Miscellaneous 7-limit temperaments

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

Below are listed some 7-limit rank-3 temperaments that do not belong to some other temperament collection, the majority of which are restrictions to the 7-limit of temperaments that emerge more fully in higher limits or subgroups; they are sorted by TE logflat badness. Most of these temperaments have low accuracy, high-complexity generators, or large number of generators for simple consonances. This is not an exhaustive list. Only expect to find a temperament here if you have not found it in:

• Individual temperament families and clans
• Very low accuracy temperaments
• Very high accuracy temperaments

See also Miscellaneous 5-limit temperaments.

Wizmic

Wizmic tempers out the wizma, the sum of the two smallest superparticular ratios in the 7-limit. It splits the septimal comma into two equal parts, which can serve as the last generator of this temperament, besides the octave and the perfect fifth. Adding a perfect fifth to this gives a sensamagic fifth, which is the generator shown in the data below. Two sensamagic fifths minus an octave give ~8/7, and five minus two octaves give a sub-octave short of the octave by a syntonic comma.

Subgroup: 2.3.5.7

Comma list: 420175/419904

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

Mapping generators: ~2, ~3, ~245/162

Optimal tunings:

• WE: ~2 = 1200.0161 ¢, ~3/2 = 701.9929 ¢, ~245/162 = 715.6723 ¢

Error map: ⟨+0.016 +0.054 -0.029 -0.106]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9981 ¢, ~245/162 = 715.6543 ¢

Error map: ⟨0.000 +0.043 -0.050 -0.134]

Optimal ET sequence: 22, 27, 45, 49, 50, 72, 99, 171, 441, 612, 2812, 3082, 3253, 3424, 3694, 3865, 4306, 4777d, 4918d, 5089d, 5701d

Badness (Sintel): 0.381

Breeze

Breeze tempers out the breeze comma in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 2460375/2458624

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

Mapping generators: ~2, ~3, ~45/28

Optimal tunings:

• WE: ~2 = 1200.0258 ¢, ~3/2 = 701.8709 ¢, ~45/28 = 821.1067 ¢

Error map: ⟨+0.026 -0.058 -0.042 +0.081]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8671 ¢, ~45/28 = 821.0917 ¢

Error map: ⟨0.000 -0.088 -0.080 +0.050]

Optimal ET sequence: 19, 41, 89, 108, 111, 130, 152, 171, 665, 795, 836, 966, 1137, 1308, 1973, 2144, 3281, 3452

Badness (Sintel): 0.520

Metric

Metric tempers out the meter, and splits the syntonic comma into three equal parts, one for the marvel comma, 225/224, and two for the starling comma, 126/125. It is therefore supported by third-comma equal temperaments, and 171edo shows an excellent example of this. 11-limit extensions of this temperament include mendel and skadi.

Subgroup: 2.3.5.7

Comma list: 703125/702464

Mapping: [⟨1 0 -1 -6], ⟨0 1 1 3], ⟨0 0 3 7]]

Mapping generators: ~2, ~3, ~112/75

Optimal tunings:

• WE: ~2 = 1200.0384 ¢, ~3/2 = 701.8990 ¢, ~112/75 = 694.7610 ¢

Error map: ⟨+0.038 -0.018 -0.132 +0.083]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8998 ¢, ~112/75 = 694.7370 ¢

Error map: ⟨0.000 -0.055 -0.203 +0.033]

Optimal ET sequence: 12, 19, 31, 81, 90, 102d, 109, 121, 140, 152, 171, 665, 836, 1007, 2185, 3192c

Badness (Sintel): 0.661

Scheme

Scheme tempers out the scheme comma in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 14348907/14336000

Mapping: [⟨1 0 0 -14], ⟨0 1 0 15], ⟨0 0 1 -3]]

Mapping generators: ~2, ~3, ~5

Optimal tunings:

• WE: ~2 = 1200.0267 ¢, ~3/2 = 701.8565 ¢, ~5/4 = 386.2912 ¢

Error map: ⟨+0.027 -0.072 +0.031 +0.015]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8494 ¢, ~5/4 = 386.3135 ¢

Error map: ⟨0.000 -0.106 -0.000 -0.025]

Optimal ET sequence: 53, 111, 118, 171, 742, 795, 913, 966, 1137, 1308, 2050, 2221, 2392, 3358, 3529, 7229b, 10758bbd

Badness (Sintel): 1.47

Canopic

For extensions, see Swetismic temperaments #Indra.

