Aberschismic temperaments: Difference between revisions

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This is a collection of [[rank-2 temperament|rank-2]] ~~[[regular temperament|~~temperaments]] [[tempering out]] the [[~~hemifamity comma~~]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].

This is a collection of [[rank-2 temperament|rank-2]] '''aberschismic temperaments''', which [[tempering out|temper out]] the [[aberschisma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].

Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, undecental, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1.

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Diaschismic is generated by the fifth with a semi-octave period. Hemififths has the fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma. Hemidromeda has the fourth sliced into two and 5/4 mapped to the hemifourth + 3d4. Rodan has the fifth sliced into three as does slendric. Alphatrimot has the twelfth sliced into three as does alphatricot. Monkey has the fifth sliced into four as does tetracot. Buzzard has the twelfth sliced into four as does vulture. Misty is generated by the fifth with a 1/3-octave period. Supers has the fifth sliced into three with a semi-octave period. Undim is generated by the fifth with a 1/4-octave period. Quinticosiennic and quintakwai have the fourth sliced into five. Amity has the eleventh sliced into five. Countercata has the twelfth sliced into six as does hanson. Warrior has the 6th harmonic sliced into seven as does sensi. Finally, alphaquarter has the fourth sliced into nine as does escapade.

Temperaments ~~considered below are septiquarter, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral and quanic, in the order of increasing [[TE logflat badness]]. Discussed~~ elsewhere are:

Temperaments discussed elsewhere are:

* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]

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

* ''[[Kwai]]'' (+16875/16807) → [[Mirkwai clan #Kwai|Mirkwai clan]]

* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]

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

* [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]]

* ''[[Alphatrimot]]'' (+2430/2401) → [[Alphatricot family #Alphatrimot|Alphatricot family]]

* ''[[Monkey]]'' (+875/864) → [[Tetracot family #Monkey|Tetracot family]]

* [[Buzzard]] (+1728/1715) → [[Vulture family #Buzzard|Vulture family]]

* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]

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

* [[Monkey]] (+875/864) → [[Tetracot family #Monkey|Tetracot family]]

* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]]

* ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]

* ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]

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* [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity family]]

* ''[[Countercata]]'' (+15625/15552) → [[Kleismic family #Countercata|Kleismic family]]

* ''[[Abergravity]]'' (+177147/175000) → [[Gravity family #Abergravity|Gravity family]]

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

* ''[[Warrior]]'' (+78732/78125) → [[Sensipent family #Warrior|Sensipent family]]

* ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]]

Considered below are septiquarter, kwai, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].

== Septiquarter ==

Septiquarter tempers out [[420175/419904]] and may be described as the {{nowrap| 94 & 99 }} temperament. Its [[ploidacot]] is epsilon-heptacot. [[99edo]] makes for an excellent tuning, and [[292edo]] an even better one. [[94edo]] and [[104edo]] in the 104c val are also among the possibilities.

[[Subgroup]]: 2.3.5.7

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Badness (Sintel): 1.44

== Kwai ==

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''

Named by [[Gene Ward Smith]] in 2004 for its "bridgeability"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10766.html Yahoo! Tuning Group | ''Kwai'']</ref>, kwai is generated by a [[3/2|perfect fifth]], and can be described as {{nowrap| 41 & 70 }}.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 5120/5103, 16875/16807

: mapping generators: ~2, ~3

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.7337{{c}}, ~3/2 = 702.4600{{c}}

: [[error map]]: {{val| -0.266 +0.239 -0.607 +1.055 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6085{{c}}

: error map: {{val| 0.000 +0.653 -0.234 +1.603 }}

[[Badness]] (Sintel): 1.38

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 5120/5103

Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}

Optimal tunings:

* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}

Badness (Sintel): 0.867

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1375/1372

Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}

Optimal tunings:

* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}

Badness (Sintel): 1.01

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088

Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}

Optimal tunings:

* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}

Badness (Sintel): 1.12

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845

Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}

Optimal tunings:

* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}

Badness (Sintel): 1.03

==== Hemikwai ====

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 676/675, 1375/1372, 5120/5103

Mapping: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}

: mapping generators: ~2, ~26/15

Optimal tunings:

* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}

Badness (Sintel): 1.82

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103

Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}

Optimal tunings:

* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}

Badness (Sintel): 1.31

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444

Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}

Optimal tunings:

* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}

Badness (Sintel): 1.16

== Ketchup ==

Ketchup may be described as the {{nowrap| 46 & 94 }} temperament. It has a semi-octave period and a generator for a syntonic~septimal comma, four of which plus a period gives the perfect fifth; its [[ploidacot]] is diploid gamma-tetracot. [[140edo]] is an obvious tuning for this temperament.

[[Subgroup]]: 2.3.5.7

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Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 352/351, 385/384, ~~561~~/~~560~~

Comma list: 289/288, 325/324, 352/351, 385/384, 442/441

Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}

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Badness (Sintel): 0.845

=== ~~19-limit~~ ===

=== 2.3.5.7.11.13.17.23 subgroup ===

Subgroup: 2.3.5.7.11.13.17.19

Subgroup: 2.3.5.7.11.13.17.23

Comma list: ~~190~~/~~189~~, ~~209~~/~~208~~, ~~289~~/~~288~~, 352/351, 385/384, ~~561~~/~~560~~

Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390

Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 ~~-12~~ }}

Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}

Optimal tunings:

* WE: ~17/12 = 600.~~1639~~{{c}}, ~66/65 = 25.~~6669~~{{c}}

* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.~~6597~~{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}

~~Badness (Sintel): 1.11~~

~~=== 23-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17.19.23~~

~~Comma list: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384~~

~~Mapping: {{mapping| 2 3 4 6 7 8 8 9 9 | 0 4 15 -9 -2 -14 4 -12 1 }}~~

Badness (Sintel): 0.772

~~Optimal tunings:~~

* WE: ~17/12 = 600.1777{{c}}, ~66/65 = 25.6682{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.6605{{c}}

~~{{Optimal ET sequence|legend=0| 46, 94, 140h }}~~

Badness (Sintel): 1.00

== Undecental ==

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: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''

Leapday tempers out the ~~leapday comma, {{monzo| 31 -21 1 }}~~, in the ~~5-limit~~, ~~mapping 5/4 to the triple-augmented unison or equivalently the minor third~~ and ~~two dieses. In the 7-limit it can~~ be described as the {{nowrap| 29 & 46 }} temperament, ~~which tempers out the hemifamity and~~ [[~~686~~/~~675~~]] (~~senga~~), and ~~extends~~ [[~~leapfrog~~]].

Leapday tempers out [[686/675]], the senga, in addition to the aberschisma, and may be described as the {{nowrap| 29 & 46 }} temperament. It extends [[leapfrog]], such that [[7/4]] is found by 15 generators up, as a double-augmented fifth (a major sixth and a diesis). 5/4 is found by a tritone above that, as a triple-augmented unison (a minor third and two dieses). [[46edo]] itself is an excellent tuning for this.

It has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell ([[6144/6125]]) rather than the ~~hemifamity comma~~ tempered out.

Leapday is more notable in the higher limits than the lower, as it nails the 13-limit pretty well from identifying [[14/11]] by a major third and [[13/11]] by a minor third, tempering out not only [[352/351]] and [[364/363]] but [[91/90]], [[121/120]], [[169/168]] and [[196/195]]. It can be further extended to include the [[17/1|17th]] and [[23/1|23rd]] [[harmonic]]s. Adding 17 would fix the valid diamond monotone tuning to 46edo, however.

Leapday has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell comma ([[6144/6125]]) rather than the aberschisma tempered out.

