Xenharmonic Wiki

Garischismic clan

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The garischismic clan of temperaments tempers out the garischisma (monzo: [25 -14 0 -1⟩, ratio: 33554432/33480783), the amount by which the Pythagorean comma falls short of the septimal comma, thus equating the two.

Gary

Gary, the head of this clan, may be viewed as the 2.3.7-subgroup counterpart of schismic. It is generated by a perfect fifth, and 7/4 is found at the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step which stands in for both the Pythagorean comma and the septimal comma. Gary can therefore use chain-of-fifths notation with an additional set of accidentals such as arrows to represent the comma step.

Just as there is the 1/8-schisma tuning for schismic, there is the 1/14-schisma tuning for gary, which tunes 7/4 pure by sharpening the perfect fifth by about 0.272 cents. Similarly, the 1/15-schisma tuning tunes 7/6 pure, and the 2/29-schisma tuning splits their difference, tuning the septimal diesis of 49/48 pure. 135edo is close to the 1/14-schisma tuning, whereas 634edo gives a tuning practically identical to 1/15-schisma. Other notable tunings not appearing in the optimal ET sequence include 311edo and 323edo.

Subgroup: 2.3.7

Comma list: 33554432/33480783

Subgroup-val mapping: [⟨1 0 25], ⟨0 1 -14]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.9155 ¢, ~3/2 = 702.1584 ¢
error map: ⟨-0.085 +0.119 +0.027]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2124 ¢
error map: ⟨0.000 +0.257 +0.201]

Optimal ET sequence: 12, 29, 41, 94, 135, 364, 499, 634, 3035bd, 3669bd, 4303bd, 4937bbdd, 5571bbdd

Badness (Sintel): 0.463

Overview to extensions

Full 11-limit extensions

The second comma of the comma list determines which full 7-limit or 11-limit family member we are looking at. Garibaldi adds the schisma, or equivalently 225/224 and finds 5/4 at the diminished fourth. Cotoneum adds 10976/10935 and finds 5/4 at the septuple-diminished octave. These are generated by the fifth as is gary.

Newt adds 2401/2400, halving the fifth. Gariwizmic adds the wizma with a 1/2-octave period. Alphatrident adds 6144/6125, slicing the twelfth in three. Satin adds 2100875/2097152, slicing the fourth in three. Vulture adds 4375/4374, slicing the twelfth in four. Sextile adds 250047/250000 with a 1/6-octave period. World calendar adds 390625/388962 with a 1/4-octave period as well as a halved fifth. Quintagar adds 3136/3125, slicing the fourth in five. Paramity adds 65625/65536, slicing the eleventh in five. Heptacot adds 703125/702464, slicing the fifth in seven. Finally, garitritonic adds 95703125/95551488 ([-17 -6 9 2⟩), slicing the 24th harmonic in nine.

Temperaments discussed elsewhere are:

Considered below are cotoneum, newt, gariwizmic, satin, sextile, and world calendar.

Subgroup extensions

Gary can be naturally extended into the no-5's 11-limit with good accuracy by equating (64/63)2 with 33/32, at the cost of doubling the complexity.

2.3.7.11 subgroup

Subgroup: 2.3.7.11

Comma list: 19712/19683, 41503/41472

Subgroup-val mapping: [⟨1 0 25 -33], ⟨0 1 -14 23]]

Optimal tunings:

  • WE: ~2 = 1199.9631 ¢, ~3/2 = 702.2077 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2290 ¢

Optimal ET sequence: 12e, 41, 94, 135, 716, 851, 986, 1121, 1256

Badness (Sintel): 0.276

Cotoneum

Main article: Cotoneum
For the 5-limit version, see Schismic–countercommatic equivalence continuum #Cotoneum (5-limit).

Cotoneum tempers out 10976/10935 (hemimage comma), and 823543/819200 (quince comma) in addition to the garischisma. This temperament can be described as 41 & 217, and is supported by 176-, 217-, and 258edo. 5/4 is found -49 generators away. In terms of chain-of-fifths notation, this is a sextuple-diminished octave, or a perfect fourth minus four generic commas.

