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.
Temperaments considered below are hemiwürschmidt, hemithirds, spell, semisept, emka, decipentic, sengagen, subpental, mowglic, and undetrita. A notable subgroup extension of didacus is roulette. Discussed elsewhere are
- Passion (+64/63 or 3125/3087) → Passion family
- Meantone (+81/80, 126/125, 225/224) → Meantone family
- Mohavila (+135/128 or 1323/1250) → Pelogic family
- Cohemimabila (+65536/64827) → Mabila family
- Sycamore (+686/675 or 875/864) → Sycamore family
- Bidia (+2048/2025) → Diaschismic family
- Hexe (+50/49 or 128/125) → Augmented family
- Misty (+5120/5103) → Misty family
- Bischismic (+32805/32768) → Schismatic family
- Clyde (+245/243) → Kleismic family
- Parakleismic (+4375/4374) → Ragismic microtemperaments
- Arch (+5250987/5242880) → Escapade family
- Subpental (+19683/19600) → Sensipent family
- Decistearn (+118098/117649) → Stearnsmic clan
- Quintagar (+33554432/33480783) → Quindromeda family
- Rubidium (+4194304/4117715) → 37th-octave temperaments
Didacus
Subgroup: 2.5.7
Comma list: 3136/3125
Sval 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 (sometimes spelled hemiwuerschmidt) is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit würschmidt, even with the rather large complexity for the fifth. It 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
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
Semihemiwüram
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
- 7-odd-limit: ~28/25 = [1/10 -1/20 0 1/20⟩
- 9-odd-limit: ~28/25 = [6/25 -2/35 0 1/35⟩
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 the tenth root of the 5th harmonic (5/1), 51/10, tuned between 75/64 and 20/17 (close to 27/23). Aside from the hemimean comma, this temperament tempers out the bronzisma, 2097152/2083725. 99edo is a good tuning for decipentic, with generator 23\99, and mos scales 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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.234
Optimal ET sequence: 49f, 50, 99ef
Badness: 0.037416
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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
Optimal ET sequence: 19, 123defgk, 142fk, 161
Badness: 0.019331
Tremka
The name tremka was 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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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, and the name undetrita is a play on the words undecimus (Latin for "eleventh") and tritave (3rd 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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.933
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
Sval 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
Sval 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