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

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

* [[Leapday]] (+686/675) → [[Sengic temperaments #Leapday|Sengic temperaments]]

* [[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]]

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

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

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

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

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

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

== Septiquarter ==

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Comma list: 5120/5103, 9801/9800, 14641/14580

~~Mapping:~~ {{~~mapping~~| 2 -8 -56 12 -25 | 0 7 38 -4 20 }}

Optimal tunings:

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Comma list: 352/351, 847/845, 1716/1715, 14641/14580

~~Mapping:~~ {{~~mapping~~| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}

Optimal tunings:

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

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

Optimal tunings:

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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:

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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 }}

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

Optimal tunings:

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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 }}

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

Optimal tunings:

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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

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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 }}

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

Optimal tunings:

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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 }}

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

Optimal tunings:

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== 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.

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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Comma list: 385/384, 1331/1323, 2200/2187

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

Optimal tunings:

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Comma list: 325/324, 352/351, 385/384, 1331/1323

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

Optimal tunings:

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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 }}

Optimal tunings:

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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~~ }}

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~~

Badness (Sintel): 0.772

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

~~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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[[Badness]] (Sintel): 2.39

~~== Leapday ==~~

~~{{Main| Leapday }}~~

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

Leapday tempers out [[686/675]], the senga, in addition to the hemifamity comma, 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.

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 ([[6144/6125]]) rather than the hemifamity comma tempered out.

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 686/675, 5120/5103~~

~~{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}~~

~~: mapping generators: ~2, ~3~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~2 = 1199.7167{{c}}, ~3/2 = 704.0971{{c}}

~~: [[error map]]: {{val| -0.283 +1.859 +2.559 -5.669 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2504{{c}}

~~: error map: {{val| 0.000 +2.295 +2.945 -5.070 }}~~

~~{{Optimal ET sequence|legend=1| 17c, 29, 46 }}~~

~~[[Badness]] (Sintel): 2.43~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 121/120, 441/440, 686/675~~

~~Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}

~~{{Optimal ET sequence|legend=0| 17c, 29, 46 }}~~

~~Badness (Sintel): 1.28~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 91/90, 121/120, 169/168, 352/351~~

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

~~Optimal tunings:~~

* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}

~~{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}~~

~~Badness (Sintel): 1.02~~

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

~~Subgroup: 2.3.5.7.11.13.17~~

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

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

~~Optimal tunings:~~

* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}

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

~~Badness (Sintel): 0.910~~

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

~~Subgroup: 2.3.5.7.11.13.17.19~~

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

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

~~Optimal tunings:~~

* WE: ~2 = 1201.0192{{c}}, ~3/2 = 704.7333{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1680{{c}}

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

~~Badness (Sintel): 1.06~~

~~===== 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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Comma list: 441/440, 896/891, 3388/3375

~~Mapping:~~ {{~~mapping~~| 29 46 0 14 33 | 0 0 1 1 1 }}

Optimal tunings:

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Comma list: 196/195, 352/351, 364/363, 676/675

~~Mapping:~~ {{~~mapping~~| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}

Optimal tunings:

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Comma list: 1331/1323, 1375/1372, 5120/5103

~~Mapping:~~ {{~~mapping~~| 1 0 38 48 32 | 0 2 -45 -57 -36 }}

Optimal tunings:

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Comma list: 352/351, 676/675, 847/845, 1331/1323

~~Mapping:~~ {{~~mapping~~| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}

Optimal tunings:

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Comma list: 352/351, 442/441, 561/560, 676/675, 715/714

~~Mapping:~~ {{~~mapping~~| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}

{{Mapping|legend=0| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}

Optimal tunings:

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Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560

~~Mapping:~~ {{~~mapping~~| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}

{{Mapping|legend=0| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}

Optimal tunings:

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Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459

~~Mapping:~~ {{~~mapping~~| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}

{{Mapping|legend=0| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}

Optimal tunings:

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Comma list: 176/175, 5120/5103, 41503/41472

~~Mapping:~~ {{~~mapping~~| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}

Optimal tunings:

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Comma list: 176/175, 351/350, 847/845, 2197/2187

~~Mapping:~~ {{~~mapping~~| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}

Optimal tunings:

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Comma list: 385/384, 2200/2187, 4000/3993

~~Mapping:~~ {{~~mapping~~| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}

Optimal tunings:

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Comma list: 325/324, 352/351, 385/384, 1575/1573

~~Mapping:~~ {{~~mapping~~| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}

Optimal tunings:

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Comma list: 325/324, 352/351, 375/374, 385/384, 595/594

~~Mapping:~~ {{~~mapping~~| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}

Optimal tunings:

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Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594

~~Mapping:~~ {{~~mapping~~| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}

{{Mapping|legend=0| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}

Optimal tunings:

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Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483

~~Mapping:~~ {{~~mapping~~| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}

{{Mapping|legend=0| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}

Optimal tunings:

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Comma list: 540/539, 1331/1323, 5120/5103

~~Mapping:~~ {{~~mapping~~| 1 1 -4 0 1 | 0 5 54 24 21 }}

Optimal tunings:

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Comma list: 352/351, 540/539, 729/728, 1331/1323

~~Mapping:~~ {{~~mapping~~| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}

Optimal tunings:

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Comma list: 352/351, 442/441, 540/539, 715/714, 847/845

~~Mapping:~~ {{~~mapping~~| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}

Optimal tunings:

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Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714

~~Mapping:~~ {{~~mapping~~| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}

Optimal tunings:

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

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

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>.

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== References ==

[[Category:Aberschismic temperaments| ]]

[[Category:Aberschismic| ]]

[[Category:Temperament collections]]

[[Category:~~Hemifamity temperaments| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 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 9: / Line 9: @@
 * [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
 * [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
+* [[Leapday]] (+686/675) → [[Sengic temperaments #Leapday|Sengic temperaments]]
 * [[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]]
+* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
 * [[Monkey]] (+875/864) → [[Tetracot family #Monkey|Tetracot family]]
 * [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]]
-* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
-* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]
 * ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]
 * ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
@@ Line 22: / Line 22: @@
 * [[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]].
+Considered below are septiquarter, kwai, ketchup, undecental, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].
 == Septiquarter ==
@@ Line 52: / Line 54: @@
 Comma list: 5120/5103, 9801/9800, 14641/14580
-Mapping: {{mapping| 2 -8 -56 12 -25 | 0 7 38 -4 20 }}
+{{Mapping|legend=0| 2 -8 -56 12 -25 | 0 7 38 -4 20 }}
 Optimal tunings:
@@ Line 67: / Line 69: @@
 Comma list: 352/351, 847/845, 1716/1715, 14641/14580
-Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}
+{{Mapping|legend=0| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}
 Optimal tunings:
@@ Line 104: / Line 106: @@
 Comma list: 540/539, 1375/1372, 5120/5103
-Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}
+{{Mapping|legend=0| 1 0 -50 -40 32 | 0 1 33 27 -18 }}
 Optimal tunings:
@@ Line 119: / Line 121: @@
 Comma list: 352/351, 540/539, 729/728, 1375/1372
-Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}
+{{Mapping|legend=0| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}
 Optimal tunings:
@@ Line 134: / Line 136: @@
 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 }}
+{{Mapping|legend=0| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}
 Optimal tunings:
@@ Line 149: / Line 151: @@
 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 }}
+{{Mapping|legend=0| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}
 Optimal tunings:
@@ Line 164: / Line 166: @@
 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|legend=0| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
 : mapping generators: ~2, ~26/15
@@ Line 180: / Line 182: @@
 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 }}
+{{Mapping|legend=0| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}
 Optimal tunings:
@@ Line 195: / Line 197: @@
 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 }}
+{{Mapping|legend=0| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}
 Optimal tunings:
@@ Line 206: / Line 208: @@
 == 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.
+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 230: / Line 232: @@
 Comma list: 385/384, 1331/1323, 2200/2187
-Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}
+{{Mapping|legend=0| 2 3 4 6 7 | 0 4 15 -9 -2 }}
 Optimal tunings:
@@ Line 245: / Line 247: @@
 Comma list: 325/324, 352/351, 385/384, 1331/1323
-Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}
+{{Mapping|legend=0| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}
 Optimal tunings:
@@ Line 258: / Line 260: @@
 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 }}
+{{Mapping|legend=0| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}
 Optimal tunings:
@@ Line 270: / Line 272: @@
 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|legend=0| 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
+Badness (Sintel): 0.772
-Mapping: {{mapping| 2 3 4 6 7 8 8 9 9 | 0 4 15 -9 -2 -14 4 -12 1 }}
-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 319: / Line 306: @@
 [[Badness]] (Sintel): 2.39
-== Leapday ==
-{{Main| Leapday }}
-: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''
-Leapday tempers out [[686/675]], the senga, in addition to the hemifamity comma, 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.
-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 ([[6144/6125]]) rather than the hemifamity comma tempered out.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 686/675, 5120/5103
-{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
-: mapping generators: ~2, ~3
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.7167{{c}}, ~3/2 = 704.0971{{c}}
-: [[error map]]: {{val| -0.283 +1.859 +2.559 -5.669 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2504{{c}}
-: error map: {{val| 0.000 +2.295 +2.945 -5.070 }}
