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This clan of temperaments tempers out the [[garischisma]], {{monzo| 25 -14 0 -1 }} = 33554432/33480783, and includes these:
{{Technical data page}}
* [[Garibaldi]], {225/224, 3125/3087} → [[Schismatic family #Garibaldi|Schismatic family]]
The '''garischismic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[garischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33554432/33480783), the amount by which the [[Pythagorean comma]] falls short of the [[septimal comma]], thus equating the two.
* ''[[Newt]]'', {2401/2400, 33554432/33480783} → [[Breedsmic temperaments #Newt|Breedsmic temperaments]]
* ''[[Quintagar]]'', {3136/3125, 33554432/33480783} → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Vulture]]'', {4375/4374, 33554432/33480783} → [[Vulture family #Vulture|Vulture family]]
* ''[[Trident]]'', {6144/6125, 14348907/14336000} → [[Tricot family #Trident|Tricot family]]
* [[Cotoneum]], {10976/10935, 823543/819200} → [[Hemimage temperaments #Cotoneum|Hemimage temperaments]]
* ''[[Paramity]]'', {65625/65536, 1600000/1594323} → [[Amity family #Paramity|Amity family]]
* ''[[Garistearn]]'', {118098/117649, 33554432/33480783} → [[Stearnsmic clan #Garistearn|Stearnsmic clan]]
* ''[[Sextile]]'', {250047/250000, 33554432/33480783} → [[Landscape microtemperaments #Sextile|Landscape microtemperaments]]
* ''[[Satin]]'', {2100875/2097152, 4802000/4782969} → [[Canousmic temperaments #Satin|Canousmic temperaments]]


== Gary ==
== Gary ==
Subgroup: 2.3.7
Gary, the head of this clan, may be viewed as the [[2.3.7 subgroup|2.3.7-subgroup]] counterpart of [[schismic]]. It is generated by a [[3/2|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
[[Comma list]]: 33554432/33480783


[[Sval]] [[mapping]]: [{{val| 1 2 -3 }}, {{val| 0 -1 14 }}]
{{Mapping|legend=2| 1 0 25 | 0 1 -14 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9155{{c}}, ~3/2 = 702.1584{{c}}
: [[error map]]: {{val| -0.085 +0.119 +0.027 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2124{{c}}
: error map: {{val| 0.000 +0.257 +0.201 }}
 
{{Optimal ET sequence|legend=1| 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 ({{monzo| -17 -6 9 2 }}), slicing the 24th harmonic in nine.


[[POTE generator]]: ~3/2 = 702.2079
Temperaments discussed elsewhere are:
* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
* ''[[Alphatrident]]'' (+6144/6125) → [[Alphatricot family #Alphatrident|Alphatricot family]]
* ''[[Vulture]]'' (+4375/4374) → [[Vulture family #Vulture|Vulture family]]
* ''[[Quintagar]]'' (+3136/3125) → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Paramity]]'' (+65625/65536) → [[Amity family #Paramity|Amity family]]
* ''[[Garistearn]]'' (+118098/117649) → [[Stearnsmic clan #Garistearn|Stearnsmic clan]]


{{Val list|legend=1| 12, 29, 41, 94, 135, 364, 499, 634, 3035bd, 3669bd, 4303bd, 4937bbdd, 5571bbdd }}
Considered below are cotoneum, newt, gariwizmic, satin, sextile, and world calendar.


[[Badness]]: 0.0135
==== Subgroup extensions ====
Gary can be naturally extended into the no-5's 11-limit with good accuracy by equating (64/63)<sup>2</sup> with 33/32, at the cost of doubling the complexity.  


=== 2.3.7.11 ===
=== 2.3.7.11 subgroup ===
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


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


Sval mapping: [{{val| 1 2 -3 13 }}, {{val| 0 -1 14 -23 }}]
Subgroup-val mapping: {{mapping| 1 0 25 -33 | 0 1 -14 23 }}
 
Optimal tunings:
* WE: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}
 
Badness (Sintel): 0.276
 
== Cotoneum ==
{{Main| 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 {{nowrap| 41 & 217 }}, and is supported by [[176edo|176-]], [[217edo|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|legend=1| 1 0 80 25 | 0 1 -49 -14 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0386{{c}}, ~3/2 = 702.3396{{c}}
: [[error map]]: {{val| +0.039 +0.423 +0.244 -1.155 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3164{{c}}
: error map: {{val| 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|legend=1| 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: {{mapping| 1 0 80 25 -33 | 0 1 -49 -14 23 }}
 
Optimal tunings:
* WE: ~2 = 1199.8629{{c}}, ~3/2 = 702.2224{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3036{{c}}
 
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|legend=0| 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: {{mapping| 1 0 80 25 -33 -93 | 0 1 -49 -14 23 61 }}
 
Optimal tunings:
* WE: ~2 = 1199.8897{{c}}, ~3/2 = 702.2415{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3061{{c}}
 
