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.

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.  

{{Mapping|legend=1| 1 3 6 5 | 0 -20 -52 -31 }}

* [[CTE]]: ~2 = 1200.0000{{c}}, ~21/20 = 84.8831{{c}}

* [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 84.880{{c}}

{{Optimal ET sequence|legend=1| 14c, …, 85c, 99, 212, 311, 721, 1032, 1753b }}

[[Badness]] (Smith): 0.045473

=== Amical ===

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

Mapping: {{mapping| 1 3 6 5 -8 | 0 -20 -52 -31 162 }}

* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8843{{c}}

* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8843{{c}}

{{Optimal ET sequence|legend=0| 99, 212e, 311, 721, 1032, 1343, 2375bc }}

Badness (Smith): 0.100668

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

Mapping: {{mapping| 1 3 6 5 -8 -5 | 0 -20 -52 -31 162 123 }}

* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8837{{c}}

* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8838{{c}}

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

Badness (Smith): 0.049893

=== Amorous ===

Comma list: 2401/2400, 6250/6237, 19712/19683

Mapping: {{mapping| 1 3 6 5 14 | 0 -20 -52 -31 -149 }}

* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8883{{c}}

* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8896{{c}}

{{Optimal ET sequence|legend=0| 99e, 212, 311, 2389bccd, 2700bccde, 3011bccde, 3322bccdde }}

Badness (Smith): 0.048924

Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647

Mapping: {{mapping| 1 3 6 5 14 17 | 0 -20 -52 -31 -149 -188 }}

* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8895{{c}}

* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8910{{c}}

{{Optimal ET sequence|legend=0| 99ef, 212, 311, 1145c, 1456cd, 1767cd }}

Badness (Smith): 0.034681

=== Pseudoamical ===

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

Mapping: {{mapping| 1 3 6 5 -1 | 0 -20 -52 -31 63 }}

* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9005{{c}}

* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9091{{c}}

{{Optimal ET sequence|legend=0| 14ce, …, 85cee, 99, 212 }}

Badness (Smith): 0.085837

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

Mapping: {{mapping| 1 3 6 5 -1 2 | 0 -20 -52 -31 63 24 }}

* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9049{{c}}

* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9127{{c}}

{{Optimal ET sequence|legend=0| 14ce, …, 85ceef, 99, 113, 212 }}

Badness (Smith): 0.047025

=== Pseudoamorous ===

Comma list: 243/242, 441/440, 980000/970299

Mapping: {{mapping| 1 3 6 5 7 | 0 -20 -52 -31 -50 }}

* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9022{{c}}

* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8917{{c}}

{{Optimal ET sequence|legend=0| 14c, …, 85ce, 99e, 113, 212e }}

Badness (Smith): 0.056583

Comma list: 243/242, 364/363, 441/440, 1875/1859

Mapping: {{mapping| 1 3 6 5 7 10 | 0 -20 -52 -31 -50 -89 }}

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

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

{{Optimal ET sequence|legend=0| 14cf, …, 85ceff, 99ef, 113, 212ef, 325ce, 537cdeef }}

Badness (Smith): 0.042826

Mapping: {{mapping| 2 6 12 10 13 | 0 -20 -52 -31 -43 }}

* CTE: ~99/70 = 600.0000{{c}}, ~21/20 = 84.8781{{c}}

* POTE: ~99/70 = 600.0000{{c}}, ~21/20 = 84.8788{{c}}

{{Optimal ET sequence|legend=0| 14c, …, 170bccde, 184c, 198, 212, 410 }}

Badness (Smith): 0.065110

Mapping: {{mapping| 2 6 12 10 13 19 | 0 -20 -52 -31 -43 -82 }}

* CTE: ~99/70 = 600.0000{{c}}, ~21/20 = 84.8759{{c}}

* POTE: ~99/70 = 600.0000{{c}}, ~21/20 = 84.8750{{c}}

{{Optimal ET sequence|legend=0| 14c, …, 184cff, 198, 410 }}

Badness (Smith): 0.037013

Mapping: {{mapping| 1 3 6 5 3 | 0 -40 -104 -62 13 }}

* CTE: ~2 = 1200.0000{{c}}, ~4096/3993 = 42.4414{{c}}

* POTE: ~2 = 1200.0000{{c}}, ~4096/3993 = 42.4391{{c}}

{{Optimal ET sequence|legend=0| 85c, 113, 198, 311, 1131, 1442, 1753be }}

Badness (Smith): 0.058249

Mapping: {{mapping| 1 3 6 5 3 6 | 0 -40 -104 -62 13 -65 }}

* CTE: ~2 = 1200.0000{{c}}, ~40/39 = 42.4425{{c}}

* POTE: ~2 = 1200.0000{{c}}, ~40/39 = 42.4391{{c}}

{{Optimal ET sequence|legend=0| 85c, 113, 198, 311, 1753beff, 2064beff, 2375bceff }}

Badness (Smith): 0.028267

Calamity tempers out 2100875/2097152, the [[rainy comma]], and may be described as the {{nowrap| 46 & 311 }} temperament. It splits the interval of two octaves plus an acute minor third into five.  

{{Mapping|legend=1| 1 13 32 -15 | 0 -25 -65 39 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~48/35 = 547.909{{c}}

[[Badness]] (Smith): 0.198130

Mapping: {{mapping| 1 13 32 -15 -18 | 0 -25 -65 39 47 }}

Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~48/35 = 547.908{{c}}

Badness (Smith): 0.060408

Mapping: {{mapping| 1 13 32 -15 -18 -31 | 0 -25 -65 39 47 76 }}

Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~48/35 = 547.907{{c}}

Badness (Smith): 0.033617

Familia tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 113 & 152 }} temperament. It splits the interval of acute minor tenth (~243/100) in five.

{{Mapping|legend=1| 1 8 19 20 | 0 -25 -65 -67 }}

* [[CTE]]: ~2 = 1200.000{{c}}, ~11907/10000 = 307.915{{c}}

* [[POTE]]: ~2 = 1200.000{{c}}, ~11907/10000 = 307.941{{c}}

{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417, 1516ccdd, 1933ccdd }}

[[Badness]] (Smith): 0.144551

Mapping: {{mapping| 1 8 19 20 5 | 0 -25 -65 -67 -6 }}

* CTE: ~2 = 1200.000{{c}}, ~3200/2673 = 307.915{{c}}

* POTE: ~2 = 1200.000{{c}}, ~3200/2673 = 307.906{{c}}

{{Optimal ET sequence|legend=0| 39d, 74cd, 113, 152, 265, 417, 1099cdee }}

Badness (Smith): 0.051740

Mapping: {{mapping| 1 8 19 20 5 25 | 0 -25 -65 -67 -6 -83 }}

* CTE: ~2 = 1200.000{{c}}, ~143/120 = 307.922{{c}}

* POTE: ~2 = 1200.000{{c}}, ~143/120 = 307.913{{c}}

Badness (Smith): 0.038473