No-fives subgroup temperaments

Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives dominant" or "no-fives superpyth". The name comes from the fact that it tempers out 64/63, the Archytas comma.

Subgroup: 2.3.7

Comma: 64/63

Gencom: [2 3/2; 64/63]

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

Sval mapping: [⟨1 2 2], ⟨0 -1 2]]

POL2 generator: ~3/2 = 709.321

Optimal ET sequence: 5, 12, 17, 22, 27, 137bd

RMS error: 1.856 cents

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

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

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

Sval mapping: [⟨1 0 6 13], ⟨0 1 -2 -6]]

POL2 generator: ~3/2 = 707.192

Optimal ET sequence: 5, 12, 17, 39d, 56d

RMS error: 1.977 cents

Supraphon

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 99/98

Gencom: [2 3/2; 64/63 78/77 99/98]

Gencom mapping: [⟨1 1 0 4 7 9], ⟨0 1 0 -2 -6 -9]]

Sval mapping: [⟨1 0 6 13 18], ⟨0 1 -2 -6 -9]]

POL2 generator: ~3/2 = 706.137

Optimal ET sequence: 5f, 12f, 17, 22, 39d, 56d

RMS error: 2.095 cents

Suhajira

Subgroup: 2.3.7.11

Comma list: 64/63, 243/242

Gencom: [2 11/9; 64/63 243/242]

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

Sval mapping: [⟨1 1 4 2], ⟨0 2 -4 5]]

POL2 generator: ~11/9 = 353.958

Optimal ET sequence: 7, 10, 17, 44e, 61de, 78de

RMS error: 1.968 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 144/143

Gencom: [2 11/9; 64/63 78/77 144/143]

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

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

POL2 generator: ~11/9 = 353.775

Optimal ET sequence: 7, 10, 17, 44e, 61de, 78de

RMS error: 1.953 cents

Flutterpyth

Subgroup: 2.3.7.11.13.19

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

Mapping: [⟨1 1 4 10 15 9], ⟨0 -1 -2 -11 -19 -8]]

Optimal tuning (CTE): ~3/2 = 713.459

Restricted to 2.3.7.11.13, this temperament is a no-5 restriction of 13-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.

Skwares

Subgroup: 2.3.7

Comma: 19683/19208

Gencom: [2 9/7; 19683/19208]

Gencom mapping: [⟨1 3 6], ⟨0 -4 -9]]

Sval mapping: [⟨1 3 6], ⟨0 -4 -9]]

POL2 generator: ~9/7 = 425.365

Optimal ET sequence: 14, 17, 31, 48, 79, 189b, 268bd, 347bd

RMS error: 1.149 cents

Related temperament: squares

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 99/98, 243/242

Gencom: [2 9/7; 99/98 243/242]

Gencom mapping: [⟨1 3 0 6 7], ⟨0 -4 0 -9 -10]]

Sval mapping: [⟨1 3 6 7], ⟨0 -4 -9 -10]]

POL2 generator: ~9/7 = 425.244

Optimal ET sequence: 14, 17, 31, 48, 79, 127, 206bde

RMS error: 1.099 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

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

Gencom: [2 9/7; 78/77, 99/98, 243/242]

Gencom mapping: [⟨1 3 0 6 7 9], ⟨0 -4 0 -9 -10 -15]]

Sval mapping: [⟨1 3 6 7 9], ⟨0 -4 -9 -10 -15]]

POL2 generator: ~9/7 = 424.457

Optimal ET sequence: 17, 48f, 65ef, 82d, 147df

RMS error: 1.769 cents

Skwairs

Subgroup: 2.3.7.11.13

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

Gencom: [2 9/7; 99/98, 144/143, 243/242]

Gencom mapping: [⟨1 3 0 6 7 3], ⟨0 -4 0 -9 -10 2]]

Sval mapping: [⟨1 3 6 7 3], ⟨0 -4 -9 -10 2]]

POL2 generator: ~9/7 = 424.702

Optimal ET sequence: 14, 17, 31

RMS error: 1.290 cents

Byhearted

For the full 19-limit version of this temperament, see Tetracot family #Byhearted.

Subgroup: 2.3.7.11.19

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

Gencom: [209/147 21/19; 99/98 243/242 363/361]

Sval mapping: [⟨2 2 3 4 5], ⟨0 4 9 10 12]]

POL2 generator: ~21/19 = 174.735

Optimal ET sequence: 14, 34dh, 48, 110e, 158e

RMS error: 0.8727 cents

Harrison

Subgroup: 2.3.7

Comma: 59049/57344

Gencom: [2 3/2; 59049/57344]

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

Sval mapping: [⟨1 1 -3], ⟨0 1 10]]

POL2 generator: ~3/2 = 696.544

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

RMS error: 1.226 cents

Related temperament: meantone

Leapfrog

See also: Gentle region

Subgroup: 2.3.7

Comma list: 14680064/14348907

Gencom: [2 3/2; 14680064/14348907]