[[Subgroup]]: 2.3.5.7

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Badness (Sintel): 0.910

===~~= 19-limit =~~===

=== 2.3.5.7.11.13.17.23 subgroup ===

Subgroup: 2.3.5.7.11.13.17.19

Subgroup: 2.3.5.7.11.13.17.23

Comma list: 91/90, 121/120~~, 133/132~~, 136/135, 154/153, 169/168

Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168

Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 9 | 0 1 21 15 11 8 24 -3 }}

Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}

Optimal tunings:

* WE: ~2 = ~~1201~~.~~0192~~{{c}}, ~3/2 = 704.~~7333~~{{c}}

* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.~~1680~~{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}

{{Optimal ET sequence|legend=0| 17cg, 29g, 46, ~~75dfgh, 121defgh~~ }}

Badness (Sintel): ~~1.06~~

Badness (Sintel): 0.872

~~===== 23-limit =====~~

~~Subgroup: 2.3.5.7.11.13.17.19.23~~

~~Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 161/160, 169/168~~

~~Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 9 -5 | 0 1 21 15 11 8 24 -3 6 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.9738{{c}}, ~3/2 = 704.7120{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1695{{c}}

~~{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 75dfgh, 121defgh }}~~

~~Badness (Sintel): 1.01~~

~~==== Leapling ====~~

~~Subgroup: 2.3.5.7.11.13.17.19~~

~~Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168~~

~~Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -37 | 0 1 21 15 11 8 24 26 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.4745{{c}}, ~3/2 = 704.4016{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1442{{c}}

~~{{Optimal ET sequence|legend=0| 17cgh, 29g, 46h, 75dfg }}~~

~~Badness (Sintel): 1.16~~

~~===== 23-limit =====~~

~~Subgroup: 2.3.5.7.11.13.17.19.23~~

~~Comma list: 77/76, 91/90, 115/114, 121/120, 136/135, 153/152, 161/160~~

~~Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -37 -5 | 0 1 21 15 11 8 24 26 6 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.5425{{c}}, ~3/2 = 704.4319{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1349{{c}}

~~{{Optimal ET sequence|legend=~~0~~| 17cgh, 29g, 46h, 75dfg }}~~

~~Badness (Sintel): 1~~.15

== Mystery ==

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: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''

Mystery tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 58 }} temperament. It has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step. [[145edo]] or [[232edo]] are good candidates for tunings.

Mystery tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 58 }} temperament. It has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step; its ploidacot is 29-ploid acot. [[145edo]] or [[232edo]] are good candidates for tunings.

[[Subgroup]]: 2.3.5.7

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Optimal tunings:

* WE: ~50/49 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}

* WE: ~45/44 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}

* CWE: ~50/49 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}

* CWE: ~45/44 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}

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Optimal tunings:

* WE: ~50/49 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}

* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}

* CWE: ~50/49 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}

* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}

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== Hemidromeda ==

Hemidromeda may be described as the {{nowrap| 29 & 111 }} temperament. ~~The name~~ ''hemidromeda'' comes from "hemi-" (Ancient Greek for "one half") and ''[[andromeda]]'', because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents).

Hemidromeda may be described as the {{nowrap| 29 & 111 }} temperament. Named by [[Xenllium]] in 2023, ''hemidromeda'' comes from ''hemi-'' (Ancient Greek for "one half") and ''[[andromeda]]'', because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents); the ploidacot for this temperament is alpha-dicot.

[[Subgroup]]: 2.3.5.7

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== Countriton ==

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

Countriton may be described as the {{nowrap| 51c & 53 }} temperament. It splits the [[24/1|24th harmonic]] into nine tritone generators; its ploidacot is thus delta-enneacot. Among the possible tunings are [[157edo]] and [[210edo]], as well as [[104edo]] in the 104c val.

Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].

[[Subgroup]]: 2.3.5.7

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== Artoneutral ==

Artoneutral is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11) ~~and can be described as~~ the ~~{{nowrap~~| ~~87 & 94 }} temperament~~. [[181edo]] may be recommended as a tuning.

Artoneutral can be described as the {{nowrap| 87 & 94 }} temperament. It is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11), nine of which make the [[12/1|12th harmonic]]; its ploidacot is thus beta-enneacot. [[181edo]] may be recommended as a tuning.

Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.

[[Subgroup]]: 2.3.5.7

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== Quanic ==

Quanic may be described as the {{nowrap| 94 & 111 }} temperament. It splits the perfect fifth into five generators which in the 13-limit extension may be taken as ~13/12; its ploidacot is thus pentacot. [[205edo]] may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

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Badness (Sintel): 1.05

== Jorgensen ==

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''

Jorgensen tempers out the [[linus comma]] in addition to the aberschisma, and may be described as the {{nowrap| 70 & 140 }} temperament, with a 70th-octave period. Its ploidacot is 70-ploid acot.

It is the natural 7-limit extension of the 5-limit temperament tempering out the 70-comma, named by [[Mike Battaglia]] in 2012 for historical interests<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_103982.html Yahoo! Tuning Group | ''Jorgensen Temperament'']</ref>.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 5120/5103, 578509309952/576650390625

: mapping generators: ~50421/50000, ~5

[[Optimal tuning]]s:

* [[WE]]: ~50421/50000 = 17.1387{{c}}, ~5/4 = 386.8071{{c}}

: [[error map]]: {{val| -0.288 +0.445 -0.084 +0.121 }}

* [[CWE]]: ~50421/50000 = 17.1429{{c}}, ~5/4 = 386.6593{{c}}

: error map: {{val| 0.000 +0.902 +0.346 +0.690 }}

[[Badness]] (Sintel): 5.40

== References ==

[[Category:Temperament collections]]

[[Category:~~Pages with mostly numerical content]]~~

[[Category:Aberschismic temperaments| ]]

