Amity family

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The amity family tempers out the 5-limit amity comma, 1600000/1594323. The generator for the amity temperament is the acute minor third, which means the 6/5 just minor third raised by an 81/80 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.

Amity

Main article: Amity

In the 5-limit amity is a genuine microtemperament, with 58\205 being a possible tuning. Another good choice is (64/5)1/13, which gives a pure classical major third. Mos scales of 11, 18, 25, 32, 39, 46 or 53 notes are available.

Subgroup: 2.3.5

Comma list: 1600000/1594323

Mapping: [⟨1 3 6], ⟨0 -5 -13]]

mapping generators: ~2, ~243/200

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.537
  • POTE: ~2 = 1\1, ~243/200 = 339.519

Optimal ET sequence: 7, 32c, 39, 46, 53, 152, 205, 258, 1085, 1343, 1601, 1859b, 2117bc

Badness: 0.021960

Overview to extensions

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.

Temperaments discussed elsewhere include:

The rest are considered below.

Septimal amity

Main article: Amity

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 5120/5103

Mapping: [⟨1 3 6 -2], ⟨0 -5 -13 17]]

Wedgie: ⟨⟨5 13 -17 9 -41 -76]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~128/105 = 339.446
  • POTE: ~2 = 1\1, ~128/105 = 339.432

Optimal ET sequence: 7, 32cd, 39, 46, 53, 99, 152, 251, 905bcdd

Badness: 0.023649

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 5120/5103

Mapping: [⟨1 3 6 -2 21], ⟨0 -5 -13 17 -62]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~128/105 = 339.485
  • POTE: ~2 = 1\1, ~128/105 = 339.464

Optimal ET sequence: 46e, 53, 99e, 152, 357d, 509dd, 661dd

Badness: 0.031506

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 625/624, 729/728

Mapping: [⟨1 3 6 -2 21 17], ⟨0 -5 -13 17 -62 -47]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~128/105 = 339.508
  • POTE: ~2 = 1\1, ~128/105 = 339.481

Optimal ET sequence: 46ef, 53, 99ef, 152f, 205

Badness: 0.028008

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 375/374, 540/539, 729/728

Mapping: [⟨1 3 6 -2 21 17 -1], ⟨0 -5 -13 17 -62 -47 18]]

Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.496

Optimal ET sequence: 46ef, 53, 99ef, 152fg, 205gg

Badness: 0.026201

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 324/323, 352/351, 375/374, 400/399, 456/455

Mapping: [⟨1 3 6 -2 21 17 -1 15], ⟨0 -5 -13 17 -62 -47 18 -38]]

Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.501

Optimal ET sequence: 46efh, 53, 99ef, 152fg, 205gg

Badness: 0.018782

Hitchcock

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 2200/2187

Mapping: [⟨1 3 6 -2 6], ⟨0 -5 -13 17 -9]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.390
  • POTE: ~2 = 1\1, ~11/9 = 339.390

Optimal ET sequence: 7, 25cdde, 32cd, 39, 46, 53, 99

Badness: 0.035187

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 176/175, 325/324

Mapping: [⟨1 3 6 -2 6 2], ⟨0 -5 -13 17 -9 6]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.411
  • POTE: ~2 = 1\1, ~11/9 = 339.419

Optimal ET sequence: 7, 25cddef, 32cd, 39, 46, 53, 99

Badness: 0.022448

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 169/168, 176/175, 273/272

Mapping: [⟨1 3 6 -2 6 2 -1], ⟨0 -5 -13 17 -9 6 18]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.366
  • POTE: ~2 = 1\1, ~11/9 = 339.366

Optimal ET sequence: 7, 25cddefgg, 32cdg, 39, 46, 99

Badness: 0.019395

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189

Mapping: [⟨1 3 6 -2 6 2 -1 0], ⟨0 -5 -13 17 -9 6 18 15]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.415
  • POTE: ~2 = 1\1, ~11/9 = 339.407

Optimal ET sequence: 7, 25cddefgghh, 32cdgh, 39h, 46, 53, 99h

Badness: 0.017513

Catamite

The catamite 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. The word "catamite" itself is a term for male homosexual, but also a play on the words "cata-" (down) and "amity."