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|legend=0| 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: {{mapping| 1 0 80 25 -33 -93 -137 | 0 1 -49 -14 23 61 89 }}
 
Optimal tunings:
* WE: ~2 = 1199.8939{{c}}, ~3/2 = 702.2445{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3064{{c}}
 
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|legend=0| 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: {{mapping| 1 0 80 25 -33 -93 -137 74 | 0 1 -49 -14 23 61 89 -44 }}
 
Optimal tunings:
* WE: ~2 = 1199.8766{{c}}, ~3/2 = 702.2355{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3077{{c}}
 
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|legend=0| 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 {{nowrap| 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|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|legend=1| 1 1 19 11 | 0 2 -57 -28 }}
: mapping generators: ~2, ~49/40
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9315{{c}}, ~49/40 = 351.0932{{c}}
: [[error map]]: {{val| -0.068 +0.163 +0.075 -0.188 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.1141{{c}}
: error map: {{val| 0.000 +0.273 +0.180 -0.022 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}
 
Optimal tunings:
* WE: ~2 = 1199.9603{{c}}, ~49/40 = 351.1038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1155{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}
 
Optimal tunings:
* WE: ~2 = 1199.9747{{c}}, ~49/40 = 351.1094{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1168{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}
 
Optimal tunings:
* WE: ~2 = 1199.9782{{c}}, ~49/40 = 351.1102{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1166{{c}}
 
{{Optimal ET sequence|legend=0| 41, 229, 270, 581, 851 }}
 
Badness (Sintel): 0.438
 
== Gariwizmic ==
Gariwizmic tempers out the [[wizma]] and the garischisma, and may be described as the {{nowrap| 94 & 176 }} temperament. It assumes a [[semioctave]] period and a [[3/2|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|legend=1| 2 0 -119 50 | 0 1 39 -14 }}
: mapping generators: ~46305/32768, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~46305/32768 = 599.9657{{c}}, ~3/2 = 702.1765{{c}}
: [[error map]]: {{val|-0.069 +0.153 -0.021 -0.053 }}
* [[CWE]]: ~46305/32768 = 600.0000{{c}}, ~3/2 = 702.2161{{c}}
: error map: {{val| 0.000 +0.261 +0.114 +0.149 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 2 0 -119 50 -66 | 0 1 39 -14 23 }}
 
Optimal tunings:
* WE: ~99/70 = 599.9790{{c}}, ~3/2 = 702.1938{{c}}
: error map: {{val| -0.042 +0.197 +0.106 -0.001 -0.440 }}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2179{{c}}
: error map: {{val| 0.000 +0.263 +0.185 +0.123 -0.306 }}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 -119 50 -66 93 | 0 1 39 -14 23 -27 }}
 
Optimal tunings:
* WE: ~99/70 = 599.9958{{c}}, ~3/2 = 702.2096{{c}}
: error map: {{val| -0.008 +0.246 +0.035 +0.146 -0.412 -0.353 }}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2145{{c}}
: error map: {{val| 0.000 +0.260 +0.054 +0.170 -0.383 -0.321 }}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 -119 50 -66 93 -131 | 0 1 39 -14 23 -27 44 }}
 
Optimal tunings:
* WE: ~99/70 = 599.9969{{c}}, ~3/2 = 702.2114{{c}}
: error map: {{val| -0.006 +0.250 +0.057 +0.147 -0.394 -0.355 -0.079 }}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2150{{c}}
: error map: {{val| 0.000 +0.260 +0.070 +0.165 -0.374 -0.332 -0.055 }}
 
{{Optimal ET sequence|legend=0| 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 {{nowrap| 94 & 217 }} temperament. It uses [[~]][[11/10]] as a generator, three of which gives a [[4/3|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|legend=1| 1 2 12 -3 | 0 -3 -70 42 }}
: mapping generators: ~2, ~8575/7776
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0198{{c}}, ~8575/7776 = 165.9161{{c}}
: [[error map]]: {{val| +0.020 +0.336 -0.200 -0.411 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8575/7776 = 165.9133{{c}}
: error map: {{val| 0.000 +0.305 -0.241 -0.469 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 2 12 -3 13 | 0 -3 -70 42 -69 }}
 
Optimal tunings:
* WE: ~2 = 1199.9931{{c}}, ~11/10 = 165.9145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9155{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 12 -3 13 -1 | 0 -3 -70 42 -69 34 }}
 
Optimal tunings:
* WE: ~2 = 1199.9607{{c}}, ~11/10 = 165.9085{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9141{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 12 -3 13 -1 11 | 0 -3 -70 42 -69 34 -50 }}
 
Optimal tunings:
* WE: ~2 = 1199.9843{{c}}, ~11/10 = 165.9110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9132{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 12 -3 13 -1 11 16 | 0 -3 -70 42 -69 34 -50 -85 }}
 
Optimal tunings:
* WE: ~2 = 1199.9875{{c}}, ~11/10 = 165.9111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9129{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~11/10 = 165.9103{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9140{{c}}
 
