Horwell 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.

This is a collection of rank-2 horwell temperaments, which temper out the horwell comma (monzo: [-16 1 5 1⟩, ratio: 65625/65536).

Temperaments discussed elsewhere are

• Pontiac (+4375/4374) → Schismatic family
• Keen (+875/864) → Diaschismic family
• Paramity (+1600000/1594323) → Amity family
• Countercata (+5120/5103) → Kleismic family
• Orwell (+1728/1715) → Semicomma family
• Worschmidt (+126/125) → Würschmidt family
• Escaped (+245/243) → Escapade family
• Semabila (+49/48) → Mabila family
• Counterpara (+100442349/100000000) → Parakleismic family
• Narayana (+321489/320000) → Vishnu family
• Hemithirds (+1029/1024) → Hemimean clan
• Bisupermajor (+10976/10935) → Hemimage temperaments
• Maquiloid (+686/675) → Maquila family
• Kaboom (+4802000/4782969) → Vavoom family
• Tertiaseptal (+2401/2400) → Breedsmic temperaments
• Eris (+16875/16807) → Canopic clan
• Soviet ferris wheel (+[-5 -9 -5 11⟩) → 20th-octave temperaments

Considered below are fifthplus, mutt, oquatonic, emkay, kastro, and bezique, in the order of increasing badness.

Fifthplus

For the 5-limit version, see Miscellaneous 5-limit temperaments #Sesesix.

Fifthplus tempers out the wizma in addition to the horwell comma, and may be described as the 22 & 171 temperament. The name fifthplus means using a sharp fifth interval (such as a superpyth fifth) as a generator.

Fifthplus may be extended to the 2.3.5.7.17 subgroup, called prime archagall, derived from the fact that it is an extension of archagall to a prime subgroup. In either the 7-limit or the add-17 subgroup, 171edo is exceptionally efficient and accurate, but in the latter case 171edo is the tuning where 2401/2400 (S49), 2500/2499 (S50), and 1701/1700 (S18/S20) all vanish, which is natural because this temperament tempers out 1225/1224 (S35, S49⋅S50) and 5832/5831 ((S18/S20)/S49) while not tempering out any of above individually.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 420175/419904

Mapping: [⟨1 -12 10 -22], ⟨0 23 -13 42]]

mapping generators: ~2, ~5488/3645

Optimal tunings:

• WE: ~2 = 1200.0934 ¢, ~5488/3645 = 708.8291 ¢

error map: ⟨+0.093 -0.007 -0.158 -0.059]

• CWE: ~2 = 1200.0000 ¢, ~5488/3645 = 708.7752 ¢

error map: ⟨0.000 -0.126 -0.391 -0.268]

Optimal ET sequence: 22, 105d, 127d, 149, 171, 1903c, 2074c, …, 3613ccd

Badness (Sintel): 0.654

Prime archagall

Subgroup: 2.3.5.7.17

Comma list: 1225/1224, 24576/24565, 57375/57344

Subgroup-val mapping: [⟨1 -12 10 -22 -3], ⟨0 23 -13 42 12]]

Optimal tunings:

• WE: ~2 = 1200.0516 ¢, ~128/85 = 708.8057 ¢
• CWE: ~2 = 1200.0000 ¢, ~128/85 = 708.7758 ¢

Optimal ET sequence: 22, 105d, 127d, 149, 171, 1219, 1390

Badness (Sintel): 0.421

Mutt

Main article: Mutt

For the 5-limit version, see Father–3 equivalence continuum #Mutt (5-limit).

Mutt tempers out the landscape comma in addition to the horwell comma, and may be described as the 84 & 87 temperament.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 250047/250000

Mapping: [⟨3 -2 6 20], ⟨0 7 1 -12]]

mapping generators: ~63/50, ~5/4

Optimal tunings:

• WE: ~63/50 = 400.0351 ¢, ~5/4 = 385.9974 ¢ (~126/125 = 14.0377 ¢)

error map: ⟨+0.105 -0.043 -0.105 -0.092]

• CWE: ~63/50 = 400.0000 ¢, ~5/4 = 385.9638 ¢ (~126/125 = 14.0362 ¢)

error map: ⟨0.000 -0.208 -0.350 -0.392]

