Hemimean clan

The hemimean clan tempers out the hemimean comma, 3136/3125, with monzo [6 0 -5 2⟩. The head of this clan is the 2.5.7 subgroup temperament didacus, generated by a tempered hemithird of 28/25. Two generator steps make 5/4 and five make 7/4.

The second comma of the comma list determines which 7-limit family member we are looking at. These extensions, in general, split the syntonic comma into two, each for 126/125~225/224, as 3136/3125 = (126/125)/(225/224). Hemiwürschmidt adds 2401/2400; hemithirds adds 1029/1024; spell adds 49/48. These all use the same nominal generator as didacus.

Septimal passion adds 64/63, splitting the hemithird into a further two. Septimal meantone adds 81/80 as well as 126/125 and 225/224, splitting an octave plus the hemithird into two perfect fifths. Sycamore adds 686/675, splitting the hemithird into three. Semisept adds 1728/1715, splitting an octave plus the hemithird into three. Mohavila adds 135/128, whereas cohemimabila adds 65536/64827, both splitting two octaves plus the hemithird into three. Emka adds 84035/82944, splitting two octaves plus the hemithird into four. Bidia adds 2048/2025 with a 1/4-octave period. Misty adds 5120/5103 with a 1/3-octave period. Bischismic adds 32805/32768 with a semioctave period. Hexe adds 50/49 with a 1/6-octave period. Clyde adds 245/243 with a generator of ~9/7, five of which make the original. Parakleismic adds 4375/4374 with a generator of ~6/5. Arch adds 5250987/5242880 with a generator of ~64/63. For these seven generators make the original. Sengagen adds 420175/419904 with a generator of ~686/675, splitting the hemithird into eight. Subpental adds 19683/19600 with a generator of ~56/45, nine of which make the original.

Discussed elsewhere are

• Passion → Passion family (+64/63 or 3125/3087)
• Meantone → Meantone family (+81/80, 126/125, 225/224)
• Mohavila → Pelogic family (+135/128 or 1323/1250)
• Cohemimabila → Mabila family (+65536/64827)
• Sycamore → Sycamore family (+686/675 or 875/864)
• Bidia → Diaschismic family (+2048/2025)
• Hexe → Augmented family (+50/49 or 128/125)
• Misty → Misty family (+5120/5103)
• Bischismic → Schismatic family (+32805/32768)
• Clyde → Kleismic family (+245/243)
• Parakleismic → Ragismic microtemperaments (+4375/4374)
• Arch → Escapade family (+5250987/5242880)
• Decistearn → Stearnsmic clan (+118098/117649)
• Quintagar → Quindromeda family (+33554432/33480783)
• Rubidium → 37th-octave temperaments (+4194304/4117715)

Considered below are hemiwürschmidt, hemithirds, spell, semisept, emka, decipentic, sengagen, subpental, mowglic, and undetrita.

A notable subgroup extension of didacus is roulette.

Didacus

Subgroup: 2.5.7

Comma list: 3136/3125

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

sval mapping generators: ~2, ~56/25

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

Gencom: [2 56/25; 3136/3125]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.772

Optimal ET sequence: 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c

Rectified hebrew

Main article: Rectified hebrew

Rectified hebrew (37 & 56) is derived from the calendar by the same name. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11. It adds harmonic 13 through tempering out 4394/4375 and spliting the generator of didacus in three.

Subgroup: 2.5.7.13

Comma list: 3136/3125, 4394/4375

Sval mapping: [⟨1 2 2 3], ⟨0 6 15 13]]

sval mapping generators: ~2, ~26/25

Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 64.6086

Optimal ET sequence: 18, 19, 37, 93, 130

Hemiwürschmidt

See also: Würschmidt family § Hemiwürschmidt

Hemiwürschmidt (sometimes spelled hemiwuerschmidt) tempers out 2401/2400, 3136/3125, and 6144/6125. 68edo, 99edo and 130edo can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, ⟨⟨ 16 2 5 40 -39 -49 -48 28 … ]].

