No-sevens subgroup temperaments

This is a collection of subgroup temperaments which omit the prime harmonic of 7.

Dicot

For full 11- and 13-limit extensions, see Dicot family.

Subgroup: 2.3.5.11

Comma list: 25/24, 45/44

Gencom: [2 5/4; 25/24 45/44]

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

Sval mapping: [⟨1 1 2 2], ⟨0 2 1 5]]

POL2 generator: ~5/4 = 346.734

Optimal ET sequence: 3e, 4e, 7, 24c, 31c, 38cc, 45cce

RMS error: 5.621 cents

Related temperaments: dicot, sharp, dichotic

2.3.5.11.13

Subgroup: 2.3.5.11.13

Comma list: 25/24, 40/39, 45/44

Gencom: [2 5/4; 25/24 40/39 45/44]

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

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

POL2 generator: ~5/4 = 350.526

Optimal ET sequence: 3e, 7, 17, 24c

RMS error: 5.916 cents

Mavila

Subgroup: 2.3.5.11

Comma list: 33/32, 45/44

Gencom: [2 4/3; 33/32, 45/44]

Gencom mapping: [⟨1 2 1 0 3], ⟨0 -1 3 0 1]]

Sval mapping: [⟨1 2 1 3], ⟨0 -1 3 1]]

POL2 generator: ~4/3 = 520.212

Optimal ET sequence: 7, 16, 23e, 30bce

RMS error: 4.705 cents

Related temperaments: mavila, pelogic, armodue

Porkypine

The S-expression-based comma list of this temperament is {S10, S11}.

Subgroup: 2.3.5.11

Comma list: 55/54, 100/99

Gencom: [2 10/9; 55/54 100/99]

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

Sval mapping: [⟨1 2 3 4], ⟨0 -3 -5 -4]]

POL2 generator: ~11/10 = 164.078

Optimal ET sequence: 7, 15, 22, 37, 73ce, 95ce, 117bce

RMS error: 2.287 cents

Related temperament: Porcupine

Mohaha

The S-expression-based comma list of this temperament is {S6/S8 = S9, S11}.

Subgroup: 2.3.5.11

Comma list: 81/80, 121/120

Gencom: [2 11/9; 81/80 121/120]

Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5]]

Sval mapping: [⟨1 1 0 2], ⟨0 2 8 5]]

POL2 generator: ~11/9 = 348.094

Optimal ET sequence: 7, 17c, 24, 31, 69e, 100e, 131bee

RMS error: 1.392 cents

Related temperament: mohajira, migration

Music

Mohaha10ping2 by Billy Stiltner

Mohoho

Subgroup: 2.3.5.11.13

Comma list: 66/65, 81/80, 121/120

Gencom: [2 11/9; 66/65 81/80 121/120]

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

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

POL2 generator: ~11/9 = 348.916

Optimal ET sequence: 7, 17c, 24, 31, 55, 86ef, 141ceff

RMS error: 1.971 cents

Tetracot

For full 11- and 13-limit extensions, see Tetracot family.

The S-expression-based comma list of this temperament is {S9/S11, S10}.

Subgroup: 2.3.5.11

Comma list: 100/99, 243/242

Gencom: [2 10/9; 100/99 243/242]

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

Sval mapping: [⟨1 1 1 2], ⟨0 4 9 10]]

POL2 generator: ~10/9 = 175.985

Optimal ET sequence: 7, 27e, 34, 41, 75e

RMS error: 1.182 cents

Related temperaments: monkey, bunya, modus, wollemia

2.3.5.11.13

The S-expression-based comma list of this temperament is {S9/S11, S10, S12(, S10/S12 = S25*S26)}.

Subgroup: 2.3.5.11.13

Comma list: 100/99, 144/143, 243/242

Gencom: [2 10/9; 100/99 243/242]

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

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

POL2 generator: ~10/9 = 176.196

Optimal ET sequence: 7, 27e, 34, 41, 75e

RMS error: 1.140 cents

Tremka

For full 11- and 13-limit extensions, see Hemimean clan #Tremka.

