Hemimean clan
The hemimean clan tempers out the no-threes hemimean comma 3136/3125. The head of this clan is the 2.5.7 subgroup temperament didacus. Full 7-limit extensions of didacus, in general, split the syntonic comma into two, each for 126/125~225/224, as 3136/3125 = (126/125)/(225/224). These include hemithirds, semisept, emka, decipentic, arch, sengagen, subpental, mowglic, and undetrita, considered below, as well as these considered elsewhere:
- Spell, {49/48, 3125/3072} → Magic family
- Hexe, {50/49, 128/125} → Augmented family
- Passion, {64/63, 3125/3087} → Passion family
- Meantone, {81/80, 126/125} → Meantone family
- Clyde, {245/243, 3136/3125} → Kleismic family
- Sycamore, {686/675, 875/864} → Sycamore family
- Bidia, {2048/2025, 3136/3125} → Diaschismic family
- Hemiwürschmidt, {2401/2400, 3136/3125} → Würschmidt family
- Parakleismic, {3136/3125, 4375/4374} → Ragismic microtemperaments
- Misty, {3136/3125, 5120/5103} → Misty family
- Bischismic, {3136/3125, 32805/32768} → Schismatic family
- Decistearn, {3136/3125, 118098/117649} → Stearnsmic clan
- Rubidium, {3136/3125, 4194304/4117715} → Fractional-octave temperaments
- Quintagar, {3136/3125, 33554432/33480783} → Quindromeda family
A notable subgroup extension of didacus is roulette.
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]
POTE generator: ~28/25 = 193.772
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 ]]
POTE generator: ~28/25 = 193.244
- [[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [11/5 -1/10 0 1/10⟩, [5/2 -1/4 0 1/4⟩]
- Eigenmonzos (unchanged intervals): 2, 7/6
- [[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [82/35 -4/35 0 2/35⟩, [20/7 -2/7 0 1/7⟩]
- Eigenmonzos (unchanged intervals): 2, 7/6
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]]
POTE generator: ~28/25 = 193.227
Minimax tuning:
- 11-odd-limit
- [[1 0 0 0 0⟩, [11/9 0 0 -5/9 5/9⟩, [64/27 0 0 2/27 -2/27⟩, [79/27 0 0 5/27 -5/27⟩, [79/27 0 0 -22/27 22/27⟩]
- Eigenmonzos (unchanged intervals): 2, 11/7
Vals: Template:Val list
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]]
POTE generator: ~28/25 = 193.166
Vals: Template:Val list
Badness: 0.021738
Semisept
- For the 5-limit version of this temperament, see High badness temperaments #Semisept.
Subgroup: 2.3.5.7
Comma list: 1728/1715, 3136/3125
Mapping: [⟨1 12 6 12], ⟨0 -17 -6 -15]]
Wedgie: ⟨⟨ 17 6 15 -30 -24 18 ]]
POTE generator: ~75/49 = 735.155
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]]
POTE generator: ~55/36 = 735.125
Vals: Template:Val list
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]]
POTE generator: ~55/36 = 735.126
Vals: Template:Val list
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]]
POTE generator: ~26/17 = 735.125
Vals: Template:Val list
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]]
POTE generator: ~26/17 = 735.116
Vals: Template:Val list
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]]
POTE generator: ~26/17 = 735.106
Vals: Template:Val list
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]]
POTE generator: ~13/10 = 464.980
Vals: Template:Val list
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]]
Wedgie: ⟨⟨ 27 8 20 -50 -44 24 ]]
POTE generator: ~48/35 = 551.782
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]]
POTE generator: ~11/8 = 551.765
Vals: Template:Val list
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]]
POTE generator: ~11/8 = 551.758
Vals: Template:Val list
Badness: 0.029741
Decipentic
The generator for the decipentic temperament (43&56) is tenth root of fifth harmonic (5/1, pentave), 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 (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
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
Vals: Template:Val list
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
Vals: Template:Val list
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
Vals: Template:Val list
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
Vals: Template:Val list
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
Vals: Template:Val list
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
Vals: Template:Val list
Badness: 0.044328
Arch
Subgroup: 2.3.5.7
Comma list: 3136/3125, 5250987/5242880
Mapping: [⟨1 2 2 2], ⟨0 -18 14 35]]
Wedgie: ⟨⟨ 18 -14 -35 -64 -106 -42 ]]
POTE generator: ~64/63 = 27.668
Badness: 0.094345
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 4000/3993
Mapping: [⟨1 2 2 2 3], ⟨0 -18 14 35 20]]
POTE generator: ~64/63 = 27.663
Vals: Template:Val list
Badness: 0.036541
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 676/675, 3136/3125
Mapping: [⟨1 2 2 2 3 4], ⟨0 -18 14 35 20 -13]]
POTE generator: ~64/63 = 27.660
Vals: Template:Val list
Badness: 0.019504
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
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
Vals: Template:Val list
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
Vals: Template:Val list
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
Vals: Template:Val list
Badness: 0.037416
Subpental
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
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
Vals: Template:Val list
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
Vals: Template:Val list
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
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
Vals: Template:Val list
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
Vals: Template:Val list
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
Vals: Template:Val list
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
Vals: Template:Val list
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
Vals: Template:Val list
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
Vals: Template:Val list
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
Vals: Template:Val list
Badness: 0.019331
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
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
Vals: Template:Val list
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
Vals: Template:Val list
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
Vals: Template:Val list
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's similar to luna.
Subgroup: 2.9.5.7
Comma list: 81/80, 126/125
Mapping: [⟨1 3 2 2], ⟨0 1 2 5]]
Mapping generators: ~2/1, ~9/8
POTE generator: ~9/8 = 192.9898