Amity family: Difference between revisions

The [[generator]] for the amity temperament is the acute minor third, which means the [[6/5]] just minor third raised by a [[syntonic comma]] to [[243/200]], and from this it derives its name. If you are looking for a different kind of neutral third, this could be the temperament for you. Its [[ploidacot]] is gamma-pentacot. It is a member of the [[syntonic–chromatic equivalence continuum]] with {{nowrap| ''n'' {{=}} 5 }}, so it equates an apotome with a stack of five syntonic commas. It is also in the [[schismic–Mercator equivalence continuum]] with {{nowrap| ''n'' {{=}} 5 }}, so unless 53edo is used as a tuning, the schisma is always observed.  

* [[CTE]]: ~2 = 1200.000{{c}}, ~243/200 = 339.537{{c}}

* [[POTE]]: ~2 = 1200.000{{c}}, ~243/200 = 339.519{{c}}

[[Badness]] (Smith): 0.021960

Septimal amity can be described as the {{nowrap| 46 & 53 }} temperament, which tempers out [[4375/4374]] and [[5120/5103]] in the 7-limit. [[99edo]] is a good tuning, with generator 28\99.  

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.000{{c}}, ~128/105 = 339.446{{c}}

* [[POTE]]: ~2 = 1200.000{{c}}, ~128/105 = 339.432{{c}}

[[Badness]] (Smith): 0.023649

* CTE: ~2 = 1200.000{{c}}, ~128/105 = 339.485{{c}}

* POTE: ~2 = 1200.000{{c}}, ~128/105 = 339.464{{c}}

{{Optimal ET sequence|legend=0| 46e, 53, 99e, 152, 357d, 509dd, 661dd }}

Badness (Smith): 0.031506

* CTE: ~2 = 1200.000{{c}}, ~128/105 = 339.508{{c}}

* POTE: ~2 = 1200.000{{c}}, ~128/105 = 339.481{{c}}

{{Optimal ET sequence|legend=0| 46ef, 53, 99ef, 152f, 205 }}

Badness (Smith): 0.028008

Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~17/14 = 339.496{{c}}

{{Optimal ET sequence|legend=0| 46ef, 53, 99ef, 152fg, 205gg }}

Badness (Smith): 0.026201

Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~17/14 = 339.501{{c}}

{{Optimal ET sequence|legend=0| 46efh, 53, 99ef, 152fg, 205gg }}

Badness (Smith): 0.018782

* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.390{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.390{{c}}

{{Optimal ET sequence|legend=0| 7, 25cdde, 32cd, 39, 46, 53, 99 }}

Badness (Smith): 0.035187

* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.411{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.419{{c}}

{{Optimal ET sequence|legend=0| 7, 25cddef, 32cd, 39, 46, 53, 99 }}

Badness (Smith): 0.022448

* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.366{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.366{{c}}

{{Optimal ET sequence|legend=0| 7, 25cddefgg, 32cdg, 39, 46, 99 }}

Badness (Smith): 0.019395

* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.415{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.407{{c}}

{{Optimal ET sequence|legend=0| 7, 25cddefgghh, 32cdgh, 39h, 46, 53, 99h }}

Badness (Smith): 0.017513

The stalagmite temperament (46 & 99ef) tempers out 441/440 (werckisma) and 896/891 (pentacircle) in the 11-limit; 196/195, 352/351 and 364/363 in the 13-limit. "-mite" in the name references amity, and stalagmites being found in caves underground references how it is ''down'' from amity.  