~~[[Category:Hemifamity~~ temperaments| ]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the [[hemifamity comma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].
+This is a collection of [[rank-2 temperament|rank-2]] '''aberschismic temperaments''', which [[tempering out|temper out]] the [[aberschisma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].
 Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, undecental, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1.
@@ Line 6: / Line 6: @@
 Diaschismic is generated by the fifth with a semi-octave period. Hemififths has the fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma. Hemidromeda has the fourth sliced into two and 5/4 mapped to the hemifourth + 3d4. Rodan has the fifth sliced into three as does slendric. Alphatrimot has the twelfth sliced into three as does alphatricot. Monkey has the fifth sliced into four as does tetracot. Buzzard has the twelfth sliced into four as does vulture. Misty is generated by the fifth with a 1/3-octave period. Supers has the fifth sliced into three with a semi-octave period. Undim is generated by the fifth with a 1/4-octave period. Quinticosiennic and quintakwai have the fourth sliced into five. Amity has the eleventh sliced into five. Countercata has the twelfth sliced into six as does hanson. Warrior has the 6th harmonic sliced into seven as does sensi. Finally, alphaquarter has the fourth sliced into nine as does escapade.
-Temperaments considered below are septiquarter, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral and quanic, in the order of increasing [[TE logflat badness]]. Discussed elsewhere are:
+Temperaments discussed elsewhere are:
 * [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
 * [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
-* ''[[Kwai]]'' (+16875/16807) → [[Mirkwai clan #Kwai|Mirkwai clan]]
 * [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
 * [[Hemififths]] (+2401/2400) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
 * [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]]
 * ''[[Alphatrimot]]'' (+2430/2401) → [[Alphatricot family #Alphatrimot|Alphatricot family]]
-* ''[[Monkey]]'' (+875/864) → [[Tetracot family #Monkey|Tetracot family]]
-* [[Buzzard]] (+1728/1715) → [[Vulture family #Buzzard|Vulture family]]
 * [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
-* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]
+* [[Monkey]] (+875/864) → [[Tetracot family #Monkey|Tetracot family]]
+* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]]
 * ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]
 * ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
@@ Line 23: / Line 21: @@
 * [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity family]]
 * ''[[Countercata]]'' (+15625/15552) → [[Kleismic family #Countercata|Kleismic family]]
+* ''[[Abergravity]]'' (+177147/175000) → [[Gravity family #Abergravity|Gravity family]]
+* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]
 * ''[[Warrior]]'' (+78732/78125) → [[Sensipent family #Warrior|Sensipent family]]
 * ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]]
+Considered below are septiquarter, kwai, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].
 == Septiquarter ==
+Septiquarter tempers out [[420175/419904]] and may be described as the {{nowrap| 94 & 99 }} temperament. Its [[ploidacot]] is epsilon-heptacot. [[99edo]] makes for an excellent tuning, and [[292edo]] an even better one. [[94edo]] and [[104edo]] in the 104c val are also among the possibilities.
 [[Subgroup]]: 2.3.5.7
@@ Line 73: / Line 77: @@
 Badness (Sintel): 1.44
+== Kwai ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''
+Named by [[Gene Ward Smith]] in 2004 for its "bridgeability"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10766.html Yahoo! Tuning Group | ''Kwai'']</ref>, kwai is generated by a [[3/2|perfect fifth]], and can be described as {{nowrap| 41 & 70 }}.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 5120/5103, 16875/16807
+{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
+: mapping generators: ~2, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7337{{c}}, ~3/2 = 702.4600{{c}}
+: [[error map]]: {{val| -0.266 +0.239 -0.607 +1.055 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6085{{c}}
+: error map: {{val| 0.000 +0.653 -0.234 +1.603 }}
+{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}
+[[Badness]] (Sintel): 1.38
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 540/539, 1375/1372, 5120/5103
+Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}
+Optimal tunings:
+* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}
+{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}
+Badness (Sintel): 0.867
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 352/351, 540/539, 729/728, 1375/1372
+Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}
+Optimal tunings:
+* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}
+{{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}
+Badness (Sintel): 1.01
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088
+Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}
+Optimal tunings:
+* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}
+{{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}
+Badness (Sintel): 1.12
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845
+Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}
