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

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

Mutt

Main article: Mutt

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

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. The name fifthplus means using a sharp fifth interval (such as a superpyth fifth) as a generator. It is a restriction of prime archagall.

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, 149, 171, 1903c, 2074c, …, 3613ccd

Badness (Sintel): 0.654

Emkay

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.

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

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

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. The card game of bezique is played with two packs of 32 cards, hence the name.

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