Xenharmonic Wiki

No-fives subgroup temperaments

Revision as of 12:59, 30 April 2026 by FloraC (talk | contribs) (Switch to Sintel's badness, WE & CWE tunings (7/))
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.

This is a collection of subgroup temperaments which omit the prime harmonic of 5.

Temperaments with a 2.3.7 gene

Archy

See Archytas clan #Archy.

Supra

See Archytas clan #Supra.

Supraphon

See Archytas clan #Supraphon.

Suhajira

See Rastmic clan #Suhajira.

Flutterpyth

Restricted to 2.3.7.11, this temperament is a no-5 restriction of 11-limit ultrapyth. This temperament was created to yield blackdye tunings where aberrisma-altered 3-limit thirds become tempered 13/11~19/16 and 14/11.

Subgroup: 2.3.7.11.13

Comma list: 64/63, 364/363, 1078/1053

Mapping: [⟨1 0 6 21 34], ⟨0 1 -2 -11 -19]]

Optimal tunings:

  • WE: ~2 = 1196.9412 ¢, ~3/2 = 711.0195 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.0039 ¢

Optimal ET sequence: 32f, 69bf, 101beff

Badness (Sintel): 1.52

2.3.7.11.13.19 subgroup

Subgroup: 2.3.7.11.13.19

Comma list: 64/63, 209/208, 343/342, 364/363

Mapping: [⟨1 0 6 21 34 17], ⟨0 1 -2 -11 -19 -8]]

Optimal tunings:

  • WE: ~2 = 1197.4072 ¢, ~3/2 = 711.2733 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 712.9612 ¢

Optimal ET sequence: 32f, 69bf

Badness (Sintel): 1.28

Semaphore

See Semaphoresmic clan #Semaphore.

Slendric

See Gamelismic clan #Slendric.

Slendroschismic

See 5th-octave temperaments #Slendroschismic.

Pentadecoid

See 15th-octave temperaments #Pentadecoid.

Navy

This temperament is the common restriction of tsaharuk and quanic.

Subgroup: 2.3.7

Comma list: 282429536481/281974669312

Subgroup-val mapping: [⟨1 1 0], ⟨0 5 24]]

mapping generators: ~2, ~243/224

Optimal tunings:

  • WE: ~2 = 1200.0302 ¢, ~243/224 = 140.3698 ¢
error map: ⟨+0.030 -0.076 +0.050]
  • CWE: ~2 = 1200.0000 ¢, ~243/224 = 140.3681 ¢
error map: ⟨0.000 -0.115 +0.008]

Optimal ET sequence: 17, 60, 77, 94, 171, 265, 436, 2351, 2787, 3223, 3659, 4095, 7754b

Badness (Sintel): 0.670

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 1331/1323, 19712/19683

Subgroup-val mapping: [⟨1 1 0 1], ⟨0 5 24 21]]

Optimal tunings:

  • WE: ~2 = 1200.1038 ¢, ~88/81 = 140.4190 ¢
  • CWE: ~2 = 1200.0000 ¢, ~88/81 = 140.4133 ¢

Optimal ET sequence: 17, 60e, 77, 94

Badness (Sintel): 0.887

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 352/351, 729/728, 1331/1323

Subgroup-val mapping: [⟨1 1 0 1 3], ⟨0 5 24 21 6]]

Optimal tunings:

  • WE: ~2 = 1199.8640 ¢, ~13/12 = 140.4206 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4292 ¢

Optimal ET sequence: 17, 60e, 77, 94

Badness (Sintel): 0.520

Lee

Subgroup: 2.3.7

Comma list: 177147/175616

Subgroup-val mapping: [⟨1 0 -3], ⟨0 3 11]]

Gencom mapping: [⟨1 0 0 -3], ⟨0 3 0 11]]

mapping generators: ~2, ~81/56

Optimal tunings:

  • WE: ~2 = 1200.2962 ¢, ~81/56 = 633.6812 ¢
error map: ⟨+0.296 -0.912 +0.778]
  • CWE: ~2 = 1200.0000 ¢, ~81/56 = 633.5658 ¢
error map: ⟨0.000 -1.258 +0.398]

Optimal ET sequence: 17, 36, 89, 125, 161, 358, 519b

Badness (Sintel): 0.741

Buzzard

See Buzzardsmic clan #Buzzard.

Hemif

Hemif is the no-5 restriction of hemififths, and the add-7 extension of namo.

Subgroup: 2.3.7

Comma list: 1605632/1594323

Subgroup-val mapping: [⟨1 1 -1], ⟨0 2 13]]

Gencom mapping: [⟨1 1 0 -1], ⟨0 2 0 13]]

mapping generators: ~2, ~2187/1792

Optimal tunings:

  • WE: ~2 = 1199.7303 ¢, ~2187/1792 = 351.4056 ¢
error map: ⟨-0.270 +0.586 -0.284]
  • CWE: ~2 = 1200.0000 ¢, ~2187/1792 = 351.4569 ¢
error map: ⟨0.000 +0.959 +0.114]

Optimal ET sequence: 17, 41, 58, 99, 239, 338, 437, 775b, 1212bb

Badness (Sintel): 0.901

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 896/891

Subgroup-val mapping: [⟨1 1 -1 2], ⟨0 2 13 5]]

Gencom mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5]]

Optimal tunings:

  • WE: ~2 = 1199.2633 ¢, ~11/9 = 351.3189 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.4593 ¢

Optimal ET sequence: 17, 41, 58, 99e

Badness (Sintel): 0.409

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 144/143, 243/242, 364/363

Sval mapping: [⟨1 1 -1 2 4], ⟨0 2 13 5 -1]]

Gencom mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1]]

Optimal tunings:

  • WE: ~2 = 1198.7603 ¢, ~11/9 = 351.3275 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.6042 ¢

Optimal ET sequence: 17, 41, 58, 331deeeffff

Badness (Sintel): 0.358

Heartful
See also: Heartland

Related temperaments: bunya.

