Xenharmonic Wiki

Amity family

Revision as of 08:31, 1 June 2026 by FloraC (talk | contribs) (Switch to Sintel's badness, WE & CWE tunings (1/))
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 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. However its functional generator is the octave complement thereof, five of which stack to the 12th harmonic. Its ploidacot is thus 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 -2 -7], ⟨0 5 13]]

mapping generators: ~2, ~400/243

Optimal tunings:

  • WE: ~2 = 1199.9135 ¢, ~400/243 = 860.4191 ¢
error map: ⟨-0.087 +0.314 -0.259]
  • CWE: ~2 = 1200.000 ¢, ~400/243 = 860.4726 ¢
error map: ⟨0.000 +0.408 -0.169]

Optimal ET sequence: 7, 32c, 39, 46, 53, 152, 205, 463, 668, 873

Badness (Sintel): 0.515

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 -2 -7 15], ⟨0 5 13 -17]]

Optimal tunings:

  • WE: ~2 = 1199.6100 ¢, ~105/64 = 860.2881 ¢
error map: ⟨-0.390 +0.266 +0.162 +0.426]
  • CWE: ~2 = 1200.0000 ¢, ~105/64 = 860.5650 ¢
error map: ⟨0.000 +0.870 +1.032 +1.568]

Optimal ET sequence: 7, 39, 46, 53, 99, 152, 251, 350, 601cd, 905bcdd

Badness (Sintel): 0.598

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 -7 15 -41], ⟨0 5 13 -17 62]]

Optimal tunings:

  • WE: ~2 = 1199.5961 ¢, ~105/64 = 860.2460 ¢
  • CWE: ~2 = 1200.0000 ¢, ~105/64 = 860.5314 ¢

Optimal ET sequence: 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee

Badness (Sintel): 1.04

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 -7 15 -41 -30], ⟨0 5 13 -17 62 47]]

Optimal tunings:

  • WE: ~2 = 1199.5437 ¢, ~105/64 = 860.1922 ¢
  • CWE: ~2 = 1200.0000 ¢, ~105/64 = 860.5140 ¢

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

* optimal patent val: 205

Badness (Sintel): 1.16

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -2 -7 15 -41 -30 17], ⟨0 5 13 -17 62 47 -18]]

Optimal tunings:

  • WE: ~2 = 1199.3376 ¢, ~28/17 = 860.0617 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/17 = 860.5317 ¢

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

Badness (Sintel): 1.33

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 -2 -7 15 -41 -30 17 -23], ⟨0 5 13 -17 62 47 -18 38]]

Optimal tunings:

  • WE: ~2 = 1199.3361 ¢, ~28/17 = 860.0605 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/17 = 860.5310 ¢

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

Badness (Sintel): 1.14

Hitchcock

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 -7 15 -3], ⟨0 5 13 -17 9]]

Optimal tunings:

  • WE: ~2 = 1199.9979 ¢, ~18/11 = 860.6089 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.6104 ¢

Optimal ET sequence: 7, 39, 46, 53, 99

Badness (Sintel): 1.16

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 -7 15 -3 8], ⟨0 5 13 -17 9 -6]]

Optimal tunings:

  • WE: ~2 = 1200.2037 ¢, ~18/11 = 860.5820 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.7268 ¢

Optimal ET sequence: 7, 39, 46, 53, 99

Badness (Sintel): 0.928

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -2 -7 15 -3 8 17], ⟨0 5 13 -17 9 -6 -18]]

Optimal tunings:

  • WE: ~2 = 1200.0179 ¢, ~18/11 = 860.6472 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.6344 ¢

Optimal ET sequence: 7, 39, 46, 99

Badness (Sintel): 0.988

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 -2 -7 15 -3 8 17 15], ⟨0 5 13 -17 9 -6 -18 -15]]

Optimal tunings:

  • WE: ~2 = 1200.1977 ¢, ~18/11 = 860.7343 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.5917 ¢

Optimal ET sequence: 7, 46, 53, 99h

Badness (Sintel): 1.07

Stalagmite

The stalagmite temperament (46 & 99ef) tempers out 441/440 (werckisma) and 896/891 (pentacircle comma) 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.