However, cotoneum can be notated like cassaschismic, where 5/4 is conceptualized as an aberschisma-up comma-down major third (C–^↓E), but with the extra equivalence that the generic aberschisma is identical to the 41-comma. In other words, we have C–^↑↑E = C–↓↓E.

It can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 823543/819200

Mapping: [⟨1 0 80 25], ⟨0 1 -49 -14]]

Optimal tunings:

  • WE: ~2 = 1200.0386 ¢, ~3/2 = 702.3396 ¢
error map: ⟨+0.039 +0.423 +0.244 -1.155]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3164 ¢
error map: ⟨0.000 +0.361 +0.182 -1.256]

Tuning ranges:

  • 7-odd-limit diamond monotone: ~4/3 = [497.14286, 498.46154] (29\70 to 27\65)
  • 9-odd-limit diamond monotone: ~4/3 = [497.14286, 498.11321] (29\70 to 22\53)
  • 7- and 9-odd-limit diamond tradeoff: ~4/3 = [497.64251, 498.04500]

Optimal ET sequence: 41, 135c, 176, 217, 258, 475

Badness (Sintel): 2.67

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 10976/10935, 16384/16335

Mapping: [⟨1 0 80 25 -33], ⟨0 1 -49 -14 23]]

Optimal tunings:

  • WE: ~2 = 1199.8629 ¢, ~3/2 = 702.2224 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3036 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~4/3 = [497.56098, 497.87234] (17\41 to 39\94)
  • 11-odd-limit diamond tradeoff: ~4/3 = [497.64251, 498.04500]

Optimal ET sequence: 41, 135c, 176, 217

Badness (Sintel): 1.68

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3584/3575, 10976/10935

Mapping: [⟨1 0 80 25 -33 -93], ⟨0 1 -49 -14 23 61]]

Optimal tunings:

  • WE: ~2 = 1199.8897 ¢, ~3/2 = 702.2415 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3061 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~4/3 = [497.56098, 497.77778] (17\41 to 56\135)
  • 13-odd-limit diamond tradeoff: ~4/3 = [497.64251, 498.04500]
  • 15-odd-limit diamond tradeoff: ~4/3 = [497.63067, 498.04500]

Optimal ET sequence: 41, 176, 217

Badness (Sintel): 1.53

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262

Mapping: [⟨1 0 80 25 -33 -93 -137], ⟨0 1 -49 -14 23 61 89]]

Optimal tunings:

  • WE: ~2 = 1199.8939 ¢, ~3/2 = 702.2445 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3064 ¢

Tuning ranges:

  • 17-odd-limit diamond monotone: ~4/3 = [497.56098, 497.72727] (17\41 to 73\176)
  • 17-odd-limit diamond tradeoff: ~4/3 = [497.63067, 498.04500]

Optimal ET sequence: 41, 176, 217

Badness (Sintel): 1.50

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728

Mapping: [⟨1 0 80 25 -33 -93 -137 74], ⟨0 1 -49 -14 23 61 89 -44]]

Optimal tunings:

  • WE: ~2 = 1199.8766 ¢, ~3/2 = 702.2355 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3077 ¢

Tuning ranges:

  • 19- and 21-odd-limit diamond monotone: ~4/3 = [497.56098, 497.72727] (17\41 to 73\176)
  • 19- and 21-odd-limit diamond tradeoff: ~4/3 = [497.62290, 498.04500]

Optimal ET sequence: 41, 176, 217

Badness (Sintel): 1.33

Newt

For the 5-limit version, see Schismic–countercommatic equivalence continuum #Newt (5-limit).

Newt tempers out the breedsma and may be described as the 41 & 270 temperament. It has a generator of a neutral third (0.2 cents flat of 49/40) with a ploidacot signature of dicot. 41 generator steps fall short of 12 octaves by a generic aberschisma step of a schisma~aberschisma. From there the intervals of 5 and 7 can be derived.

Like cotoneum, newt can be notated in the same way as cassaschismic, but with half-sharps and half-flats and the extra equivalence that two comma steps and an aberschisma step make a half-apotome step. In other words, C–^↑↑E = C–v↓↓E = C–Ed.