-{{Optimal ET sequence|legend=1| 17c, 29, 46 }}
-[[Badness]] (Sintel): 2.43
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 121/120, 441/440, 686/675
-Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}
-Optimal tunings:
-* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}
-{{Optimal ET sequence|legend=0| 17c, 29, 46 }}
-Badness (Sintel): 1.28
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 121/120, 169/168, 352/351
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}
-Optimal tunings:
-* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}
-{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}
-Badness (Sintel): 1.02
-=== 17-limit ===
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}
-Optimal tunings:
-* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}
-{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
-Badness (Sintel): 0.910
-==== 19-limit ====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 9 | 0 1 21 15 11 8 24 -3 }}
-Optimal tunings:
-* WE: ~2 = 1201.0192{{c}}, ~3/2 = 704.7333{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1680{{c}}
-{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 75dfgh, 121defgh }}
-Badness (Sintel): 1.06
-===== 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 480: / Line 335: @@
 Comma list: 441/440, 896/891, 3388/3375
-Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}
+{{Mapping|legend=0| 29 46 0 14 33 | 0 0 1 1 1 }}
 Optimal tunings:
@@ Line 495: / Line 350: @@
 Comma list: 196/195, 352/351, 364/363, 676/675
-Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}
+{{Mapping|legend=0| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}
 Optimal tunings:
@@ Line 530: / Line 385: @@
 Comma list: 1331/1323, 1375/1372, 5120/5103
-Mapping: {{mapping| 1 0 38 48 32 | 0 2 -45 -57 -36 }}
+{{Mapping|legend=0| 1 0 38 48 32 | 0 2 -45 -57 -36 }}
 Optimal tunings:
@@ Line 545: / Line 400: @@
 Comma list: 352/351, 676/675, 847/845, 1331/1323
-Mapping: {{mapping| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}
+{{Mapping|legend=0| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}
 Optimal tunings:
@@ Line 560: / Line 415: @@
 Comma list: 352/351, 442/441, 561/560, 676/675, 715/714
-Mapping: {{mapping| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}
+{{Mapping|legend=0| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}
 Optimal tunings:
@@ Line 575: / Line 430: @@
 Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560
-Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}
+{{Mapping|legend=0| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}
 Optimal tunings:
@@ Line 590: / Line 445: @@
 Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459
-Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}
+{{Mapping|legend=0| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}
 Optimal tunings:
@@ Line 629: / Line 484: @@
 Comma list: 176/175, 5120/5103, 41503/41472
-Mapping: {{mapping| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}
+{{Mapping|legend=0| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}
 Optimal tunings:
@@ Line 644: / Line 499: @@
 Comma list: 176/175, 351/350, 847/845, 2197/2187
-Mapping: {{mapping| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}
+{{Mapping|legend=0| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}
 Optimal tunings:
@@ Line 681: / Line 536: @@
 Comma list: 385/384, 2200/2187, 4000/3993
-Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}
+{{Mapping|legend=0| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}
 Optimal tunings:
@@ Line 696: / Line 551: @@
 Comma list: 325/324, 352/351, 385/384, 1575/1573
-Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}
+{{Mapping|legend=0| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}
 Optimal tunings:
@@ Line 711: / Line 566: @@
 Comma list: 325/324, 352/351, 375/374, 385/384, 595/594
-Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}
+{{Mapping|legend=0| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}
 Optimal tunings:
@@ Line 726: / Line 581: @@
 Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594
-Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}
+{{Mapping|legend=0| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}
 Optimal tunings:
@@ Line 741: / Line 596: @@
 Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483
-Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}
+{{Mapping|legend=0| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}
 Optimal tunings:
@@ Line 776: / Line 631: @@
 Comma list: 540/539, 1331/1323, 5120/5103
-Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}
+{{Mapping|legend=0| 1 1 -4 0 1 | 0 5 54 24 21 }}
 Optimal tunings:
@@ Line 791: / Line 646: @@
 Comma list: 352/351, 540/539, 729/728, 1331/1323
-Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}
+{{Mapping|legend=0| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}
 Optimal tunings:
@@ Line 806: / Line 661: @@
 Comma list: 352/351, 442/441, 540/539, 715/714, 847/845
-Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}
+{{Mapping|legend=0| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}
 Optimal tunings:
@@ Line 821: / Line 676: @@
 Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
-Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}
+{{Mapping|legend=0| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}
 Optimal tunings:
@@ Line 834: / Line 689: @@
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''
-Jorgensen tempers out the [[linus comma]] in addition to the hemifamity comma, and may be described as the {{nowrap| 70 & 140 }} temperament, with a 70th-octave period. Its ploidacot is 70-ploid acot.
+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>.
@@ Line 857: / Line 712: @@
 == References ==
+[[Category:Aberschismic temperaments| ]] <!-- main article -->
+[[Category:Aberschismic| ]] <!-- key article -->
 [[Category:Temperament collections]]
-[[Category:Hemifamity temperaments| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]

Aberschismic temperaments: Difference between revisions

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