Subgroup: 2.3.7.11.19

Comma list: 243/242, 896/891, 1083/1078

Subgroup-val mapping: [⟨1 1 -1 2 0], ⟨0 4 26 10 29]]

mapping generators: ~2, ~21/19

Optimal tunings:

  • WE: ~2 = 1199.2636 ¢, ~21/19 = 175.6963 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.7665 ¢

Optimal ET sequence: 34dh, 41, 116e, 157e

Badness (Sintel): 0.984

Hearts

See also: Heartland

This temperament is the common restriction of monkey and sesquiquartififths.

Subgroup: 2.3.7

Comma list: 34451725707/34359738368

Subgroup-val mapping: [⟨1 1 5], ⟨0 4 -15]]

mapping generators: ~2, ~567/512

Optimal tunings:

  • WE: ~2 = 1200.0845 ¢, ~567/512 = 175.4449 ¢
error map: ⟨+0.085 -0.091 -0.076]
  • CWE: ~2 = 1200.0000 ¢, ~567/512 = 175.4307 ¢
error map: ⟨0.000 -0.232 -0.286]

Optimal ET sequence: 7, 27d, 34, 41, 89, 130, 171, 643, 814, 985, 1156

Badness (Sintel): 0.959

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 65536/65219

Subgroup-val mapping: [⟨1 1 5 2], ⟨0 4 -15 10]]

Optimal tunings:

  • WE: ~2 = 1199.8467 ¢, ~256/231 = 175.3468 ¢
  • CWE: ~2 = 1200.0000 ¢, ~256/231 = 175.3691 ¢

Optimal ET sequence: 7, 34, 41, 89, 130, 349e, 479e

Badness (Sintel): 0.801

2.3.7.11.19

Subgroup: 2.3.7.11.19

Comma list: 243/242, 513/512, 1083/1078

Subgroup-val mapping: [⟨1 1 5 2 6], ⟨0 4 -15 10 -12]]

Optimal tunings:

  • WE: ~2 = 1199.9531 ¢, ~21/19 = 175.3344 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.3417 ¢

Optimal ET sequence: 7, 34, 41, 89, 130, 219

Badness (Sintel): 0.529

Magi

This temperament is the no-5 restriction of magic, tempering out the septimagic comma.

Subgroup: 2.3.7

Comma list: 537824/531441

Subgroup-val mapping: [⟨1 0 -1], ⟨0 5 12]]

mapping generators: ~2, ~243/196

Optimal tunings:

  • WE: ~2 = 1199.8224 ¢, ~243/196 = 380.6043 ¢
error map: ⟨-0.178 +1.066 -1.397]
  • CWE: ~2 = 1200.0000 ¢, ~243/196 = 380.6378 ¢
error map: ⟨0.000 +1.234 -1.173]

Optimal ET sequence: 19, 22, 41, 104, 145, 186, 331

Badness (Sintel): 1.30

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 896/891, 26411/26244

Subgroup-val mapping: [⟨1 0 -1 6], ⟨0 5 12 -8]]

Optimal tunings:

  • WE: ~2 = 1199.4843 ¢, ~96/77 = 380.6040 ¢
  • CWE: ~2 = 1200.0000 ¢, ~96/77 = 380.7490 ¢

Optimal ET sequence: 19, 22, 41, 63, 104

Badness (Sintel): 0.661

Caspar

Subgroup: 2.3.7.11.13

Comma list: 144/143, 343/338, 729/728

Subgroup-val mapping: [⟨1 0 -1 6 -2], ⟨0 5 12 -8 18]]

Optimal tunings:

  • WE: ~2 = 1199.3353 ¢, ~26/21 = 380.3206 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/21 = 380.5041 ¢

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 1.09

Twenothology

Subgroup: 2.3.7.11.13.29

Comma list: 144/143, 232/231, 343/338, 729/728

Subgroup-val mapping: [⟨1 0 -1 6 -2 2], ⟨0 5 12 -8 18 9]]

Optimal tunings:

  • WE: ~2 = 1199.6175 ¢, ~26/21 = 380.4049 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/21 = 380.5103 ¢

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 0.964

Melchior

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363, 26411/26244

Subgroup-val mapping: [⟨1 0 -1 6 11], ⟨0 5 12 -8 -23]]

Optimal tunings:

  • WE: ~2 = 1199.4887 ¢, ~96/77 = 380.6034 ¢
  • CWE: ~2 = 1200.0000 ¢, ~96/77 = 380.7669 ¢

Optimal ET sequence: 19f, 22, 41, 63, 104

Badness (Sintel): 0.710

Balthazar

Subgroup: 2.3.7.11.13

Comma list: 169/168, 896/891, 26411/26244

Subgroup-val mapping: [⟨1 0 -1 6 1], ⟨0 10 24 -16 17]]

mapping generators: ~2, ~143/128

Optimal tunings:

  • WE: ~2 = 1199.7322 ¢, ~143/128 = 190.3647 ¢
  • CWE: ~2 = 1200.0000 ¢, ~143/128 = 190.4016 ¢

Optimal ET sequence: 19, 44, 63, 145f

Badness (Sintel): 1.82

Hogwarts

Subgroup: 2.3.7.29

Comma list: 784/783, 5887/5832

Subgroup-val mapping: [⟨1 0 -1 2], ⟨0 5 12 9]]

Optimal tunings:

  • WE: ~2 = 1200.1518 ¢, ~36/29 = 380.6661 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/29 = 380.6375 ¢

Optimal ET sequence: 19, 22, 41, 145, 186j, 227j

Badness (Sintel): 0.424

Skwares

Main article: Squares

Skwares is the no-5 restriction of squares.

