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
- Doubloon (+33756345/33554432) → Vavoom family
- Decistearn (+118098/117649) → Stearnsmic clan
- Quintagar (+33554432/33480783) → Quindromeda family
- Rubidium (+4194304/4117715) → 37th-octave temperaments
Didacus
In the no-3's 11-limit, there is a natural extension with prime 11 by equating 25/16 (which is already tuned sharp anyways) with 11/7 by tempering out 176/175, which is the same route that undecimal meantone uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~28/25 at 2 generators (as a flat ~9/8) while here it is the generator. This is equivalent to finding 11/4 as (7/5)3 In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest mediants of 9/8 and 10/9, namely 19/17 and 28/25, while in didacus and its extension to the no-3's 13-limit called roulette only the latter interpretation is relevant.
Subgroup: 2.5.7
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
RMS error: 0.2138 cents
Badness (Dirichlet): 0.091
2.5.7.11 subgroup
Subgroup: 2.5.7.11
Comma list: 176/175, 1375/1372
Sval mapping: [⟨1 0 -3 -7], ⟨0 2 5 9]]
- sval mapping generators: ~2, ~56/25
Optimal tuning (CWE): 2 = 1\1, ~28/25 = 194.428
Optimal ET sequence: 6, 19e, 25, 31, 37
RMS error: 0.5567 cents
Badness (Dirichlet): 0.195
Roulette
Roulette is equivalent to hemiwur or grosstone with no mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators (and a large amount of them), but it can be rationalized in a variety of ways, such as that because ~8/7 is already tuned 4 ¢ flat, it makes sense to equate two of it with ~13/10 (tempering out the 8 ¢ huntma). This mapping of 13 increases the badness of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension to didacus if prime 3 is not included.
Subgroup: 2.5.7.11.13
Comma list: 176/175, 640/637, 1375/1372
Sval mapping: [⟨1 0 -3 -7 13], ⟨0 2 5 9 -8]]
- sval mapping generators: ~2, ~56/25
Gencom mapping: [⟨1 0 2 2 2 5], ⟨0 0 2 5 9 -8]]
- gencom: [2 28/25; 176/175 1375/1372 640/637]
Optimal tuning (POTE): 2 = 1\1, ~28/25 = 194.594
Optimal ET sequence: 6, 25, 31, 37
Badness (Dirichlet): 0.324
Mediantone
Mediantone is named after its whole tone generator serving as the mediant of 9/8 and 10/9, namely 19/17, in addition to 28/25, as well as by the observation that this temperament seems to have been repeatedly rediscovered in parts in a variety of contexts, so that it seems to exist as a "median" of all of these temperaments' logics. It is also an intentional play on "meantone", as the context one is most likely to first discover this logic is when the tone also represents ~10/9~9/8.
In the full no-3's 19-limit, this temperament is a structure common to quite a few temperaments. It is a rank-2 version of orion with a mapping for primes 11 and 13. It is a no-3's version of 19-limit grosstone which can be seen as an extension of undecimal meantone according to the "mediant-tone" logic of this temperament, and which as aforementioned effectively doubles the complexity of the temperament as a result of finding the generator of ~19/17~28/25 as (~3/2)2/2. It does not work so well as an extension for hemiwur to the full 19-limit, but if you want to try anyway (at the cost of primes 17 and 19), a notable patent-val tuning is 37edo, which finds prime 3 through the würschmidt mapping so that 6/1 is found at 16 generators.
Subgroup: 2.5.7.11.13.17
Comma list: 176/175, 640/637, 221/220, 1375/1372
Sval mapping: [⟨1 0 -3 -7 13 -18], ⟨0 2 5 9 -8 19]]
- sval mapping generators: ~2, ~56/25
Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.887
Optimal ET sequence: 6h, 31gh, 37, 80, 117d
Badness (Dirichlet): 0.612
2.5.7.11.13.17.19 subgroup
Subgroup: 2.5.7.11.13.17.19
Comma list: 176/175, 640/637, 221/220, 476/475, 1375/1372
Sval mapping: [⟨1 0 -3 -7 13 -18 -19], ⟨0 2 5 9 -8 19 20]]
- sval mapping generators: ~2, ~56/25
Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927
Optimal ET sequence: 6h, 31gh, 37, 80
Badness (Dirichlet): 0.618
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
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]]
Mapping generators: ~2, ~25/14
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
2.3.5.7.23 subgroup
As described at the page for würschmidt, there is an extension to prime 23 with essentially no damage, which maps the prime to 28 generators (or 14 generators of würschmidt).
Subgroup: 2.3.5.7.23
Comma list: 576/575, 736/735, 1127/1125
Mapping: [⟨1 15 4 7 28], ⟨0 -16 -2 -5 -28]]
Mapping generators: ~2, ~25/14
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.901
Optimal ET sequence: 31, 68, 99, 229, 328, 557c, 885cc
 |
Todo: verify this & get badness |
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
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⟩
- [[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⟩]
- eigenmonzo (unchanged-interval) basis: 2.7/3
- 9-odd-limit: ~28/25 = [6/35 -2/35 0 1/35⟩
- [[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⟩]
- 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⟩
- [[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
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
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 the 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.
7-limit
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
Deutone is every other step of meanpop.
Subgroup: 2.9.5.7.13
Comma list: 65/64, 81/80, 91/90
Sval mapping: [⟨1 0 -4 -13 10], ⟨0 1 2 5 -2]]
Gencom mapping: [⟨1 3/2 2 2 0 4], ⟨0 1/2 2 5 0 -2]]
- gencom: [2 9/8; 65/64 81/80 91/90]
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 191.059
Optimal ET sequence: 6, 7, 13, 19, 25e, 44de
RMS error: 2.003 cents
Leantone
Leantone is every other step of meanenneadecal.
Subgroup: 2.9.5.7.11
Comma list: 45/44, 56/55, 81/80
Sval mapping: [⟨1 0 -4 -13 -6], ⟨0 1 2 5 3]]
Gencom mapping: [⟨1 3/2 2 2 3], ⟨0 1/2 2 5 3]]
- gencom: [2 9/8; 45/44 56/55 81/80]
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 192.500
Optimal ET sequence: 6, 7, 13, 19, 25, 31, 56, 81
RMS error: 3.882 cents