Amity family: Difference between revisions

Latest revision as of 00:26, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The amity family of temperaments tempers out the amity comma (monzo: [9 -13 15⟩, ratio: 1600000/1594323).

Amity

Main article: Amity

The generator for the amity temperament is the acute minor third, which means the 6/5 just minor third raised by a syntonic comma 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 n = 5, so it equates an apotome with a stack of five syntonic commas. It is also in the schismic–Mercator equivalence continuum with n = 5, so unless 53edo is used as a tuning, the schisma is always observed.

Amity is a genuine microtemperament in the 5-limit, 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:

Chromat → Hemimage temperaments #Chromat (+10976/10935)
Witcher → Wizmic microtemperaments #Witcher (+420175/419904)

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]]

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]]

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]]

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

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

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

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]]

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]]

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]]

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

@@ Line 1: / Line 1: @@
-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. It can also be described as the 46&amp;53 temperament, which tempers out [[4375/4374]] and [[5120/5103]] in the 7-limit. [[99edo|99EDO]] is a good tuning for amity, with generator 28\99, and MOS of 11, 18, 25, 32, 39, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.
+{{Technical data page}}
+The '''amity family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[amity comma]] ({{monzo|legend=1| 9 -13 15 }}, [[ratio]]: 1600000/1594323).
 == Amity ==
 {{Main| Amity }}
-{{See also| Ragismic microtemperaments #Amity }}
+The [[generator]] for the amity temperament is the acute minor third, which means the [[6/5]] just minor third raised by a [[81/80|syntonic comma]] 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.
+Amity is a genuine [[microtemperament]] in the 5-limit, with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives a pure classical major third. [[Mos scale]]s of 11, 18, 25, 32, 39, 46 or 53 notes are available.
 [[Subgroup]]: 2.3.5
@@ Line 9: / Line 13: @@
 [[Comma list]]: 1600000/1594323
-[[Mapping]]: [{{val| 1 3 6 }}, {{val| 0 -5 -13 }}]
+{{Mapping|legend=1| 1 3 6 | 0 -5 -13 }}
-Mapping generators: ~2, ~243/200
+: mapping generators: ~2, ~243/200
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.519
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1\1, ~243/200 = 339.537
+* [[POTE]]: ~2 = 1\1, ~243/200 = 339.519
-{{Val list|legend=1| 7, 39, 46, 53, 152, 205, 463, 668, 873 }}
+{{Optimal ET sequence|legend=1| 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 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, and 16875/16807 for familia.
+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:
-* ''[[Amicable]]'', {2401/2400, 1600000/1594323} → [[Breedsmic temperaments #Amicable]]
+* ''[[Chromat]]'' → [[Hemimage temperaments #Chromat]] (+10976/10935)
-* ''[[Chromat]]'', {10976/10935, 235298/234375} → [[Hemimage temperaments #Chromat]]
+* ''[[Witcher]]'' → [[Wizmic microtemperaments #Witcher]] (+420175/419904)
-* ''[[Witch]]'', {420175/419904, 1600000/1594323} → [[Wizmic microtemperaments #Witch]]
+The rest are considered below.
 == Septimal amity ==
+{{Main| Amity }}
+Septimal amity can be described as the 46 &amp; 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]]: [{{val| 1 3 6 -2 }}, {{val| 0 -5 -13 17 }}]
+{{Mapping|legend=1| 1 3 6 -2 | 0 -5 -13 17 }}