Canopic (formerly mirkwai) tempers out the canopic comma in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 16875/16807

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

Mapping generators: ~2, ~3, ~10/7

Optimal tunings:

• WE: ~2 = 1199.9999 ¢, ~3/2 = 701.7827 ¢, ~10/7 = 616.0944 ¢

error map: ⟨-0.000 -0.172 -0.493 +0.900]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7827 ¢, ~10/7 = 616.0945 ¢

error map: ⟨0.000 -0.172 -0.493 +0.900]

Minimax tuning:

• 7-odd-limit

[[1 0 0 0⟩, [0 4/7 -4/7 5/7⟩, [0 -3/7 3/7 5/7⟩, [0 0 0 1⟩]

eigenmonzo (unchanged-interval) basis: 2.5/3.7

• 9-odd-limit

[[1 0 0 0⟩, [0 8/11 -4/11 5/11⟩, [0 -6/11 3/11 10/11⟩, [0 0 0 1⟩]

eigenmonzo (unchanged-interval) basis: 2.9/5.7

Optimal ET sequence: 31, 41, 72, 152, 224

Badness (Sintel): 1.51

Projection pairs: 5 84375/16807 7 16875/2401 to 2.3.7/5

Greenwoodmic

Greenwoodmic tempers out the greenwoodma in the 7-limit. It equates 5/2 with a stack of two 14/9's. This implies primes 3 and 5 should be tuned flat, and 7 should be tuned sharp. A rank-2 temperament that does that is injera, which introduces little extra damage over greenwoodmic.

In contrast to sensamagic, where two 9/7's stack to 5/3, here two 9/7's stack to 8/5. As such, greenwoodmic induces essentially tempered chords in the 9-odd-limit. An obvious 11-limit extension then equates 5/4 with 11/9 and equates 9/7 with 14/11, tempering out 45/44 as well as 99/98 using the identity 405/392 = (45/44)⋅(99/98).

Subgroup: 2.3.5.7

Comma list: 405/392

Mapping: [⟨1 0 1 -1], ⟨0 1 0 2], ⟨0 0 2 1]]

Mapping generators: ~2, ~3, ~14/9

Optimal tunings:

• WE: ~2 = 1201.9369 ¢, ~3/2 = 693.3783 ¢, ~14/9 = 790.3845 ¢

Error map: ⟨+1.937 -6.640 -3.608 +10.252]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.3443 ¢, ~14/9 = 790.0724 ¢

Error map: ⟨0.000 -8.611 -6.169 +7.935]

Optimal ET sequence: 9, 12, 26, 38, 73bc

Badness (Sintel): 1.82

Avicennmic

Avicennmic tempers out the avicennma in the 7-limit. It equates 32/21 with a stack of two 5/4's, and 12/7 with a stack of two 15/8's octave reduced. This implies primes 3, 5 and 7 should all be tuned flat. A rank-2 temperament that does that is flattone, which introduces little extra damage over avicennmic.

One possible extension of avicennmic to the 11-limit is via 45/44 and 385/384, using the identity 525/512 = (45/44)⋅(385/384), but the result is somewhat less accurate. Instead, it is more natural to extend it to the 2.3.5.7.13 subgroup by tempering out 65/64 and 105/104, using the identity 525/512 = (65/64)⋅(105/104).