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3136/3125

Mapping: [⟨1 15 4 7], ⟨0 -16 -2 -5]]

Wedgie: ⟨⟨ 16 2 5 -34 -37 6 ]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.898

Optimal ET sequence: 31, 68, 99, 229, 328, 557c, 885cc

Badness: 0.020307

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 3136/3125

Mapping: [⟨1 15 4 7 37], ⟨0 -16 -2 -5 -40]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.840

Optimal ET sequence: 31, 99e, 130, 650ce, 811ce

Badness: 0.021069

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 441/440, 3584/3575

Mapping: [⟨1 15 4 7 37 -29], ⟨0 -16 -2 -5 -40 39]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829

Optimal ET sequence: 31, 99e, 130, 291, 421e, 551ce

Badness: 0.023074

Hemithir

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 275/273

Mapping: [⟨1 15 4 7 37 -3], ⟨0 -16 -2 -5 -40 8]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918

Optimal ET sequence: 31, 68e, 99ef

Badness: 0.031199

Hemiwur

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 1375/1372

Mapping: [⟨1 15 4 7 11], ⟨0 -16 -2 -5 -9]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884

Optimal ET sequence: 31, 68, 99, 130e, 229e

Badness: 0.029270

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 275/273

Mapping: [⟨1 15 4 7 11 -3], ⟨0 -16 -2 -5 -9 8]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004

Optimal ET sequence: 31, 68, 99f, 167ef

Badness: 0.028432

Hemiwar

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 121/120, 1375/1372

Mapping: [⟨1 15 4 7 11 23], ⟨0 -16 -2 -5 -9 -23]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698

Optimal ET sequence: 6f, 31

Badness: 0.044886

Quadrawürschmidt

This has been documented in Graham Breed's temperament finder as semihemiwürschmidt, but quadrawürschmidt arguably makes more sense.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 3136/3125

Mapping: [⟨1 15 4 7 24], ⟨0 -32 -4 -10 -49]]

mapping generators: ~2, ~147/110

Optimal tuning (POTE): ~2 = 1\1, ~147/110 = 503.0404

Optimal ET sequence: 31, 105be, 136e, 167, 198, 427c

Badness: 0.034814

Semihemiwür

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3136/3125, 9801/9800

Mapping: [⟨2 14 6 9 -10], ⟨0 -16 -2 -5 25]]

mapping generators: ~99/70, ~495/392

Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9021

Optimal ET sequence: 62e, 68, 130, 198, 328

Badness: 0.044848

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125

Mapping: [⟨2 14 6 9 -10 25], ⟨0 -16 -2 -5 25 -26]]

Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035

Optimal ET sequence: 62e, 68, 130, 198, 328

Badness: 0.023388

Semihemiwürat

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625

Mapping: [⟨2 14 6 9 -10 25 19], ⟨0 -16 -2 -5 25 -26 -16]]

Optimal tuning (POTE): ~17/12 = 1\2, ~28/25 = 193.9112

Optimal ET sequence: 62e, 68, 130, 198, 328g, 526cfgg

Badness: 0.028987

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625

Mapping: [⟨2 14 6 9 -10 25 19 20], ⟨0 -16 -2 -5 25 -26 -16 -17]]

Optimal tuning (POTE): ~17/12 = 1\2, ~19/17 = 193.9145

Optimal ET sequence: 62e, 68, 130, 198, 328g, 526cfgg

Badness: 0.021707

Semihemiwürand

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224

Mapping: [⟨2 14 6 9 -10 25 -4], ⟨0 -16 -2 -5 25 -26 18]]

Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9112

Optimal ET sequence: 62eg, 68, 130g, 198g

Badness: 0.029718

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224

Mapping: [⟨2 14 6 9 -10 25 -4 -3], ⟨0 -16 -2 -5 25 -26 18 17]]

Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428

Optimal ET sequence: 62egh, 68, 130gh, 198gh

Badness: 0.029545

Hemithirds

Main article: Hemithirds

See also: Luna family § Hemithirds

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3136/3125

Mapping: [⟨1 4 2 2], ⟨0 -15 2 5]]