-{{Multival|legend=1| 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 tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 339.432
+{{Optimal ET sequence|legend=1| 7, 32cd, 39, 46, 53, 99, 152, 251, 905bcdd }}
-{{Val list|legend=1| 7, 32c, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd }}
 [[Badness]]: 0.023649
@@ Line 47: / Line 58: @@
 Comma list: 540/539, 4375/4374, 5120/5103
-Mapping: [{{val| 1 3 6 -2 21 }}, {{val| 0 -5 -13 17 -62 }}]
+Mapping: {{mapping| 1 3 6 -2 21 | 0 -5 -13 17 -62 }}
-Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.464
+Optimal tunings:
+* CTE: ~2 = 1\1, ~128/105 = 339.485
+* POTE: ~2 = 1\1, ~128/105 = 339.464
-Optimal GPV sequence: {{Val list| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}
+Optimal ET sequence: {{Optimal ET sequence| 46e, 53, 99e, 152, 357d, 509dd, 661dd }}
 Badness: 0.031506
@@ Line 58: / Line 71: @@
 Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 540/539, 625/624, 847/845
+Comma list: 352/351, 540/539, 625/624, 729/728
+Mapping: {{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: {{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: [{{val| 1 3 6 -2 21 17 }}, {{val| 0 -5 -13 17 -62 -47 }}]
+Mapping: {{mapping| 1 3 6 -2 21 17 -1 | 0 -5 -13 17 -62 -47 18 }}
-Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.481
+Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.496
-Optimal GPV sequence: {{Val list| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>
+Optimal ET sequence: {{Optimal ET sequence| 46ef, 53, 99ef, 152fg, 205gg }}
-<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
+Badness: 0.026201
-Badness: 0.028008
+==== 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: {{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: {{Optimal ET sequence| 46efh, 53, 99ef, 152fg, 205gg }}
+Badness: 0.018782
 === Hitchcock ===
@@ Line 75: / Line 114: @@
 Comma list: 121/120, 176/175, 2200/2187
-Mapping: [{{val| 1 3 6 -2 6 }}, {{val| 0 -5 -13 17 -9 }}]
+Mapping: {{mapping| 1 3 6 -2 6 | 0 -5 -13 17 -9 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.390
+Optimal tunings:
+* CTE: ~2 = 1\1, ~11/9 = 339.390
+* POTE: ~2 = 1\1, ~11/9 = 339.390
-Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
+Optimal ET sequence: {{Optimal ET sequence| 7, 25cdde, 32cd, 39, 46, 53, 99 }}
 Badness: 0.035187
@@ Line 88: / Line 129: @@
 Comma list: 121/120, 169/168, 176/175, 325/324
-Mapping: [{{val| 1 3 6 -2 6 2 }}, {{val| 0 -5 -13 17 -9 6 }}]
+Mapping: {{mapping| 1 3 6 -2 6 2 | 0 -5 -13 17 -9 6 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.419
+Optimal tunings:
+* CTE: ~2 = 1\1, ~11/9 = 339.411
+* POTE: ~2 = 1\1, ~11/9 = 339.419
-Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
+Optimal ET sequence: {{Optimal ET sequence| 7, 25cddef, 32cd, 39, 46, 53, 99 }}
 Badness: 0.022448
@@ Line 101: / Line 144: @@
 Comma list: 121/120, 154/153, 169/168, 176/175, 273/272
-Mapping: [{{val| 1 3 6 -2 6 2 -1 }}, {{val| 0 -5 -13 17 -9 6 18 }}]
+Mapping: {{mapping| 1 3 6 -2 6 2 -1 | 0 -5 -13 17 -9 6 18 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.366
+Optimal tunings:
+* CTE: ~2 = 1\1, ~11/9 = 339.366
+* POTE: ~2 = 1\1, ~11/9 = 339.366
-Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
+Optimal ET sequence: {{Optimal ET sequence| 7, 25cddefgg, 32cdg, 39, 46, 99 }}
 Badness: 0.019395
@@ Line 114: / Line 159: @@
 Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189
-Mapping: [{{val| 1 3 6 -2 6 2 -1 0 }}, {{val| 0 -5 -13 17 -9 6 18 15 }}]
+Mapping: {{mapping| 1 3 6 -2 6 2 -1 0 | 0 -5 -13 17 -9 6 18 15 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.407
+Optimal tunings:
+* CTE: ~2 = 1\1, ~11/9 = 339.415
+* POTE: ~2 = 1\1, ~11/9 = 339.407
-Optimal GPV sequence: {{Val list| 7, 39h, 46, 53, 99h }}
+Optimal ET sequence: {{Optimal ET sequence| 7, 25cddefgghh, 32cdgh, 39h, 46, 53, 99h }}
 Badness: 0.017513