Gencom mapping: [⟨1 1 0 -6], ⟨0 1 0 15]]

Sval mapping: [⟨1 0 -21], ⟨0 1 15]]

POL2 generator: ~3/2 = 704.721 cents

Optimal ET sequence: 17, 46, 63

RMS error: 0.6202 cents

Related temperaments: leapday, leapweek, srutal

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 896/891, 1331/1323

Gencom: [2 3/2; 896/891 1331/1323]

Gencom mapping: [⟨1 1 0 -6 -3], ⟨0 1 0 15 11]]

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

POL2 generator: ~3/2 = 704.753 cents

Optimal ET sequence: 17, 46, 63

RMS error: 0.6047 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

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

Gencom: [2 3/2; 169/169 352/351 364/363]

Gencom mapping: [⟨1 1 0 -6 -3 -1], ⟨0 1 0 15 11 8]]

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

POL2 generator: ~3/2 = 704.745 cents

Optimal ET sequence: 17, 46, 63

RMS error: 0.7541 cents

Skidoo

Subgroup: 2.3.7.11.13.23

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

Gencom: [2 3/2; 169/169 208/207 352/351 364/363]

Gencom mapping: [⟨1 1 0 -6 -3 -1 0 0 1], ⟨0 1 0 15 11 8 0 0 6]]

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

POL2 generator: ~3/2 = 704.729 cents

Optimal ET sequence: 17, 46, 63

RMS error: 0.6265 cents

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

Gencom: [2 3/2; 169/169 208/207 352/351 364/363]

Gencom mapping: [⟨1 1 0 -6 -3 -1 0 0 1 -11], ⟨0 1 0 15 11 8 0 0 6 27]]

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

POL2 generator: ~3/2 = 704.729 cents

Optimal ET sequence: 17, 46, 63

Music

Suite for Harpsichord in A Locrian, tuning: Eb-G# in 46EDO by IlL (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

Subgroup: 2.3.7

Comma list: 5764801/5668704

Gencom: [2 343/324; 5764801/5668704]

Sval mapping: [⟨1 1 2], ⟨0 8 11]]

POL2 generator: ~343/324 = 87.8304

Optimal ET sequence: 14, 27, 41

RMS error: 0.5041 cents

Related temperaments: octacot

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 2401/2376

Gencom: [2 22/21; 243/242 2401/2376]

Sval mapping: [⟨1 1 2 2], ⟨0 8 11 20]]

POL2 generator: ~22/21 = 87.6512

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

RMS error: 0.7147 cents

Related temperaments: octacot

2.3.7.11.19

Subgroup: 2.3.7.11.19

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

Gencom: [2 19/18; 133/132 243/242 343/342]

Sval mapping: [⟨1 1 2 2 3], ⟨0 8 11 20 17]]

POL2 generator: ~19/18 = 87.6684

Optimal ET sequence: 14, 27e, 41

RMS error: 0.7065 cents

Related temperaments: octacot

Magi

Subgroup: 2.3.7

Comma list: 537824/531441

Gencom: [2 243/196; 537824/531441]

Sval mapping: [⟨1 0 -1], ⟨0 5 12]]

POL2 generator: ~243/196 = 380.661

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

RMS error: 0.4277 cents

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 896/891, 26411/26244

Gencom: [2 96/77; 896/891 26411/26244]

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

POL2 generator: ~96/77 = 380.768

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

RMS error: 0.4262 cents

Balthazar

Subgroup: 2.3.7.11.13

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

Gencom: [2 143/128; 169/168 896/891 26411/26244]

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

POL2 generator: ~143/128 = 190.407

Optimal ET sequence: 19, 44, 63, 145f

RMS error: 0.6937 cents

Caspar

Subgroup: 2.3.7.11.13

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

Gencom: [2 26/21; 144/143 343/338 729/728]

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

POL2 generator: ~26/21 = 380.531

Optimal ET sequence: 19, 22f, 41

RMS error: 1.032 cents

Melchior

Subgroup: 2.3.7.11.13

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

Gencom: [2 96/77; 352/351 364/363 26411/26244]

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

POL2 generator: ~96/77 = 380.766

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

RMS error: 0.3891 cents

Hogwarts

Subgroup: 2.3.7.29

Comma list: 784/783, 5887/5832

Gencom: [2 36/29; 784/783 5887/5832]

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

POL2 generator: ~36/29 = 380.618

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

Twenothology

Subgroup: 2.3.7.11.13.29

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

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

POL2 generator: ~26/21 = 380.526

Optimal ET sequence: 19, 22f, 41

Lee

Subgroup: 2.3.7

Comma: 177147/175616

Gencom: [2 81/56; 177147/175616]

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

Sval mapping: [⟨1 0 -3], ⟨0 3 11]]

POL2 generator: ~81/56 = 633.525

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

RMS error: 0.3519 cents

Slendric

Subgroup: 2.3.7

Comma: 1029/1024

Gencom: [2 8/7; 1029/1024]