+Optimal tunings:
+* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}
+{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}
+Badness (Sintel): 1.03
+==== Hemikwai ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 540/539, 676/675, 1375/1372, 5120/5103
+Mapping: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
+: mapping generators: ~2, ~26/15
+Optimal tunings:
+* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}
+{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
+Badness (Sintel): 1.82
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103
+Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}
+Optimal tunings:
+* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}
+{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
+Badness (Sintel): 1.31
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444
+Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}
+Optimal tunings:
+* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}
+{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}
+Badness (Sintel): 1.16
 == Ketchup ==
+Ketchup may be described as the {{nowrap| 46 & 94 }} temperament. It has a semi-octave period and a generator for a syntonic~septimal comma, four of which plus a period gives the perfect fifth; its [[ploidacot]] is diploid gamma-tetracot. [[140edo]] is an obvious tuning for this temperament.
 [[Subgroup]]: 2.3.5.7
@@ Line 125: / Line 259: @@
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 289/288, 325/324, 352/351, 385/384, 561/560
+Comma list: 289/288, 325/324, 352/351, 385/384, 442/441
 Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}
@@ Line 137: / Line 271: @@
 Badness (Sintel): 0.845
-=== 19-limit ===
+=== 2.3.5.7.11.13.17.23 subgroup ===
-Subgroup: 2.3.5.7.11.13.17.19
+Subgroup: 2.3.5.7.11.13.17.23
-Comma list: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560
+Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390
-Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 -12 }}
+Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}
 Optimal tunings:
-* WE: ~17/12 = 600.1639{{c}}, ~66/65 = 25.6669{{c}}
+* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
-* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.6597{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}
-{{Optimal ET sequence|legend=0| 46, 94, 140h }}
+{{Optimal ET sequence|legend=0| 46, 94, 140 }}
-Badness (Sintel): 1.11
-=== 23-limit ===
-Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384
-Mapping: {{mapping| 2 3 4 6 7 8 8 9 9 | 0 4 15 -9 -2 -14 4 -12 1 }}
+Badness (Sintel): 0.772
-Optimal tunings:
-* WE: ~17/12 = 600.1777{{c}}, ~66/65 = 25.6682{{c}}
-* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.6605{{c}}
-{{Optimal ET sequence|legend=0| 46, 94, 140h }}
-Badness (Sintel): 1.00
 == Undecental ==
@@ Line 191: / Line 310: @@
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''
-Leapday tempers out the leapday comma, {{monzo| 31 -21 1 }}, in the 5-limit, mapping 5/4 to the triple-augmented unison or equivalently the minor third and two dieses. In the 7-limit it can be described as the {{nowrap| 29 & 46 }} temperament, which tempers out the hemifamity and [[686/675]] (senga), and extends [[leapfrog]].
+Leapday tempers out [[686/675]], the senga, in addition to the aberschisma, and may be described as the {{nowrap| 29 & 46 }} temperament. It extends [[leapfrog]], such that [[7/4]] is found by 15 generators up, as a double-augmented fifth (a major sixth and a diesis). 5/4 is found by a tritone above that, as a triple-augmented unison (a minor third and two dieses). [[46edo]] itself is an excellent tuning for this.
-It has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell ([[6144/6125]]) rather than the hemifamity comma tempered out.
+Leapday is more notable in the higher limits than the lower, as it nails the 13-limit pretty well from identifying [[14/11]] by a major third and [[13/11]] by a minor third, tempering out not only [[352/351]] and [[364/363]] but [[91/90]], [[121/120]], [[169/168]] and [[196/195]]. It can be further extended to include the [[17/1|17th]] and [[23/1|23rd]] [[harmonic]]s. Adding 17 would fix the valid diamond monotone tuning to 46edo, however.
+Leapday has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell comma ([[6144/6125]]) rather than the aberschisma tempered out.
 [[Subgroup]]: 2.3.5.7
@@ Line 257: / Line 378: @@
 Badness (Sintel): 0.910
-==== 19-limit ====
+=== 2.3.5.7.11.13.17.23 subgroup ===
-Subgroup: 2.3.5.7.11.13.17.19
+Subgroup: 2.3.5.7.11.13.17.23
-Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168
+Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 9 | 0 1 21 15 11 8 24 -3 }}
+Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}
 Optimal tunings:
-* WE: ~2 = 1201.0192{{c}}, ~3/2 = 704.7333{{c}}
+* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1680{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}
-{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 75dfgh, 121defgh }}
+{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
-Badness (Sintel): 1.06
+Badness (Sintel): 0.872
-===== 23-limit =====
-Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 161/160, 169/168
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 9 -5 | 0 1 21 15 11 8 24 -3 6 }}
-Optimal tunings:
-* WE: ~2 = 1200.9738{{c}}, ~3/2 = 704.7120{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1695{{c}}