Wedgie: ⟨⟨ 15 -2 -5 -38 -50 -6 ]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.244

Minimax tuning:

• 7-odd-limit: ~28/25 = [1/10 -1/20 0 1/20⟩

Eigenmonzo (unchanged-interval) basis: 2.7/3

• 9-odd-limit: ~28/25 = [6/25 -2/35 0 1/35⟩

Eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence: 25, 31, 87, 118

Badness: 0.044284

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 3136/3125

Mapping: [⟨1 4 2 2 7], ⟨0 -15 2 5 -22]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.227

Minimax tuning:

• 11-odd-limit: ~28/25 = [5/27 0 0 1/27 -1/27⟩

Eigenmonzo (unchanged-interval) basis: 2.11/7

Optimal ET sequence: 25e, 31, 87, 118

Badness: 0.019003

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384, 625/624

Mapping: [⟨1 4 2 2 7 0], ⟨0 -15 2 5 -22 23]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166

Optimal ET sequence: 31, 56, 87, 118, 205d

Badness: 0.021738

Spell

See also: Magic family

Subgroup: 2.3.5.7

Comma list: 49/48, 3125/3072

Mapping: [⟨1 0 2 2], ⟨0 10 2 5]]

Wedgie: ⟨⟨ 10 2 5 -20 -20 6 ]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 189.927

Optimal ET sequence: 6, 19, 82dd

Badness: 0.080958

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 125/121

Mapping: [⟨1 0 2 2 3], ⟨0 10 2 5 3]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.285

Optimal ET sequence: 6, 19, 44de, 63dee, 82ddee

Badness: 0.059791

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 78/77, 125/121

Mapping: [⟨1 0 2 2 3 4], ⟨0 10 2 5 3 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 189.928

Optimal ET sequence: 6, 19, 82ddeeff

Badness: 0.045591

Cantrip

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 125/121

Mapping: [⟨1 0 2 2 3 1], ⟨0 10 2 5 3 17]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.360

Optimal ET sequence: 19, 44de, 63dee, 82ddee

Badness: 0.041603

Semisept

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

The minimal generator of semisept is half a tempered septimal major sixth (12/7), hence the name. Three such generator steps minus an octave give the hemithird, and six give the classical major third. It can be described as the 31 & 80 temperament, and as one may expect, 111edo makes for a great tuning.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 3136/3125

Mapping: [⟨1 12 6 12], ⟨0 -17 -6 -15]]

mapping generators: ~2, ~75/49

Wedgie: ⟨⟨ 17 6 15 -30 -24 18 ]]

Optimal tuning (POTE): ~2 = 1\1, ~75/49 = 735.155

Optimal ET sequence: 18, 31, 80, 111

Badness: 0.050472

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 1331/1323

Mapping: [⟨1 12 6 12 20], ⟨0 -17 -6 -15 -27]]

Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.125

Optimal ET sequence: 18e, 31, 80, 111, 364cd

Badness: 0.022476

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 540/539, 1375/1372

Mapping: [⟨1 12 6 12 20 -11], ⟨0 -17 -6 -15 -27 24]]

Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.126

Optimal ET sequence: 31, 80, 111

Badness: 0.025204

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 640/637, 715/714

Mapping: [⟨1 12 6 12 20 -11 -10], ⟨0 -17 -6 -15 -27 24 23]]

Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.125

Optimal ET sequence: 31, 80, 111

Badness: 0.019919

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323

Mapping: [⟨1 12 6 12 20 -11 -10 -8], ⟨0 -17 -6 -15 -27 24 23 20]]

Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.116

Optimal ET sequence: 31, 80, 111

Badness: 0.016301

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459

Mapping: [⟨1 12 6 12 20 -11 -10 -8 18], ⟨0 -17 -6 -15 -27 24 23 20 -22]]

Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.106

Optimal ET sequence: 31, 80, 111, 191cdh, 302cdgh

Badness: 0.014957

Semishly

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 196/195, 275/273

Mapping: [⟨1 12 6 12 20 8], ⟨0 -17 -6 -15 -27 -7]]

Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 464.980

Optimal ET sequence: 31, 49f, 80f

Badness: 0.028408

Emka

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

Emka tempers out [-50 -8 27⟩ in the 5-limit. This temperament can be described as 37 & 50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension emkay (87 & 224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 84035/82944

Mapping: [⟨1 14 6 12], ⟨0 -27 -8 -20]]

mapping generators: ~2, ~48/35

Wedgie: ⟨⟨ 27 8 20 -50 -44 24 ]]

Optimal tuning (POTE): ~2 = 1\1, ~48/35 = 551.782

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

Badness: 0.144338

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2401/2376, 3136/3125

Mapping: [⟨1 14 6 12 3], ⟨0 -27 -8 -20 1]]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.765

Optimal ET sequence: 37, 50, 87, 224d, 311d

Badness: 0.054744

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 364/363, 385/384, 625/624

Mapping: [⟨1 14 6 12 3 6], ⟨0 -27 -8 -20 1 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.758

Optimal ET sequence: 37, 50, 87, 224d, 311d, 398d

Badness: 0.029741

Decipentic

The generator for the decipentic temperament (43&56) is tenth root of fifth harmonic (5/1, pentave), 5^1/10, tuned between 75/64 and 20/17 (close to 27/23). Aside from the hemimean comma, this temperament tempers out the bronzisma, 2097152/2083725 (satriru-agugu). 99EDO is a good tuning for decipentic, with generator 23\99, and MOS of 9, 13, 17, 30, 43 or 56 notes are available.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 2097152/2083725

Mapping: [⟨1 6 0 -3], ⟨0 -19 10 25]]

Wedgie: ⟨⟨ 19 -10 -25 -60 -93 -30 ]]

POTE generator: ~75/64 = 278.800

Optimal ET sequence: 13, 43, 56, 99

Badness: 0.087325

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 1344/1331, 3136/3125

Mapping: [⟨1 6 0 -3 3], ⟨0 -19 10 25 2]]

POTE generator: ~75/64 = 278.799

Optimal ET sequence: 13, 43, 56, 99e

Badness: 0.061413

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 441/440, 832/825, 975/968

Mapping: [⟨1 6 0 -3 3 3], ⟨0 -19 10 25 2 3]]

POTE generator: ~13/11 = 278.802

Optimal ET sequence: 13, 43, 56, 99e

Badness: 0.047611

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 256/255, 273/272, 375/374

Mapping: [⟨1 6 0 -3 3 3 2], ⟨0 -19 10 25 2 3 9]]

POTE generator: ~13/11 = 278.798

Optimal ET sequence: 13, 43, 56, 99e

Badness: 0.031191

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285

Mapping: [⟨1 6 0 -3 3 3 2 1], ⟨0 -19 10 25 2 3 9 14]]

POTE generator: ~13/11 = 278.790

Optimal ET sequence: 13, 43, 56, 99e

Badness: 0.023899

Quasijerome

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 15488/15435, 16384/16335

Mapping: [⟨1 6 0 -3 3], ⟨0 -38 20 50 47]]

POTE generator: ~896/825 = 139.403

Optimal ET sequence: 43, 112, 155, 198, 439cd, 637cd

Badness: 0.092996

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435

Mapping: [⟨1 6 0 -3 3 8], ⟨0 -38 20 50 47 -37]]

POTE generator: ~13/12 = 139.403

Optimal ET sequence: 43, 155, 198, 439cdf, 637cdf

Badness: 0.044328

Sengagen

Subgroup: 2.3.5.7

Comma list: 3136/3125, 420175/419904

Mapping: [⟨1 1 2 2], ⟨0 29 16 40]]

Wedgie: ⟨⟨ 29 16 40 -42 -18 48 ]]

POTE generator: ~686/675 = 24.217

Optimal ET sequence: 49, 50, 99, 248, 347, 446

Badness: 0.057978

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1344/1331, 3136/3125

Mapping: [⟨1 1 2 2 3], ⟨0 29 16 40 23]]