Subgroup: 2.3.5.11

Comma list: 6912/6875, 526153617/524288000

Gencom: [2 363/320; 6912/6875 526153617/524288000]

Gencom mapping: [⟨1 -4 -2 0 4], ⟨0 31 24 0 -3]]

Sval mapping: [⟨1 -4 -2 4], ⟨0 31 24 -3]]

POL2 generator: ~363/320 = 216.182

Optimal ET sequence: 11b, 28bbcc, 39bc, 50, 111, 161, 272

RMS error: 0.3717 cents

2.3.5.11.13

Subgroup: 2.3.5.11.13

Comma list: 2200/2197, 3267/3250, 6656/6655

Gencom: [2 192/169; 2200/2197 3267/3250 6656/6655]

Gencom mapping: [⟨1 -4 -2 0 4 1], ⟨0 31 24 0 -3 15]]

Sval mapping: [⟨1 -4 -2 4 1], ⟨0 31 24 -3 15]]

POL2 generator: ~192/169 = 216.186

Optimal ET sequence: 11b, 28bbccf, 39bc, 50, 111, 272, 383cf

RMS error: 0.3705 cents

Larry

Subgroup: 2.3.5.11

Comma list: 243/242, 4000/3993

Gencom: [2 40/27; 243/242 4000/3993]

Gencom mapping: [⟨1 5 12 0 12], ⟨0 -6 -17 0 -15]]

Sval mapping: [⟨1 5 12 12], ⟨0 -6 -17 -15]]

POL2 generator: ~40/27 = 683.166

Optimal ET sequence: 7, 58, 65, 137, 202

RMS error: 0.3025 cents

Related temperaments: gravity, harry

Twentcufo

Subgroup: 2.3.5.11

Comma list: 8019/8000, 14641/14580

Gencom: [2 400/363; 8019/8000 14641/14580]

Gencom mapping: [⟨1 0 -2 0 0], ⟨0 11 30 0 24]]

Sval mapping: [⟨1 0 -2 0], ⟨0 11 30 24]]

POL2 generator: ~400/363 = 172.8796

Optimal ET sequence: 7, 111, 118

RMS error: 0.2393 cents

Related temperament: undetrita

Yarman

For full 11- and 13-limit extensions, see Turkish maqam music temperaments.

Subgroup: 2.3.5.11

Comma list: 4000/3993, 390625000/387420489

Gencom: [2 100/99; 4000/3993 390625000/387420489]

Gencom mapping: [⟨1 2 3 0 4], ⟨0 -33 -54 0 -43]]

Sval mapping: [⟨1 2 3 4], ⟨0 -33 -54 -43]]

POL2 generator: ~100/99 = 15.0773

Optimal ET sequence: 79, 80, 159, 239, 398

RMS error: 0.1682 cents

2.3.5.11.13

Subgroup: 2.3.5.11.13

Comma list: 325/324, 4000/3993, 85683/85184

Gencom: [2 100/99; 325/324 4000/3993 85683/85184]

Gencom mapping: [⟨1 2 3 0 4 4], ⟨0 -33 -54 0 -43 -24]]

Sval mapping: [⟨1 2 3 4 4], ⟨0 -33 -54 -43 -24]]

POL2 generator: ~100/99 = 15.0885

Optimal ET sequence: 79, 80, 159

RMS error: 0.2830 cents

Emka

For full 11- and 13-limit extensions, see Hemimean clan #Emka or Horwell temperaments #Emkay.