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 4375/4374

Mapping: [⟨1 3 6 -2 -7], ⟨0 -5 -13 17 37]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~128/105 = 339.314
  • POTE: ~2 = 1\1, ~128/105 = 339.340

Optimal ET sequence: 46, 99e, 145

Badness: 0.040976

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 4375/4374

Mapping: [⟨1 3 6 -2 -7 -11], ⟨0 -5 -13 17 37 52]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~128/105 = 339.277
  • POTE: ~2 = 1\1, ~128/105 = 339.313

Optimal ET sequence: 46, 99ef, 145, 191c, 336cef, 527bccef

Badness: 0.034215

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155

Mapping: [⟨1 3 6 -2 -7 -11 -1], ⟨0 -5 -13 17 37 52 18]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~17/14 = 339.272
  • POTE: ~2 = 1\1, ~17/14 = 339.313

Optimal ET sequence: 46, 99ef, 145, 191c

Badness: 0.021193

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475

Mapping: [⟨1 3 6 -2 -7 -11 -1 -13], ⟨0 -5 -13 17 37 52 18 61]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~17/14 = 339.282
  • POTE: ~2 = 1\1, ~17/14 = 339.325

Optimal ET sequence: 46, 99ef, 145, 191c, 336cefg

Badness: 0.018864

Hemiamity

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 5120/5103

Mapping: [⟨2 1 -1 13 13], ⟨0 5 13 -17 -14]]

mapping generators: ~99/70, ~64/55

Optimal tunings:

  • CTE: ~99/70 = 1\2, ~64/55 = 260.566
  • POTE: ~99/70 = 1\2, ~64/55 = 260.561

Optimal ET sequence: 14cde, 32cde, 46, 106, 152, 350

Badness: 0.031307

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1716/1715, 3025/3024

Mapping: [⟨2 1 -1 13 13 20], ⟨0 5 13 -17 -14 -29]]

Optimal tunings:

  • CTE: ~99/70 = 1\2, ~64/55 = 260.607
  • POTE: ~99/70 = 1\2, ~64/55 = 260.583

Optimal ET sequence: 46, 106f, 152f, 198

Badness: 0.025784

Accord

Subgroup: 2.3.5.7

Comma list: 126/125, 100352/98415

Mapping: [⟨1 3 6 11], ⟨0 -5 -13 -29]]

Wedgie: ⟨⟨5 13 29 9 32 31]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.154
  • POTE: ~2 = 1\1, ~243/200 = 338.993

Optimal ET sequence: 7d, 25cddd, 32cdd, 39d, 46

Badness: 0.095612

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125, 896/891

Mapping: [⟨1 3 6 11 6], ⟨0 -5 -13 -29 -9]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.136
  • POTE: ~2 = 1\1, ~11/9 = 339.047

Optimal ET sequence: 7d, 25cddde, 32cdd, 39d, 46

Badness: 0.042468

Houborizic

The houborizic temperament (53 & 60) tempers out the marvel comma, 225/224. It is so named because it is closely related to the houboriz tuning (generator: 339.774971 cents).

Subgroup: 2.3.5.7

Comma list: 225/224, 1250000/1240029

Mapping: [⟨1 3 6 13], ⟨0 -5 -13 -36]]

Wedgie: ⟨⟨5 13 36 9 43 47]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.711
  • POTE: ~2 = 1\1, ~243/200 = 339.763

Optimal ET sequence: 7d, 32cdddd, 39ddd, 46d, 53, 166, 219c, 272c

Badness: 0.066638

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 1250000/1240029

Mapping: [⟨1 3 6 13 -9], ⟨0 -5 -13 -36 44]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.751
  • POTE: ~2 = 1\1, ~243/200 = 339.763

Optimal ET sequence: 53, 113, 166, 551ccee

Badness: 0.067891

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 2200/2197

Mapping: [⟨1 3 6 13 -9 2], ⟨0 -5 -13 -36 44 6]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~39/32 = 339.754
  • POTE: ~2 = 1\1, ~39/32 = 339.764