Newt continues to be significant as an 11-limit temperament, where it tempers out the lehmerisma (3025/3024). This extends into a very strong 13-limit temperament and eventually a very strong no-17 19-limit temperament, a.k.a. neonewt. 270edo and 311edo are obvious tuning choices, but 581edo and especially 851edo are more accurate.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 33554432/33480783

Mapping: [⟨1 1 19 11], ⟨0 2 -57 -28]]

mapping generators: ~2, ~49/40

Optimal tunings:

  • WE: ~2 = 1199.9315 ¢, ~49/40 = 351.0932 ¢
error map: ⟨-0.068 +0.163 +0.075 -0.188]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1141 ¢
error map: ⟨0.000 +0.273 +0.180 -0.022]

Optimal ET sequence: 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201

Badness (Sintel): 1.06

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 19712/19683

Mapping: [⟨1 1 19 11 -10], ⟨0 2 -57 -28 46]]

Optimal tunings:

  • WE: ~2 = 1199.9603 ¢, ~49/40 = 351.1038 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1155 ¢

Optimal ET sequence: 41, 188, 229, 270, 581, 851, 1121, 1972

Badness (Sintel): 0.643

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095

Mapping: [⟨1 1 19 11 -10 -20], ⟨0 2 -57 -28 46 81]]

Optimal tunings:

  • WE: ~2 = 1199.9747 ¢, ~49/40 = 351.1094 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1168 ¢

Optimal ET sequence: 41, 229, 270, 581, 851, 2283b

Badness (Sintel): 0.571

2.3.5.7.11.13.19 subgroup (neonewt)

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400

Mapping: [⟨1 1 19 11 -10 -20 18], ⟨0 2 -57 -28 46 81 -47]]

Optimal tunings:

  • WE: ~2 = 1199.9782 ¢, ~49/40 = 351.1102 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1166 ¢

Optimal ET sequence: 41, 229, 270, 581, 851

Badness (Sintel): 0.438

Gariwizmic

Gariwizmic tempers out the wizma and the garischisma, and may be described as the 94 & 176 temperament. It assumes a semioctave period and a perfect fifth generator that is slightly sharp of just. It finds 5/4 39 fifths away, shifted by a semioctave. It extends extremely well to the 2.3.5.7.11.13.19 subgroup. Notable tunings not appearing in the optimal ET sequence include 364edo and 634edo.

Gariwizmic was named by Eufalesio in 2026 as a concatenation of gary and wizmic.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 33554432/33480783

Mapping: [⟨2 0 -119 50], ⟨0 1 39 -14]]

mapping generators: ~46305/32768, ~3

Optimal tunings:

  • WE: ~46305/32768 = 599.9657 ¢, ~3/2 = 702.1765 ¢
error map: ⟨-0.069 +0.153 -0.021 -0.053]
  • CWE: ~46305/32768 = 600.0000 ¢, ~3/2 = 702.2161 ¢
error map: ⟨0.000 +0.261 +0.114 +0.149]

Optimal ET sequence: 94, 176, 270, 904, 1174, 1444, 1714, 3158b, 4872bbcd

Badness (Sintel): 2.22

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 19712/19683, 41503/41472

Mapping: [⟨2 0 -119 50 -66], ⟨0 1 39 -14 23]]

Optimal tunings:

  • WE: ~99/70 = 599.9790 ¢, ~3/2 = 702.1938 ¢
error map: ⟨-0.042 +0.197 +0.106 -0.001 -0.440]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.2179 ¢
error map: ⟨0.000 +0.263 +0.185 +0.123 -0.306]

Optimal ET sequence: 94, 176, 270, 1174, 1444, 1714, 1984e

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 19712/19683

Mapping: [⟨2 0 -119 50 -66 93], ⟨0 1 39 -14 23 -27]]

Optimal tunings:

  • WE: ~99/70 = 599.9958 ¢, ~3/2 = 702.2096 ¢
error map: ⟨-0.008 +0.246 +0.035 +0.146 -0.412 -0.353]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.2145 ¢
error map: ⟨0.000 +0.260 +0.054 +0.170 -0.383 -0.321]

Optimal ET sequence: 94, 176, 270, 634, 904, 1174

Badness (Sintel): 0.822

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1716/1715, 1729/1728, 2080/2079

Mapping: [⟨2 0 -119 50 -66 93 -131], ⟨0 1 39 -14 23 -27 44]]

Optimal tunings:

  • WE: ~99/70 = 599.9969 ¢, ~3/2 = 702.2114 ¢
error map: ⟨-0.006 +0.250 +0.057 +0.147 -0.394 -0.355 -0.079]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.2150 ¢
error map: ⟨0.000 +0.260 +0.070 +0.165 -0.374 -0.332 -0.055]

Optimal ET sequence: 94, 176, 270, 634, 904, 1174

Badness (Sintel): 0.655

Satin

For the 5-limit version, see Miscellaneous 5-limit temperaments #Satin.