@@ Line 129: / Line 176: @@
 Comma list: 441/440, 896/891, 4375/4374
-Mapping: [{{val| 1 3 6 -2 -7 }}, {{val| 0 -5 -13 17 37 }}]
+Mapping: {{mapping| 1 3 6 -2 -7 | 0 -5 -13 17 37 }}
-Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.340
+Optimal tunings:
+* CTE: ~2 = 1\1, ~128/105 = 339.314
+* POTE: ~2 = 1\1, ~128/105 = 339.340
-Optimal GPV sequence: {{Val list| 46, 99e, 145, 244e }}
+Optimal ET sequence: {{Optimal ET sequence| 46, 99e, 145 }}
 Badness: 0.040976
@@ Line 142: / Line 191: @@
 Comma list: 196/195, 352/351, 364/363, 4375/4374
-Mapping: [{{val| 1 3 6 -2 -7 -11 }}, {{val| 0 -5 -13 17 37 52 }}]
+Mapping: {{mapping| 1 3 6 -2 -7 -11 | 0 -5 -13 17 37 52 }}
-Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.313
+Optimal tunings:
+* CTE: ~2 = 1\1, ~128/105 = 339.277
+* POTE: ~2 = 1\1, ~128/105 = 339.313
-Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
+Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c, 336cef, 527bccef }}
 Badness: 0.034215
@@ Line 155: / Line 206: @@
 Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155
-Mapping: [{{val| 1 3 6 -2 -7 -11 -1 }}, {{val| 0 -5 -13 17 37 52 18 }}]
+Mapping: {{mapping| 1 3 6 -2 -7 -11 -1 | 0 -5 -13 17 37 52 18 }}
-Optimal tuning (POTE): ~2 = 1\1, ~17/14 = 339.313
+Optimal tunings:
+* CTE: ~2 = 1\1, ~17/14 = 339.272
+* POTE: ~2 = 1\1, ~17/14 = 339.313
-Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
+Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c }}
 Badness: 0.021193
@@ Line 168: / Line 221: @@
 Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475
-Mapping: [{{val| 1 3 6 -2 -7 -11 -1 -13 }}, {{val| 0 -5 -13 17 37 52 18 61 }}]
+Mapping: {{mapping| 1 3 6 -2 -7 -11 -1 -13 | 0 -5 -13 17 37 52 18 61 }}
-Optimal tuning (POTE): ~2 = 1\1, ~17/14 = 339.325
+Optimal tunings:
+* CTE: ~2 = 1\1, ~17/14 = 339.282
+* POTE: ~2 = 1\1, ~17/14 = 339.325
-Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
+Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c, 336cefg }}
 Badness: 0.018864
@@ Line 181: / Line 236: @@
 Comma list: 3025/3024, 4375/4374, 5120/5103
-Mapping: [{{val| 2 1 -1 13 13 }}, {{val| 0 5 13 -17 -14 }}]
+Mapping: {{mapping| 2 1 -1 13 13 | 0 5 13 -17 -14 }}
-Mapping generators: ~99/70, ~64/55
+: mapping generators: ~99/70, ~64/55
-Optimal tuning (POTE): ~99/70 = 1\2, ~64/55 = 260.561
+Optimal tunings:
+* CTE: ~99/70 = 1\2, ~64/55 = 260.566
+* POTE: ~99/70 = 1\2, ~64/55 = 260.561
-Optimal GPV sequence: {{Val list| 14cde, 46, 106, 152, 350, 502d }}
+Optimal ET sequence: {{Optimal ET sequence| 14cde, 32cde, 46, 106, 152, 350 }}
 Badness: 0.031307
@@ Line 196: / Line 253: @@
 Comma list: 352/351, 847/845, 1716/1715, 3025/3024
-Mapping: [{{val| 2 1 -1 13 13 20 }}, {{val| 0 5 13 -17 -14 -29 }}]
+Mapping: {{mapping| 2 1 -1 13 13 20 | 0 5 13 -17 -14 -29 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~64/55 = 260.583
+Optimal tunings:
+* CTE: ~99/70 = 1\2, ~64/55 = 260.607
+* POTE: ~99/70 = 1\2, ~64/55 = 260.583
-Optimal GPV sequence: {{Val list| 46, 106f, 152f, 198, 350f, 548cdff }}
+Optimal ET sequence: {{Optimal ET sequence| 46, 106f, 152f, 198 }}
 Badness: 0.025784
@@ Line 209: / Line 268: @@
 [[Comma list]]: 126/125, 100352/98415
-[[Mapping]]: [{{val| 1 3 6 11 }}, {{val| 0 -5 -13 -29 }}]
+{{Mapping|legend=1| 1 3 6 11 | 0 -5 -13 -29 }}
-{{Multival|legend=1| 5 13 29 9 32 31 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 338.993
+Optimal tunings:
+* CTE: ~2 = 1\1, ~243/200 = 339.154
+* POTE: ~2 = 1\1, ~243/200 = 338.993
-{{Val list|legend=1| 7d, 39d, 46, 131c, 177c }}
+{{Optimal ET sequence|legend=1| 7d, 25cddd, 32cdd, 39d, 46 }}
 [[Badness]]: 0.095612
@@ Line 224: / Line 283: @@
 Comma list: 121/120, 126/125, 896/891
-Mapping: [{{val| 1 3 6 11 6 }}, {{val| 0 -5 -13 -29 -9 }}]
+Mapping: {{mapping| 1 3 6 11 6 | 0 -5 -13 -29 -9 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.047
+Optimal tunings:
+* CTE: ~2 = 1\1, ~11/9 = 339.136
+* POTE: ~2 = 1\1, ~11/9 = 339.047
-Optimal GPV sequence: {{Val list| 7d, 39d, 46, 177c, 223bc, 269bce }}
+Optimal ET sequence: {{Optimal ET sequence| 7d, 25cddde, 32cdd, 39d, 46 }}
 Badness: 0.042468
 == Houborizic ==
-The ''houborizic'' temperament (53&amp;113) tempers out the [[marvel comma]], 225/224. It is so named because it is closely related to the '''houboriz tuning''' (generator: 339.774971 cents).
+The ''houborizic'' temperament (53 &amp; 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
@@ Line 239: / Line 300: @@
 [[Comma list]]: 225/224, 1250000/1240029
-[[Mapping]]: [{{val| 1 3 6 13 }}, {{val| 0 -5 -13 -36 }}]
+{{Mapping|legend=1| 1 3 6 13 | 0 -5 -13 -36 }}
-{{Multival|legend=1| 5 13 36 9 43 47 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.763
+Optimal tunings:
+* CTE: ~2 = 1\1, ~243/200 = 339.711
+* POTE: ~2 = 1\1, ~243/200 = 339.763
-{{Val list|legend=1| 7d, 46d, 53, 113, 166 }}
+{{Optimal ET sequence|legend=1| 7d, 32cdddd, 39ddd, 46d, 53, 166, 219c, 272c }}
 [[Badness]]: 0.066638
@@ Line 254: / Line 315: @@
 Comma list: 225/224, 385/384, 1250000/1240029
-Mapping: [{{val| 1 3 6 13 -9 }}, {{val| 0 -5 -13 -36 44 }}]
+Mapping: {{mapping| 1 3 6 13 -9 | 0 -5 -13 -36 44 }}
-Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.763
+Optimal tunings:
+* CTE: ~2 = 1\1, ~243/200 = 339.751
+* POTE: ~2 = 1\1, ~243/200 = 339.763
-Optimal GPV sequence: {{Val list| 53, 113, 166 }}
+Optimal ET sequence: {{Optimal ET sequence| 53, 113, 166, 551ccee }}
 Badness: 0.067891
@@ Line 267: / Line 330: @@
 Comma list: 225/224, 325/324, 385/384, 2200/2197
-Mapping: [{{val| 1 3 6 13 -9 2 }}, {{val| 0 -5 -13 -36 44 6 }}]
+Mapping: {{mapping| 1 3 6 13 -9 2 | 0 -5 -13 -36 44 6 }}
-Optimal tuning (POTE): ~2 = 1\1, ~39/32 = 339.764
+Optimal tunings:
+* CTE: ~2 = 1\1, ~39/32 = 339.754
+* POTE: ~2 = 1\1, ~39/32 = 339.764
-Optimal GPV sequence: {{Val list| 53, 113, 166 }}
+Optimal ET sequence: {{Optimal ET sequence| 53, 113, 166 }}
 Badness: 0.032996
@@ Line 282: / Line 347: @@
 Mapping: [{{val| 1 3 6 13 6 }}, {{val| 0 -5 -13 -36 -9 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.814
+Optimal tunings:
+* CTE: ~2 = 1\1, ~11/9 = 339.680
+* POTE: ~2 = 1\1, ~11/9 = 339.814
-Optimal GPV sequence: {{Val list| 7d, 46d, 53, 60e, 113e }}
+Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}
 Badness: 0.045232
@@ Line 295: / Line 362: @@
 Mapping: [{{val| 1 3 6 13 6 2 }}, {{val| 0 -5 -13 -36 -9 6 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.784
+Optimal tunings:
+* CTE: ~2 = 1\1, ~11/9 = 339.685
+* POTE: ~2 = 1\1, ~11/9 = 339.784
-Optimal GPV sequence: {{Val list| 7d, 46d, 53, 60e, 113e }}
+Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}
 Badness: 0.031331
@@ Line 308: / Line 377: @@
 [[Comma list]]: 65625/65536, 1600000/1594323
-[[Mapping]]: [{{val|1 3 6 -17}}, {{val|0 -5 -13 70}}]
+{{Mapping|legend=1| 1 3 6 -17 | 0 -5 -13 70 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.553
+Optimal tunings:
+* CTE: ~2 = 1\1, ~243/200 = 339.554
+* POTE: ~2 = 1\1, ~243/200 = 339.553
-{{Val list|legend=1| 53, 205d, 258, 311, 675, 986, 1297c, 2283bc }}
+{{Optimal ET sequence|legend=1| 53, 205d, 258, 311, 675, 986 }}
 [[Badness]]: 0.113655
@@ Line 321: / Line 392: @@
 Comma list: 6250/6237, 19712/19683, 41503/41472
-Mapping: [{{val| 1 3 6 -17 36 }}, {{val| 0 -5 -13 70 -115 }}]
+Mapping: {{mapping| 1 3 6 -17 36 | 0 -5 -13 70 -115 }}
-Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.554
+Optimal tunings:
+* CTE: ~2 = 1\1, ~243/200 = 339.554
+* POTE: ~2 = 1\1, ~243/200 = 339.554
-Optimal GPV sequence: {{Val list| 53, 205de, 258, 311, 675, 986 }}
+Optimal ET sequence: {{Optimal ET sequence| 53, 205de, 258, 311, 675, 986 }}
 Badness: 0.064853
@@ Line 334: / Line 407: @@
 Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683
