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.

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

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

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

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.

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

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 equates a 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