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
- 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. 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
- 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
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
- 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")[1].
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
- 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.
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
- 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[1].
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
- 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
- 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