7
Badness: 0.0547
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 8019/8000
Mapping: [⟨2 0 30 69 102], ⟨0 1 -8 -20 -30]]
Mapping generators: ~99/70, ~3
POTE generator: ~3/2 = 701.612
Badness: 0.0282
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 729/728, 1001/1000, 3136/3125
Mapping generators: ~99/70, ~3
Mapping: [⟨2 0 30 69 102 -75], ⟨0 1 -8 -20 -30 26]]
POTE generator: ~3/2 = 701.590
Badness: 0.0287
Kleischismic
Subgroup: 2.3.5.7
Comma list: 32805/32768, 1500625/1492992
Mapping: [⟨2 1 22 -15], ⟨0 2 -16 19]]
Mapping generators: ~1225/864, ~35/24
Wedgie: ⟨⟨ 4 -32 38 -60 49 178 ]]
POTE generator: ~36/35 = 50.920
Badness: 0.1106
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 9801/9800, 14641/14580
Mapping: [⟨2 1 22 -15 8], ⟨0 2 -16 19 -1]]
Mapping generators: ~99/70, ~16/11
POTE generator: ~36/35 = 50.918
Badness: 0.0367
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 729/728, 1575/1573
Mapping: [⟨2 1 22 -15 8 15], ⟨0 2 -16 19 -1 -7]]
Mapping generators: ~99/70, ~16/11
POTE generator: ~36/35 = 50.938
Badness: 0.0376
Pogo
Subgroup: 2.3.5.7
Comma list: 32805/32768, 118098/117649
Mapping: [⟨2 1 22 2], ⟨0 3 -24 5]]
Mapping generators: ~343/243, ~9/7
POTE generator: ~9/7 = 433.901
Badness: 0.0796
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4000/3993, 32805/32768
Mapping: [⟨2 1 22 2 25], ⟨0 3 -24 5 -25]]
Mapping generators: ~99/70, ~9/7
POTE generator: ~9/7 = 433.911
Badness: 0.0319
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 4000/3993, 4225/4224
Mapping: [⟨2 1 22 2 25 -2], ⟨0 3 -24 5 -25 13]]
Mapping generators: ~99/70, ~9/7
POTE generator: ~9/7 = 433.911
Badness: 0.0175
Squirrel
Subgroup: 2.3.5.7
Comma list: 686/675, 32805/32768
Mapping: [⟨1 2 -1 1], ⟨0 -3 24 13]]
POTE generator: ~160/147 = 166.140
Badness: 0.1747
11-limit
Subgroup: 2.3.5.7.11
Comma list: 245/242, 686/675, 896/891
Mapping: [⟨1 2 -1 1 0], ⟨0 -3 24 13 25]]
POTE generator: ~11/10 = 166.097
Badness: 0.0683
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 169/168, 245/242, 896/891
Mapping: [⟨1 2 -1 1 0 3], ⟨0 -3 24 13 25 5]]
POTE generator: ~11/10 = 166.054
Badness: 0.0437
Term
Subgroup: 2.3.5.7
Comma list: 32805/32768, 250047/250000
Mapping: [⟨3 0 45 94], ⟨0 1 -8 -18]]
Mapping generators: ~63/50, ~3
Wedgie: ⟨⟨ 3 -24 -54 -45 -94 -58 ]]
POTE generator: ~3/2 = 701.742
- 7-odd-limit eigenmonzos: 2, 6/5
- 9-odd-limit eigenmonzos: 2, 9/7
Badness: 0.019950
Sesquiquartififths
Subgroup: 2.3.5.7
Comma list: 2401/2400, 32805/32768
Mapping: [⟨1 1 7 5], ⟨0 4 -32 -15]]
Mapping generators: ~2, ~448/405
Wedgie: ⟨⟨ 4 -32 -15 -60 -35 55 ]]
POTE generator: ~448/405 = 175.434
- 7-odd-limit eigenmonzos: 2, 7/6
- 9-odd-limit eigenmonzos: 2, 9/7
Badness: 0.0112
Sesquart
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 16384/16335
Mapping: [⟨1 1 7 5 2], ⟨0 4 -32 -15 10]]
Mapping generators: ~2, ~256/231
POTE generator: ~256/231 = 175.406
Badness: 0.0293
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 364/363, 441/440, 3584/3575
Mapping: [⟨1 1 7 5 2 -2], ⟨0 4 -32 -15 10 39]]
POTE generator: ~256/231 = 175.409
Badness: 0.0224
Bisesqui
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 9801/9800, 32805/32768
Mapping: [⟨2 2 14 10 23], ⟨0 4 -32 -15 -55]]
POTE generator: ~448/405 = 175.435
Badness: 0.0170
Sextilififths
Subgroup: 2.3.5.7
Comma list: 32768/32805, 235298/234375
Mapping: [⟨1 2 -1 -1], ⟨0 -6 48 55]]
Mapping generators: ~2, ~21/20
POTE generator: ~21/20 = 83.053
Badness: 0.1088
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4000/3993, 235298/234375
Mapping: [⟨1 2 -1 -1 0], ⟨0 -6 48 55 50]]
Mapping generators: ~2, ~21/20
POTE generator: ~21/20 = 83.049
Badness: 0.0455
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 676/675, 10985/10976
Mapping: [⟨1 2 -1 -1 0 1], ⟨0 -6 48 55 50 39]]
Mapping generators: ~2, ~21/20
POTE generator: ~21/20 = 83.049
Badness: 0.0253
Tsaharuk
Subgroup: 2.3.5.7
Comma list: 32805/32768, 420175/419904
Mapping: [⟨1 1 7 0], ⟨0 5 -40 24]]
Mapping generators: ~2, ~243/224
Wedgie: ⟨⟨ 5 -40 24 -75 24 168 ]]
POTE generator: ~243/224 = 140.350
Badness: 0.0307
Quanharuk
Subgroup: 2.3.5.7
Comma list: 16875/16807, 32805/32768
Mapping: [⟨1 0 15 12], ⟨0 5 -40 -29]]
Mapping generators: ~2, ~56/45
Wedgie: ⟨⟨ 5 -40 -29 -75 -60 45 ]]
POTE generator: ~56/45 = 380.355
Badness: 0.0720
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 32805/32768
Mapping: [⟨1 0 15 12 -7], ⟨0 5 -40 -29 33]]
Mapping generators: ~2, ~56/45
POTE generator: ~56/45 = 380.352
Badness: 0.0315
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1375/1372, 4096/4095
Mapping: [⟨1 0 15 12 -7 -15], ⟨0 5 -40 -29 33 59]]
Mapping generators: ~2, ~56/45
POTE generator: ~56/45 = 380.351
Badness: 0.0214
Octant
Subgroup: 2.3.5.7
Comma list: 32805/32768, 2259436291848/2251875390625
Mapping: [⟨8 0 120 -117], ⟨0 1 -8 11]]
Mapping generators: ~42875/39366, ~3
Wedgie: ⟨⟨ 8 -64 88 -120 117 384 ]]
POTE generator: ~3/2 = 701.713
Badness: 0.1572
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 32805/32768, 46656/46585
Mapping: [⟨8 0 120 -117 15], ⟨0 1 -8 11 1]]
Mapping generators: ~12/11, ~3
POTE generator: ~3/2 = 701.713
Badness: 0.0448
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
Mapping: [⟨8 0 120 -117 15 93], ⟨0 1 -8 11 1 -5]]
Mapping generators: ~12/11, ~3
POTE generator: ~3/2 = 701.725
Badness: 0.0304