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 -7 15 30], ⟨0 5 13 -17 -37]]

Optimal tunings:

  • WE: ~2 = 1199.3915 ¢, ~105/64 = 860.2240 ¢
  • CWE: ~2 = 1200.0000 ¢, ~105/64 = 860.6649 ¢

Optimal ET sequence: 46, 99e, 145, 244e

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 -7 15 30 41], ⟨0 5 13 -17 -37 -52]]

Optimal tunings:

  • WE: ~2 = 1199.3577 ¢, ~105/64 = 860.2259 ¢
  • CWE: ~2 = 1200.0000 ¢, ~105/64 = 860.6929 ¢

Optimal ET sequence: 46, 99ef, 145

Badness (Sintel): 1.41

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -2 -7 15 30 41 17], ⟨0 5 13 -17 -37 -52 -18]]

Optimal tunings:

  • WE: ~2 = 1199.3555 ¢, ~28/17 = 860.2244 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/17 = 860.6932 ¢

Optimal ET sequence: 46, 99ef, 145

Badness (Sintel): 1.08

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 -2 -7 15 30 41 17 48], ⟨0 5 13 -17 -37 -52 -18 -61]]

Optimal tunings:

  • WE: ~2 = 1199.3599 ¢, ~28/17 = 860.2161 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/17 = 860.6820 ¢

Optimal ET sequence: 46, 99ef, 145

Badness (Sintel): 1.15

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:

  • WE: ~99/70 = 599.8218 ¢, ~64/55 = 260.4833 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~64/55 = 260.5615 ¢

Optimal ET sequence: 46, 106, 152, 350, 502d, 852bcdde

Badness (Sintel): 1.04

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:

  • WE: ~99/70 = 599.7965 ¢, ~64/55 = 260.4951 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~64/55 = 260.5872 ¢

Optimal ET sequence: 46, 106f, 152f, 198, 350f, 548cdff

Badness (Sintel): 1.07

Accord

Subgroup: 2.3.5.7

Comma list: 126/125, 100352/98415

Mapping: [⟨1 -2 -7 -18], ⟨0 5 13 29]]

Optimal tunings:

  • WE: ~2 = 1198.7458 ¢, ~224/135 = 860.1071 ¢
error map: ⟨-1.254 +1.089 +3.858 -3.144]
  • CWE: ~2 = 1200.0000 ¢, ~224/135 = 860.9433 ¢
error map: ⟨0.000 +2.761 +5.949 -1.471]

Optimal ET sequence: 7d, …, 39d, 46, 131c

Badness (Sintel): 2.42

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 -7 -18 -3], ⟨0 5 13 29 9]]

Optimal tunings:

  • WE: ~2 = 1199.3576 ¢, ~18/11 = 860.4919 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.9224 ¢

Optimal ET sequence: 7d, 39d, 46

Badness (Sintel): 1.40

Houborizic

Houborizic tempers out 225/224, the marvel comma, and may be described as the 53 & 60 temperament. 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 -2 -7 -23], ⟨0 5 13 36]]

Optimal tunings:

  • WE: ~2 = 1200.4959 ¢, ~400/243 = 860.5922 ¢
error map: ⟨+0.496 +0.014 -2.086 +1.090]
  • CWE: ~2 = 1200.0000 ¢, ~400/243 = 860.2558 ¢
error map: ⟨0.000 -0.676 -2.989 +0.382]

Optimal ET sequence: 7d, …, 46d, 53, 113, 166

Badness (Sintel): 1.69

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 -7 -23 35], ⟨0 5 13 36 -44]]

Optimal tunings:

  • WE: ~2 = 1200.4973 ¢, ~400/243 = 860.5930 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/243 = 860.2386 ¢

Optimal ET sequence: 53, 113, 166

Badness (Sintel): 2.24

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 -7 -23 35 8], ⟨0 5 13 36 -44 -6]]

Optimal tunings:

  • WE: ~2 = 1200.5022 ¢, ~64/39 = 860.5964 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/39 = 860.2378 ¢

