Xenharmonic Wiki

Amity family

Revision as of 09:29, 3 June 2026 by FloraC (talk | contribs) (+ more intro)
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

Accord tempers out 126/125, the starling comma, and may be described as the 39d & 46 temperament.

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 was named by Xenllium in 2021 for its close relation 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. It was named by Xenllium in 2021, with the prefix para- signifying its relation to amity.

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:

  • WE: ~343/240 = 600.3379 ¢, ~7/6 = 260.5488 ¢
error map: ⟨+0.676 +1.127 +0.483 -4.519]
  • CWE: ~343/240 = 600.0000 ¢, ~7/6 = 260.4686 ¢
error map: ⟨0.000 +0.388 -0.222 -6.014]

Optimal ET sequence: 14c, 32c, 46, 60, 106d

Badness (Sintel): 2.12

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:

  • WE: ~99/70 = 600.5039 ¢, ~7/6 = 260.6119 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~7/6 = 260.4948 ¢

Optimal ET sequence: 14c, 32c, 46, 60e, 106de

Badness (Sintel): 1.17

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:

  • WE: ~55/39 = 600.3896 ¢, ~7/6 = 260.7874 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~7/6 = 260.6927 ¢

Optimal ET sequence: 14cf, 32cf, 46, 106deff

Badness (Sintel): 1.28

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 -2 -7 -4], ⟨0 10 26 19]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • WE: ~2 = 1199.5499 ¢, ~9/7 = 430.0579 ¢
error map: ⟨-0.450 -0.476 -1.657 +4.075]
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 430.1958 ¢
error map: ⟨0.000 +0.003 -1.222 +4.895]

Optimal ET sequence: 14c, 39d, 53

Badness (Sintel): 1.87

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 -7 -4 -3], ⟨0 10 26 19 18]]

Optimal tunings:

  • WE: ~2 = 1200.2215 ¢, ~9/7 = 430.2713 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 430.2036 ¢

Optimal ET sequence: 14c, 39d, 53

Badness (Sintel): 1.42

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:

  • WE: ~2 = 1200.1379 ¢, ~9/7 = 430.2658 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 430.2227 ¢

Optimal ET sequence: 14cf, 39df, 53

Badness (Sintel): 1.23

Gamity

Gamity tempers out 1029/1024, the gamelisma, and may be described as the 46 & 113 temperament. It splits the interval of the grave major sixth in three. It was named by Xenllium in 2021 as a contraction of gamelismic and amity.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1071875/1062882

Mapping: [⟨1 -2 -7 4], ⟨0 15 39 -5]]

mapping generators: ~2, ~189/160

Optimal tunings:

  • WE: ~2 = 1200.4434 ¢, ~189/160 = 286.8930 ¢
error map: ⟨+0.443 +0.553 -0.591 -1.571]
  • CWE: ~2 = 1200.0000 ¢, ~189/160 = 286.7955 ¢
error map: ⟨0.000 -0.022 -1.288 -2.804]

Optimal ET sequence: 46, 113, 159

Badness (Sintel): 3.18

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 -7 4 8], ⟨0 15 39 -5 -19]]

Optimal tunings:

  • WE: ~2 = 1200.3315 ¢, ~33/28 = 286.8764 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/28 = 286.8002 ¢

Optimal ET sequence: 46, 113, 159

Badness (Sintel): 1.69

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 -7 4 8 8], ⟨0 15 39 -5 -19 -18]]

Optimal tunings:

  • WE: ~2 = 1200.4248 ¢, ~13/11 = 286.8906 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/11 = 286.7912 ¢

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

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -2 -7 4 8 8 6], ⟨0 15 39 -5 -19 -18 -8]]

Optimal tunings:

  • WE: ~2 = 1200.3287 ¢, ~13/11 = 286.8731 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/11 = 286.7957 ¢

Optimal ET sequence: 46, 113, 159

Badness (Sintel): 1.12

Trinity

Trinity tempers out 703125/702464, the meter, and may be described as the 152 & 159 temperament. 178\311 is a recommendable generator. It was named by Xenllium in 2022 for the fact that it splits amity's generator in three.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 1600000/1594323

Mapping: [⟨1 -7 -20 -55], ⟨0 15 39 101]]

mapping generators: ~2, ~125/84

Optimal tunings:

  • WE: ~2 = 1199.9474 ¢, ~125/84 = 686.7921 ¢
error map: ⟨-0.053 +0.295 -0.369 +0.071]
  • CWE: ~2 = 1200.0000 ¢, ~125/84 = 686.8215 ¢
error map: ⟨0.000 +0.367 -0.276 +0.144]

Optimal ET sequence: 7d, …, 152, 311, 463, 774

Badness (Sintel): 3.02

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -7 -20 -55 -16], ⟨0 15 39 101 34]]

Optimal tunings:

  • WE: ~2 = 1199.9090 ¢, ~125/84 = 686.7710 ¢
  • CWE: ~2 = 1200.0000 ¢, ~125/84 = 686.8218 ¢

Optimal ET sequence: 7d, …, 152, 311, 463, 774, 1237e

Badness (Sintel): 1.03

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -7 -20 -55 -16 -77], ⟨0 15 39 101 34 141]]

Optimal tunings:

  • WE: ~2 = 1199.9531 ¢, ~52/35 = 686.7909 ¢
  • CWE: ~2 = 1200.0000 ¢, ~52/35 = 686.8172 ¢

Optimal ET sequence: 7df, …, 152f, 311

Badness (Sintel): 1.09

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -7 -20 -55 -16 -77 39], ⟨0 15 39 101 34 141 -61]]

Optimal tunings:

  • WE: ~2 = 1200.0347 ¢, ~52/35 = 686.8342 ¢
  • CWE: ~2 = 1200.0000 ¢, ~52/35 = 686.8146 ¢

Optimal ET sequence: 152f, 159, 311, 1092cdg

Badness (Sintel): 1.30

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 -7 -20 -55 -16 -77 39 -61], ⟨0 15 39 101 34 141 -61 114]]

Optimal tunings:

  • WE: ~2 = 1200.0406 ¢, ~52/35 = 686.8382 ¢
  • CWE: ~2 = 1200.0000 ¢, ~52/35 = 686.8152 ¢

Optimal ET sequence: 152f, 159, 311

Badness (Sintel): 1.12

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 -7 -20 -55 -16 -77 39 -61 -55], ⟨0 15 39 101 34 141 -61 114 104]]

Optimal tunings:

  • WE: ~2 = 1200.0311 ¢, ~52/35 = 686.8324 ¢
  • CWE: ~2 = 1200.0000 ¢, ~52/35 = 686.8148 ¢

Optimal ET sequence: 152f, 159, 311

Badness (Sintel): 1.03

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 -7 -20 -55 -16 -77 39 -61 -55 -85], ⟨0 15 39 101 34 141 -61 114 104 157]]

Optimal tunings:

  • WE: ~2 = 1200.0311 ¢, ~52/35 = 686.8318 ¢
  • CWE: ~2 = 1200.0000 ¢, ~52/35 = 686.8142 ¢

Optimal ET sequence: 152fj, 159, 311, 781dh

Badness (Sintel): 1.01

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 -17 -46 -26], ⟨0 20 52 31]]

mapping generators: ~2, ~40/21

Optimal tunings:

  • WE: ~2 = 1199.9463 ¢, ~40/21 = 1115.0701 ¢
error map: ⟨-0.054 +0.361 -0.195 -0.254]
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1115.1187 ¢
error map: ⟨0.000 +0.419 -0.142 -0.146]

Optimal ET sequence: 14c, …, 85c, 99, 212, 311, 410, 1131, 1541b

Badness (Sintel): 1.15

Amical

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -17 -46 -26 154], ⟨0 20 52 31 -162]]

Optimal tunings:

  • WE: ~2 = 1200.0008 ¢, ~40/21 = 1115.1165 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1115.1157 ¢

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

Badness (Sintel): 3.33

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -17 -46 -26 154 118], ⟨0 20 52 31 -162 -123]]

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~40/21 = 1115.1094 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1115.1162 ¢

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

Badness (Sintel): 2.06

Amorous

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -17 -46 -26 -135], ⟨0 20 52 31 149]]

Optimal tunings:

  • WE: ~2 = 1200.0380 ¢, ~40/21 = 1115.1458 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1115.1109 ¢

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

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

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

apping: [⟨1 -17 -46 -26 -135 -171], ⟨0 20 52 31 149 188]]

Optimal tunings:

  • WE: ~2 = 1200.0543 ¢, ~40/21 = 1115.1594 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1115.1095 ¢

Optimal ET sequence: 99ef, 212, 311, 834, 1145c

Badness (Sintel): 1.43

Pseudoamical

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -17 -46 -26 62], ⟨0 20 52 31 -63]]

Optimal tunings:

  • WE: ~2 = 1200.3436 ¢, ~40/21 = 1115.4102 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1115.0925 ¢

Optimal ET sequence: 99, 113, 212, 961ccdeee

Badness (Sintel): 2.84

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -17 -46 -26 62 26], ⟨0 20 52 31 -63 -24]]

Optimal tunings:

  • WE: ~2 = 1200.4212 ¢, ~40/21 = 1115.4787 ¢
  • CWE: ~2 = 1200.0000 ¢, 40/21 = 1115.0885 ¢

Optimal ET sequence: 99, 113, 212, 537cdeff, 749ccdeefff

Badness (Sintel): 1.94

Pseudoamorous

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -17 -46 -26 -43], ⟨0 20 52 31 50]]

Optimal tunings:

  • WE: ~2 = 1199.8269 ¢, ~21/11 = 1114.9475 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/11 = 1115.1038 ¢

Optimal ET sequence: 99e, 212e *

  • optimal patent val: 113

Badness (Sintel): 1.87

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -17 -46 -26 -43 -79], ⟨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: 99ef, 113, 212ef

Badness (Sintel): 1.77

Floral

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 -14 -40 -21 -30], ⟨0 20 52 31 43]]

mapping generators: ~99/70, ~66/49

Optimal tunings:

  • WE: ~99/70 = 600.0000 ¢, ~66/49 = 515.1257 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~66/49 = 515.1215 ¢

Optimal ET sequence: 14c, … 198, 212, 410

Badness (Sintel): 2.15

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 -14 -40 -21 -30 -63], ⟨0 20 52 31 43 82]]

Optimal tunings:

  • WE: ~99/70 = 599.9923 ¢, ~66/49 = 515.1184 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~66/49 = 515.1246 ¢

Optimal ET sequence: 14cf, …, 198, 410

Badness (Sintel): 1.53

Humorous

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -37 -98 -57 16], ⟨0 40 104 62 -13]]

mapping generators: ~2, ~3993/2048

Optimal tunings:

  • WE: ~2 = 1199.9189 ¢, ~3993/2048 = 1157.4826 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3993/2048 = 1157.5602 ¢

Optimal ET sequence: 85c, 113, 198, 311, 509, 820

Badness (Sintel): 1.93

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -37 -98 -57 16 -59], ⟨0 40 104 62 -13 65]]

Optimal tunings:

  • WE: ~2 = 1199.8903 ¢, ~39/20 = 1157.4551 ¢
  • CWE: ~2 = 1200.0000 ¢, ~39/20 = 1157.5600 ¢

Optimal ET sequence: 85c, 113, 198, 311, 509, 820f

Badness (Sintel): 1.17

Calamity

Named by Flora Canou in 2023, calamity tempers out 2100875/2097152, the rainy comma, and may be described as the 46 & 311 temperament. It splits the interval of grave major sixth plus an octave into five.

Subgroup: 2.3.5.7

Comma list: 1600000/1594323, 2100875/2097152

Mapping: [⟨1 -12 -33 24], ⟨0 25 65 -39]]

mapping generators: ~2, ~35/24

Optimal tunings:

  • WE: ~2 = 1200.0332 ¢, ~35/24 = 652.1081 ¢
error map: ⟨+0.033 +0.348 -0.387 -0.243]
  • CWE: ~2 = 1200.0000 ¢, ~35/24 = 652.0902 ¢
error map: ⟨0.000 +0.300 -0.452 -0.343]

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

Badness (Sintel): 5.01

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -12 -33 24 29], ⟨0 25 65 -39 -47]]

Optimal tunings:

  • WE: ~2 = 1199.9938 ¢, ~35/24 = 652.0887 ¢
  • CWE: ~2 = 1200.0000 ¢, ~35/24 = 652.0920 ¢

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

Badness (Sintel): 2.00

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -12 -33 24 29 45], ⟨0 25 65 -39 -47 -76]]

Optimal tunings:

  • WE: ~2 = 1199.9742 ¢, ~35/24 = 652.0794 ¢
  • CWE: ~2 = 1200.0000 ¢, ~35/24 = 652.0934 ¢

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

Badness (Sintel): 1.39

Familia

Named by Xenllium in 2021, familia tempers out 16875/16807, the mirkwai comma, and may be described as the 113 & 152 temperament. It is generated by a major sixth, five of which octave reduced make the original generator of amity.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 1600000/1594323

Mapping: [⟨1 -17 -46 -47], ⟨0 25 65 67]]

mapping generators: ~2, ~6561/3920

Optimal tunings:

  • WE: ~2 = 1199.9165 ¢, ~6561/3920 = 892.0267 ¢
error map: ⟨-0.084 +0.132 -0.736 +0.888]
  • CWE: ~2 = 1200.0000 ¢, ~6561/3920 = 892.0871 ¢
error map: ⟨0.000 +0.224 -0.649 +1.013]

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

Badness (Sintel): 3.66

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -17 -46 -47 -1], ⟨0 25 65 67 6]]

Optimal tunings:

  • WE: ~2 = 1199.7927 ¢, ~2673/1600 = 891.9397 ¢
  • CWE: ~2 = 1200.0000 ¢, ~2673/1600 = 892.0909 ¢

Optimal ET sequence: 39d, 74cd, 113, 152, 417, 569de, 721de

Badness (Sintel): 1.71

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -17 -46 -47 -1 -58], ⟨0 25 65 67 6 83]]

Optimal tunings:

  • WE: ~2 = 1199.7844 ¢, ~240/143 = 892.9268 ¢
  • CWE: ~2 = 1200.0000 ¢, ~240/143 = 892.0839 ¢

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

Badness (Sintel): 1.59

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