Subgroup: 2.3.7

Comma list: 19683/19208

Subgroup-val mapping: [⟨1 -1 -3], ⟨0 4 9]]

Gencom mapping: [⟨1 -1 0 -3], ⟨0 4 0 9]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1200.3703 ¢, ~14/9 = 774.8736 ¢
error map: ⟨+0.370 -2.831 +3.925]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.6974 ¢
error map: ⟨0.000 -3.166 +3.450]

Optimal ET sequence: 14, 17, 31, 48, 79

Badness (Sintel): 1.55

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 99/98, 243/242

Subgroup-val mapping: [⟨1 -1 -3 -3], ⟨0 4 9 10]]

Gencom mapping: [⟨1 -1 0 -3 -3], ⟨0 4 0 9 10]]

Optimal tunings:

  • WE: ~2 = 1200.3726 ¢, ~14/9 = 774.9970 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.8197 ¢

Optimal ET sequence: 14, 17, 31, 48, 79, 127

Badness (Sintel): 0.405

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 243/242

Subgroup-val mapping: [⟨1 -1 -3 -3 -6], ⟨0 4 9 10 15]]

Gencom mapping: [⟨1 -1 0 -3 -3 -6], ⟨0 4 0 9 10 15]]

Optimal tunings:

  • WE: ~2 = 1199.3264 ¢, ~14/9 = 775.1081 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 775.4463 ¢

Optimal ET sequence: 14f, 17, 48f

Badness (Sintel): 0.587

Skwairs

Subgroup: 2.3.7.11.13

Comma list: 99/98, 144/143, 243/242

Subgroup-val mapping: [⟨1 -1 -3 -3 5], ⟨0 4 9 10 -2]]

Gencom mapping: [⟨1 -1 0 -3 -3 5], ⟨0 4 0 9 10 -2]]

Optimal tunings:

  • WE: ~2 = 1198.8812 ¢, ~14/9 = 775.5748 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 775.1930 ¢

Optimal ET sequence: 14, 17, 31, 48, 65d, 113df

Badness (Sintel): 0.538

Byhearted

This temperament is the restriction of weasel to the 2.3.7.11.19 subgroup.

Subgroup: 2.3.7.11.19

Comma list: 99/98, 243/242, 363/361

Subgroup-val mapping: [⟨2 2 3 4 5], ⟨0 4 9 10 12]]

mapping generators: ~209/147, ~21/19

Optimal tunings:

  • WE: ~2 = 600.1836 ¢, ~21/19 = 174.7882 ¢
  • CWE: ~2 = 600.0000 ¢, ~21/19 = 174.7975 ¢

Optimal ET sequence: 14, 34dh, 48, 110e

Badness (Sintel): 0.893

Harrison

Harrison is the no-5 restriction of meantone. As such, there is little reason to consider this temperament in practice – since intervals of 5 in meantone are as accurate as intervals of 7, only simpler, they are always available by the time intervals of 7 are generated.

Subgroup: 2.3.7

Comma list: 59049/57344

Subgroup-val mapping: [⟨1 0 -13], ⟨0 1 10]]

Gencom mapping: [⟨1 0 0 -13], ⟨0 1 0 10]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1201.5353 ¢, ~3/2 = 697.4352 ¢
error map: ⟨+1.535 -2.984 +0.920]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.7289 ¢
error map: ⟨0.000 -5.226 -1.537]

Optimal ET sequence: 12, 19, 31, 112b, 143b, 174b

Badness (Sintel): 2.35

Bleu

Bleu can be described as the 8d & 9 temperament in the no-5 13-limit, and is the common restriction of progression and jerome.

Subgroup: 2.3.7

Comma list: 17496/16807

Subgroup-val mapping: [⟨1 1 2], ⟨0 5 7]]

Gencom mapping: [⟨1 1 0 2], ⟨0 5 0 7]]

mapping generators: ~2, ~54/49

Optimal tunings:

  • WE: ~2 = 1199.3538 ¢, ~54/49 = 139.848 ¢
error map: ⟨-0.646 -3.736 +8.293]
  • CWE: ~2 = 1200.0000 ¢, ~54/49 = 139.848 ¢
error map: ⟨0.000 -3.270 +9.333]

Optimal ET sequence: 8d, 9, 17, 43, 60d, 103d

Badness (Sintel): 2.48

2.3.7.11 subgroup

Subgroup: 2.3.7.11

Comma list: 99/98, 864/847

Subgroup-val mapping: [⟨1 1 2 3], ⟨0 5 7 4]]

Gencom mapping: [⟨1 1 0 2 3], ⟨0 5 0 7 4]]

Optimal tunings:

  • WE: ~2 = 1198.6613 ¢, ~12/11 = 139.8489 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 139.7839 ¢

Optimal ET sequence: 8d, 9, 17, 43, 60d

Badness (Sintel): 0.624

2.3.7.11.13 subgroup

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 144/143

Subgroup-val mapping: [⟨1 1 2 3 3], ⟨0 5 7 4 6]]

Gencom mapping: [⟨1 1 0 2 3 3], ⟨0 5 0 7 4 6]]

Optimal tunings:

  • WE: ~2 = 1198.9768 ¢, ~13/12 = 139.8704 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.8166 ¢

Optimal ET sequence: 8d, 9, 17, 43, 60d

Badness (Sintel): 0.400

Music

Doublehearted

See also: Heartland

This temperament is the no-5 restriction of octacot.

