No-fives subgroup temperaments

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

Semaphore

See Semaphoresmic clan #Semaphore.

Bleu

Bleu can be described as the 8d & 9 temperament in the no-5 13-limit.

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

• On a Well Worn Riff (2011) by Chris Vaisvil – blog | play – in Bleu[17]

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

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

Leapfrog

For extensions, see Leapday, Leapweek, Supermean, and Srutal.

See also: Gentle region

Leapfrog is generated by a perfect fifth and the interval class of 7 is found at +15 steps, as an double-augmented fifth (C–G𝄪).

In regular 13-limit leapday, the mapping for prime 5 is very complex at +21 generator steps. Furthermore, adding prime 5 to rank-3 parapythic is arguably against the original vision of it as a 2.3.7.11.13-subgroup temperament, so avoiding prime 5 may be preferred for this reason also. This results in no-5's leapday, or leapfrog, which as aforementioned is much lower in badness, but it also allows more tunings to be used: a notable patent val 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). In other words, the only reason 80edo was "disqualified" from leapday is that the mapping for prime 5 constrains the tuning range which is naturally more flexible as a no-5's 13-limit temperament, which is also a sign of leapfrog being very efficient.

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

• Suite for Harpsichord in A Locrian, tuning: Eb–G# in 46edo by Inthar (in progress):
  • I. Prelude
  • II. Allemande
  • III. Courante
  • IV. Sarabande (score, 17edo version)
  • V. Menuet and Trio
  • VI. Gavotte I and II
  • VII. Gigue

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

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

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.

Slendric

See Gamelismic clan #Slendric.

Hemif

Related temperaments: hemififths, 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]]

gencom: [2 2187/1792; 1605632/1594323]

Optimal tuning (POTE): ~2 = 1\1, ~2187/1792 = 351.485

Optimal ET sequence: 7, 17, 41, 58, 99

RMS error: 0.2344 cents

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 896/891

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

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

gencom: [2 11/9; 243/242 896/891]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.535

Optimal ET sequence: 7, 17, 41, 58, 99e

RMS error: 0.6108 cents

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]]

gencom: [2 11/9; 144/143 243/242 364/363]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.691

Optimal ET sequence: 7, 10, 17, 24, 41, 58

RMS error: 0.7167 cents

Heartful

See also: Heartland

Related temperaments: bunya

Subgroup: 2.3.7.11.19

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

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

gencom: [2 21/19; 243/242 896/891 1083/1078]

Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.804

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

RMS error: 0.5360 cents

Hearts

See also: Heartland

Subgroup: 2.3.7

Comma list: 34451725707/34359738368 (trila-quadzo comma)

Gencom: [2 567/512; 34451725707/34359738368]

Sval mapping: [⟨1 1 5], ⟨0 4 -15]]

POL2 generator: ~567/512 = 175.433

Optimal ET sequence: 7, 27d, 34, 41, 89, 130, 171

RMS error: 0.0529 cents

Related temperaments: monkey, sesquiquartififths

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 65536/65219

Gencom: [2 256/231; 243/242 65536/65219]

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

POL2 generator: ~256/231 = 175.369

Optimal ET sequence: 7, 27de, 34, 41, 89, 130

RMS error: 0.3224 cents

Related temperaments: monkey, sesquart

2.3.7.11.19

Subgroup: 2.3.7.11.19

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

Gencom: [2 21/19; 243/242 513/512 1083/1078]

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

POL2 generator: ~21/19 = 175.341

Optimal ET sequence: 7, 27deh, 34, 41, 89, 130, 219

RMS error: 0.3121 cents

Related temperaments: monkey, sesquart

Navy

Subgroup: 2.3.7

Comma list: 282429536481/281974669312

Mapping: [⟨1 1 0], ⟨0 5 24]]

POL2 generator: ~243/224 = 140.366

Optimal ET sequence: 17, 60, 77, 94, 171, 265, 436

RMS error: 0.0296 cents

Related temperaments: tsaharuk, quanic

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 1331/1323, 19712/19683

Mapping: [⟨1 1 0 1], ⟨0 5 24 21]]

POL2 generator: ~88/81 = 140.407

Optimal ET sequence: 17, 60e, 77, 94, 359e, 453ee, 547ee, 641ee

RMS error: 0.3778 cents

Related temperaments: tsaharuk, quanic

2.3.7.11.13

Subgroup: 2.3.7.11.13

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

Mapping: [⟨1 1 0 1 3], ⟨0 5 24 21 6]]

POL2 generator: ~13/12 = 140.437

Optimal ET sequence: 17, 60e, 77, 94

RMS error: 0.4044 cents

Related temperaments: tsaharuk, quanic

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 7/4s.