-Mapping: [{{val| 1 3 6 -17 36 17 }}, {{val| 0 -5 -13 70 -115 -47 }}]
+Mapping: {{mapping| 1 3 6 -17 36 17 | 0 -5 -13 70 -115 -47 }}
-Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.554
+Optimal tunings:
+* CTE: ~2 = 1\1, ~243/200 = 339.554
+* POTE: ~2 = 1\1, ~243/200 = 339.554
-Optimal GPV sequence: {{Val list| 53, 205de, 258, 311, 675, 986, 1661cf }}
+Optimal ET sequence: {{Optimal ET sequence| 53, 205de, 258, 311, 675, 986, 1661cf }}
 Badness: 0.030347
@@ Line 347: / Line 422: @@
 Comma list: 625/624, 1225/1224, 2080/2079, 2200/2197, 2431/2430
-Mapping: [{{val| 1 3 6 -17 36 17 -31 }}, {{val| 0 -5 -13 70 -115 -47 124 }}]
+Mapping: {{mapping| 1 3 6 -17 36 17 -31 | 0 -5 -13 70 -115 -47 124 }}
-Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.555
+Optimal tunings:
+* CTE: ~2 = 1\1, ~243/200 = 339.555
+* POTE: ~2 = 1\1, ~243/200 = 339.555
-Optimal GPV sequence: {{Val list| 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf }}
+Optimal ET sequence: {{Optimal ET sequence| 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf }}
 Badness: 0.024118
@@ Line 360: / Line 437: @@
 Comma list: 625/624, 1225/1224, 1540/1539, 1729/1728, 2080/2079, 2200/2197
-Mapping: [{{val| 1 3 6 -17 36 17 -31 15 }}, {{val| 0 -5 -13 70 -115 -47 124 -38 }}]
+Mapping: {{mapping| 1 3 6 -17 36 17 -31 15 | 0 -5 -13 70 -115 -47 124 -38 }}
-Optimal tuning (POTE): ~2 = 1\1, ~208/171 = 339.555
+Optimal tunings:
+* CTE: ~2 = 1\1, ~208/171 = 339.555
+* POTE: ~2 = 1\1, ~208/171 = 339.555
-Optimal GPV sequence: {{Val list| 53, 205deg, 258g, 311, 675, 986, 1661cfh }}
+Optimal ET sequence: {{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 play on the words ''bi-'' and ''amity''.
+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
@@ Line 375: / Line 454: @@
 [[Comma list]]: 245/243, 64827/64000
-[[Mapping]]: [{{val| 2 1 -1 3 }}, {{val| 0 5 13 6 }}]
+{{Mapping|legend=1| 2 1 -1 3 | 0 5 13 6 }}
-Mapping generators: ~343/240, ~7/6
+: mapping generators: ~343/240, ~7/6
-{{Multival|legend=1| 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 tuning]] ([[POTE]]): ~2 = 1\1, ~7/6 = 260.402
+{{Optimal ET sequence|legend=1| 14c, 32c, 46, 106d, 152d }}
-{{Val list|legend=1| 14c, 32c, 46, 60, 106d }}
 [[Badness]]: 0.083601
@@ Line 392: / Line 471: @@
 Comma list: 121/120, 245/243, 441/440
-Mapping: [{{val| 2 1 -1 3 3 }}, {{val| 0 5 13 6 9 }}]
+Mapping: {{mapping| 2 1 -1 3 3 | 0 5 13 6 9 }}
-Mapping generators: ~99/70, ~7/6
+Optimal tunings:
+* CTE: ~99/70 = 1\2, ~7/6 = 260.653
+* POTE: ~99/70 = 1\2, ~7/6 = 260.393
-POTE generator: ~7/6 = 260.393
+Optimal ET sequence: {{Optimal ET sequence| 14c, 32c, 46, 152de, 198, 244dee }}
-Optimal GPV sequence: {{Val list| 14c, 32c, 46, 60e, 106de }}
 Badness: 0.035504
@@ Line 407: / Line 486: @@
 Comma list: 91/90, 121/120, 245/243, 441/440
-Mapping: [{{val| 2 1 -1 3 3 0 }}, {{val| 0 5 13 6 9 17 }}]
+Mapping: {{mapping| 2 1 -1 3 3 0 | 0 5 13 6 9 17 }}
-Mapping generators: ~55/39, ~7/6
+Optimal tunings:
+* CTE: ~55/39 = 1\2, ~7/6 = 260.811
+* POTE: ~55/39 = 1\2, ~7/6 = 260.618
-POTE generator: ~7/6 = 260.618
+Optimal ET sequence: {{Optimal ET sequence| 14cf, 32cf, 46 }}
-Optimal GPV sequence: {{Val list| 14cf, 32cf, 46, 106def, 152def }}
 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 play on the words ''half'' and ''amity''.
+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
@@ Line 424: / Line 503: @@
 [[Comma list]]: 2430/2401, 4000/3969
-[[Mapping]]: [{{val| 1 -2 -7 -4 }}, {{val| 0 10 26 19 }}]
+{{Mapping|legend=1| 1 8 19 15 | 0 -10 -26 -19 }}
-{{Multival|legend=1| 10 26 19 18 2 -29 }}
+: mapping generators: ~2, ~14/9
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 430.219
+Optimal tunings:
+* CTE: ~2 = 1\1, ~9/7 = 430.168
+* POTE: ~2 = 1\1, ~9/7 = 430.219
-{{Val list|legend=1| 14c, 39d, 53 }}
+{{Optimal ET sequence|legend=1| 14c, 39d, 53 }}
 [[Badness]]: 0.073956
@@ Line 439: / Line 520: @@