Optimal ET sequence: 84, 87, 171

Badness (Sintel): 0.719

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4356, 16384/16335

Mapping: [⟨3 -2 6 20 21], ⟨0 7 1 -12 -11]]

Optimal tunings:

• WE: ~44/35 = 399.9783 ¢, ~5/4 = 385.9993 ¢ (~126/125 = 13.9790 ¢)
• CWE: ~44/35 = 400.0000 ¢, ~5/4 = 386.0208 ¢ (~126/125 = 13.9792 ¢)

Optimal ET sequence: 84, 87, 171, 258

Badness (Sintel): 1.93

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 625/624, 2200/2197

Mapping: [⟨3 -2 6 20 21 14], ⟨0 7 1 -12 -11 -3]]

Optimal tunings:

• WE: ~44/35 = 399.9610 ¢, ~5/4 = 385.9842 ¢ (~126/125 = 13.9768 ¢)
• CWE: ~44/35 = 400.0000 ¢, ~5/4 = 386.0231 ¢ (~126/125 = 13.9769 ¢)

Optimal ET sequence: 84, 87, 171, 258, 429ef

Badness (Sintel): 1.20

Oquatonic

For the 5-limit version, see 28th-octave temperaments #Oquatonic (5-limit).

Oquatonic has a period of 1/28 octave and tempers out the horwell (65625/65536) and the dimcomp comma (390625/388962). In this temperament, the 5/4 major third is mapped to 9\28.

The name oquatonic was given by Petr Pařízek in 2011 as an abbreviation of the Italian ottantaquatro ("eighty-four").

Subgroup: 2.3.5.7

Comma list: 65625/65536, 390625/388962

Mapping: [⟨28 0 65 123], ⟨0 1 0 -1]]

mapping generators: ~128/125, ~3

Optimal tunings:

• WE: ~128/125 = 42.8570 ¢, ~3/2 = 702.1112 ¢

error map: ⟨-0.004 +0.152 -0.609 +0.477]

• CWE: ~128/125 = 42.8571 ¢, ~3/2 = 702.1132 ¢

error map: ⟨0.000 +0.158 -0.599 +0.489]

Optimal ET sequence: 28, 56, 84, 140, 224, 364, 588, 952

Badness (Sintel): 2.23

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 6250/6237, 65625/65536

Mapping: [⟨28 0 65 123 230], ⟨0 1 0 -1 -3]]

Optimal tunings:

• WE: ~128/125 = 42.8577 ¢, ~3/2 = 702.0275 ¢
• CWE: ~128/125 = 42.8571 ¢, ~3/2 = 702.0174 ¢

Optimal ET sequence: 84, 140, 224, 364, 588

Badness (Sintel): 1.58

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197

Mapping: [⟨28 0 65 123 230 148], ⟨0 1 0 -1 -3 -1]]

Optimal tunings:

• WE: ~40/39 = 42.8571 ¢, ~3/2 = 702.0289 ¢
• CWE: ~40/39 = 42.8571 ¢, ~3/2 = 702.0288 ¢

Optimal ET sequence: 84, 140, 224, 364, 588

Badness (Sintel): 0.908

Emkay

For the 5-limit version, see Miscellaneous 5-limit temperaments #Emka.

Scale tree graph for emkay.

Emkay may be described as the 87 & 224 temperament. It tempers out the same 5-limit comma as the emka (37 & 50), but with the horwell comma (65625/65536) rather than the hemimean comma (3136/3125) tempered out.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 244140625/243045684

Mapping: [⟨1 -13 -2 39], ⟨0 27 8 -67]]

mapping generators: ~2, ~4536/3125

Optimal tunings:

• WE: ~2 = 1200.0279 ¢, ~4536/3125 = 648.2405 ¢

error map: ⟨+0.028 +0.177 -0.445 +0.146]

• CWE: ~2 = 1200.0000 ¢, ~4536/3125 = 648.2254 ¢

error map: ⟨0.000 +0.133 -0.510 +0.069]

Optimal ET sequence: 87, 137, 224, 311, 535, 1381c, 1916c

Badness (Sintel): 3.43

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 65625/65536

Mapping: [⟨1 -13 -2 39 4], ⟨0 27 8 -67 -1]]

Optimal tunings:

• WE: ~2 = 1199.9958 ¢, ~16/11 = 648.2231 ¢
• CWE: ~2 = 1200.0000 ¢, ~16/11 = 648.2254 ¢

Optimal ET sequence: 87, 137, 224, 311, 535

Badness (Sintel): 1.18

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197

Mapping: [⟨1 -13 -2 39 4 1], ⟨0 27 8 -67 -1 5]]

Optimal tunings:

• WE: ~2 = 1199.9694 ¢, ~16/11 = 648.2085 ¢
• CWE: ~2 = 1200.0000 ¢, ~16/11 = 648.2251 ¢

Optimal ET sequence: 87, 137, 224, 311, 535

Badness (Sintel): 0.738

Kastro

For the 5-limit version, see Very high accuracy temperaments #Astro.

Kastro may be described as the 109 & 118 temperament, named by Petr Pařízek in 2011 as a variation of astro.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 117649/116640

Mapping: [⟨1 -26 13 -23], ⟨0 31 -12 29]]

mapping generators: ~2, ~6272/3375

Optimal tunings:

• WE: ~2 = 1200.1529 ¢, ~6272/3375 = 1067.9515 ¢

error map: ⟨+0.153 +0.567 +0.256 -1.749]

• CWE: ~2 = 1200.0000 ¢, ~6272/3375 = 1067.8174 ¢

error map: ⟨0.000 +0.384 -0.122 -2.122]

Optimal ET sequence: 109, 118, 345d, 463d, 581dd

Badness (Sintel): 4.64

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 12005/11979

Mapping: [⟨1 -26 13 -23 -9], ⟨0 31 -12 29 14]]

Optimal tunings:

• WE: ~2 = 1200.2427 ¢, ~224/121 = 1068.0296 ¢
• CWE: ~2 = 1200.0000 ¢, ~224/121 = 1067.8166 ¢

Optimal ET sequence: 109, 118, 345de, 463de, 581dde

Badness (Sintel): 1.74

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 364/363, 385/384, 3388/3375

Mapping: [⟨1 -26 13 -23 -9 -23], ⟨0 31 -12 29 14 30]]

Optimal tunings:

• WE: ~2 = 1200.4303 ¢, ~13/7 = 1068.2040 ¢
• CWE: ~2 = 1200.0000 ¢, ~13/7 = 1067.8267 ¢

Optimal ET sequence: 109, 118f, 227f

Badness (Sintel): 1.93

Bezique

Bezique splits the octave into 32 equal parts and reaches 3/2, 8/5 and 11/8 in just one generator with the 64-tone mos. A notable edo tuning overshadowed by 224edo is 320edo. Bezique was named by Eliora in 2023 for the fact that the card game of bezique is played with two packs of 32 cards.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 847288609443/843308032000

Mapping: [⟨32 0 125 -113], ⟨0 1 -1 4]]

mapping generators: ~100352/98415, ~3

Optimal tunings:

• WE: ~100352/98415 = 37.5038 ¢, ~3/2 = 701.6058 ¢

error map: ⟨+0.120 -0.229 -0.071 +0.154]

• CWE: ~100352/98415 = 37.5000 ¢, ~3/2 = 701.5544 ¢

error map: ⟨0.000 -0.401 -0.368 -0.108]

Optimal ET sequence: 96d, 224, 544, 768, 1312, 2080bc

Badness (Sintel): 6.82

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 46656/46585, 65625/65536

Mapping: [⟨32 0 125 -113 60], ⟨0 1 -1 4 1]]

Optimal tunings:

• WE: ~45/44 = 37.5025 ¢, ~3/2 = 701.5912 ¢
• CWE: ~45/44 = 37.5000 ¢, ~3/2 = 701.5566 ¢

Optimal ET sequence: 96d, 224, 544, 768

Badness (Sintel): 2.25

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1575/1573, 4225/4224, 6656/6655

Mapping: [⟨32 0 125 -113 60 17], ⟨0 1 -1 4 1 2]]

Optimal tunings:

• WE: ~45/44 = 37.5021 ¢, ~3/2 = 701.5769 ¢
• CWE: ~45/44 = 37.5000 ¢, ~3/2 = 701.5490 ¢

Optimal ET sequence: 96d, 224, 544, 768, 1312

Badness (Sintel): 1.23