Subgroup: 2.3.5.7

Comma list: 525/512

Mapping: [⟨1 0 0 9], ⟨0 1 0 -1], ⟨0 0 1 -2]]

Mapping generators: ~2, ~3, ~5

Optimal tunings:

• WE: ~2 = 1203.4446 ¢, ~3/2 = 697.5230 ¢, ~5/4 = 375.2486 ¢

Error map: ⟨+3.445 -0.987 -4.176 -3.068]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.1860 ¢, ~5/4 = 373.6255 ¢

Error map: ⟨0.000 -5.579 -12.687 -12.265]

Optimal ET sequence: 7, 9, 10, 16, 19, 45, 64cd, 93cdd, 119bccdd, 138bccdd

Badness (Sintel): 2.10

Varuna

For extensions, see Werckismic temperaments #Varuna.

Varuna tempers out the varunisma in the 7-limit, and splits the octave in two. It then finds 7/4 by a stack of two 10/9's and a semi-octave period. The obvious 11-limit extension tempers out the kalisma, 9801/9800.

Subgroup: 2.3.5.7

Comma list: 321489/320000

Mapping: [⟨2 0 0 9], ⟨0 1 0 -4], ⟨0 0 1 2]]

Mapping generators: ~567/400, ~3, ~5

Optimal tunings:

• WE: ~567/400 = 600.1005 ¢, ~3/2 = 701.3045 ¢, ~5/4 = 386.3934 ¢

Error map: ⟨+0.201 -0.449 +0.482 -0.353]

• CWE: ~567/400 = 600.0000 ¢, ~3/2 = 701.2691 ¢, ~5/4 = 386.5935 ¢

Error map: ⟨0.000 -0.686 +0.280 -0.715]

Optimal ET sequence: 12, 26, 34, 46, 58, 72, 118, 130, 202, 320, 450, 522, 972bd, 1174bd

Badness (Sintel): 2.21

Nuwell

For extensions, see Biyatismic clan #Big brother.

Nuwell tempers out the nuwell comma in the 7-limit, and identifies 15/8 by a stack of four 7/6's. An obvious 11-limit extension then finds 11/8~15/11 as an exact half of it, tempering out 99/98 and 121/120.

Subgroup: 2.3.5.7

Comma list: 2430/2401

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

Mapping generators: ~2, ~3, ~14/9

Optimal tunings:

• WE: ~2 = 1199.8917 ¢, ~3/2 = 700.7235 ¢, ~14/9 = 770.8371 ¢

Error map: ⟨-0.108 -1.340 -0.578 +3.350]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.6977 ¢, ~14/9 = 770.9417 ¢

Error map: ⟨0.000 -1.257 -0.454 +3.511]

Optimal ET sequence: 8d, 9, 14c, 17c, 22, 31, 53, 84, 137, 221d

Badness (Sintel): 2.29

Projection pair: 5 2401/486 to 2.3.7

Aberdietic

Aberdietic tempers out the aberdiesis in the 7-limit. It equates 7/5 with a stack of three 9/8's.

Subgroup: 2.3.5.7

Comma list: 3645/3584

Mapping: [⟨1 0 0 -9], ⟨0 1 0 6], ⟨0 0 1 1]]

Mapping generators: ~2, ~3, ~5

Optimal tunings:

• WE: ~2 = 1201.420 ¢, ~3/2 = 698.145 ¢, ~5/4 = 382.612 ¢
• CWE: ~2 = 1200.000 ¢, ~3/2 = 697.459 ¢, ~5/4 = 383.104 ¢

Optimal ET sequence: 5c, 7d, 12, 19, 31, 81, 98c, 117c, 129c, 148bc, 160bc, 179bcd

Badness (Sintel): 2.96

Keegic

Keegic tempers out the keega in the 7-limit, and finds the 3rd harmonic by a stack of three 10/7's.