Subgroup: 2.3.5.11

Comma list: 4000/3993, 9453125/9437184

Gencom: [2 11/8; 4000/3993 9453125/9437184]

Gencom mapping: [⟨1 14 6 0 3], ⟨0 -27 -8 0 1]]

Sval mapping: [⟨1 14 6 3], ⟨0 -27 -8 1]]

POL2 generator: ~11/8 = 551.7778

Optimal ET sequence: 37, 50, 87, 137, 224

RMS error: 0.1188 cents

2.3.5.11.13

Subgroup: 2.3.5.11.13

Comma list: 625/624, 2200/2197, 4000/3993

Gencom: [2 11/8; 625/624 2200/2197 4000/3993]

Gencom mapping: [⟨1 14 6 0 3 6], ⟨0 -27 -8 0 1 -5]]

Sval mapping: [⟨1 14 6 3 6], ⟨0 -27 -8 1 -5]]

POL2 generator: ~11/8 = 551.7753

Optimal ET sequence: 37, 50, 87, 137, 224

RMS error: 0.1250 cents

Tricot

For full 13-limit extensions, see Tricot family.

Subgroup: 2.3.5.13

Comma list: 2197/2187, 41067/40960

Gencom: [2 13/9; 2197/2187, 41067/40960]

Gencom mapping: [⟨1 0 -13 0 0 0], ⟨0 3 29 0 0 7]]

Sval mapping: [⟨1 0 -13 0], ⟨0 3 29 7]]

POL2 generator: ~13/9 = 633.997

Optimal ET sequence: 17c, 36c, 53

RMS error: 0.2342 cents

Taylor

Subgroup: 2.3.5.13

Comma list: 676/675, 32805/32768

Gencom: [2 15/13; 676/675 32805/32768]

Gencom mapping: [⟨1 2 -1 0 0 1], ⟨0 -2 16 0 0 13]]

Sval mapping: [⟨1 2 -1 1], ⟨0 -2 16 13]]

POL2 generator: ~15/13 = 249.145

Optimal ET sequence: 24, 53, 130, 183, 236

RMS error: 0.1485 cents

Related temperament: hemischis

Majvam

For full 13- and 17-limit extensions, see Parkleiness temperaments #Majvamic or Cataharry temperaments #Majvamoid.

Subgroup: 2.3.5.13

Comma list: 676/675, 127401984/126953125

Gencom: [2 125/96; 676/675 127401984/126953125]

Gencom mapping: [⟨1 10 5 0 0 19], ⟨0 -22 -7 0 0 -40]]

Sval mapping: [⟨1 10 5 19], ⟨0 -22 -7 -40]]

POL2 generator: ~125/96 = 458.988

Optimal ET sequence: 34, 149, 183, 217, 400

RMS error: 0.1171 cents

2.3.5.13.17

Subgroup: 2.3.5.13.17

Comma list: 676/675, 2601/2600, 24576/24565

Gencom: [2 125/96; 676/675 2601/2600 24576/24565]

Gencom mapping: [⟨1 10 5 0 0 19 6], ⟨0 -22 -7 0 0 -40 -5]]

Sval mapping: [⟨1 10 5 19 6], ⟨0 -22 -7 -40 -5]]

POL2 generator: ~125/96 = 458.991

Optimal ET sequence: 34, 149, 183, 217, 400

RMS error: 0.1129 cents

Photia

See also: No-elevens subgroup temperaments #Garibaldia

Subgroup: 2.3.5.17

Comma list: 256/255, 1458/1445

Gencom: [2 4/3; 256/255 1458/1445]

Gencom mapping: [⟨1 2 -1 0 0 0 7], ⟨0 -1 8 0 0 0 -7]]

Sval mapping: [⟨1 2 -1 7], ⟨0 -1 8 -7]]

POL2 generator: ~3/2 = 701.491

Optimal ET sequence: 12, 41, 53, 65

RMS error: 0.4842 cents

Related temperament: Schismic

2.3.5.17.19

Subgroup: 2.3.5.17.19

Comma list: 171/170, 256/255, 324/323

Gencom: [2 4/3; 171/170 256/255 324/323]

Gencom mapping: [⟨1 2 -1 0 0 0 7 3], ⟨0 -1 8 0 0 0 -7 3]]

Sval mapping: [⟨1 2 -1 7 3], ⟨0 -1 8 -7 3]]

POL2 generator: ~3/2 = 701.470

Optimal ET sequence: 12, 41, 53, 65

RMS error: 0.5374 cents

Quintaleap

For full 17- and 19-limit extensions, see Quintaleap family.

