Aberschismic temperaments: Difference between revisions

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.

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 discussed elsewhere are:

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

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

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

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

[[Comma list]]: 5120/5103, 420175/419904

{{Mapping|legend=1| 1 -4 -28 6 | 0 7 38 -4 }}

: mapping generators: ~2, ~243/140

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.7212{{c}}, ~243/140 = 957.3250{{c}}

: [[error map]]: {{val| -0.279 +0.435 -0.158 +0.201 }}

: error map: {{val| 0.000 +0.842 +0.298 +1.004 }}

{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}

[[Badness]] (Sintel): 1.36

=== Semiseptiquarter ===

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:  

* WE: ~99/70 = 599.8953{{c}}, ~210/121 = 957.3819{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5449{{c}}

{{Optimal ET sequence|legend=0| 94, 198, 292, 490 }}

Badness (Sintel): 2.12

Subgroup: 2.3.5.7.11.13

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:  

* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5508{{c}}

{{Optimal ET sequence|legend=0| 94, 198, 490f }}

Badness (Sintel): 1.44

: ''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

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

: 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:  

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

{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}

Badness (Sintel): 0.867

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

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 }}

* 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

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 }}

* 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

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

* 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

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 }}

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

[[Comma list]]: 5120/5103, 1071875/1062882

: mapping generators: ~1225/864, ~64/63

* [[WE]]: ~1225/864 = 599.9685{{c}}, ~64/63 = 25.7181{{c}}

* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~64/63 = 25.7181{{c}}

{{Optimal ET sequence|legend=1| 46, 94, 140 }}

[[Badness]] (Sintel): 2.14

Subgroup: 2.3.5.7.11

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

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

* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}

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

Badness (Sintel): 1.31

Subgroup: 2.3.5.7.11.13

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:  

* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}

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

Badness (Sintel): 1.03

Subgroup: 2.3.5.7.11.13.17

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:  

* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}

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

Badness (Sintel): 0.845

=== 2.3.5.7.11.13.17.23 subgroup ===

Subgroup: 2.3.5.7.11.13.17.23

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 1 }}

Optimal tunings:  

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

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

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

Badness (Sintel): 0.772

Undecental adds the triwellisma to the comma list and may be described as the {{nowrap| 29 & 70 }} temperament. 5/4 is mapped to the quintuple-diminished seventh or equivalently the perfect fourth minus three [[diesis (scale theory)|dieses]]. [[99edo|58\99]] is an almost perfect generator, just as the name suggests. Another interesting tuning choice is the argent fifth, {{nowrap| 2^{(2 - sqrt (2))} }}.  

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 5120/5103, 235298/234375

: mapping generators: ~2, ~3

* [[WE]]: ~2 = 1199.6543{{c}}, ~3/2 = 702.8370{{c}}

: [[error map]]: {{val| -0.346 +0.536 +0.423 -0.494 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.0465{{c}}

: error map: {{val| 0.000 +1.092 +0.966 +0.175 }}

{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}

[[Badness]] (Sintel): 2.39

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

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.  

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

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

: mapping generators: ~2, ~3

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

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

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

[[Badness]] (Sintel): 2.43

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 }}

* 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

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:  

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

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

Badness (Sintel): 1.02

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

Subgroup: 2.3.5.7.11.13.17.23

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

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

Optimal tunings:  

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

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

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

Badness (Sintel): 0.872

: ''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; its ploidacot is 29-ploid acot. [[145edo]] or [[232edo]] are good candidates for tunings.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 5120/5103, 50421/50000

: mapping generators: ~50/49, ~5

* [[WE]]: ~50/49 = 41.3652{{c}}, ~5/4 = 388.5128{{c}}

: [[error map]]: {{val| -0.410 +0.842 +1.378 -2.022 }}

* [[CWE]]: ~50/49 = 41.3793{{c}}, ~5/4 = 388.3030{{c}}

: error map: {{val| 0.000 +1.493 +1.989 -1.213 }}

{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}

[[Badness]] (Sintel): 2.63

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}

Badness (Sintel): 1.13

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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:  

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

{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}

Badness (Sintel): 0.768

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

[[Comma list]]: 5120/5103, 52734375/52706752

{{Mapping|legend=1| 1 0 38 48 | 0 2 -45 -57 }}

: mapping generator: ~2, ~12500/7203

[[Optimal tuning]]s:  

* [[CWE]]: ~2 = 1200.0000{{c}}, ~12500/7203 = 951.4098{{c}}

: error map: {{val| 0.000 +0.865 +0.243 +0.813 }}

{{Optimal ET sequence|legend=1| 29, 82cd, 111, 140, 251, 391, 1424bbcdd }}

[[Badness]] (Sintel): 2.93

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.4063{{c}}

{{Optimal ET sequence|legend=0| 29, 82cd, 111, 140, 251, 391e }}

Badness (Sintel): 2.01

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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:  

* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}

{{Optimal ET sequence|legend=0| 29, 82cdf, 111, 140, 251, 391e }}

Badness (Sintel): 1.18

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

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 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}

{{Optimal ET sequence|legend=0| 29g, 82cdfg, 111, 140, 251, 391e }}

Badness (Sintel): 0.971

Subgroup: 2.3.5.7.11.13.17.19

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 }}

* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}

{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}

Badness (Sintel): 1.01

Subgroup: 2.3.5.7.11.13.17.19.23

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 }}

Optimal tunings:  

* WE: ~2 = 1199.9128{{c}}, ~26/15 = 951.3371{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4076{{c}}

{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}

Badness (Sintel): 1.10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Badness (Sintel): 1.94

 

 

 

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

 

* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}