Septischismic clan: Difference between revisions

(11 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]], thus equating the two.

The '''septischismic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[septischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33 554 432 / 33 480 783), the amount by which the [[Pythagorean comma]] falls short of the [[septimal comma]], thus equating the two.

== Gary ==

Line 28:

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]], halving the fifth~~. 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.

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 39:

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.

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, acting as a temperament of [[olympic clan #Olympian|olympian]] which accurately represents the entire [[2.3.7.11 subgroup]] into the chain of fifths. The major third additionally is extremely close to [[19/15]], and is a natural interpretation in extensions with 5 and 19.

=== 2.3.7.11 subgroup ===

Line 53:

Line 52:

Optimal tunings:

* WE: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}}

: error map: {{val| -0.037 0.216 -0.139 -0.173 }}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}

: error map: {{val| -0.000 0.274 -0.032 -0.051 }}

{{Optimal ET sequence|legend=0| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}

Line 61:

Line 62:

== 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.

Cotoneum tempers out 10976/10935 ([[hemimage comma]]) and 823543/819200 ([[quince comma]]) in addition to the septischisma. 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 ~~where~~ the generic aberschisma is identical to the [[41-comma]].

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.

Line 165:

Line 167:

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. [[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 ===

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 ==

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 tempers out the [[wizma]] and the septischisma, 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''.

Line 188:

Line 261:

[[Badness]] (Sintel): 2.22

==== 11-limit ====

=== 11-limit ===

Subgroup: 2.3.5.7.11

Line 205:

Line 278:

Badness (Sintel): 1.01

==== 13-limit ====

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Line 222:

Line 295:

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 242:

Line 315:

: ''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.

Satin tempers out the [[rainy comma]] and the [[cathartic comma]] in addition to the septischisma, 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

Line 389:

Line 462:

=== 2.3.5.7.11.13.19 subgroup ===

Subgroup: 2.3.5.7.11.13~~.17~~.19

Subgroup: 2.3.5.7.11.13.19

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

Line 404:

Line 477:

== 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.

World calendar tempers out the [[dimcomp comma]] and the septischisma, 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

Line 453:

Line 526:

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:Septischismic clan| ]]

[[Category:Temperament clans]]

[[Category:~~Garischismic clan| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[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]], thus equating the two.
+The '''septischismic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[septischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33 554 432 / 33 480 783), the amount by which the [[Pythagorean comma]] falls short of the [[septimal comma]], thus equating the two.
 == Gary ==
@@ Line 28: / Line 28: @@
 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]], halving the fifth. 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.
+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 39: / 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.
+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, acting as a temperament of [[olympic clan #Olympian|olympian]] which accurately represents the entire [[2.3.7.11 subgroup]] into the chain of fifths. The major third additionally is extremely close to [[19/15]], and is a natural interpretation in extensions with 5 and 19.
 === 2.3.7.11 subgroup ===
@@ Line 53: / Line 52: @@
 Optimal tunings:
 * WE: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}}
+: error map: {{val| -0.037 0.216 -0.139 -0.173 }}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}
+: error map: {{val| -0.000 0.274 -0.032 -0.051 }}
 {{Optimal ET sequence|legend=0| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}
@@ Line 61: / Line 62: @@
 == 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.
+Cotoneum tempers out 10976/10935 ([[hemimage comma]]) and 823543/819200 ([[quince comma]]) in addition to the septischisma. 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 where the generic aberschisma is identical to the [[41-comma]].
+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.
@@ Line 165: / Line 167: @@
 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. [[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 ===
+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 tempers out the [[wizma]] and the septischisma, 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''.
@@ Line 188: / Line 261: @@
 [[Badness]] (Sintel): 2.22
-==== 11-limit ====
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 205: / Line 278: @@
 Badness (Sintel): 1.01
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 222: / Line 295: @@
 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 242: / Line 315: @@
 : ''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.
+Satin tempers out the [[rainy comma]] and the [[cathartic comma]] in addition to the septischisma, 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
@@ Line 389: / Line 462: @@
 === 2.3.5.7.11.13.19 subgroup ===
-Subgroup: 2.3.5.7.11.13.17.19
+Subgroup: 2.3.5.7.11.13.19
 Comma list: 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625
@@ Line 404: / Line 477: @@
 == 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.
+World calendar tempers out the [[dimcomp comma]] and the septischisma, 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
@@ Line 453: / Line 526: @@
 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:Septischismic clan| ]] <!-- main article -->
 [[Category:Temperament clans]]
-[[Category:Garischismic clan| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]