* CTE: ~2 = 1200.000{{c}}, ~128/105 = 339.314{{c}}

* POTE: ~2 = 1200.000{{c}}, ~128/105 = 339.340{{c}}

{{Optimal ET sequence|legend=0| 46, 99e, 145 }}

Badness (Smith): 0.040976

* CTE: ~2 = 1200.000{{c}}, ~128/105 = 339.277{{c}}

* POTE: ~2 = 1200.000{{c}}, ~128/105 = 339.313{{c}}

{{Optimal ET sequence|legend=0| 46, 99ef, 145, 191c, 336cef, 527bccef }}

Badness (Smith): 0.034215

* CTE: ~2 = 1200.000{{c}}, ~17/14 = 339.272{{c}}

* POTE: ~2 = 1200.000{{c}}, ~17/14 = 339.313{{c}}

{{Optimal ET sequence|legend=0| 46, 99ef, 145, 191c }}

Badness (Smith): 0.021193

* CTE: ~2 = 1200.000{{c}}, ~17/14 = 339.282{{c}}

* POTE: ~2 = 1200.000{{c}}, ~17/14 = 339.325{{c}}

{{Optimal ET sequence|legend=0| 46, 99ef, 145, 191c, 336cefg }}

Badness (Smith): 0.018864

* CTE: ~99/70 = 600.000{{c}}, ~64/55 = 260.566{{c}}

* POTE: ~99/70 = 600.000{{c}}, ~64/55 = 260.561{{c}}

{{Optimal ET sequence|legend=0| 14cde, 32cde, 46, 106, 152, 350 }}

Badness (Smith): 0.031307

* CTE: ~99/70 = 600.000{{c}}, ~64/55 = 260.607{{c}}

* POTE: ~99/70 = 600.000{{c}}, ~64/55 = 260.583{{c}}

{{Optimal ET sequence|legend=0| 46, 106f, 152f, 198 }}

Badness (Smith): 0.025784

* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.154{{c}}

* POTE: ~2 = 1200.000{{c}}, ~243/200 = 338.993{{c}}

[[Badness]] (Smith): 0.095612

* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.136{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.047{{c}}

{{Optimal ET sequence|legend=0| 7d, 25cddde, 32cdd, 39d, 46 }}

Badness (Smith): 0.042468

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).

* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.711{{c}}

* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.763{{c}}

[[Badness]] (Smith): 0.066638

* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.751{{c}}

* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.763{{c}}

{{Optimal ET sequence|legend=0| 53, 113, 166, 551ccee }}

Badness (Smith): 0.067891

* CTE: ~2 = 1200.000{{c}}, ~39/32 = 339.754{{c}}

* POTE: ~2 = 1200.000{{c}}, ~39/32 = 339.764{{c}}

{{Optimal ET sequence|legend=0| 53, 113, 166 }}

Badness (Smith): 0.032996

Mapping: {{mapping| 1 3 6 13 6 | 0 -5 -13 -36 -9 }}

* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.680{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.814{{c}}

{{Optimal ET sequence|legend=0| 7d, 32cdddd, 39ddd, 46d, 53 }}

Badness (Smith): 0.045232

Mapping: {{mapping| 1 3 6 13 6 2 | 0 -5 -13 -36 -9 6 }}

* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.685{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.784{{c}}

{{Optimal ET sequence|legend=0| 7d, 32cdddd, 39ddd, 46d, 53 }}

Badness (Smith): 0.031331

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

* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}

* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.553{{c}}

[[Badness]] (Smith): 0.113655

* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}

* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}

{{Optimal ET sequence|legend=0| 53, 205de, 258, 311, 675, 986 }}

Badness (Smith): 0.064853

* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}

* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}

{{Optimal ET sequence|legend=0| 53, 205de, 258, 311, 675, 986, 1661cf }}

Badness (Smith): 0.030347

* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.555{{c}}

* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.555{{c}}

{{Optimal ET sequence|legend=0| 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf }}