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

Sval mapping: [⟨1 1 3], ⟨0 3 -1]]

POL2 generator: ~8/7 = 233.688

Optimal ET sequence: 5, 21, 26, 31, 36, 77, 113, 190

RMS error: 0.3202 cents

Baladic

Subgroup: 2.3.7.13

Comma list: 169/168, 1029/1024

Gencom: [91/64 8/7; 169/168 1029/1024]

Sval mapping: [⟨2 2 6 7], ⟨0 3 -1 1]]

POL2 generator: ~8/7 = 233.6044

Optimal ET sequence: 10, 26, 36, 154…, 190…, 226…, 262…

RMS error: 0.5452 cents

2.3.7.13.17

Subgroup: 2.3.7.13.17

Comma list: 169/168, 273/272, 289/288

Gencom: [17/12 8/7; 169/168 273/272 289/288]

Sval mapping: [⟨2 2 6 7 7], ⟨0 3 -1 1 3]]

POL2 generator: ~8/7 = 233.6155

Optimal ET sequence: 10, 26, 36, 154…, 190…, 226…

RMS error: 0.5073 cents

Gigapyth

Subgroup: 2.3.85

Comma list: 2.3.85 ⟨-40 1 6]

Sval 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

Sval 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

Hemif

Subgroup: 2.3.7

Comma: 1605632/1594323

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

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

Sval mapping: [⟨1 1 -1], ⟨0 2 13]]

POL2 generator: ~2187/1792 = 351.485

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

RMS error: 0.2344 cents

Related temperaments: hemififths, namo

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 896/891

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

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

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

POL2 generator: ~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

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

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

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

POL2 generator: ~11/9 = 351.691

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

RMS error: 0.7167 cents

Heartful

See also: Heartland

Subgroup: 2.3.7.11.19

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

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

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

POL2 generator: ~21/19 = 175.804

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

RMS error: 0.5360 cents

Related temperaments: bunya

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

Slendrismic

See also: 5th-octave temperaments #Slendrismic

Subgroup: 2.3.7

Comma list: 68719476736/68641485507

Mapping: [⟨5 8 14], ⟨0 -2 1]]

POL2 generator: ~1029/1024 = 8.9906

Optimal ET sequence: 130, 135, 265, 400, 935, 1335, 1735

RMS error: 0.0212 cents

Related temperaments: hemipental

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

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

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

Paralimmal

Subgroup: 2.3.11

Comma list: 4096/3993

Sval 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

Neutral can be thought of as the 2.3.11 version of either mohajira or neutrominant, as well as suhajira and ringo. Among other things, it is the temperament optimizing the neutral tetrad.

Subgroup: 2.3.11

Comma: 243/242

Gencom: [2 11/9; 243/242]

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

Sval mapping: [⟨1 1 2], ⟨0 2 5]]

POL2 generator: ~11/9 = 350.525

Optimal ET sequence: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380

RMS error: 0.3021 cents

Scales

Namo

Subgroup: 2.3.11.13

Comma list: 144/143, 243/242

Gencom: [2 11/9; 144/143 243/242]

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

Sval mapping: [⟨1 1 2 4], ⟨0 2 5 -1]]

POL2 generator: ~11/9 = 351.488

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

RMS error: 0.7038 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

Aerophore

Subgroup: 2.3.11.19

Comma list: 363/361, 729/704

Mapping: [⟨1 0 -6 -6], ⟨0 2 12 13]]

POTE generator: ~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

Mapping: [⟨1 0 2 -6 -6], ⟨0 2 1 12 13]]

POTE generator: ~7/4 = 944.667

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

Superflat aka tridecimal

Superflat temperament, or alternatively, tridecimal temperament, 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

Sval 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 temperamenets intersect in 7edo, where major sixths and minor sixths are not distinguished.

Subgroup: 2.3.13

Comma list: 27/26

Sval 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

Sval 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

Semitonic

Subgroup: 2.3.17

Comma list: 289/288

Sval 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

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 19/16 is a minor third. It can be viewed as a 2.3.19 subgroup analogue of schismic temperament.

Subgroup: 2.3.19

Comma list: 513/512

Mapping: [⟨1 0 9], ⟨0 1 -3]]

Mapping generators: ~2, ~3

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.3288

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

Badness: 0.000374

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

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

Io aka undecimal

Io is a very low-complexity temperament which tempers out the undecimal quartertone 33/32, and with a generator representing both 3/2 and 16/11. It may be considered an exotemperament by some definitions and not one by others. 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

Sval mapping: [⟨1 1 4], ⟨0 1 -1]]

mapping generators: ~2/1, ~3/2

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

Optimal ET sequence: 2, 5, 7, 12e

Badness: 0.185

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

Sval mapping: [⟨1 2 6], ⟨0 -2 -7]]

Optimal tuning (TE): 2/1 = 1200.5469, ~27/23 = 251.9737

Supporting ETs: 5, 9, 14, 19, 24, 43, 62, 81, 100