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]].  

* ''[[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 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: 289/288, 325/324, 352/351, 385/384, 561/560

=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

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): 1.00

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 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.

==== 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 }}

 

 

 

 

 

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

 

Badness (Sintel): 1.01

 

 

 

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

 

 

 

 

 

 

 

 

 

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