Subgroup: 2.3.5.7

Comma list: 1029/1000

Mapping: [⟨1 0 0 1], ⟨0 3 0 -1], ⟨0 0 1 1]]

Mapping generators: ~2, ~10/7, ~5

Optimal tunings:

• WE: ~2 = 1201.1181 ¢, ~10/7 = 633.6603 ¢, ~5/4 = 390.1534 ¢

Error map: ⟨+1.118 -0.974 +6.076 -8.979]

• CWE: ~2 = 1200.0000 ¢, ~10/7 = 633.3435 ¢, ~5/4 = 391.2567 ¢

Error map: ⟨0.000 -1.924 +4.943 -10.913]

Optimal ET sequence: 15, 19, 53d, 55, 74d

Badness (Sintel): 2.99

Uniwiz

For extensions, see Keenanismic temperaments #Uniwiz.

Uniwiz tempers out the uniwiz comma in the 7-limit, equating the whole tone with a stack of four septimal quartertones of 36/35, and splits the octave in two. This means the quartertone should be sharpened a bit, leading to the natural 11-limit extension where 385/384 and 9801/9800 are tempered out.

Subgroup: 2.3.5.7

Comma list: 1500625/1492992

Mapping: [⟨2 1 0 7], ⟨0 2 0 3], ⟨0 0 1 -1]]

Mapping generators: ~1225/864, ~35/24, ~5

Optimal tunings:

• WE: ~1225/864 = 600.1145 ¢, ~35/24 = 651.0771 ¢, ~5/4 = 385.4061 ¢

Error map: ⟨+0.229 +0.314 -0.450 -0.657]

• CWE: ~1225/864 = 600.1145 ¢, ~35/24 = 651.0546 ¢, ~5/4 = 385.4793 ¢

Error map: ⟨0.000 +0.154 -0.834 -1.141]

Optimal ET sequence: 22, 46, 68, 72, 118, 140, 212, 330, 470, 542d, 872cdd, 1012cdd, 1414ccddd

Badness (Sintel): 3.11

Stearnsmic

For extensions, see Swetismic temperaments #Hades.

Stearnsmic tempers out the stearnsma, and splits the octave in two. A stack of three ~9/7 generators and a semi-octave period give the 3rd harmonic.

Subgroup: 2.3.5.7

Comma list: 118098/117649

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

Mapping generators: ~343/243, ~9/7, ~5

Optimal tunings:

• WE: ~343/243 = 599.9938 ¢, ~9/7 = 433.8840 ¢, ~5/4 = 386.3383 ¢

Error map: ⟨-0.012 -0.309 -0.000 +0.582]

• CWE: ~343/243 = 600.0000 ¢, ~9/7 = 433.8851 ¢, ~5/4 = 386.3279 ¢

Error map: ⟨0.000 -0.300 +0.014 +0.600]

Optimal ET sequence: 22, 50, 58, 72, 130, 152, 202, 224, 354

Badness (Sintel): 3.30

Mirwomo

For extensions, see Rastmic rank-3 clan #Mirwomo.

Mirwomo tempers out the mirwomo comma in the 7-limit, equating the Pythagorean apotome with a stack of two septimal quartertones of 36/35, and splits the fifth in two. This means the fifth should be flattened a bit and the quartertone should be sharpened, leading to a natural 11-limit extension where 243/242 and 385/384 are tempered out.

Subgroup: 2.3.5.7

Comma list: 33075/32768

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

Mapping generators: ~2, ~128/105, ~5

Optimal tunings:

• WE: ~2 = 1200.8046 ¢, ~128/105 = 350.3723 ¢, ~5/4 = 384.1239 ¢

Error map: ⟨+0.805 -0.406 -0.581 -0.848]

• CWE: ~2 = 1200.0000 ¢, ~128/105 = 350.1448 ¢, ~5/4 = 383.8961 ¢

Error map: ⟨0.000 -1.665 -2.418 -3.157]

Optimal ET sequence: 17, 21, 24, 31, 41, 72, 281d, 322cd, 353cd, 425bcdd, 497bcdd

Badness (Sintel): 3.40

Triwellismic

Triwellismic tempers out the triwellisma. Besides the octave and perfect fifth, it is generated by a 15/14 semitone, with six of them making a retroptolemaic fifth, 243/160, and seven making a septimal augmented fifth, 729/448. An alternative generator is the 10/7 tritone, with two minus an octave giving the jubilisma and three jubilismas being equated with 16/15.