Badness (Smith): 0.024118

* CTE: ~2 = 1200.000{{c}}, ~208/171 = 339.555{{c}}

* POTE: ~2 = 1200.000{{c}}, ~208/171 = 339.555{{c}}

{{Optimal ET sequence|legend=0| 53, 205deg, 258g, 311, 675, 986, 1661cfh }}

Badness (Smith): 0.017420

* CTE: ~343/240 = 600.000{{c}}, ~7/6 = 260.563{{c}}

* POTE: ~343/240 = 600.000{{c}}, ~7/6 = 260.402{{c}}

[[Badness]] (Smith): 0.083601

* CTE: ~99/70 = 600.000{{c}}, ~7/6 = 260.653{{c}}

* POTE: ~99/70 = 600.000{{c}}, ~7/6 = 260.393{{c}}

{{Optimal ET sequence|legend=0| 14c, 32c, 46, 152de, 198, 244dee }}

Badness (Smith): 0.035504

* CTE: ~55/39 = 600.000{{c}}, ~7/6 = 260.811{{c}}

* POTE: ~55/39 = 600.000{{c}}, ~7/6 = 260.618{{c}}

{{Optimal ET sequence|legend=0| 14cf, 32cf, 46 }}

Badness (Smith): 0.030885

* CTE: ~2 = 1200.000{{c}}, ~9/7 = 430.168{{c}}

* POTE: ~2 = 1200.000{{c}}, ~9/7 = 430.219{{c}}

[[Badness]] (Smith): 0.073956

* CTE: ~2 = 1200.000{{c}}, ~9/7 = 430.220{{c}}

* POTE: ~2 = 1200.000{{c}}, ~9/7 = 430.192{{c}}

{{Optimal ET sequence|legend=0| 14c, 39d, 53 }}

Badness (Smith): 0.042947

* CTE: ~2 = 1200.000{{c}}, ~9/7 = 430.233{{c}}

* POTE: ~2 = 1200.000{{c}}, ~9/7 = 430.216{{c}}

{{Optimal ET sequence|legend=0| 14cf, 39df, 53 }}

Badness (Smith): 0.029753

Gamity tempers out 1029/1024, the [[gamelisma]], and may be described as the {{nowrap| 46 & 113 }} temperament. It splits the interval of grave major sixth (~400/243, an octave minus acute minor third) in three.

* CTE: ~2 = 1200.000{{c}}, ~189/160 = 286.816{{c}}

* POTE: ~2 = 1200.000{{c}}, ~189/160 = 286.787{{c}}

[[Badness]] (Smith): 0.125733

* CTE: ~2 = 1200.000{{c}}, ~33/28 = 286.813{{c}}

* POTE: ~2 = 1200.000{{c}}, ~33/28 = 286.797{{c}}

{{Optimal ET sequence|legend=0| 46, 113, 159, 205d, 364d }}

Badness (Smith): 0.051111

* CTE: ~2 = 1200.000{{c}}, ~13/11 = 286.803{{c}}

* POTE: ~2 = 1200.000{{c}}, ~13/11 = 286.789{{c}}

{{Optimal ET sequence|legend=0| 46, 113, 159, 364df, 523ddff }}

Badness (Smith): 0.030297

* CTE: ~2 = 1200.000{{c}}, ~13/11 = 286.804{{c}}

* POTE: ~2 = 1200.000{{c}}, ~13/11 = 286.795{{c}}

{{Optimal ET sequence|legend=0| 46, 113, 159, 364df, 523ddff }}

Badness (Smith): 0.022036

Trinity tempers out 703125/702464, the [[meter]], and may be described as the {{nowrap| 152 & 159 }} temperament. It splits the acute minor tenth (~243/100, an octave plus acute minor third) 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.

* CTE: ~2 = 1200.000{{c}}, ~168/125 = 513.180{{c}}

* POTE: ~2 = 1200.000{{c}}, ~168/125 = 513.178{{c}}

[[Badness]] (Smith): 0.119453

* CTE: ~2 = 1200.000{{c}}, ~121/90 = 513.181{{c}}

* POTE: ~2 = 1200.000{{c}}, ~121/90 = 513.177{{c}}

{{Optimal ET sequence|legend=0| 7d, …, 145d, 152, 311, 774, 1085e, 1396e }}

Badness (Smith): 0.031296

* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.184{{c}}

* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.182{{c}}

{{Optimal ET sequence|legend=0| 152f, 159, 311 }}

Badness (Smith): 0.026418

* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}

* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.186{{c}}

{{Optimal ET sequence|legend=0| 152f, 159, 311, 1714cdeg, 2025cdefgg, 2336bccdefgg }}

Badness (Smith): 0.025588

* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.184{{c}}

* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}

{{Optimal ET sequence|legend=0| 152f, 159, 311 }}

Badness (Smith): 0.018412

* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.184{{c}}

* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}

{{Optimal ET sequence|legend=0| 152f, 159, 311, 1714cdeghi, 2025cdefgghhi, 2336bccdefgghhi }}

Badness (Smith): 0.014343

* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}

* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.186{{c}}

{{Optimal ET sequence|legend=0| 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij }}

Badness (Smith): 0.012038

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.  

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

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

[[Badness]] (Smith): 0.045473

* 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

* 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

* 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

* 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

* 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

* 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

* 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

* 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

* 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

* 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

* 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

* 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.  

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

[[Badness]] (Smith): 0.198130

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

{{Optimal ET sequence|legend=0| 46, 219c, 265, 311, 979, 1290 }}

Badness (Smith): 0.060408

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

{{Optimal ET sequence|legend=0| 46, 265, 311, 668, 979, 1290 }}

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.

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

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

[[Badness]] (Smith): 0.144551

* 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

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

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

{{Optimal ET sequence|legend=0| 39df, 74cdf, 113, 152f, 265 }}

Badness (Smith): 0.038473