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Garischismic clan

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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.

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 generic 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

The second comma of the comma list determines which full 7-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.

Gariwizmic adds the wizma with a 1/2-octave period. Newt adds 2401/2400, slicing the fifth in two. Sextile adds 250047/250000 with a 1/3-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. World calendar adds 390625/388962 with a 1/4-octave period as well as a bisect generator. Quintagar adds 3136/3125, slicing the fourth in five. Paramity adds 65625/65536, slicing the eleventh in five.

Temperaments discussed elsewhere are:

Considered below are cotoneum and world calendar.

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

The cotoneum temperament 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 at the septuple-diminished octave (C-Cbbbbbbb) or equivalently at the perfect fourth minus four Pythagorean commas. 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]

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 ¢

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 ¢

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 ¢

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 ¢

Optimal ET sequence: 41, 176, 217

Badness (Sintel): 1.33

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

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