Subgroup: 2.3.5.7

Comma list: 235298/234375

Mapping: [⟨1 0 1 1], ⟨0 1 5 6], ⟨0 0 -6 -7]]

Mapping generators: ~2, ~3, ~15/14

Optimal tunings:

• WE: ~2 = 1199.9876 ¢, ~3/2 = 701.3045 ¢, ~15/14 = 120.5354 ¢

Error map: ⟨-0.012 +0.031 +0.466 -0.584]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9958 ¢, ~15/14 = 120.5308 ¢

Error map: ⟨0.000 +0.041 +0.480 -0.567]

Optimal ET sequence: 29, 31, 60, 68, 91, 99, 229, 289, 388, 518, 906

Badness (Sintel): 3.55

Decovulture

For extensions, see Olympic clan #Baffin.

Decovulture tempers out the decovulture comma and splits the 3rd harmonic into two semitwelfths, separated from ~7/4 and ~12/7 each by a diaschisma.

Subgroup: 2.3.5.7

Comma list: 67108864/66976875

Mapping: [⟨1 0 0 13], ⟨0 2 0 -7], ⟨0 0 1 -2]]

mapping generators: ~2, ~8192/4725, ~5

Optimal tunings:

• WE: ~2 = 1199.9033 ¢, ~8192/4725 = 951.0102 ¢, ~5/4 = 386.5872 ¢

error map: ⟨-0.097 +0.065 +0.080 +0.059]

• CWE: ~2 = 1200.0000 ¢, ~8192/4725 = 951.0899 ¢, ~5/4 = 386.6184 ¢

error map: ⟨0.000 +0.225 +0.305 +0.308]

Optimal ET sequence: 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd

Badness (Sintel): 3.82

Trimyna

For extensions, see Werckismic temperaments #Trimyna.

Trimyna tempers out the trimyna comma in the 7-limit, and finds the 6th harmonic by a stack of five 10/7's.

Subgroup: 2.3.5.7

Comma list: 50421/50000

Mapping: [⟨1 -1 0 1], ⟨0 5 0 -1], ⟨0 0 1 1]]

Mapping generators: ~2, ~10/7, ~5

Optimal tunings:

• WE: ~2 = 1200.1652 ¢, ~10/7 = 620.4031 ¢, ~5/4 = 387.0990 ¢

Error map: ⟨+0.165 -0.105 +1.116 -1.634]

• CWE: ~2 = 1200.0000 ¢, ~10/7 = 620.3427 ¢, ~5/4 = 387.2591 ¢

Error map: ⟨0.000 -0.241 +0.945 -1.910]

Optimal ET sequence: 27, 31, 58, 87, 118, 267d, 385d, 412d *

* optimal patent val: 294

Badness (Sintel): 3.84

Projection pair: 3 50000/16807 to 2.5.7

Squalentine

For extensions, see Biyatismic clan #Aphrodite.

Squalentine tempers out the squalentine comma in the 7-limit. Its generators can be taken to be 2, 3, and 21/20, and it equates (21/20)³ with 8/7. An obvious 11-limit extension then equates the last generator with 22/21, tempering out 121/120 and 441/440. Notice also 64827/64000 = (121/120)⋅(441/440)².

Subgroup: 2.3.5.7

Comma list: 64827/64000

Mapping: [⟨1 0 1 3], ⟨0 1 1 0], ⟨0 0 -4 -3]]

Mapping generators: ~2, ~3, ~21/20

Optimal tunings:

• WE: ~2 = 1200.7175 ¢, ~3/2 = 700.6331 ¢, ~21/20 = 78.6164 ¢

Error map: ⟨+0.718 -0.604 +1.289 -2.523]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.5831 ¢, ~21/20 = 78.4341 ¢

Error map: ⟨0.000 -1.372 +0.533 -4.128]

Optimal ET sequence: 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d

Badness (Sintel): 4.16

Projection pairs: 5 320000/64827 7 64000/9261 to 2.3.7/5

Quasiorwellismic

For extensions, see Lehmerismic temperaments #Ganesha.