Subgroup: 2.3.5.17

Comma list: 256/255, 1419857/1417176

Gencom: [2 18/17; 256/255 1419857/1417176]

Gencom mapping: [⟨1 2 1 0 0 0 5], ⟨0 -5 16 0 0 0 -11]]

Sval mapping: [⟨1 2 1 5], ⟨0 -5 16 -11]]

POL2 generator: ~18/17 = 99.272

Optimal ET sequence: 12, 109, 121, 133

RMS error: 0.3051 cents

2.3.5.17.19

Subgroup: 2.3.5.17.19

Comma list: 256/255, 361/360, 4624/4617

Gencom: [2 18/17; 256/255 361/360 4624/4617]

Gencom mapping: [⟨1 2 1 0 0 0 5 4], ⟨0 -5 16 0 0 0 -11 3]]

Sval mapping: [⟨1 2 1 5 4], ⟨0 -5 16 -11 3]]

POL2 generator: ~18/17 = 99.276

Optimal ET sequence: 12, 109, 121, 133

RMS error: 0.3427 cents

Quindromeda

For full 17- and 19-limit extensions, see Quindromeda family.

Subgroup: 2.3.5.17

Comma list: 24576/24565, 295936/295245

Gencom: [2 18/17; 24576/24565 295936/295245]

Gencom mapping: [⟨1 2 0 0 0 0 5], ⟨0 -5 28 0 0 0 -11]]

Sval mapping: [⟨1 2 0 5], ⟨0 -5 28 -11]]

POL2 generator: ~18/17 = 99.524

Optimal ET sequence: 12, 169, 181, 193, 205, 422

RMS error: 0.0757 cents

2.3.5.17.19

Subgroup: 2.3.5.17.19

Comma list: 1216/1215, 1445/1444, 6144/6137

Gencom: [2 18/17; 1216/1215 1445/1444 6144/6137]

Gencom mapping: [⟨1 2 0 0 0 0 5 4], ⟨0 -5 28 0 0 0 -11 3]]

Sval mapping: [⟨1 2 0 5 4], ⟨0 -5 28 -11 3]]

POL2 generator: ~18/17 = 99.524

Optimal ET sequence: 12, 169, 181, 193, 205, 422

RMS error: 0.0813 cents

Quintilischis

For full 17- and 19-limit extensions, see Schismatic family #Quintilipyth or Schismatic family #Quintaschis.

Subgroup: 2.3.5.17

Comma list: 32805/32768, 1419857/1417176

Gencom: [2 18/17; 32805/32768 1419857/1417176]

Gencom mapping: [⟨1 2 -1 0 0 0 5], ⟨0 -5 40 0 0 0 -11]]

Sval mapping: [⟨1 2 -1 5], ⟨0 -5 40 -11]]

POL2 generator: ~18/17 = 99.649

Optimal ET sequence: 12, 253, 265, 277, 289

RMS error: 0.0719 cents

2.3.5.17.19

Subgroup: 2.3.5.17.19

Comma list: 4624/4617, 6144/6137, 6885/6859

Gencom: [2 18/17; 4624/4617 6144/6137 6885/6859]

Gencom mapping: [⟨1 2 -1 0 0 0 5 4], ⟨0 -5 40 0 0 0 -11 3]]

Sval mapping: [⟨1 2 -1 5 4], ⟨0 -5 40 -11 3]]