Satin tempers out the rainy comma and the canousma in addition to the garischisma, and may be described as the 94 & 217 temperament. It uses ~11/10 as a generator, three of which gives a perfect fourth, tempering out 4000/3993 in the 11-limit and onwards. Its ploidacot is omega-tricot.

Subgroup: 2.3.5.7

Comma list: 2100875/2097152, 4802000/4782969

Mapping: [⟨1 2 12 -3], ⟨0 -3 -70 42]]

mapping generators: ~2, ~8575/7776

Optimal tunings:

  • WE: ~2 = 1200.0198 ¢, ~8575/7776 = 165.9161 ¢
error map: ⟨+0.020 +0.336 -0.200 -0.411]
  • CWE: ~2 = 1200.0000 ¢, ~8575/7776 = 165.9133 ¢
error map: ⟨0.000 +0.305 -0.241 -0.469]

Optimal ET sequence: 94, 217, 311, 839, 1150

Badness (Sintel): 4.99

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 19712/19683, 41503/41472

Mapping: [⟨1 2 12 -3 13], ⟨0 -3 -70 42 -69]]

Optimal tunings:

  • WE: ~2 = 1199.9931 ¢, ~11/10 = 165.9145 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9155 ¢

Optimal ET sequence: 94, 217, 311

Badness (Sintel): 1.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689

Mapping: [⟨1 2 12 -3 13 -1], ⟨0 -3 -70 42 -69 34]]

Optimal tunings:

  • WE: ~2 = 1199.9607 ¢, ~11/10 = 165.9085 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9141 ¢

Optimal ET sequence: 94, 217, 311, 839e

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095

Mapping: [⟨1 2 12 -3 13 -1 11], ⟨0 -3 -70 42 -69 34 -50]]

Optimal tunings:

  • WE: ~2 = 1199.9843 ¢, ~11/10 = 165.9110 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9132 ¢

Optimal ET sequence: 94, 217, 311, 839e

Badness (Sintel): 1.02

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573

Mapping: [⟨1 2 12 -3 13 -1 11 16], ⟨0 -3 -70 42 -69 34 -50 -85]]

Optimal tunings:

  • WE: ~2 = 1199.9875 ¢, ~11/10 = 165.9111 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9129 ¢

Optimal ET sequence: 94, 217, 311, 839e

Badness (Sintel): 0.881

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155

Mapping: [⟨1 2 12 -3 13 -1 11 16 16], ⟨0 -3 -70 42 -69 34 -50 -85 -83]]

Optimal tunings:

  • WE: ~2 = 1199.9745 ¢, ~11/10 = 165.9103 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9140 ¢

Optimal ET sequence: 94, 217, 311

Badness (Sintel): 0.871

Sextile

For the 5-limit version, see Schismic–commatic equivalence continuum #Sextile (5-limit).

Sextile tempers out the landscape comma with a 1/6-octave period and is the 12 & 270 temperament.

Subgroup: 2.3.5.7

Comma list: 250047/250000, 33554432/33480783

Mapping: [⟨6 0 71 150], ⟨0 1 -6 -14]]

mapping generators: ~4096/3645, ~3

Optimal tunings:

  • WE: ~4096/3645 = 199.9828 ¢, ~3/2 = 702.1521 ¢
error map: ⟨-0.103 +0.094 +0.173 -0.088]
  • CWE: ~4096/3645 = 200.0000 ¢, ~3/2 = 702.2187 ¢
error map: ⟨0.000 +0.264 +0.374 +0.112]