Quasiorwellismic tempers out the quasiorwellisma in the 7-limit, and finds 7/6 by a stack of ten 5/4's octave reduced. A natural 11-limit extension thus arises from mapping 11/9 to a stack of four 5/4's octave reduced, leading to ganesha, which tempers out 3025/3024 and 5632/5625.

Subgroup: 2.3.5.7

Comma list: 29360128/29296875

Mapping: [⟨1 0 0 -22], ⟨0 1 0 1], ⟨0 0 1 10]]

Mapping generators: ~2, ~3, ~5

Optimal tunings:

• WE: ~2 = 1199.9205 ¢, ~3/2 = 702.0435 ¢, ~5/4 = 386.6674 ¢

Error map: ⟨-0.079 +0.009 +0.195 -0.029]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0491 ¢, ~5/4 = 386.6885 ¢

Error map: ⟨0.000 +0.094 +0.375 +0.108]

Optimal ET sequence: 31, 87, 118, 152, 239, 270, 571, 723, 841, 993, 1263, 1564c, 1834c, 2104c

Badness (Sintel): 5.00

Buzzardsmic

Buzzardsmic tempers out the buzzardsma and gives buzzard an independent generator for the 5th harmonic.

Subgroup: 2.3.5.7

Comma list: 65536/64827

Mapping: [⟨1 0 0 4], ⟨0 4 0 -3], ⟨0 0 1 0]]

Mapping generators: ~2, ~21/16, ~5

Optimal tunings:

• WE: ~2 = 1199.2548 ¢, ~21/16 = 475.5761 ¢, ~5/4 = 387.8025 ¢

Error map: ⟨-0.745 +0.350 -0.002 +1.465]

• CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.8328 ¢, ~5/4 = 387.5778 ¢

Error map: ⟨0.000 +1.376 +1.264 +3.676]

Optimal ET sequence: 5, 10, 15, 33, 38, 43, 53, 111, 121, 164d, 174d, 179, 232d

Badness (Sintel): 6.18

Tolerant

For extensions, see Pentacircle clan #Tolerant.

Subgroup: 2.3.5.7

Comma list: 179200/177147

Mapping: [⟨1 0 0 -10], ⟨0 1 0 11], ⟨0 0 1 -2]]

Mapping generators: ~2, ~3, ~5

Optimal tunings:

• WE: ~2 = 1199.539 ¢, ~3 = 1903.226 ¢, ~5 = 2785.816 ¢
• CWE: ~2 = 1200.000 ¢, ~3 = 1903.805 ¢, ~5 = 2786.356 ¢

Optimal ET sequence: 34d, 39d, 41, 80, 87, 121, 167, 208, 329b, 375b, 496bd

Badness (Sintel): 7.26

History

For extensions, see Werckismic temperaments #History.

History tempers out the historisma in the 7-limit, and splits the fourth in six.

Comma list: 257298363/256000000

Mapping: [⟨1 2 0 0], ⟨0 -6 0 7], ⟨0 0 1 1]]

Mapping generators: ~2, ~21/20, ~5

Optimal tunings:

• WE: ~2 = 1200.154 ¢, ~21/20 = 83.091 ¢, ~5 = 2786.669 ¢
• CWE: ~2 = 1200.000 ¢, ~21/20 = 83.067 ¢, ~5 = 2786.515 ¢

Optimal ET sequence: 14c, 15, 29, 43, 58, 72, 130, 202

Badness (Sintel): 7.95

Sensibeta

See also: Sensibeta comma

Sensibeta tempers out the sensibeta comma in the 7-limit.