Optimal ET sequence: 53, 113, 166

Badness (Sintel): 1.36

Houbor

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 -7 -23 -3], ⟨0 5 13 36 9]]

Optimal tunings:

  • WE: ~2 = 1201.1684 ¢, ~18/11 = 861.0240 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.2273 ¢

Optimal ET sequence: 7d, 53, 60e, 113e

Badness (Sintel): 1.49

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 -7 -23 -3 8], ⟨0 -5 -13 -36 -9 6]]

Optimal tunings:

  • WE: ~2 = 1200.9339 ¢, ~18/11 = 860.8859 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.2376 ¢

Optimal ET sequence: 7d, 53, 113e

Badness (Sintel): 1.29

Paramity

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

Subgroup: 2.3.5.7

Comma list: 65625/65536, 1600000/1594323

Mapping: [⟨1 -2 -7 53], ⟨0 5 13 -70]]

Optimal tunings:

  • WE: ~2 = 1200.0199 ¢, ~400/243 = 860.4612 ¢
error map: ⟨+0.020 +0.311 -0.458 -0.055]
  • CWE: ~2 = 1200.0000 ¢, ~400/243 = 860.4468 ¢
error map: ⟨0.000 +0.279 -0.505 -0.103]

Optimal ET sequence: 53, 205d, 258, 311, 675, 986, 1297c

Badness (Sintel): 2.88

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 -7 53 -79], ⟨0 5 13 -70 115]]

Optimal tunings:

  • WE: ~2 = 1200.0139 ¢, ~400/243 = 860.4562 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/243 = 860.4462 ¢

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

Badness (Sintel): 2.14

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 -7 53 -79 -30], ⟨0 5 13 -70 115 47]]

Optimal tunings:

  • WE: ~2 = 1199.9945 ¢, ~385/234 = 860.4417 ¢
  • CWE: ~2 = 1200.0000 ¢, ~385/234 = 860.4456 ¢

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

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -2 -7 53 -79 -30 93], ⟨0 5 13 -70 115 47 -124]]

Optimal tunings:

  • WE: ~2 = 1200.0026 ¢, ~385/234 = 860.4468 ¢
  • CWE: ~2 = 1200.0000 ¢, ~385/234 = 860.4450 ¢

Optimal ET sequence: 53, 311, 675

Badness (Sintel): 1.23

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 -2 -7 53 -79 -30 93 -23], ⟨0 5 13 -70 115 47 -124 38]]

Optimal tunings:

  • WE: ~2 = 1200.0196 ¢, ~208/171 = 860.4594 ¢
  • CWE: ~2 = 1200.0000 ¢, ~208/171 = 860.4454 ¢

Optimal ET sequence: 53, 311, 675, 986

Badness (Sintel): 1.06

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 = 600.000 ¢, ~7/6 = 260.563 ¢
  • POTE: ~343/240 = 600.000 ¢, ~7/6 = 260.402 ¢

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

Badness (Smith): 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 = 600.000 ¢, ~7/6 = 260.653 ¢
  • POTE: ~99/70 = 600.000 ¢, ~7/6 = 260.393 ¢

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

Badness (Smith): 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 = 600.000 ¢, ~7/6 = 260.811 ¢
  • POTE: ~55/39 = 600.000 ¢, ~7/6 = 260.618 ¢

Optimal ET sequence: 14cf, 32cf, 46

Badness (Smith): 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 = 1200.000 ¢, ~9/7 = 430.168 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 430.219 ¢

Optimal ET sequence: 14c, 39d, 53

Badness (Smith): 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 = 1200.000 ¢, ~9/7 = 430.220 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 430.192 ¢

Optimal ET sequence: 14c, 39d, 53

Badness (Smith): 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 = 1200.000 ¢, ~9/7 = 430.233 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 430.216 ¢