Subgroup: 2.3.7

Comma list: 5764801/5668704

Subgroup-val mapping: [⟨1 1 2], ⟨0 8 11]]

mapping generators: ~2, ~343/342

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~343/324 = 87.8431 ¢
error map: ⟨+0.174 +0.964 -2.204]
  • CWE: ~2 = 1200.0000 ¢, ~343/324 = 87.8492 ¢
error map: ⟨0.000 +0.838 -2.485]

Optimal ET sequence: 14, 27, 41

Badness (Sintel): 2.62

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 2401/2376

Subgroup-val mapping: [⟨1 1 2 2], ⟨0 8 11 20]]

Optimal tunings:

  • WE: ~2 = 1200.4071 ¢, ~22/21 = 87.6809 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/21 = 87.6902 ¢

Optimal ET sequence: 14, 27e, 41, 96d, 137d, 178d

Badness (Sintel): 0.815

2.3.7.11.19

Subgroup: 2.3.7.11.19

Comma list: 133/132, 243/242, 343/342

Subgroup-val mapping: [⟨1 1 2 2 3], ⟨0 8 11 20 17]]

Optimal tunings:

  • WE: ~2 = 1200.6100 ¢, ~19/18 = 87.7129 ¢
  • CWE: ~2 = 1200.0000 ¢, ~19/18 = 87.7285 ¢

Optimal ET sequence: 14, 27e, 41, 137dh

Badness (Sintel): 0.560

Purpleheart

See also: Whitewood

Subgroup: 2.3.7

Comma list: 2187/2048

Subgroup-val mapping: [⟨7 11 0], ⟨0 0 1]]

mapping generators: ~9/8, ~7

Optimal tunings:

  • WE: ~9/8 = 172.1541 ¢, ~7/4 = 958.5433 ¢ (~64/63 = 74.3805 ¢)
error map: ⟨+5.079 -8.260 -0.124]
  • CWE: ~9/8 = 171.4286 ¢, ~7/4 = 959.2372 ¢ (~64/63 = 69.3373 ¢)
error map: ⟨0.000 -16.241 -9.589]

Optimal ET sequence: 7, 14, 35, 49bd

Badness (Sintel): 3.00

Chrysanthemum

This microtemperament extends amaranthine to prime 3 by tempering out 43923/43904, the chrysia, to find 3 at 29 steps down on the chain of nearly pure 7/4's.

Subgroup: 2.3.7

Comma list: [83 -1 -29⟩

Subgroup-val mapping: [⟨1 -4 3], ⟨0 29 -1]]

mapping generators: ~2, ~8/7

Optimal tunings:

  • WE: ~2 = 1199.9871 ¢, ~8/7 = 231.1001 ¢
error map: ⟨-0.013 +0.000 +0.035]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.1024 ¢
error map: ⟨0.000 +0.014 +0.072]

Optimal ET sequence: 26, 83, 109, 135, 566, 701, 836, 971, 1106, 2077, 5260, 7337, 9414d

Badness (Sintel): 3.06

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 43923/43904, 5767168/5764801

Subgroup-val mapping: [⟨1 -4 3 5], ⟨0 29 -1 -8]]

Optimal tunings:

  • WE: ~2 = 1200.0050 ¢, ~8/7 = 231.1024 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.1015 ¢

Optimal ET sequence: 26, 83, 109, 135, 566, 701, 836, 971, 1807, 2778, 4585

Badness (Sintel): 0.324

Leapfrog

See also: Gentle region

Leapfrog is generated by a perfect fifth and the interval class of 7 is found at +15 steps, as a double-augmented fifth (C–G𝄪). For this to work, it entails a fifth about 2–3 cents sharp of just; as a result the major third lands comfortably at a near-just 14/11 so that it can be extended to the 2.3.7.11 subgroup via tempering out 896/891. The minor third can then be identified with 13/11, tempering out 352/351 and 364/363, which implies 169/168 is tempered out as well in this case. Leapfrog is most naturally treated as such, in which it is very efficient.

A notable patent-val edo tuning not appearing in the optimal ET sequence is 80edo, which is approximately the just-13's tuning (as 10edo is used as a consistent circle of ~16/13's therein), with 13/8 still tuned slightly flat so qualifying a reasonable tuning for the 2.3.13 subgroup (as evidenced by appearing in the sequence for tetris).

Strong extensions for prime 5 include leapday (29 & 46), leapweek (46 & 63), and leapmonth (63 & 80), all of which are more complex than vanilla leapfrog. A low-complexity low-accuracy extension is given by supermean (5de & 17c), where it is merged with meantone. Srutal (46 & 80), usually considered as a strong extension of diaschismic, is a weak extension of leapfrog, and yet another weak extension is immune (29 & 63), which is in turn a strong extension of 5-limit immunity.