POTE generator: ~99/98 = 24.235

Optimal ET sequence: 49, 50, 99e

Badness: 0.053828

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 975/968, 1344/1331

Mapping: [⟨1 1 2 2 3 4], ⟨0 29 16 40 23 -15]]

POTE generator: ~99/98 = 24.181

Optimal ET sequence: 49, 50, 99e, 149e

Badness: 0.053531

Sengage

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 364/363, 625/624

Mapping: [⟨1 1 2 2 3 3], ⟨0 29 16 40 23 35]]

POTE generator: ~99/98 = 24.234

Optimal ET sequence: 49f, 50, 99ef

Badness: 0.037416

Subpental

See also: Sensipent family

Subgroup: 2.3.5.7

Comma list: 3136/3125, 19683/19600

Mapping: [⟨1 6 8 17], ⟨0 -14 -18 -45]]

Wedgie: ⟨⟨ 14 18 45 -4 32 54 ]]

POTE generator: ~56/45 = 378.467

Optimal ET sequence: 19, 111, 130, 929c, 1059c, 1189bc, 1319bc

Badness: 0.054303

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 8019/8000

Mapping: [⟨1 6 8 17 -6], ⟨0 -14 -18 -45 30]]

POTE generator: ~56/45 = 378.440

Optimal ET sequence: 19, 111, 130, 241, 371ce, 501cde, 872cde

Badness: 0.045352

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 3136/3125

Mapping: [⟨1 6 8 17 -6 16], ⟨0 -14 -18 -45 30 -39]]

POTE generator: ~56/45 = 378.437

Optimal ET sequence: 19, 111, 130, 241, 371ce

Badness: 0.023940

Mowglic

The mowglic temperament (19&161) is an extension of the mowgli temperament which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 177147/175000

Mapping: [⟨1 0 0 -3], ⟨0 15 22 55]]

Wedgie: ⟨⟨ 15 22 55 0 45 66 ]]

POTE generator: ~27/25 = 126.706

Optimal ET sequence: 19, 123d, 142, 161

Badness: 0.129915

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 72171/71680

Mapping: [⟨1 0 0 -3 8], ⟨0 15 22 55 -43]]

POTE generator: ~27/25 = 126.711

Optimal ET sequence: 19, 123de, 142, 161

Badness: 0.094032

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 1701/1690, 3136/3125

Mapping: [⟨1 0 0 -3 8 -2], ⟨0 15 22 55 -43 54]]

POTE generator: ~14/13 = 126.705

Optimal ET sequence: 19, 123def, 142f, 161

Badness: 0.051571

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 833/832, 1701/1690, 3136/3125

Mapping: [⟨1 0 0 -3 8 -2 10], ⟨0 15 22 55 -43 54 -56]]

POTE generator: ~14/13 = 126.703

Optimal ET sequence: 19, 123defg, 142f, 161

Badness: 0.041918

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 476/475, 495/494, 513/512, 540/539, 1701/1690

Mapping: [⟨1 0 0 -3 8 -2 10 9], ⟨0 15 22 55 -43 54 -56 -45]]

POTE generator: ~14/13 = 126.705

Optimal ET sequence: 19, 123defg, 142f, 161

Badness: 0.032168

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: [⟨1 0 0 -3 8 -2 10 9 6], ⟨0 15 22 55 -43 54 -56 -45 -14]]

POTE generator: ~14/13 = 126.703

Optimal ET sequence: 19, 123defg, 142f, 161

Badness: 0.026117

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 261/260, 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: [⟨1 0 0 -3 8 -2 10 9 6 0], ⟨0 15 22 55 -43 54 -56 -45 -14 46]]

POTE generator: ~14/13 = 126.704

Optimal ET sequence: 19, 123defg, 142f, 161

Badness: 0.021398

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 261/260, 276/275, 351/350, 435/434, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: [⟨1 0 0 -3 8 -2 10 9 6 0 2], ⟨0 15 22 55 -43 54 -56 -45 -14 46 28]]

POTE generator: ~14/13 = 126.703

Optimal ET sequence: 19, 123defgk, 142fk, 161

Badness: 0.019331

Tremka

The name "tremka" is initially used for no-sevens version of 50&111 (especially in the 2.3.5.11.13 subgroup), but extending to full 13-limit or higher prime limit does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 2125764/2100875