Optimal ET sequence: 53, 113, 166

Badness: 0.032996

Houbor

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 2200/2187

Mapping: [⟨1 3 6 13 6], ⟨0 -5 -13 -36 -9]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.680
  • POTE: ~2 = 1\1, ~11/9 = 339.814

Optimal ET sequence: 7d, 32cdddd, 39ddd, 46d, 53

Badness: 0.045232

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273, 325/324

Mapping: [⟨1 3 6 13 6 2], ⟨0 -5 -13 -36 -9 6]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.685
  • POTE: ~2 = 1\1, ~11/9 = 339.784

Optimal ET sequence: 7d, 32cdddd, 39ddd, 46d, 53

Badness: 0.031331

Paramity

The paramity temperament (53 & 311) tempers out the horwell comma (65625/65536) and garischisma (33554432/33480783).

Subgroup: 2.3.5.7

Comma list: 65625/65536, 1600000/1594323

Mapping: [⟨1 3 6 -17], ⟨0 -5 -13 70]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.554
  • POTE: ~2 = 1\1, ~243/200 = 339.553

Optimal ET sequence: 53, 205d, 258, 311, 675, 986

Badness: 0.113655

11-limit

Subgroup: 2.3.5.7.11

Comma list: 6250/6237, 19712/19683, 41503/41472

Mapping: [⟨1 3 6 -17 36], ⟨0 -5 -13 70 -115]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.554
  • POTE: ~2 = 1\1, ~243/200 = 339.554

Optimal ET sequence: 53, 205de, 258, 311, 675, 986

Badness: 0.064853

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683

Mapping: [⟨1 3 6 -17 36 17], ⟨0 -5 -13 70 -115 -47]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.554
  • POTE: ~2 = 1\1, ~243/200 = 339.554

Optimal ET sequence: 53, 205de, 258, 311, 675, 986, 1661cf

Badness: 0.030347

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 1225/1224, 2080/2079, 2200/2197, 2431/2430

Mapping: [⟨1 3 6 -17 36 17 -31], ⟨0 -5 -13 70 -115 -47 124]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.555
  • POTE: ~2 = 1\1, ~243/200 = 339.555

Optimal ET sequence: 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf

Badness: 0.024118

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 625/624, 1225/1224, 1540/1539, 1729/1728, 2080/2079, 2200/2197

Mapping: [⟨1 3 6 -17 36 17 -31 15], ⟨0 -5 -13 70 -115 -47 124 -38]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~208/171 = 339.555
  • POTE: ~2 = 1\1, ~208/171 = 339.555

Optimal ET sequence: 53, 205deg, 258g, 311, 675, 986, 1661cfh

Badness: 0.017420

Bamity

Bamity has a period of half octave and tempers out the sensamagic comma, 245/243. The name bamity is a contraction of bi- and amity.

Subgroup: 2.3.5.7

Comma list: 245/243, 64827/64000

Mapping: [⟨2 1 -1 3], ⟨0 5 13 6]]

mapping generators: ~343/240, ~7/6

Wedgie: ⟨⟨10 26 12 18 -9 -45]]

Optimal tunings:

  • CTE: ~343/240 = 1\2, ~7/6 = 260.563
  • POTE: ~343/240 = 1\2, ~7/6 = 260.402

Optimal ET sequence: 14c, 32c, 46, 106d, 152d

Badness: 0.083601

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 245/243, 441/440

Mapping: [⟨2 1 -1 3 3], ⟨0 5 13 6 9]]

Optimal tunings:

  • CTE: ~99/70 = 1\2, ~7/6 = 260.653
  • POTE: ~99/70 = 1\2, ~7/6 = 260.393

Optimal ET sequence: 14c, 32c, 46, 152de, 198, 244dee

Badness: 0.035504

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 245/243, 441/440

Mapping: [⟨2 1 -1 3 3 0], ⟨0 5 13 6 9 17]]

Optimal tunings:

  • CTE: ~55/39 = 1\2, ~7/6 = 260.811
  • POTE: ~55/39 = 1\2, ~7/6 = 260.618

Optimal ET sequence: 14cf, 32cf, 46

Badness: 0.030885

Hamity

Hamity has a generator of about 430 cents which represents 9/7. It is also generated by half of acute minor "tenth" (acute minor third of 243/200 plus an octave), and its name is a contraction of half and amity.

