Garischismic clan: Difference between revisions

(9 intermediate revisions by 3 users not shown)

Line 1:

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]].

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.

== Gary ==

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 ~~generic~~ comma step.

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]].

Line 25:

=== 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.

==== 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.

Gariwizmic adds the [[wizma]] with a 1/2-octave period~~. Newt adds [[2401/2400]], slicing the fifth in two~~. 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 ~~bisect generator~~. Quintagar adds [[3136/3125]], slicing the fourth in five. Paramity adds [[65625/65536]], slicing the eleventh in five.

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.

Temperaments discussed elsewhere are:

* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]

* ''[[Newt]]'' (+2401/2400) → [[Breedsmic temperaments #Newt|Breedsmic temperaments]]

* ''[[Alphatrident]]'' (+6144/6125) → [[Alphatricot family #Alphatrident|Alphatricot family]]

* ''[[Vulture]]'' (+4375/4374) → [[Vulture family #Vulture|Vulture family]]

Line 38:

* ''[[Garistearn]]'' (+118098/117649) → [[Stearnsmic clan #Garistearn|Stearnsmic clan]]

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

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)<sup>2</sup> with 33/32, at the cost of doubling the complexity.

=== 2.3.7.11 subgroup ===

Line 57:

Line 60:

== 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.

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 {{nowrap| 41 & 217 }}, and is supported by [[176edo|176-]], [[217edo|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.

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

Line 71:

Line 79:

* [[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]

Line 86:

Line 99:

* 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]

Line 101:

Line 118:

* 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]

Line 116:

Line 138:

* 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]

Line 131:

Line 157:

* 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]

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 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}}

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}}

Badness (Sintel): 0.438

== Gariwizmic ==

Line 158:

Line 259:

[[Badness]] (Sintel): 2.22

==== 11-limit ====

=== 11-limit ===

Subgroup: 2.3.5.7.11

Line 175:

Line 276:

Badness (Sintel): 1.01

==== 13-limit ====

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Line 192:

Line 293:

Badness (Sintel): 0.822

==== 2.3.5.7.11.13.19 subgroup ====

=== 2.3.5.7.11.13.19 subgroup ===

Subgroup: 2.3.5.7.11.13.19

Line 422:

Line 523:

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 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}}

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}}

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}}

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 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}}

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}}

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 }}

Optimal tunings:

* WE: ~2 = 1199.9884{{c}}, ~500/351 = 611.3531{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~500/351 = 611.3590{{c}}

Badness (Sintel): 1.22

[[Category:Temperament clans]]

[[Category:Garischismic clan| ]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-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]].
+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.
 == Gary ==
-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 generic comma step.
+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]].
@@ Line 25: / Line 25: @@
 === 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.
+==== 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.
-Gariwizmic adds the [[wizma]] with a 1/2-octave period. Newt adds [[2401/2400]], slicing the fifth in two. 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 bisect generator. Quintagar adds [[3136/3125]], slicing the fourth in five. Paramity adds [[65625/65536]], slicing the eleventh in five.
+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.
 Temperaments discussed elsewhere are:
 * [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
-* ''[[Newt]]'' (+2401/2400) → [[Breedsmic temperaments #Newt|Breedsmic temperaments]]
 * ''[[Alphatrident]]'' (+6144/6125) → [[Alphatricot family #Alphatrident|Alphatricot family]]
 * ''[[Vulture]]'' (+4375/4374) → [[Vulture family #Vulture|Vulture family]]
@@ Line 38: / Line 38: @@
 * ''[[Garistearn]]'' (+118098/117649) → [[Stearnsmic clan #Garistearn|Stearnsmic clan]]
-Considered below are cotoneum, gariwizmic, satin, sextile, and world calendar.
+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)<sup>2</sup> with 33/32, at the cost of doubling the complexity.
 === 2.3.7.11 subgroup ===
@@ Line 57: / Line 60: @@
 == 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.
-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 {{nowrap| 41 & 217 }}, and is supported by [[176edo|176-]], [[217edo|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.
+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
@@ Line 71: / Line 79: @@
 * [[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 }}
@@ Line 86: / Line 99: @@
 * 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 }}
@@ Line 101: / Line 118: @@
 * 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 }}
@@ Line 116: / Line 138: @@
 * 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 }}
@@ Line 131: / Line 157: @@
 * 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 ==
@@ Line 158: / Line 259: @@
 [[Badness]] (Sintel): 2.22
-==== 11-limit ====
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 175: / Line 276: @@
 Badness (Sintel): 1.01
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 192: / Line 293: @@
 Badness (Sintel): 0.822
-==== 2.3.5.7.11.13.19 subgroup ====
+=== 2.3.5.7.11.13.19 subgroup ===
 Subgroup: 2.3.5.7.11.13.19
@@ Line 422: / Line 523: @@
 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 }}
+Optimal tunings:
+* WE: ~2 = 1199.9884{{c}}, ~500/351 = 611.3531{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~500/351 = 611.3590{{c}}
+{{Optimal ET sequence|legend=0| 53, 528, 581, 1796, 2377b }}
+Badness (Sintel): 1.22
 [[Category:Temperament clans]]
 [[Category:Garischismic clan| ]] <!-- main article -->
 [[Category:Rank 2]]