 Comma list: 99/98, 121/120, 2200/2187
-Mapping: [{{val| 1 -2 -7 -4 -3 }}, {{val| 0 10 26 19 18 }}]
+Mapping: {{mapping| 1 8 19 15 15 | 0 -10 -26 -19 -18 }}
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 430.192
+Optimal tunings:
+* CTE: ~2 = 1\1, ~9/7 = 430.220
+* POTE: ~2 = 1\1, ~9/7 = 430.192
-Optimal GPV sequence: {{Val list| 14c, 39d, 53 }}
+Optimal ET sequence: {{Optimal ET sequence| 14c, 39d, 53 }}
 Badness: 0.042947
@@ Line 452: / Line 535: @@
 Comma list: 99/98, 121/120, 275/273, 572/567
-Mapping: [{{val| 1 -2 -7 -4 -3 -11 }}, {{val| 0 10 26 19 18 41 }}]
+Mapping: {{mapping| 1 8 19 15 15 30 | 0 -10 -26 -19 -18 -41 }}
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 430.216
+Optimal tunings:
+* CTE: ~2 = 1\1, ~9/7 = 430.233
+* POTE: ~2 = 1\1, ~9/7 = 430.216
-Optimal GPV sequence: {{Val list| 14cf, 39df, 53 }}
+Optimal ET sequence: {{Optimal ET sequence| 14cf, 39df, 53 }}
 Badness: 0.029753
@@ Line 467: / Line 552: @@
 [[Comma list]]: 1029/1024, 1071875/1062882
-[[Mapping]]: [{{val| 1 -2 -7 4 }}, {{val| 0 15 39 -5 }}]
+{{Mapping|legend=1| 1 13 32 -1 | 0 -15 -39 5 }}
-{{Multival|legend=1| 15 39 -5 27 -50 -121 }}
+: mapping generators: ~2, ~320/189
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 286.787
+Optimal tunings:
+* CTE: ~2 = 1\1, ~189/160 = 286.816
+* POTE: ~2 = 1\1, ~189/160 = 286.787
-{{Val list|legend=1| 46, 113, 159 }}
+{{Optimal ET sequence|legend=1| 46, 113, 159, 205d, 364d }}
 [[Badness]]: 0.125733
@@ Line 482: / Line 569: @@
 Comma list: 385/384, 441/440, 1071875/1062882
-Mapping: [{{val| 1 -2 -7 4 8 }}, {{val| 0 15 39 -5 -19 }}]
+Mapping: {{mapping| 1 13 32 -1 -11 | 0 -15 -39 5 19 }}
-Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 286.797
+Optimal tunings:
+* CTE: ~2 = 1\1, ~33/28 = 286.813
+* POTE: ~2 = 1\1, ~33/28 = 286.797
-Optimal GPV sequence: {{Val list| 46, 113, 159 }}
+Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 205d, 364d }}
 Badness: 0.051111
@@ Line 495: / Line 584: @@
 Comma list: 325/324, 364/363, 385/384, 10985/10976
-Mapping: [{{val| 1 -2 -7 4 8 8 }}, {{val| 0 15 39 -5 -19 -18 }}]
+Mapping: {{mapping| 1 13 32 -1 -11 -10 | 0 -15 -39 5 19 18 }}
-Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 286.789
+Optimal tunings:
+* CTE: ~2 = 1\1, ~13/11 = 286.803
+* POTE: ~2 = 1\1, ~13/11 = 286.789
-Optimal GPV sequence: {{Val list| 46, 113, 159 }}
+Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 364df, 523ddff }}
 Badness: 0.030297
@@ Line 508: / Line 599: @@
 Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757
-Mapping: [{{val| 1 -2 -7 4 8 8 6 }}, {{val| 0 15 39 -5 -19 -18 -8 }}]
+Mapping: {{mapping| 1 13 32 -1 -11 -10 -2 | 0 -15 -39 5 19 18 8 }}
-Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 286.795
+Optimal tunings:
+* CTE: ~2 = 1\1, ~13/11 = 286.804
+* POTE: ~2 = 1\1, ~13/11 = 286.795
-Optimal GPV sequence: {{Val list| 46, 113, 159 }}
+Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 364df, 523ddff }}
 Badness: 0.022036
 == Trinity ==
-The ''trinity'' temperament (152 &amp; 159) tempers out the [[meter]], 703125/702464. It splits the interval of 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.
+The trinity temperament (152 &amp; 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
@@ Line 523: / Line 616: @@
 [[Comma list]]: 703125/702464, 1600000/1594323
-[[Mapping]]: [{{val| 1 8 19 46 }}, {{val| 0 -15 -39 -101 }}]
+{{Mapping|legend=1| 1 8 19 46 | 0 -15 -39 -101 }}
-{{Multival|legend=1| 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 tuning]] ([[POTE]]): ~2 = 1\1, ~168/125 = 513.178
+{{Optimal ET sequence|legend=1| 7d, …, 145d, 152, 311, 774, 1085 }}
-{{Val list|legend=1| 152, 311, 463, 774 }}
 [[Badness]]: 0.119453
@@ Line 538: / Line 631: @@
 Comma list: 3025/3024, 4000/3993, 19712/19683
-Mapping: [{{val| 1 8 19 46 18 }}, {{val| 0 -15 -39 -101 -34 }}]
+Mapping: {{mapping| 1 8 19 46 18 | 0 -15 -39 -101 -34 }}
