Miscellaneous 7-limit temperaments

From Xenharmonic Wiki
Revision as of 11:51, 24 December 2025 by FloraC (talk | contribs) (Move stearnsmic here)
Jump to navigation Jump to search
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:

See also Miscellaneous 5-limit temperaments.

Metric

For extensions, see Lehmerismic temperaments #Skadi.

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.

Subgroup: 2.3.5.7

Comma list: 703125/702464

Mapping: [⟨1 0 2 1], ⟨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

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.

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

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.

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

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

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

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

Retrieved from "https://en.xen.wiki/index.php?title=Miscellaneous_7-limit_temperaments&oldid=219612"