Optimal ET sequence: 14cf, 39df, 53

Badness (Smith): 0.029753

Gamity

Gamity tempers out 1029/1024, the gamelisma, and may be described as the 46 & 113 temperament. 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 = 1200.000 ¢, ~189/160 = 286.816 ¢
  • POTE: ~2 = 1200.000 ¢, ~189/160 = 286.787 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~33/28 = 286.813 ¢
  • POTE: ~2 = 1200.000 ¢, ~33/28 = 286.797 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~13/11 = 286.803 ¢
  • POTE: ~2 = 1200.000 ¢, ~13/11 = 286.789 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~13/11 = 286.804 ¢
  • POTE: ~2 = 1200.000 ¢, ~13/11 = 286.795 ¢

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

Badness (Smith): 0.022036

Trinity

Trinity tempers out 703125/702464, the meter, and may be described as the 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 ¢ 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 = 1200.000 ¢, ~168/125 = 513.180 ¢
  • POTE: ~2 = 1200.000 ¢, ~168/125 = 513.178 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~121/90 = 513.181 ¢
  • POTE: ~2 = 1200.000 ¢, ~121/90 = 513.177 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~35/26 = 513.184 ¢
  • POTE: ~2 = 1200.000 ¢, ~35/26 = 513.182 ¢

Optimal ET sequence: 152f, 159, 311

Badness (Smith): 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 = 1200.000 ¢, ~35/26 = 513.185 ¢
  • POTE: ~2 = 1200.000 ¢, ~35/26 = 513.186 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~35/26 = 513.184 ¢
  • POTE: ~2 = 1200.000 ¢, ~35/26 = 513.185 ¢

Optimal ET sequence: 152f, 159, 311

Badness (Smith): 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 = 1200.000 ¢, ~35/26 = 513.184 ¢
  • POTE: ~2 = 1200.000 ¢, ~35/26 = 513.185 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~35/26 = 513.185 ¢
  • POTE: ~2 = 1200.000 ¢, ~35/26 = 513.186 ¢

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

Badness (Smith): 0.012038

Amicable

Amicable tempers out the breedsma as well as the canousma, and may be described as the 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.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1600000/1594323

Mapping: [⟨1 3 6 5], ⟨0 -20 -52 -31]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8831 ¢
  • POTE: ~2 = 1200.000 ¢, ~21/20 = 84.880 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~21/20 = 84.8843 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8843 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~21/20 = 84.8837 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8838 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~21/20 = 84.8883 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8896 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~21/20 = 84.8895 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8910 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~21/20 = 84.9005 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.9091 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~21/20 = 84.9049 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.9127 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~21/20 = 84.9022 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8917 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~21/20 = 84.9153 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.9164 ¢

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

Badness (Smith): 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 = 600.0000 ¢, ~21/20 = 84.8781 ¢
  • POTE: ~99/70 = 600.0000 ¢, ~21/20 = 84.8788 ¢

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

Badness (Smith): 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 = 600.0000 ¢, ~21/20 = 84.8759 ¢
  • POTE: ~99/70 = 600.0000 ¢, ~21/20 = 84.8750 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~4096/3993 = 42.4414 ¢
  • POTE: ~2 = 1200.0000 ¢, ~4096/3993 = 42.4391 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~40/39 = 42.4425 ¢
  • POTE: ~2 = 1200.0000 ¢, ~40/39 = 42.4391 ¢

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

Badness (Smith): 0.028267

Calamity

Calamity tempers out 2100875/2097152, the rainy comma, and may be described as the 46 & 311 temperament. It splits 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 = 1200.000 ¢, ~48/35 = 547.909 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~48/35 = 547.908 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~48/35 = 547.907 ¢

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

Badness (Smith): 0.033617

Familia

Familia tempers out 16875/16807, the mirkwai comma, and may be described as the 113 & 152 temperament. 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 = 1200.000 ¢, ~11907/10000 = 307.915 ¢
  • POTE: ~2 = 1200.000 ¢, ~11907/10000 = 307.941 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~3200/2673 = 307.915 ¢
  • POTE: ~2 = 1200.000 ¢, ~3200/2673 = 307.906 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~143/120 = 307.922 ¢
  • POTE: ~2 = 1200.000 ¢, ~143/120 = 307.913 ¢

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

Badness (Smith): 0.038473

Retrieved from "https://en.xen.wiki/index.php?title=Amity_family&oldid=231402"