Subgroup: 2.3.7

Comma list: 14680064/14348907

Subgroup-val mapping: [⟨1 0 -21], ⟨0 1 15]]

Gencom mapping: [⟨1 0 0 -21], ⟨0 1 0 15]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.1807 ¢, ~3/2 = 704.2400 ¢
error map: ⟨-0.819 +1.466 -0.311]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.6600 ¢
error map: ⟨0.000 +2.705 +1.074]

Optimal ET sequence: 17, 46, 63, 235b, 298b, 361bd, 424bd, 487bbd

Badness (Sintel): 4.33

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 896/891, 1331/1323

Subgroup-val mapping: [⟨1 0 -21 -14], ⟨0 1 15 11]]

Gencom mapping: [⟨1 0 0 -21 -14], ⟨0 1 0 15 11]]

Optimal tunings:

  • WE: ~2 = 1199.2683 ¢, ~3/2 = 704.3230 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.6926 ¢

Optimal ET sequence: 17, 46, 63

Badness (Sintel): 0.629

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 169/168, 352/351, 364/363

Subgroup-val mapping: [⟨1 0 -21 -14 -9], ⟨0 1 15 11 8]]

Gencom mapping: [⟨1 0 0 -21 -14 -9], ⟨0 1 0 15 11 8]]

Optimal tunings:

  • WE: ~2 = 1199.5654 ¢, ~3/2 = 704.4898 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7084 ¢

Optimal ET sequence: 17, 46, 63

Badness (Sintel): 0.436

Skidoo

Subgroup: 2.3.7.11.13.23

Comma list: 169/168, 208/207, 352/351, 364/363

Subgroup-val mapping: [⟨1 0 -21 -14 -9 -5], ⟨0 1 15 11 8 6]]

Gencom mapping: [⟨1 0 0 -21 -14 -9 0 0 -5], ⟨0 1 0 15 11 8 0 0 6]]

Optimal tunings:

  • WE: ~2 = 1199.6639 ¢, ~3/2 = 704.5315 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7021 ¢

Optimal ET sequence: 17, 46, 63

Badness (Sintel): 0.356

2.3.7.11.13.23.29

Subgroup: 2.3.7.11.13.23.29

Comma list: 169/168, 208/207, 232/231, 352/351, 364/363

Subgroup-val mapping: [⟨1 0 -21 -14 -9 -5 -38], ⟨0 1 15 11 8 6 27]]

Gencom mapping: [⟨1 0 0 -21 -14 -9 -5 0 0 -38], ⟨0 1 0 15 11 8 0 0 6 27]]

Optimal tunings:

  • WE: ~2 = 1199.5755 ¢, ~3/2 = 704.5533 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7750 ¢

Optimal ET sequence: 17, 46, 63

Badness (Sintel): 0.441

Music

Superslendric

In superslendric, eight 8/7's are equated to 3/1. This relates it to 8edt.

Subgroup: 2.3.7

Comma list: 17294403/16777216

Subgroup-val mapping: [⟨1 0 3], ⟨0 8 -1]]

mapping generators: ~2, ~8/7

Optimal tunings:

  • WE: ~2 = 1201.1628 ¢, ~8/7 = 237.7287 ¢
error map: ⟨+1.163 -0.125 -3.066]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 237.5664 ¢
error map: ⟨0.000 -1.424 -6.392]

Optimal ET sequence: 5, …, 66b, 71b, 76, 81, 86, 91, 96d

Badness (Sintel): 6.15

Hectosaros leap week

This temperament may be described as the 320 & 1803 temperament, in the 2.3.7.13.17.19 on the basis of the fact that 1803 tropical years make up almost exactly 100 saros cycles.

Subgroup: 2.3.7

Comma list: [-50 -746 439⟩

Subgroup-val mapping: [⟨1 -126 -214], ⟨0 439 746]]

mapping generators: ~2, ~[-16 -243 143⟩

Optimal tunings:

  • WE: ~2 = 1200.0010 ¢, ~[-16 -243 143⟩ = 348.7520 ¢
error map: ⟨+0.001 +0.036 -0.067]
  • CWE: ~2 = 1200.0000 ¢, ~[-16 -243 143⟩ = 348.7517 ¢
error map: ⟨0.000 +0.035 -0.068]

Optimal ET sequence: 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689

Badness (Sintel): 17.7 × 103

2.3.7.13 subgroup

Subgroup: 2.3.7.13

Comma list: [-42 -2 -5 16⟩, [10 -46 29 -5⟩

Subgroup-val mapping: [⟨1 -126 -214 -80], ⟨0 439 746 288]]

Optimal tunings:

  • WE: ~2 = 1200.0058 ¢, ~[18 -9 8 -7⟩ = 348.7534 ¢
  • CWE: ~2 = 1200.0000 ¢, ~[18 -9 8 -7⟩ = 348.8517 ¢

Optimal ET sequence: 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689, 11255d

Badness (Sintel): 53.2

2.3.7.13.17 subgroup

Subgroup: 2.3.7.13.17

Comma list: 39337984/39328497, [0 -14 7 4 -3⟩, [-18 -24 14 -1 5⟩

Subgroup-val mapping: [⟨1 -126 -214 -80 -18], ⟨0 439 746 288 76]]

Optimal tunings:

  • WE: ~2 = 1200.9870 ¢, ~3757/3072 = 348.7480 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3757/3072 = 348.7517 ¢

Optimal ET sequence: 320, 1483bd, 1803, 2123

Badness (Sintel): 13.4

2.3.7.13.17.19 subgroup

Subgroup: 2.3.7.13.17.19

Comma list: 10081799/10077696, 10754912/10744731, 39337984/39328497, 480024727/480020256

Subgroup-val mapping: [⟨1 -126 -214 -80 -18 -171], ⟨0 439 746 288 76 603]]

Optimal tunings:

  • WE: ~2 = 1200.9961 ¢, ~3757/3072 = 348.7506 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3757/3072 = 348.7517 ¢

Optimal ET sequence: 320, 1483bd, 1803, 2123

Badness (Sintel): 7.46

Heartland (rank 3)

Main article: Heartland

Heartland, with a generator of ~21/19, is named for its tempering of the heartlandisma, 3971/3969. Aside from the heartlandisma, the heartland temperament tempers out 243/242 (rastma) and 1083/1078 (bihendrixma), and slices the fifth in four (the number of chambers of the heart).