Subgroup: 2.3.5.7

Comma list: 2430/2401, 4000/3969

Mapping: [⟨1 8 19 15], ⟨0 -10 -26 -19]]

mapping generators: ~2, ~14/9

Wedgie: ⟨⟨10 26 19 18 2 -29]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~9/7 = 430.168
  • POTE: ~2 = 1\1, ~9/7 = 430.219

Optimal ET sequence: 14c, 39d, 53

Badness: 0.073956

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 2200/2187

Mapping: [⟨1 8 19 15 15], ⟨0 -10 -26 -19 -18]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~9/7 = 430.220
  • POTE: ~2 = 1\1, ~9/7 = 430.192

Optimal ET sequence: 14c, 39d, 53

Badness: 0.042947

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 275/273, 572/567

Mapping: [⟨1 8 19 15 15 30], ⟨0 -10 -26 -19 -18 -41]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~9/7 = 430.233
  • POTE: ~2 = 1\1, ~9/7 = 430.216

Optimal ET sequence: 14cf, 39df, 53

Badness: 0.029753

Gamity

The gamity temperament (46 & 113) tempers out the gamelisma, 1029/1024. It splits the interval of grave major sixth (~400/243, an octave minus acute minor third) in three.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1071875/1062882

Mapping: [⟨1 13 32 -1], ⟨0 -15 -39 5]]

mapping generators: ~2, ~320/189

Wedgie: ⟨⟨15 39 -5 27 -50 -121]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~189/160 = 286.816
  • POTE: ~2 = 1\1, ~189/160 = 286.787

Optimal ET sequence: 46, 113, 159, 205d, 364d

Badness: 0.125733

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 1071875/1062882

Mapping: [⟨1 13 32 -1 -11], ⟨0 -15 -39 5 19]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~33/28 = 286.813
  • POTE: ~2 = 1\1, ~33/28 = 286.797

Optimal ET sequence: 46, 113, 159, 205d, 364d

Badness: 0.051111

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384, 10985/10976

Mapping: [⟨1 13 32 -1 -11 -10], ⟨0 -15 -39 5 19 18]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~13/11 = 286.803
  • POTE: ~2 = 1\1, ~13/11 = 286.789

Optimal ET sequence: 46, 113, 159, 364df, 523ddff

Badness: 0.030297

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757

Mapping: [⟨1 13 32 -1 -11 -10 -2], ⟨0 -15 -39 5 19 18 8]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~13/11 = 286.804
  • POTE: ~2 = 1\1, ~13/11 = 286.795

Optimal ET sequence: 46, 113, 159, 364df, 523ddff

Badness: 0.022036

Trinity

The trinity temperament (152 & 159) tempers out the meter, 703125/702464. 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¢ flat), three of which makes acute minor third of ~243/200 with octave reduction.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 1600000/1594323

Mapping: [⟨1 8 19 46], ⟨0 -15 -39 -101]]

Wedgie: ⟨⟨15 39 101 27 118 125]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~168/125 = 513.180
  • POTE: ~2 = 1\1, ~168/125 = 513.178

Optimal ET sequence: 7d, …, 145d, 152, 311, 774, 1085

Badness: 0.119453

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 19712/19683

Mapping: [⟨1 8 19 46 18], ⟨0 -15 -39 -101 -34]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~121/90 = 513.181
  • POTE: ~2 = 1\1, ~121/90 = 513.177

Optimal ET sequence: 7d, …, 145d, 152, 311, 774, 1085e, 1396e

Badness: 0.031296

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689

Mapping: [⟨1 8 19 46 18 64], ⟨0 -15 -39 -101 -34 -141]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~35/26 = 513.184
  • POTE: ~2 = 1\1, ~35/26 = 513.182