{{Optimal ET sequence|legend=0| 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 {{nowrap| 12 & 270 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250047/250000, 33554432/33480783
 
{{Mapping|legend=1| 6 0 71 150 | 0 1 -6 -14 }}
: mapping generators: ~4096/3645, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~4096/3645 = 199.9828{{c}}, ~3/2 = 702.1521{{c}}
: [[error map]]: {{val| -0.103 +0.094 +0.173 -0.088 }}
* [[CWE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2187{{c}}
: error map: {{val| 0.000 +0.264 +0.374 +0.112 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 6 0 71 150 230 | 0 1 -6 -14 -22 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9817{{c}}, ~3/2 = 702.1383{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2080{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 6 0 71 150 230 279 | 0 1 -6 -14 -22 -27 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9804{{c}}, ~3/2 = 702.1260{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2001{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 6 0 71 150 230 279 35 | 0 1 -6 -14 -22 -27 -1 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9826{{c}}, ~3/2 = 702.1359{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2003{{c}}
 
{{Optimal ET sequence|legend=0| 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 {{nowrap| 12 & 364 }} temperament. The name derives from a {{w|World Calendar|certain calendar layout}} by the same name.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 390625/388962, 33554432/33480783
 
{{Mapping|legend=1| 4 1 -44 86 | 0 2 -13 -28 }}
: mapping generators: ~25/21, ~91125/57344
 
[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 299.9938{{c}}, ~91125/57344 = 801.0780{{c}}
: [[error map]]: {{val| -0.025 +0.195 -0.603 +0.452 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~91125/57344 = 801.0955{{c}}
: error map: {{val| 0.000 +0.236 -0.555 +0.501 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 4 1 -44 86 3 | 0 2 -13 -28 5 }}
 
Optimal tunings:
* WE: ~25/21 = 299.9861{{c}}, ~27/17 = 801.0536{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~27/17 = 801.0919{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 4 1 -44 86 3 25 | 0 2 -13 -28 5 -3 }}
 
Optimal tunings:
* WE: ~25/21 = 299.9982{{c}}, ~27/17 = 801.0898{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~27/17 = 801.0946{{c}}
 
{{Optimal ET sequence|legend=0| 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 {{nowrap| 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 {{nowrap| 299 & 311 }} and {{nowrap| 323 & 335d }} make for simpler but less accurate alternative extensions.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 703125/702464, 33554432/33480783
 
{{Mapping|legend=1| 1 1 6 11 | 0 7 -44 -98 }}
: mapping generators: ~2, ~1323/1250
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9434{{c}}, ~1323/1250 = 100.3096{{c}}
: [[error map]]: {{val| -0.057 +0.155 -0.274 +0.215 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1250 = 100.3148{{c}}
: error map: {{val| 0.000 +0.249 -0.165 +0.324 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 1 6 11 -10 | 0 7 -44 -98 161 }}
: mapping generators: ~2, ~1323/1250
 
Optimal tunings:
* WE: ~2 = 1199.9981{{c}}, ~1323/1250 = 100.3174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1323/1250 = 100.3176{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 6 11 -10 -7 | 0 7 -44 -98 161 128 }}
: mapping generators: ~2, ~1323/1250
 
Optimal tunings:
* WE: ~2 = 1199.9938{{c}}, ~675/637 = 100.3169{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~675/637 = 100.3174{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~675/637 = 100.3179{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~675/637 = 100.3173{{c}}
 
{{Optimal ET sequence|legend=0| 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 {{nowrap| 53 & 581 }} temperament, splitting the [[24/1|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|legend=1| 1 -3 -15 67 | 0 9 34 -126 }}
: mapping generators: ~2, ~4375/3072
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9678{{c}}, ~4375/3072 = 611.3417{{c}}
: [[error map]]: {{val| -0.032 +0.217 -0.213 -0.036 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~4375/3072 = 611.3582{{c}}
: error map: {{val| 0.000 +0.268 -0.136 +0.045 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 -3 -15 67 -102 | 0 9 34 -126 207 }}
 
Optimal tunings:
* WE: ~2 = 1199.9795{{c}}, ~4375/3072 = 611.3485{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~4375/3072 = 611.3589{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -3 -15 67 -102 -34 | 0 9 34 -126 207 74 }}
 
Optimal tunings:
* WE: ~2 = 1199.9813{{c}}, ~500/351 = 611.3494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~500/351 = 611.3589{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -3 -15 67 -102 -34 -36 | 0 9 34 -126 207 74 79 }}


POTE generator: ~3/2 = 702.2292
Optimal tunings:
* WE: ~2 = 1199.9884{{c}}, ~500/351 = 611.3531{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~500/351 = 611.3590{{c}}


Vals: {{Val list| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}
{{Optimal ET sequence|legend=0| 53, 528, 581, 1796, 2377b }}


Badness: 0.00731
Badness (Sintel): 1.22


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Garischismic clan| ]] <!-- main article -->
[[Category:Garischismic clan| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Garischismic_clan"