Comma list: 1071875/1062882

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

Mapping generators: ~2, ~3, ~175/162

Optimal tunings:

• WE: ~2 = 1200.025 ¢, ~3 = 1902.728 ¢, ~175/162 = 128.524 ¢
• CWE: ~2 = 1200.000 ¢, ~3 = 1902.709 ¢, ~175/162 = 128.530 ¢

Optimal ET sequence: 19, 27, 46, 94, 113, 121, 140

Badness (Sintel): 7.97

Parahemif

For extensions, see Rastmic rank-3 clan #Parahemif.

Parahemif tempers out the parahemif comma in the 7-limit, equating a Pythagorean apotome with a stack of two septimal third-tones of 28/27, and splits the fifth in two. It also equates the large septimal diesis of 49/48 with the Pythagorean comma. This means the fifth should be tuned sharp and the septimal third-tone should be flattened to a somewhat large quartertone which can be used as the undecimal quartertone of 33/32, leading to a natural 11-limit extension where 243/242 and 896/891 are tempered out.

Subgroup: 2.3.5.7

Comma list: 1605632/1594323

Mapping: [⟨1 1 0 -1], ⟨0 2 0 13], ⟨0 0 1 0]]

Mapping generators: ~2, ~896/729, ~5

Optimal tunings:

• WE: ~2 = 1199.7303 ¢, ~896/729 = 351.4056 ¢, ~5/4 = 386.8527 ¢

Error map: ⟨-0.270 +0.586 -0.000 -0.284]

• CWE: ~2 = 1200.0000 ¢, ~896/729 = 351.4569 ¢, ~5/4 = 386.6884 ¢

Error map: ⟨0.000 +0.959 +0.375 +0.114]

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99, 239, 338

Badness (Sintel): 8.77

Aristoschismic

Aristoschismic tempers out the aristoschisma, which is the difference between the septischisma and the schisma. It sets the Pythagorean comma to the mean of the syntonic comma and the septimal comma, so like orthoschismic and septischismic, it functions as a detemperament of garibaldi. 441edo shows an excellent example of an edo tuning, but it can be tuned even more accurately than that.

Subgroup: 2.3.5.7

Comma list: [40 -22 -1 -1⟩

Mapping: [⟨1 0 0 40], ⟨0 1 0 -22], ⟨0 0 1 -1]]

Mapping generators: ~2, ~3, ~5

Optimal tunings:

• WE: ~2 = 1199.9738 ¢, ~3/2 = 702.0173 ¢, ~5/4 = 386.3696 ¢

Error map: ⟨-0.026 +0.036 +0.004 +0.005]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0343 ¢, ~5/4 = 386.3610 ¢

Error map: ⟨0.000 +0.079 +0.047 +0.058]

Optimal ET sequence: 41, 53, 94, …, 294, 306, 335, 347, 388, 400, 441, 935, 1376, 3193, 4087, 4528, 5463, 5904, 7280, 12743bcd, 13184bcd, 18647bbccdd

Badness (Sintel): 11.2

Septimagic

Septimagic tempers out the septimagic comma in the 7-limit and gives the 2.3.7-subgroup magic restriction an independent generator for the 5th harmonic.

Subgroup: 2.3.5.7

Comma list: 537824/531441

Mapping: [⟨1 0 0 -1], ⟨0 5 0 12], ⟨0 0 1 0]]

Mapping generators: ~2, ~243/196, ~5

Optimal tunings:

• WE: ~2 = 1199.8224 ¢, ~243/196 = 380.6043 ¢, ~5/4 = 386.6676 ¢

error map: ⟨-0.178 +1.066 -0.001 -1.397]

• CWE: ~2 = 1200.0000 ¢, ~243/196 = 380.6378 ¢, ~5/4 = 386.5230 ¢

error map: ⟨0.000 +1.234 +0.209 -1.173]

Optimal ET sequence: 19, 41, 104c, 123, 126, 145, 167, 186, 394b, 413

Badness (Sintel): 11.8

Compass

For extensions, see Moctdelismic clan #Compass.