Subgroup: 2.3.7.11.19

Comma list: 243/242, 1083/1078

Subgroup-val mapping: [⟨1 1 0 2 1], ⟨0 4 0 10 3], ⟨0 0 1 0 1]]

mapping generators: ~2, ~21/19, ~7

Optimal tunings:

  • WE: ~2 = 1200.0983 ¢, ~21/19 = 175.2856 ¢, ~7/4 = 969.4578 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.2894 ¢, ~7/4 = 969.5203 ¢

Optimal ET sequence: 14, 27e, 34dh, 41, 89, 130, 219

Badness (Sintel): 0.615

Temperaments with a 2.3.11 gene

Neutral

See Rastmic clan #Neutral.

Namo

See Rastmic clan #Namo.

Io

Io is a very low-complexity temperament which tempers out the undecimal quartertone 33/32. This equates very different intervals (for example, the generator itself represents both 3/2 and 16/11), and as such some consider it to be an exotemperament. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of flattone temperament.

The name io was coined by CompactStar in 2024 based on the color name ilo, prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.

Subgroup: 2.3.11

Comma list: 33/32

Subgroup-val mapping: [⟨1 0 5], ⟨0 1 -1]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1206.6866 ¢, ~3/2 = 691.7837 ¢
error map: ⟨+6.687 -3.485 -16.355]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 689.2066 ¢
error map: ⟨0.000 -12.748 -40.525]

Optimal ET sequence: 2, 5, 7, 12e, 40ee, 47eee, 54beee, 61beeee

Badness (Sintel): 0.185

Paralimmal

Subgroup: 2.3.11

Comma list: 4096/3993

Subgroup-val mapping: [⟨1 0 4], ⟨0 3 -1]]

mapping generators: ~2, ~16/11

Optimal tunings:

  • WE: ~2 = 1197.9124 ¢, ~16/11 = 634.1269 ¢
error map: ⟨-2.088 +0.426 +6.205]
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 634.9546 ¢
error map: ⟨0.000 +2.909 +13.727]

Optimal ET sequence: 2, 11b, 13, 15, 17

Badness (Sintel): 0.984

Huxley

Main article: Huxley

Huxley, the 4 & 13 temperament in the 2.3.11.13 subgroup, extends lovecraft. Specifically it tunes the ~13/8 to exactly half of ~8/3.

Subgroup: 2.3.11.13

Comma list: 512/507, 1352/1331

Subgroup-val mapping: [⟨1 -3 5 6], ⟨0 6 -2 -3]]

mapping generators: ~2, ~22/13

Optimal tunings:

  • WE: ~2 = 1198.0036 ¢, ~22/13 = 916.0595 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/13 = 917.5184 ¢

Optimal ET sequence: 4, 13, 17

Badness (Sintel): 1.31

Aerophore

Subgroup: 2.3.11.19

Comma list: 363/361, 729/704

Subgroup-val mapping: [⟨1 0 -6 -6], ⟨0 2 12 13]]

mapping generators: ~2, ~19/11

Optimal tunings:

  • WE: ~2 = 1202.6380 ¢, ~19/11 = 947.4782 ¢
  • CWE: ~2 = 1200.0000 ¢, ~19/11 = 945.7779 ¢

Optimal ET sequence: 14, 19, 33

Badness (Sintel): 1.59

Semaerophore

Subgroup: 2.3.7.11

Comma list: 49/48, 729/704

Subgroup-val mapping: [⟨1 0 2 -6], ⟨0 2 1 12]]

Optimal tunings:

  • WE: ~2 = 1204.9027 ¢, ~7/4 = 948.7772 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 945.4959 ¢

Optimal ET sequence: 14, 33d, 47de

Badness (Sintel): 1.27

2.3.7.11.19

Subgroup: 2.3.7.11.19

Comma list: 49/48, 77/76, 729/704

Subgroup-val mapping: [⟨1 0 2 -6 -6], ⟨0 2 1 12 13]]

Optimal tunings:

  • WE: ~2 = 1204.9645 ¢, ~7/4 = 948.5749 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 945.2236 ¢

Optimal ET sequence: 14, 33d, 47deh

Badness (Sintel): 1.08

Temperaments with a 2.3.13 gene

Superflat

Superflat is a diatonic-based temperament that makes 1053/1024 vanish, so 13/8 is a minor sixth, and 16/13 is a major third. The more accurate tunings for this temperament are generated by a fifth at least as flat as those of flattone, although often even flatter (such as 40edo's fifth). Superflat can be viewed as a 2.3.13 subgroup analogue of meantone and archy. Superflat diatonic scales have a character somewhere between neutral third scales (or mosh) and meantone diatonic scales.

Subgroup: 2.3.13

Comma list: 1053/1024

Subgroup-val mapping: [⟨1 1 6], ⟨0 1 -4]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 692.939

Optimal ET sequence: 5f, 7, 12, 19, 45f, 64f, 147bfff

RMS error: 1.591 cents

2.3.11.13

Subgroup: 2.3.11.13

Comma list: 144/143, 729/704

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 692.247

Optimal ET sequence: 7, 19, 26, 59b

Ultraflat

Ultraflat is the much more inaccurate cousin of superflat, with even flatter fifths. 27/26 is tempered out rather than 1053/1024, so 13/8 is a major sixth. These temperaments intersect in 7edo, where major sixths and minor sixths are not distinguished.