-POTE generator: ~121/90 = 513.177
+Optimal tunings:
+* CTE: ~2 = 1\1, ~121/90 = 513.181
+* POTE: ~2 = 1\1, ~121/90 = 513.177
-Optimal GPV sequence: {{Val list| 152, 311, 463, 774, 1237e }}
+Optimal ET sequence: {{Optimal ET sequence| 7d, …, 145d, 152, 311, 774, 1085e, 1396e }}
 Badness: 0.031296
@@ Line 551: / Line 646: @@
 Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689
-Mapping: [{{val| 1 8 19 46 18 64 }}, {{val| 0 -15 -39 -101 -34 -141 }}]
+Mapping: {{mapping| 1 8 19 46 18 64 | 0 -15 -39 -101 -34 -141 }}
-Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.182
+Optimal tunings:
+* CTE: ~2 = 1\1, ~35/26 = 513.184
+* POTE: ~2 = 1\1, ~35/26 = 513.182
-Optimal GPV sequence: {{Val list| 152f, 311 }}
+Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311 }}
 Badness: 0.026418
@@ Line 564: / Line 661: @@
 Comma list: 595/594, 625/624, 833/832, 1575/1573, 8624/8619
-Mapping: [{{val| 1 8 19 46 18 64 -22 }}, {{val| 0 -15 -39 -101 -34 -141 61 }}]
+Mapping: {{mapping| 1 8 19 46 18 64 -22 | 0 -15 -39 -101 -34 -141 61 }}
-Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.186
+Optimal tunings:
+* CTE: ~2 = 1\1, ~35/26 = 513.185
+* POTE: ~2 = 1\1, ~35/26 = 513.186
-Optimal GPV sequence: {{Val list| 152f, 159, 311, 1092cdg, 1403cdg, 1714cdeg }}
+Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311, 1714cdeg, 2025cdefgg, 2336bccdefgg }}
 Badness: 0.025588
@@ Line 577: / Line 676: @@
 Comma list: 595/594, 625/624, 833/832, 969/968, 1216/1215, 1575/1573
-Mapping: [{{val| 1 8 19 46 18 64 -22 53 }}, {{val| 0 -15 -39 -101 -34 -141 61 -114 }}]
+Mapping: {{mapping| 1 8 19 46 18 64 -22 53 | 0 -15 -39 -101 -34 -141 61 -114 }}
-Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.185
+Optimal tunings:
+* CTE: ~2 = 1\1, ~35/26 = 513.184
+* POTE: ~2 = 1\1, ~35/26 = 513.185
-Optimal GPV sequence: {{Val list| 152f, 159, 311, 1403cdgh, 1714cdegh, 2025cdefgghh, 2336bccdefgghh }}
+Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311 }}
 Badness: 0.018412
@@ Line 590: / Line 691: @@
 Comma list: 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104
-Mapping: [{{val| 1 8 19 46 18 64 -22 53 49 }}, {{val| 0 -15 -39 -101 -34 -141 61 -114 -104 }}]
+Mapping: {{mapping| 1 8 19 46 18 64 -22 53 49 | 0 -15 -39 -101 -34 -141 61 -114 -104 }}
-Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.185
+Optimal tunings:
+* CTE: ~2 = 1\1, ~35/26 = 513.184
+* POTE: ~2 = 1\1, ~35/26 = 513.185
-Optimal GPV sequence: {{Val list| 152f, 159, 311, 1092cdgh, 1403cdgh, 1714cdeghi }}
+Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311, 1714cdeghi, 2025cdefgghhi, 2336bccdefgghhi }}
 Badness: 0.014343
@@ Line 603: / Line 706: @@
 Comma list: 595/594, 625/624, 760/759, 784/783, 833/832, 875/874, 969/968, 1045/1044
-Mapping: [{{val| 1 8 19 46 18 64 -22 53 49 72 }}, {{val| 0 -15 -39 -101 -34 -141 61 -114 -104 -157 }}]
+Mapping: {{mapping| 1 8 19 46 18 64 -22 53 49 72 | 0 -15 -39 -101 -34 -141 61 -114 -104 -157 }}
-Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.186
+Optimal tunings:
+* CTE: ~2 = 1\1, ~35/26 = 513.185
+* POTE: ~2 = 1\1, ~35/26 = 513.186
-Optimal GPV sequence: {{Val list| 152fj, 159, 311, 781dh, 1092cdgh, 1403cdgh }}
+Optimal ET sequence: {{Optimal ET sequence| 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij }}
 Badness: 0.012038
+== Amicable ==
+The amicable temperament (99 &amp; 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 &amp; 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 &amp; 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 &amp; 113) and pseudoamorous (14cf &amp; 99ef) are the corresponding low-complexity interpretations. Floral (198 &amp; 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 &amp; 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|legend=1| 1 3 6 5 | 0 -20 -52 -31 }}
+Optimal tunings:
+* CTE: ~2 = 1\1, ~21/20 = 84.8831