Optimal ET sequence: 152f, 159, 311

Badness: 0.026418

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 625/624, 833/832, 1575/1573, 8624/8619

Mapping: [⟨1 8 19 46 18 64 -22], ⟨0 -15 -39 -101 -34 -141 61]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~35/26 = 513.185
  • POTE: ~2 = 1\1, ~35/26 = 513.186

Optimal ET sequence: 152f, 159, 311, 1714cdeg, 2025cdefgg, 2336bccdefgg

Badness: 0.025588

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 625/624, 833/832, 969/968, 1216/1215, 1575/1573

Mapping: [⟨1 8 19 46 18 64 -22 53], ⟨0 -15 -39 -101 -34 -141 61 -114]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~35/26 = 513.184
  • POTE: ~2 = 1\1, ~35/26 = 513.185

Optimal ET sequence: 152f, 159, 311

Badness: 0.018412

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104

Mapping: [⟨1 8 19 46 18 64 -22 53 49], ⟨0 -15 -39 -101 -34 -141 61 -114 -104]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~35/26 = 513.184
  • POTE: ~2 = 1\1, ~35/26 = 513.185

Optimal ET sequence: 152f, 159, 311, 1714cdeghi, 2025cdefgghhi, 2336bccdefgghhi

Badness: 0.014343

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 595/594, 625/624, 760/759, 784/783, 833/832, 875/874, 969/968, 1045/1044

Mapping: [⟨1 8 19 46 18 64 -22 53 49 72], ⟨0 -15 -39 -101 -34 -141 61 -114 -104 -157]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~35/26 = 513.185
  • POTE: ~2 = 1\1, ~35/26 = 513.186

Optimal ET sequence: 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij

Badness: 0.012038

Amicable

The amicable temperament (99 & 311) tempers out the amity comma and the canousma in addition to the breedsma, and is closely associated with the canou 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.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1600000/1594323

Mapping: [⟨1 3 6 5], ⟨0 -20 -52 -31]]

Wedgie: ⟨⟨20 52 31 36 -7 -74]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.8831
  • POTE: ~2 = 1\1, ~21/20 = 84.880

Optimal ET sequence: 14c, …, 85c, 99, 212, 311, 721, 1032, 1753b

Badness: 0.045473

Amical

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 5 -8], ⟨0 -20 -52 -31 162]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.8843
  • POTE: ~2 = 1\1, ~21/20 = 84.8843

Optimal ET sequence: 99, 212e, 311, 721, 1032, 1343, 2375bc

Badness: 0.100668

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 6 5 -8 -5], ⟨0 -20 -52 -31 162 123]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.8837
  • POTE: ~2 = 1\1, ~21/20 = 84.8838

Optimal ET sequence: 99, 212ef, 311, 721, 1032

Badness: 0.049893

Amorous

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 5 14], ⟨0 -20 -52 -31 -149]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.8883
  • POTE: ~2 = 1\1, ~21/20 = 84.8896

Optimal ET sequence: 99e, 212, 311, 2389bccd, 2700bccde, 3011bccde, 3322bccdde

Badness: 0.048924

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 6 5 14 17], ⟨0 -20 -52 -31 -149 -188]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.8895
  • POTE: ~2 = 1\1, ~21/20 = 84.8910

Optimal ET sequence: 99ef, 212, 311, 1145c, 1456cd, 1767cd

Badness: 0.034681

Pseudoamical

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 5 -1], ⟨0 -20 -52 -31 63]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.9005
  • POTE: ~2 = 1\1, ~21/20 = 84.9091

Optimal ET sequence: 14ce, …, 85cee, 99, 212

Badness: 0.085837

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 6 5 -1 2], ⟨0 -20 -52 -31 63 24]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.9049
  • POTE: ~2 = 1\1, ~21/20 = 84.9127

Optimal ET sequence: 14ce, …, 85ceef, 99, 113, 212

Badness: 0.047025

Pseudoamorous

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 5 7], ⟨0 -20 -52 -31 -50]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.9022
  • POTE: ~2 = 1\1, ~21/20 = 84.8917