Mapping: [⟨1 -4 -2 -8], ⟨0 31 24 60]]

Wedgie: ⟨⟨ 31 24 60 -34 8 72 ]]

POTE generator: ~4375/3888 = 216.173

Optimal ET sequence: 50, 111, 161, 272

Badness: 0.179925

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 35937/35840

Mapping: [⟨1 -4 -2 -8 4], ⟨0 31 24 60 -3]]

POTE generator: ~112/99 = 216.168

Optimal ET sequence: 50, 111, 161, 272, 433c

Badness: 0.068825

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845, 3136/3125

Mapping: [⟨1 -4 -2 -8 4 1], ⟨0 31 24 60 -3 15]]

POTE generator: ~112/99 = 216.172

Optimal ET sequence: 50, 111, 161, 272

Badness: 0.036070

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 561/560, 847/845, 1089/1088

Mapping: [⟨1 -4 -2 -8 4 1 -6], ⟨0 31 24 60 -3 15 56]]

POTE generator: ~17/15 = 216.172

Optimal ET sequence: 50, 111, 161, 272

Badness: 0.022528

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 351/350, 456/455, 476/455, 495/494, 540/539

Mapping: [⟨1 -4 -2 -8 4 1 -6 -8], ⟨0 31 24 60 -3 15 56 68]]

POTE generator: ~17/15 = 216.170

Optimal ET sequence: 50, 111, 161, 272h, 433cfh, 705ccdffhh

Badness: 0.016900

Undetrita

The undetrita temperament (111&118) tempers out the hemimean comma (3136/3125) and skeetsma (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to 11EDT tuning, and the name undetrita is a play on the words "undecimus" (Latin for "eleventh") and "tritave" (third harmonic). It is also related to the twentcufo temperament, which is no-sevens version of 111&118.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 14348907/14336000

Mapping: [⟨1 0 -2 -8], ⟨0 11 30 75]]

Wedgie: ⟨⟨ 11 30 75 22 88 90 ]]

POTE generator: ~448/405 = 172.917

Optimal ET sequence: 111, 118, 229, 347, 576c

Badness: 0.114188

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 8019/8000

Mapping: [⟨1 0 -2 -8 0], ⟨0 11 30 75 24]]

POTE generator: ~400/363 = 172.912

Optimal ET sequence: 111, 118, 229, 347

Badness: 0.043883

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 729/728, 1001/1000, 3025/3024

Mapping: [⟨1 0 -2 -8 0 5], ⟨0 11 30 75 24 -9]]

POTE generator: ~72/65 = 172.930

Optimal ET sequence: 111, 229f

Badness: 0.038771

Undetritoid

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125

Mapping: [⟨1 0 -2 -8 0 -11], ⟨0 11 30 75 24 102]]

POTE generator: ~400/363 = 172.933

Optimal ET sequence: 111, 229

Badness: 0.042744

Isra

Isra results from taking every other generator of septimal meantone. It is named after the Isrāʾ (iss-RAH) night journey in the Qur'an, because it is similar to luna.

Subgroup: 2.9.5.7

Comma list: 81/80, 126/125

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

sval mapping generators: ~2, ~9

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 192.9898

Optimal ET sequence: 6, 19, 25, 31, 56b, 87b

Tutone

Tutone is every other step of undecimal meantone.

Subgroup: 2.9.5.7.11

Comma list: 81/80, 99/98, 126/125

Subgroup-val mapping: [⟨1 0 -4 -13 -25], ⟨0 1 2 5 9]]

Gencom mapping: [⟨1 3/2 2 2 2], ⟨0 1/2 2 5 9]]

gencom: [2 9/8; 81/80 99/98 126/125]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 193.937

Optimal ET sequence: 6, 19e, 25, 31, 68b, 99b

Badness: 0.00536

Deutone

See also: Chromatic pairs § Deutone

Leantone

See also: Chromatic pairs § Leantone