Aberschismic temperaments: Difference between revisions

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]].  

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

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

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

== 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.  

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

{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}

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

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

: [[error map]]: {{val| -0.063 +0.823 -0.668 -0.478 }}

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

: error map: {{val| 0.000 +0.917 -0.543 -0.288 }}

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

[[Badness]] (Sintel): 2.14

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

=== 13-limit ===

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

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

* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}

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

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

Badness (Sintel): 1.03

=== 17-limit ===

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

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

* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}

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

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

Badness (Sintel): 0.845

=== 19-limit ===

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

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

* WE: ~17/12 = 600.1639{{c}}, ~66/65 = 25.6669{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.6597{{c}}

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

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 }}

* 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 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|legend=1| 1 0 61 71 | 0 1 -37 -43 }}

* [[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

== 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.

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

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

: mapping generators: ~2, ~3

* [[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

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

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

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

* 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

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 }}

* 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 }}

* 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 }}

* 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

 

 

 

 

 

Badness (Sintel): 1.16

 

 

 

 

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

 

 

Badness (Sintel): 1.15

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); the ploidacot for this temperament is alpha-dicot.  

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.  

[[Category:Pages with mostly numerical content]]

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

[[Category:Rank 2]]