Optimal ET sequence: 12, …, 258, 270, 1362c, 1632c, …, 2442bc, 2712bc

Badness (Sintel): 1.77

11-limit

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 9801/9800, 151263/151250

Mapping: [⟨6 0 71 150 230], ⟨0 1 -6 -14 -22]]

Optimal tunings:

  • WE: ~55/49 = 199.9817 ¢, ~3/2 = 702.1383 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2080 ¢

Optimal ET sequence: 12, …, 258e, 270, 822, 1092, 1362c

Badness (Sintel): 0.981

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647

Mapping: [⟨6 0 71 150 230 279], ⟨0 1 -6 -14 -22 -27]]

Optimal tunings:

  • WE: ~55/49 = 199.9804 ¢, ~3/2 = 702.1260 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2001 ¢

Optimal ET sequence: 12f, …, 258ef, 270, 552, 822, 1092, 1914cde

Badness (Sintel): 0.788

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625

Mapping: [⟨6 0 71 150 230 279 35], ⟨0 1 -6 -14 -22 -27 -1]]

Optimal tunings:

  • WE: ~55/49 = 199.9826 ¢, ~3/2 = 702.1359 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2003 ¢

Optimal ET sequence: 12f, 258ef, 270, 552, 822, 1092

Badness (Sintel): 0.634

World calendar

World calendar tempers out the dimcomp comma and the garischisma, and can be described as the 12 & 364 temperament. The name derives from a certain calendar layout by the same name.

Subgroup: 2.3.5.7

Comma list: 390625/388962, 33554432/33480783

Mapping: [⟨4 1 -44 86], ⟨0 2 -13 -28]]

mapping generators: ~25/21, ~91125/57344

Optimal tunings:

  • WE: ~25/21 = 299.9938 ¢, ~91125/57344 = 801.0780 ¢
error map: ⟨-0.025 +0.195 -0.603 +0.452]
  • CWE: ~25/21 = 300.0000 ¢, ~91125/57344 = 801.0955 ¢
error map: ⟨0.000 +0.236 -0.555 +0.501]

Optimal ET sequence: 12, …, 352, 364

Badness (Sintel): 7.39

2.3.5.7.17 subgroup

Subgroup: 2.3.5.7.17

Comma list: 2025/2023, 24576/24565, 390625/388962

Subgroup-val mapping: [⟨4 1 -44 86 3], ⟨0 2 -13 -28 5]]

Optimal tunings:

  • WE: ~25/21 = 299.9861 ¢, ~27/17 = 801.0536 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~27/17 = 801.0919 ¢

Optimal ET sequence: 12, …, 352, 364

Badness (Sintel): 2.74

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 1216/1215, 2025/2023, 8075/8064, 48013/48000

Subgroup-val mapping: [⟨4 1 -44 86 3 25], ⟨0 2 -13 -28 5 -3]]

Optimal tunings:

  • WE: ~25/21 = 299.9982 ¢, ~27/17 = 801.0898 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~27/17 = 801.0946 ¢

Optimal ET sequence: 12, …, 352, 364

Badness (Sintel): 1.82

Heptacot

For the 5-limit version, see Schismic–commatic equivalence continuum #Heptacot (5-limit).

Heptacot tempers out the meter and may be described as the 12 & 311 temperament, splitting the perfect fifth into seven semitones. It is the natural 7-limit extension of the 5-limit temperament named by Tristan Bay in 2024. 311edo and 323edo are obvious tuning choices, as well as anything in between such as 634edo.

Heptacot extends to the 11-limit in the same way as does gary, which best preserves its accuracy, though it should be noted that 299 & 311 and 323 & 335d make for simpler but less accurate alternative extensions.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 33554432/33480783

Mapping: [⟨1 1 6 11], ⟨0 7 -44 -98]]

mapping generators: ~2, ~1323/1250

Optimal tunings:

  • WE: ~2 = 1199.9434 ¢, ~1323/1250 = 100.3096 ¢
error map: ⟨-0.057 +0.155 -0.274 +0.215]
  • CWE: ~2 = 1200.0000 ¢, ~1323/1250 = 100.3148 ¢
error map: ⟨0.000 +0.249 -0.165 +0.324]