-{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 75dfgh, 121defgh }}
-Badness (Sintel): 1.01
-==== Leapling ====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -37 | 0 1 21 15 11 8 24 26 }}
-Optimal tunings:
-* WE: ~2 = 1200.4745{{c}}, ~3/2 = 704.4016{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1442{{c}}
-{{Optimal ET sequence|legend=0| 17cgh, 29g, 46h, 75dfg }}
-Badness (Sintel): 1.16
-===== 23-limit =====
-Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 77/76, 91/90, 115/114, 121/120, 136/135, 153/152, 161/160
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -37 -5 | 0 1 21 15 11 8 24 26 6 }}
-Optimal tunings:
-* WE: ~2 = 1200.5425{{c}}, ~3/2 = 704.4319{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1349{{c}}
-{{Optimal ET sequence|legend=0| 17cgh, 29g, 46h, 75dfg }}
-Badness (Sintel): 1.15
 == Mystery ==
@@ Line 321: / Line 397: @@
 : ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''
-Mystery tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 58 }} temperament. It has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step. [[145edo]] or [[232edo]] are good candidates for tunings.
+Mystery tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 58 }} temperament. It has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step; its ploidacot is 29-ploid acot. [[145edo]] or [[232edo]] are good candidates for tunings.
 [[Subgroup]]: 2.3.5.7
@@ Line 348: / Line 424: @@
 Optimal tunings:
-* WE: ~50/49 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
+* WE: ~45/44 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
-* CWE: ~50/49 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}
+* CWE: ~45/44 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}
 {{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}
@@ Line 363: / Line 439: @@
 Optimal tunings:
-* WE: ~50/49 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
+* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
-* CWE: ~50/49 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}
+* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}
 {{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}
@@ Line 371: / Line 447: @@
 == Hemidromeda ==
-Hemidromeda may be described as the {{nowrap| 29 & 111 }} temperament. The name ''hemidromeda'' comes from "hemi-" (Ancient Greek for "one half") and ''[[andromeda]]'', because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents).
+Hemidromeda may be described as the {{nowrap| 29 & 111 }} temperament. Named by [[Xenllium]] in 2023, ''hemidromeda'' comes from ''hemi-'' (Ancient Greek for "one half") and ''[[andromeda]]'', because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents); the ploidacot for this temperament is alpha-dicot.
 [[Subgroup]]: 2.3.5.7
@@ Line 467: / Line 543: @@
 == Countriton ==
 : ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
+Countriton may be described as the {{nowrap| 51c & 53 }} temperament. It splits the [[24/1|24th harmonic]] into nine tritone generators; its ploidacot is thus delta-enneacot. Among the possible tunings are [[157edo]] and [[210edo]], as well as [[104edo]] in the 104c val.
+Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].
 [[Subgroup]]: 2.3.5.7
@@ Line 516: / Line 596: @@
 == Artoneutral ==
-Artoneutral is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11) and can be described as the {{nowrap| 87 & 94 }} temperament. [[181edo]] may be recommended as a tuning.
+Artoneutral can be described as the {{nowrap| 87 & 94 }} temperament. It is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11), nine of which make the [[12/1|12th harmonic]]; its ploidacot is thus beta-enneacot. [[181edo]] may be recommended as a tuning.
+Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.
 [[Subgroup]]: 2.3.5.7
@@ Line 611: / Line 693: @@
 == Quanic ==
+Quanic may be described as the {{nowrap| 94 & 111 }} temperament. It splits the perfect fifth into five generators which in the 13-limit extension may be taken as ~13/12; its ploidacot is thus pentacot. [[205edo]] may be recommended as a tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 687: / Line 771: @@
 Badness (Sintel): 1.05
+== Jorgensen ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''
+Jorgensen tempers out the [[linus comma]] in addition to the aberschisma, and may be described as the {{nowrap| 70 & 140 }} temperament, with a 70th-octave period. Its ploidacot is 70-ploid acot.
+It is the natural 7-limit extension of the 5-limit temperament tempering out the 70-comma, named by [[Mike Battaglia]] in 2012 for historical interests<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_103982.html Yahoo! Tuning Group | ''Jorgensen Temperament'']</ref>.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 5120/5103, 578509309952/576650390625
+{{Mapping|legend=1| 70 111 0 34 | 0 0 1 1 }}
+: mapping generators: ~50421/50000, ~5
+[[Optimal tuning]]s:
+* [[WE]]: ~50421/50000 = 17.1387{{c}}, ~5/4 = 386.8071{{c}}
+: [[error map]]: {{val| -0.288 +0.445 -0.084 +0.121 }}
+* [[CWE]]: ~50421/50000 = 17.1429{{c}}, ~5/4 = 386.6593{{c}}
+: error map: {{val| 0.000 +0.902 +0.346 +0.690 }}
+{{Optimal ET sequence|legend=1| 70, 140, 350, 490 }}
+[[Badness]] (Sintel): 5.40
+== References ==
 [[Category:Temperament collections]]
-[[Category:Pages with mostly numerical content]]
+[[Category:Aberschismic temperaments| ]] <!-- main article -->
-[[Category:Hemifamity temperaments| ]] <!-- main article -->
 [[Category:Rank 2]]