Compass tempers out the compass comma in the 7-limit, and splits the fourth in five. The obvious 11-limit extension tempers out the moctdelisma, 1375/1372.

Subgroup: 2.3.5.7

Comma list: 9765625/9680832

Mapping: [⟨1 2 0 -2], ⟨0 -5 0 2], ⟨0 0 1 2]]

Mapping generators: ~2, ~625/588, ~5

Optimal tunings:

• WE: ~2 = 1200.1156 ¢, ~625/588 = 99.6359 ¢, ~5/4 = 385.0414 ¢

error map: ⟨+0.116 +0.097 -1.041 +0.760]

• CWE: ~2 = 1200.0000 ¢, ~625/588 = 99.6080 ¢, ~5/4 = 385.1108 ¢

error map: ⟨0.000 +0.005 -1.203 +0.612]

Optimal ET sequence: 12, …, 37, 48d, 49, 60, 72, 181, 193, 265

Badness (Sintel): 13.3

Cloudy

Cloudy tempers out the cloudy comma, a.k.a. the 5-7-comma, equating the octave with a stack of five 8/7's.

Subgroup: 2.3.5.7

Comma list: 16807/16384

Mapping: [⟨5 0 0 14], ⟨0 1 0 0], ⟨0 0 1 0]]

Mapping generators: ~8/7, ~3, ~5

Optimal tunings:

• WE: ~8/7 = 240.3136 ¢, ~3/2 = 700.3807 ¢, ~5/4 = 383.1686 ¢

error map: ⟨+1.568 -0.007 -0.010 -4.436]

• CWE: ~8/7 = 240.0000 ¢, ~3/2 = 700.2940 ¢, ~5/4 = 383.8804 ¢

error map: ⟨0.000 -1.661 -2.433 -8.826]

Optimal ET sequence: 5, 10, 15, 35, 40, 45, 50, 60, 110d, 170cdd, 175dd, 235cddd

Badness (Sintel): 13.9

Linus

For extensions, see Kalismic temperaments #Linus.

Linus tempers out the linus comma in the 7-limit, and splits the octave into twelve equal parts of ~15/14. The obvious 11-limit extension tempers out the kalisma, 9801/9800.

Subgroup: 2.3.5.7

Comma list: 578509309952/576650390625

Mapping: [⟨10 0 0 -11], ⟨0 1 0 1], ⟨0 0 1 1]]

Mapping generators: ~15/14, ~3, ~5

Optimal tunings:

• WE: ~15/14 = 119.9964 ¢, ~3/2 = 702.0734 ¢, ~5/4 = 386.5626 ¢

error map: ⟨-0.036 +0.082 +0.177 -0.258]

• CWE: ~15/14 = 120.0000 ¢, ~3/2 = 702.0700 ¢, ~5/4 = 386.5404 ¢

error map: ⟨0.000 +0.115 +0.227 -0.215]

Optimal ET sequence: 50, 60, 80, 130, 270, 1270, 1540, 1810, 1940, 2080, 2210c, 2480c

Badness (Sintel): 15.7

Naiad

For extensions, see Wizardharry clan #Naiad.

Naiad tempers out the naiadisma in the 7-limit. An obvious 13-limit interpretation of one generator (~98/75) is 13/10.

Subgroup: 2.3.5.7

Comma list: 161414428/158203125

Mapping: [⟨1 5 0 2], ⟨0 -9 0 -4], ⟨0 0 1 1]]

Mapping generators: ~2, ~98/75, ~5

Optimal tunings:

• WE: ~2 = 1199.937 ¢, ~98/75 = 455.269 ¢, ~5/4 = 387.946 ¢

error map: ⟨-0.063, +0.311, +1.506, -2.207]

• CWE: ~2 = 1200.000 ¢, ~98/75 = 455.298 ¢, ~5/4 = 387.896 ¢

error map: ⟨0.000, +0.362, +1.582, -2.123]

Optimal ET sequence: 8d, 21, 29, 37, 50, 58, 87, 145

Badness (Sintel): 52.5