Amity family: Difference between revisions

The second comma to extend the 5-limit amity include [[4375/4374]] for septimal amity, [[225/224]] for houborizic, 65625/65536 for paramity, [[126/125]] for accord, [[245/243]] for bamity, [[2430/2401]] for hamity, [[1029/1024]] for gamity, [[10976/10935]] for chromat, [[703125/702464]] for trinity, [[2401/2400]] for amicable, [[2100875/2097152]] for calamity, [[420175/419904]] for witcher, and [[16875/16807]] for familia.  

* ''[[Chromat]]'' → [[Hemimage temperaments #Chromat]] (+10976/10935)

* ''[[Witcher]]'' → [[Wizmic microtemperaments #Witcher]] (+420175/419904)

 

The rest are considered below.  

Houborizic tempers out 225/224, the [[marvel comma]], and may be described as the {{nowrap| 53 & 60 }} temperament. It is so named because it is closely related to the ''houboriz tuning'' (generator: 339.774971 cents).

Paramity tempers out the [[horwell comma]] (65625/65536) and [[garischisma]] (33554432/33480783), and may be described as the {{nowrap| 53 & 311 }} temperament.  

Gamity tempers out 1029/1024, the [[gamelisma]], and may be described as the {{nowrap| 46 & 113 }} temperament. It splits the interval of the grave major sixth in three.

Trinity tempers out 703125/702464, the [[meter]], and may be described as the {{nowrap| 152 & 159 }} temperament. It splits the grave major sixth plus an octave in three. It was so named for the following reason – 133\311 (133 steps of 311edo) is a possible generator, which is placed around 3\7 (1.1{{c}} flat), three of which makes acute minor third of ~243/200 with octave reduction.

== Amicable ==

Amicable tempers out the [[breedsma]] as well as the [[canousma]], and may be described as the {{nowrap| 99 & 311 }} temperament.

 

While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.  

[[Comma list]]: 2401/2400, 1600000/1594323

{{Mapping|legend=1| 1 -17 -46 -26 | 0 20 52 31 }}

: mapping generators: ~2, ~40/21

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9463{{c}}, ~40/21 = 1115.0701{{c}}

: [[error map]]: {{val| -0.054 +0.361 -0.195 -0.254 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1187{{c}}

: error map: {{val| 0.000 +0.419 -0.142 -0.146 }}

{{Optimal ET sequence|legend=1| 14c, …, 85c, 99, 212, 311, 410, 1131, 1541b }}

[[Badness]] (Sintel): 1.15

=== Amical ===

Comma list: 2401/2400, 131072/130977, 1600000/1594323

Mapping: {{mapping| 1 -17 -46 -26 154 | 0 20 52 31 -162 }}

Optimal tunings:

* WE: ~2 = 1200.0008{{c}}, ~40/21 = 1115.1165{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1157{{c}}

{{Optimal ET sequence|legend=0| 99, 212e, 311, 721, 1032, 1343 }}

Badness (Sintel): 3.33

Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206

Mapping: {{mapping| 1 -17 -46 -26 154 118 | 0 20 52 31 -162 -123 }}

Optimal tunings:

* WE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1094{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1162{{c}}

{{Optimal ET sequence|legend=0| 99, 212ef, 311, 721, 1032 }}

=== Pseudoamical ===

Comma list: 385/384, 1375/1372, 1600000/1594323

Mapping: {{mapping| 1 -17 -46 -26 62 | 0 20 52 31 -63 }}

* WE: ~2 = 1200.3436{{c}}, ~40/21 = 1115.4102{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.0925{{c}}

{{Optimal ET sequence|legend=0| 99, 113, 212, 961ccdeee }}

Badness (Sintel): 2.84

Comma list: 325/324, 385/384, 1375/1372, 19773/19712

Mapping: {{mapping| 1 -17 -46 -26 62 26 | 0 20 52 31 -63 -24 }}

* WE: ~2 = 1200.4212{{c}}, ~40/21 = 1115.4787{{c}}

* CWE: ~2 = 1200.0000{{c}}, 40/21 = 1115.0885{{c}}

{{Optimal ET sequence|legend=0| 99, 113, 212, 537cdeff, 749ccdeefff }}

Badness (Sintel): 1.94

* optimal patent val: [[113edo|113]]

* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9153{{c}}

* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9164{{c}}

* WE: ~99/70 = 600.0000{{c}}, ~66/49 = 515.1257{{c}}

Calamity tempers out 2100875/2097152, the [[rainy comma]], and may be described as the {{nowrap| 46 & 311 }} temperament. It splits the interval of grave major sixth plus an octave into five.  

Familia tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 113 & 152 }} temperament. It is generated by a major sixth, five of which octave reduced make the original generator of amity.