+* POTE: ~2 = 1\1, ~21/20 = 84.880
+{{Optimal ET sequence|legend=1| 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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{Optimal ET sequence| 85c, 113, 198, 311, 1753beff, 2064beff, 2375bceff }}
+Badness: 0.028267
+== Calamity ==
+The calamity temperament (46 &amp; 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|legend=1| 1 13 32 -15 | 0 -25 -65 39 }}
+[[Optimal tuning]] (CTE): ~2 = 1\1, ~48/35 = 547.909
+{{Optimal ET sequence|legend=1| 46, 219c, 265, 311 }}
+[[Badness]]: 0.198130
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 3025/3024, 12005/11979, 131072/130977
+Mapping: {{mapping| 1 13 32 -15 -18 | 0 -25 -65 39 47 }}
+Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.908
+Optimal ET sequence: {{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: {{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: {{Optimal ET sequence| 46, 265, 311, 668, 979, 1290 }}
+Badness: 0.033617
 == Familia ==
-The familia temperament (113 &amp; 152) tempers out the mirkwai comma, 16875/16807. It splits the interval of acute minor tenth (~243/100) in five.
+The familia temperament (113 &amp; 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
@@ Line 618: / Line 963: @@
 [[Comma list]]: 16875/16807, 1600000/1594323
-[[Mapping]]: [{{val| 1 8 19 20 }}, {{val| 0 -25 -65 -67 }}]
+{{Mapping|legend=1| 1 8 19 20 | 0 -25 -65 -67 }}
-{{Multival|legend=1| 25 65 67 45 36 -27 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11907/10000 = 307.941
+Optimal tunings:
+* CTE: ~2 = 1\1, ~11907/10000 = 307.915
+* POTE: ~2 = 1\1, ~11907/10000 = 307.941
-{{Val list|legend=1| 39d, 74cd, 113, 152, 265, 417, 986d }}
+{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417, 1516ccdd, 1933ccdd }}
 [[Badness]]: 0.144551
@@ Line 633: / Line 978: @@
 Comma list: 540/539, 1375/1372, 1600000/1594323
-Mapping: [{{val| 1 8 19 20 5 }}, {{val| 0 -25 -65 -67 -6 }}]
+Mapping: {{mapping| 1 8 19 20 5 | 0 -25 -65 -67 -6 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3200/2673 = 307.906
+Optimal tunings:
+* CTE: ~2 = 1\1, ~3200/2673 = 307.915
+* POTE: ~2 = 1\1, ~3200/2673 = 307.906
-Optimal GPV sequence: {{Val list| 39d, 74cd, 113, 152, 417, 569de, 721de }}
+Optimal ET sequence: {{Optimal ET sequence| 39d, 74cd, 113, 152, 265, 417, 1099cdee }}
 Badness: 0.051740
@@ Line 646: / Line 993: @@
 Comma list: 540/539, 729/728, 1375/1372, 2205/2197
-Mapping: [{{val| 1 8 19 20 5 25 }}, {{val| 0 -25 -65 -67 -6 -83 }}]
+Mapping: {{mapping| 1 8 19 20 5 25 | 0 -25 -65 -67 -6 -83 }}
-Optimal tuning (POTE): ~2 = 1\1, ~143/120 = 307.913
+Optimal tunings:
+* CTE: ~2 = 1\1, ~143/120 = 307.922
+* POTE: ~2 = 1\1, ~143/120 = 307.913
-Optimal GPV sequence: {{Val list| 39df, 74cdf, 113, 152f, 265, 417f }}
+Optimal ET sequence: {{Optimal ET sequence| 39df, 74cdf, 113, 152f, 265 }}
 Badness: 0.038473
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Amity family| ]] <!-- main article -->
-[[Category:Amity|#]] <!-- key article -->
+[[Category:Amity| ]] <!-- key article -->
 [[Category:Rank 2]]

Amity family: Difference between revisions

Latest revision as of 00:26, 24 June 2025

Amity

Overview to extensions

Septimal amity

11-limit

13-limit

17-limit

19-limit

Hitchcock

13-limit

17-limit

19-limit

Catamite

13-limit

17-limit

19-limit

Hemiamity

13-limit

Accord

11-limit

Houborizic

11-limit

13-limit

Houbor

13-limit

Paramity

11-limit

13-limit

17-limit

19-limit

Bamity

11-limit

13-limit

Hamity

11-limit

13-limit

Gamity

11-limit

13-limit

17-limit

Trinity

11-limit

13-limit

17-limit

19-limit

23-limit

29-limit

Amicable

Amical

13-limit

Amorous

13-limit

Pseudoamical

13-limit

Pseudoamorous

13-limit

Floral

13-limit

Humorous

13-limit

Calamity

11-limit

13-limit

Familia

11-limit

13-limit

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