Subgroup: 2.3.7.11

Comma list: 43923/43904, 5767168/5764801

Subgroup-val mapping: [⟨1 25 2 -3], ⟨0 -29 1 8]]

Optimal tuning (CTE): ~2 = 1\1, ~7/4 = 968.898

Optimal ET sequence: 26, 83, 109, 135, 566, 701, 836, 971, 1807, 2778, 4585, 11948dee

Badness (Sintel): 0.324

Slendroschismic

Main article: 5th-octave temperaments § Slendroschismic

Pentadecoid

Main article: 15th-octave temperaments § Pentadecoid

Hectosaros leap week

Defined 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⟩

Mapping: [⟨1 313 532], ⟨0 -439 -746]]

Optimal tuning (CTE): ~[17 343 143⟩ = 851.248

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

RMS error: 0.0164 cents

2.3.7.13 subgroup

Subgroup: 2.3.7.13

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

Mapping: [⟨1 313 532 208], ⟨0 -439 -746 -288]]

Optimal tuning (CTE): ~1235079060111/755603996672 = 851.248

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

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⟩

Mapping: [⟨1 313 532 208 58], ⟨0 -439 -746 -288 -76]]

Optimal tuning (CTE): ~6144/3757 = 851.248

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

2.3.7.13.17.19 subgroup

Subgroup: 2.3.7.13.17.19

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

Mapping: [⟨1 313 532 208 58 432], ⟨0 -439 -746 -288 -76 -603]]

Optimal tuning (CTE): ~6144/3757 = 851.248

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

Purpleheart

Subgroup: 2.3.7

Comma list: 2187/2048

Mapping: [⟨7 11 0], ⟨0 0 1]]

mapping generators: ~9/8, ~7

Optimal tuning (CTE): ~9/8 = 1\7, ~7/4 = 968.826 (~64/63 = 59.746)

Optimal ET sequence: 7, 14, 35, 49bd

Badness: 0.0875

Ennea

Subgroup: 2.3.7.11

Comma list: 41503/41472, 43923/43904

Gencom: [2 99/98; 41503/41472, 43923/43904]

Gencom mapping: [⟨1 14/9 0 25/9 31/9], ⟨0 2 0 2 1]]

Sval mapping: [⟨9 0 11 24], ⟨0 2 2 1]]

POL2 generator: ~99/98 = 17.6258

Optimal ET sequence: 54, 63, 72, 135, 342, 477, 1089, 1566

RMS error: 0.0383 cents

Superslendric

In superslendric, 8 8/7s are equated to 3/1 (related to 8edt).

Subgroup: 2.3.7

Comma list: 17294403/16777216

Gencom: [2 8/7; 17294403/16777216]

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

Optimal tuning (CTE): ~2 = 1\1, ~8/7 = 237.712

Parapyth (rank 3)

See also: Pentacircle temperaments § Parapyth

Subgroup: 2.3.7.11

Comma list: 896/891

Gencom: [2 3/2 28/27; 896/891]

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

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

POL2 tuning: ~3 = 1903.834, ~7 = 3369.872

Optimal ET sequence: 17, 36, 41, 58, 63, 104

RMS error: 0.4149 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363

The gencom below gives Margo Schulter's favored basis

Gencom: [2 3/2 28/27; 352/351 364/363]

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

Sval mapping: [⟨1 0 0 7 12], ⟨0 1 0 -4 -7], ⟨0 0 1 1 1]]

POL2 tuning: ~3 = 1903.856, ~7 = 3369.907

Optimal ET sequence: 17, 41, 46, 58, 87, 104

RMS error: 0.3789 cents

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

Gencom: [2 21/19 7; 243/242 1083/1078]

Sval mapping: [⟨1 1 0 2 1], ⟨0 4 0 10 3], ⟨0 0 1 0 1]]

POL2 generator: ~21/19 = 175.2713, ~7 = 3369.3784

Optimal ET sequence: 7, 14, 27e, 34dh, 41, 89, 130

RMS error: 0.3066 cents

Temperaments with a 2.3.11 gene

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 tuning (CTE): ~2 = 1\1, ~3/2 = 692.713

Optimal ET sequence: 2, 5, 7, 12e

Badness: 0.185

Paralimmal

Subgroup: 2.3.11

Comma list: 4096/3993

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

Optimal tuning (CTE): ~2 = 1/1, ~16/11 = 634.320

Optimal ET sequence: 11b, 13, 15, 17

RMS error: 1.237 cents

Neutral

See Rastmic clan #Neutral

Namo

See Rastmic clan #Namo

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 3 3], ⟨0 -6 2 3]]

mapping generators: ~2, ~13/11

Optimal tunings:

• CTE: ~2 = 1\1, ~13/11 = 282.726
• CWE: ~2 = 1\1, ~13/11 = 282.482

Optimal ET sequence: 4, 13, 17

Badness: 0.0263

Aerophore

Subgroup: 2.3.11.19

Comma list: 363/361, 729/704

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

Optimal tuning (POTE): ~2 = 1\1, ~19/11 = 945.4

Optimal ET sequence: 9eehh, 14, 19, 33

Semaerophore

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 tuning (POTE): ~2 = 1\1, ~7/4 = 944.667

Optimal ET sequence: 9eehh, 14, 33d, 47deh

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:

• Shoal17

Music:

• Moody Improvisation in the Shoal Temperament - Budjarn Lambeth (2025)

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