Subgroup: 2.3.13

Comma list: 27/26

Subgroup-val mapping: [⟨1 1 2], ⟨0 1 3]]

Optimal tuning (CTE): ~2 = 1/1, ~3/2 = 688.391

Optimal ET sequence: 5, 7

RMS error: 4.367 cents

Threedic

Subgroup: 2.3.13

Comma list: 2197/2187

Subgroup-val mapping: [⟨1 0 0], ⟨0 3 7]]

Optimal tuning (CTE): ~2 = 1/1, ~13/9 = 634.173

Optimal ET sequence: 11bff, 13f, 15, 17, 36, 53, 70, 123, 193, 316, 755f

RMS error: 0.2054 cents

Shoal

The 2.3.13.23 subgroup is remarkable for containing not one but two superparticular intervals as small as 3888/3887 and 12168/12167. Tempering out both of them gives us a rank-2 temperament where a sharp whole tone of 26/23 is the generator, two of which stack to a 23/18 supermajor third, and eight of which stack to a 8/3 perfect eleventh. 17edo is a trivial tuning where 26/23 is equated to 9/8, tempering out the comma 208/207. More accurate tunings of shoal create a 17-note MOS scale, serving as a circulating temperament of 17edo, where 208/207 is the chroma between large and small steps.

Subgroup: 2.3.13.23

Comma list: 3888/3887, 12168/12167

Subgroup-val mapping: [⟨1 3 6 7], ⟨0 -8 -13 -14]]

Optimal tuning (CTE): ~2 = 1/1, ~26/23 = 212.261

Optimal ET sequence: 11fi, 17, 79, 96, 113, 130, 147, 424, 571, 1289, 1860, 3149

Badness (Sintel): 0.021

Scales:

Music:

Glacier

The 2.3.13 gene subgroup is not nearly as good as Shoal, but it can extend extremely well to other no-5 subgroups. It is very well represented in 26edo, where a pure 13/12 can serve as the generator, but 94edo provides a much better tuning in higher subgroups.

Subgroup: 2.3.13

Comma list: 373248/371293

Subgroup-val mapping: [⟨1 1 3], ⟨0 5 6]]

Optimal tuning (CTE): ~2 = 1/1, ~13/12 = 140.360

Optimal ET sequence: 9, 17, 26, 43, 60, 77, 94, 111, 137, 171

Badness (Sintel): 0.383

Glaishur

Subgroup: 2.3.11.13

Comma list: 352/351, 531674/531441

Subgroup-val mapping: [⟨1 1 0 3], ⟨0 5 21 6]]

Optimal tuning (CTE): ~2 = 1/1, ~13/12 = 140.537

Optimal ET sequence: 17, 77, 94, 111, 128, 145, 205

Badness (Sintel): 0.400

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 352/351, 729/728, 1573/1568

Subgroup-val mapping: [⟨1 1 0 1 3], ⟨0 5 24 21 6]]

Optimal tuning (CTE): ~2 = 1/1, ~13/12 = 140.429

Optimal ET sequence: 17, 77, 94, 111, 128, 145, 205

Badness (Sintel): 0.415

2.3.7.11.13.23

Subgroup: 2.3.7.11.13.23

Comma list: 352/351, 729/728, 253/252, 1288/1287

Subgroup-val mapping: [⟨1 1 0 1 3 3], ⟨0 5 24 21 6 13]]

Optimal tuning (CTE): ~2 = 1/1, ~13/12 = 140.426

Optimal ET sequence: 17, 77, 94, 111, 205

Badness (Sintel): 0.452

2.3.7.11.13.23.29

Subgroup: 2.3.7.11.13.23.29

Comma list: 352/351, 729/728, 253/252, 1288/1287, 378/377

Subgroup-val mapping: [⟨1 1 0 1 3 3 1], ⟨0 5 24 21 6 13 33]]

Optimal tuning (CTE): ~2 = 1/1, ~13/12 = 140.426

Optimal ET sequence: 17, 77, 94, 111j

Badness (Sintel): 0.511

2.3.7.11.13.19.23.29

Subgroup: 2.3.7.11.13.19.23.29

Comma list: 352/351, 729/728, 209/208, 253/252, 1288/1287, 378/377

Subgroup-val mapping: [⟨1 1 0 1 3 -6 3 1], ⟨0 5 24 21 6 -15 13 33]]

Optimal tuning (CTE): ~2 = 1/1, ~13/12 = 140.426

Optimal ET sequence: 17, 77, 94, 111j

Badness (Sintel): 0.699

Temperaments with a higher-limit gene

Semitonic

Subgroup: 2.3.17

Comma list: 289/288

Subgroup-val mapping: [⟨2 0 5], ⟨0 1 1]]

sval mapping generators: ~17/12, ~3
gencom: [17/12 3; 289/288]

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)

Optimal ET sequence: 12, 58, 70, 82, 94, 106, 118, 224g

RMS error: 0.2247 cents

Gigapyth

Subgroup: 2.3.85

Comma list: 2.3.85 ⟨-40 1 6]

Subgroup-val mapping: [⟨1 4 6], ⟨0 -6 1]]

mapping generators: ~2, ~85/64

Optimal tuning (CTE): ~2 = 1\1, ~85/64 = 483.034

Supporting ETs: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159**

*Wart for 85

2.3.7.85 subgroup

Subgroup: 2.3.7.85

Comma list: 1029/1024, 7225/7203

Subgroup-val mapping: [⟨1 4 2 6], ⟨0 -6 2 1]]

mapping generators: ~2, ~85/64

Optimal tuning (CTE): ~2 = 1\1, ~85/64 = 483.031

Supporting ETs: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159**

*Wart for 85

Dog

The dog temperament is based by 2L 5s or 7L 2s scale that makes 81/76 vanish, so 19/16 is a major third. It can be viewed as a 2.3.19 subgroup analogue of mavila.

