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

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

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

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

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

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

Neutral

Neutral can be thought of as the 2.3.11 version of either mohajira or maqamic, 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

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

Magi

Subgroup: 2.3.7.11

Comma list: 896/891, 537824/531441

Gencom: [2 96/77; 896/891, 537824/531441]

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

Balthazar

Subgroup: 2.3.7.11.13

Comma list: 896/891, 537824/531441, 169/168

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

Caspar

Subgroup: 2.3.7.11.13

Comma list: 896/891, 537824/531441, 144/143

Optimal ET sequence: 19, 22f, 41

Melchior

Subgroup: 2.3.7.11.13

Comma list: 896/891, 537824/531441, 364/363

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

Hogwarts

Subgroup: 2.3.7.29

Comma list: 537824/531441, 784/783

Gencom: [2 36/29; 537824/531441, 784/783]

Optimal ET sequence: 16dj, 19, 22, 41, 145, 186j, 227j

Twenothology

Subgroup: 2.3.7.11.13.29

Comma list: 537824/531441, 896/891, 144/143, 784/783

Optimal ET sequence: 19, 22f, 41

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

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

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

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

Superflat temperament, or alternatively, tridecimal temperament makes 1053/1024 vanish, so 13/8 is a minor sixth, and 16/13 is a major triad. 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).

Subgroup: 2.3.13

Comma list: 1053/1024

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

Preimage: 2, 3/2

CTE generator: ~3/2 = 692.939

Supporting ETs: 7, 19, 12, 5f, 26, 9f, 31f, 33, 17f, 45f, 16f, 40, 50f, 43f

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.

Subgroup: 2.3.13

Comma list: 27/26

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

CTE generator: ~3/2 = 688.391

Optimal ET sequence: 5, 7