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

Main article: 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

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

RMS error: 0.2138 cents

Badness (Dirichlet): 0.091

Undecimal didacus

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

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]]

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

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

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

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

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⟩

[[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

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), 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. 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

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

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

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

See also: Chromatic pairs § 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