Subgroup: 2.3.19

Comma list: 81/76

Gencom: [2 4/3; 81/76]

Sval mapping: [⟨1 2 6], ⟨0 -1 -4]]

POL2 generator: ~4/3 = 521.403

Optimal ET sequence: 5h, 7, 16, 23

RMS error: 4.943 cents

Boethian

Boethian is a diatonic-based temperament that makes 513/512 vanish, so that the major third (C–E) is ~24/19 and the minor third (C–E♭) is ~19/16. As such, it functions as a 2.3.19-subgroup analogue of meantone, though the small size of the comma puts it at schismic level of accuracy. In particular, the equal temperaments in the tuning spectrum up to 1/2-comma (flattened) boethian temperament (very close to 12edo) are included in the schismic tuning spectrum in the 5-limit, so boethian intersects with schismic in the prime-5 infill extension thereof, called nestoria, which also tempers out 361/360, the difference between 19/18 and 20/19 or between 19/15 and 24/19.

Subgroup: 2.3.19

Comma list: 513/512

Subgroup-val mapping: [⟨1 0 9], ⟨0 1 -3]]

mapping generators: ~2, ~3

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 701.3288 ¢

Optimal ET sequence: 5, 7, 12, 41, 53, 65, 77, 219, 296

Badness (Smith): 0.000374

Lipsett

Lipsett temperament is a pleasantly melodic little temperament with a highly useable 5-tone and 9-tone mos. It is audibly similar to semaphore temperament, so it could be thought of as semaphore but for the 23rd harmonic instead of the 7th. It is named for Arthur Lipsett, the director off the Canadian short film ’21-87’. Leia’s prison cell in Star Wars is numbered ‘2187’, as a nod to the influence the film had on George Lucas.

Subgroup: 2.3.23

Comma list: 2187/2116

Subgroup-val mapping: [⟨1 0 -1], ⟨0 2 7]]

Optimal tuning (CTE): ~2 = 1\1, ~46/27 = 948.526

Optimal ET sequence: 5, 14, 19, 43, 62i, 81i

Badness (Smith): 8.998 × 10-3

Porpoise

Subgroup: 2.3.29

Comma list: 24576/24389

Mapping: [⟨1 2 5], ⟨0 3 -1]]

CTE generator: ~32/29 = 166.067

Optimal ET sequence: 7, 22, 29, 94, 123, 152j, 275jj, 427jjj

Sematology

This temperament tempers out 4107/4096 and thus equates 2 37/32's with 4/3.

Subgroup: 2.3.37

Comma list: 4107/4096

Gencom: [2 37/32; 4107/4096]

Mapping: [⟨1 1 5], ⟨0 -2 1]]

POTE generator: ~37/32 = 249.075

Optimal ET sequence: 5, 14, 19, 24, 53, 77, 130

2.3.7.37 subgroup

Subgroup: 2.3.7.37

Comma list: 4107/4096, 259/256

Gencom: [2 37/32; 4107/4096 259/256]

Mapping: [⟨1 1 1 5], ⟨0 -2 -1 1]]

POTE generator: ~37/32 = 247.782

Optimal ET sequence: 5, 14, 19, 24, 53d

2.3.5.37 subgroup

It is difficult to extend sematology to include 5, due the 5th harmonic being quite high-complexity.

Subgroup: 2.3.5.37

Comma list: 4107/4096, 17592186044416/17562397269605

Gencom: [2 37/32; 4107/4096 17592186044416/17562397269605]

Mapping: [⟨1 1 4 5], ⟨0 -2 -8 1]]

POTE generator: ~37/32 = 251.393

Optimal ET sequence: 5, 14c, 19, 43, 62

2.3.5.7.37 subgroup

Subgroup: 2.3.5.7.37

Comma list: 4107/4096, 17592186044416/17562397269605, 259/256

Gencom: [2 37/32; 4107/4096 17592186044416/17562397269605 259/256]

Mapping: [⟨1 1 4 1 5], ⟨0 -2 -8 -1 1]]

POTE generator: ~37/32 = 251.204

Optimal ET sequence: 5, 14c, 19

Reversed mavila

Subgroup: 2.3.37

Comma list: 81/74

Gencom: [2 4/3; 81/74]

Mapping: [⟨1 1 0], ⟨0 -1 12]]

POTE generator: ~4/3 = 521.397

Optimal ET sequence: 5l, 7l, 9, 16l

Reversed meantone

Main article: Reversed meantone

Subgroup: 2.3.41

Comma list: 82/81

Gencom: [2 4/3; 82/81]

Sval mapping: [⟨1 2 7], ⟨0 -1 -4]]

POL2 generator: ~4/3 = 494.509

Optimal ET sequence: 5, 12, 17

2.3.7.41 subgroup

Subgroup: 2.3.7.41

Comma list: 64/63, 82/81

Gencom: [2 4/3; 64/63 82/81]

Sval mapping: [⟨1 2 2 7], ⟨0 -1 2 -4]]

POTE generator: ~4/3 = 490.0323

TOP generators: ~2 = 1197.2342, ~4/3 = 488.9029

Optimal ET sequence: 5, 12, 17, 22, 49

2.3.7.11.41 subgroup

Subgroup: 2.3.7.11.41

Comma list: 64/63, 82/81, 99/98

Gencom: [2 4/3; 64/63 82/81 99/98]

Sval mapping: [⟨1 2 2 1 7], ⟨0 -1 2 6 -4]]

POTE generator: ~4/3 = 492.1787

TOP generators: ~2 = 1197.9683, ~4/3 = 491.3454

Optimal ET sequence: 5, 12, 17, 22, 39d

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