Optimal ET sequence: 12, …, 299, 311, 323, 634, 957, 1591

Badness (Sintel): 3.06

11-limit

Subgroup: 2.3.5.7.11

Comma list: 19712/19683, 41503/41472, 703125/702464

Mapping: [⟨1 1 6 11 -10], ⟨0 7 -44 -98 161]]

mapping generators: ~2, ~1323/1250

Optimal tunings:

  • WE: ~2 = 1199.9981 ¢, ~1323/1250 = 100.3174 ¢
  • CWE: ~2 = 1200.0000 ¢, ~1323/1250 = 100.3176 ¢

Optimal ET sequence: 12e, 311, 634, 945

Badness (Sintel): 3.21

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095, 19712/19683, 31250/31213

Mapping: [⟨1 1 6 11 -10 -7], ⟨0 7 -44 -98 161 128]]

mapping generators: ~2, ~1323/1250

Optimal tunings:

  • WE: ~2 = 1199.9938 ¢, ~675/637 = 100.3169 ¢
  • CWE: ~2 = 1200.0000 ¢, ~675/637 = 100.3174 ¢

Optimal ET sequence: 12e, 311, 634, 945

Badness (Sintel): 1.89

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 31250/31213

Mapping: [⟨1 1 6 11 -10 -7 5], ⟨0 7 -44 -98 161 128 -9]]

mapping generators: ~2, ~1323/1250

Optimal tunings:

  • WE: ~2 = 1200.0076 ¢, ~675/637 = 100.3179 ¢
  • CWE: ~2 = 1200.0000 ¢, ~675/637 = 100.3173 ¢

Optimal ET sequence: 12e, 311, 634, 945

Badness (Sintel): 1.38

Garitritonic

For the 5-limit version, see Schismic–Mercator equivalence continuum #Countritonic.

Garitritonic may be described as the 53 & 581 temperament, splitting the 24th harmonic into nine tritone generators; its ploidacot is thus delta-enneacot. 634edo makes for a strong 7-limit tuning, though in the higher limits one may prefer sticking to 581edo.

Garitritonic was named by Flora Canou in 2026 as a contraction of gary and tritonic.

Subgroup: 2.3.5.7

Comma list: 33554432/33480783, 95703125/95551488

Mapping: [⟨1 -3 -15 67], ⟨0 9 34 -126]]

mapping generators: ~2, ~4375/3072

Optimal tunings:

  • WE: ~2 = 1199.9678 ¢, ~4375/3072 = 611.3417 ¢
error map: ⟨-0.032 +0.217 -0.213 -0.036]
  • CWE: ~2 = 1200.0000 ¢, ~4375/3072 = 611.3582 ¢
error map: ⟨0.000 +0.268 -0.136 +0.045]

Optimal ET sequence: 53, 422d, 475, 528, 581, 634, 1215

Badness (Sintel): 6.12

11-limit

Subgroup: 2.3.5.7.11

Comma list: 19712/19683, 41503/41472, 1953125/1948617

Mapping: [⟨1 -3 -15 67 -102], ⟨0 9 34 -126 207]]

Optimal tunings:

  • WE: ~2 = 1199.9795 ¢, ~4375/3072 = 611.3485 ¢
  • CWE: ~2 = 1200.0000 ¢, ~4375/3072 = 611.3589 ¢

Optimal ET sequence: 53, 528, 581, 1796, 2377b

Badness (Sintel): 3.60

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095, 19712/19683, 78125/78078

Mapping: [⟨1 -3 -15 67 -102 -34], ⟨0 9 34 -126 207 74]]

Optimal tunings:

  • WE: ~2 = 1199.9813 ¢, ~500/351 = 611.3494 ¢
  • CWE: ~2 = 1200.0000 ¢, ~500/351 = 611.3589 ¢

Optimal ET sequence: 53, 528, 581, 1796, 2377b

Badness (Sintel): 1.73

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 59375/59319

Mapping: [⟨1 -3 -15 67 -102 -34 -36], ⟨0 9 34 -126 207 74 79]]

Optimal tunings:

  • WE: ~2 = 1199.9884 ¢, ~500/351 = 611.3531 ¢
  • CWE: ~2 = 1200.0000 ¢, ~500/351 = 611.3590 ¢

Optimal ET sequence: 53, 528, 581, 1796, 2377b

Badness (Sintel): 1.22

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