Optimal ET sequence: 14c, …, 85ce, 99e, 113, 212e

Badness: 0.056583

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 6 5 7 10], ⟨0 -20 -52 -31 -50 -89]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.9153
  • POTE: ~2 = 1\1, ~21/20 = 84.9164

Optimal ET sequence: 14cf, …, 85ceff, 99ef, 113, 212ef, 325ce, 537cdeef

Badness: 0.042826

Floral

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 14641/14580

Mapping: [⟨2 6 12 10 13], ⟨0 -20 -52 -31 -43]]

Optimal tunings:

  • CTE: ~99/70 = 1\2, ~21/20 = 84.8781
  • POTE: ~99/70 = 1\2, ~21/20 = 84.8788

Optimal ET sequence: 14c, …, 170bccde, 184c, 198, 212, 410

Badness: 0.065110

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580

Mapping: [⟨2 6 12 10 13 19], ⟨0 -20 -52 -31 -43 -82]]

Optimal tunings:

  • CTE: ~99/70 = 1\2, ~21/20 = 84.8759
  • POTE: ~99/70 = 1\2, ~21/20 = 84.8750

Optimal ET sequence: 14c, …, 184cff, 198, 410

Badness: 0.037013

Humorous

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 1600000/1594323

Mapping: [⟨1 3 6 5 3], ⟨0 -40 -104 -62 13]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~4096/3993 = 42.4414
  • POTE: ~2 = 1\1, ~4096/3993 = 42.4391

Optimal ET sequence: 85c, 113, 198, 311, 1131, 1442, 1753be

Badness: 0.058249

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024

Mapping: [⟨1 3 6 5 3 6], ⟨0 -40 -104 -62 13 -65]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~40/39 = 42.4425
  • POTE: ~2 = 1\1, ~40/39 = 42.4391

Optimal ET sequence: 85c, 113, 198, 311, 1753beff, 2064beff, 2375bceff

Badness: 0.028267

Calamity

The calamity temperament (46 & 311) tempers out the rainy comma, 2100875/2097152, splitting the interval of two octaves plus an acute minor third into five.

Subgroup: 2.3.5.7

Comma list: 1600000/1594323, 2100875/2097152

Mapping: [⟨1 13 32 -15], ⟨0 -25 -65 39]]

Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.909

Optimal ET sequence: 46, 219c, 265, 311

Badness: 0.198130

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 12005/11979, 131072/130977

Mapping: [⟨1 13 32 -15 -18], ⟨0 -25 -65 39 47]]

Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.908

Optimal ET sequence: 46, 219c, 265, 311, 979, 1290

Badness: 0.060408

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3025/3024, 4096/4095, 12005/11979

Mapping: [⟨1 13 32 -15 -18 -31], ⟨0 -25 -65 39 47 76]]

Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.907

Optimal ET sequence: 46, 265, 311, 668, 979, 1290

Badness: 0.033617

Familia

The familia temperament (113 & 152) tempers out the mirkwai comma, 16875/16807. It splits the interval of acute minor tenth (~243/100) in five.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 1600000/1594323

Mapping: [⟨1 8 19 20], ⟨0 -25 -65 -67]]

Wedgie: ⟨⟨25 65 67 45 36 -27]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11907/10000 = 307.915
  • POTE: ~2 = 1\1, ~11907/10000 = 307.941

Optimal ET sequence: 39d, 74cd, 113, 152, 265, 417, 1516ccdd, 1933ccdd

Badness: 0.144551

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 1600000/1594323

Mapping: [⟨1 8 19 20 5], ⟨0 -25 -65 -67 -6]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~3200/2673 = 307.915
  • POTE: ~2 = 1\1, ~3200/2673 = 307.906

Optimal ET sequence: 39d, 74cd, 113, 152, 265, 417, 1099cdee

Badness: 0.051740

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372, 2205/2197

Mapping: [⟨1 8 19 20 5 25], ⟨0 -25 -65 -67 -6 -83]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~143/120 = 307.922
  • POTE: ~2 = 1\1, ~143/120 = 307.913

Optimal ET sequence: 39df, 74cdf, 113, 152f, 265

Badness: 0.038473

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