POL2 generator: ~18/17 = 99.652

Optimal ET sequence: 12, 253, 265, 277, 289

RMS error: 0.1636 cents

Stützel

Subgroup: 2.3.5.19

Comma list: 81/80, 96/95

Gencom: [2 4/3; 81/80 96/95]

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

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

POL2 generator: ~3/2 = 697.867

Optimal ET sequence: 5, 7, 12, 31, 43

RMS error: 1.378 cents

Related temperament: meantone

Dequarter

Subgroup: 2.3.5.11

Comma list: 33/32, 55/54

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

sval mapping generators: ~2, ~3

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.0387

Optimal ET sequence: 5, 7, 19e, 26e

Badness: 0.0145

Related temperament: meantone

Dreamtone

Subgroup: 2.3.5.11.13

Comma list: 33/32, 55/54, 975/968

Sval mapping: [⟨1 0 -4 5 21], ⟨0 1 4 -1 -11]]

sval mapping generators: ~2, ~3

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 689.6993

Optimal ET sequence: 7, 19eff, 26eff, 33ceeff, 40ceeff

Badness: 0.0353

Hypnotone

Subgroup: 2.3.5.11

Comma list: 45/44, 81/80

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

sval mapping generators: ~2, ~3

Optimal tuning (CTE): ~2/1 = 1\1, ~3/2 = 694.6998

Optimal ET sequence: 7, 12, 19, 26, 45

Badness: 0.0104

Related temperament: flattone. (Hypnotone is no-sevens flattone.)

2.3.5.11.13 subgroup

Subgroup: 2.3.5.11.13

Comma list: 45/44, 65/64, 81/80

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

sval mapping generators: ~2, ~3

Optimal tuning (CTE): ~2/1 = 1\1, ~3/2 = 693.9513

Optimal ET sequence: 7, 12, 19, 26, 45f

Badness: 0.0141

Devisemi

See also: No-elevens subgroup temperaments #Devisemi

Subgroup: 2.3.5.19

Comma list: 361/360, 20000/19683

Gencom: [2 19/18; 361/360 20000/19683]

Gencom mapping: [⟨1 1 1 0 0 0 0 3], ⟨0 8 18 0 0 0 0 17]]

Sval mapping: [⟨1 1 1 3], ⟨0 8 18 17]]

POL2 generator: ~19/18 = 88.077

Optimal ET sequence: 14c, 27, 41, 68, 109

RMS error: 0.5701 cents

Related temperament: octacot

Nestoria

See also: No-elevens subgroup temperaments #Garibaldia and #Pontia

The S-expression-based comma list of this temperament is {S16/S18, S19(, S15/S20)}.

Subgroup: 2.3.5.19

Comma list: 361/360, 513/512

Sval mapping: [⟨1 0 15 9], ⟨0 1 -8 -3]]

mapping generators: ~2, ~3

Gencom mapping: [⟨1 2 -1 0 0 0 0 3], ⟨0 -1 8 0 0 0 0 3]]

gencom: [2 4/3; 361/360 513/512]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.746

Optimal ET sequence: 12, 29, 41, 53, 118, 171

RMS error: 0.1763 cents

Related temperament: schismic

Rarity

For the 5-limit version of this temperament, see High badness temperaments #Rarity.

Rarity generator is so close to 23/19, and some edos supporting it have good 19th and 23rd harmonics that it is natural to prescribe a 2.3.5.19.23 extension. Since the generator is also mapped to 368/225, this means that 4275/4232 is tempered out.

Subgroup: 2.3.5.19.23

Comma list: 1035/1024, 16875/16606, 192375/188416

Sval mapping: [⟨1 11 -10 -3 -2], ⟨0 -13 17 10 9]]

Sval mapping generators: ~1\1 = 2, ~368/225 = 869.420

Optimal tuning (CTE): ~368/225 = 869.420

Optimal ET sequence: 11, 29, 69, 98c

Badness: 0.0656