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
Subgroup: 2.3.5
Comma list: [-44 -3 21⟩
Mapping: [⟨3 -2 6], ⟨0 7 1]]
- mapping generators: ~98304/78125, ~5/4
- WE: ~98304/78125 = 400.0227 ¢, ~5/4 = 386.0017 ¢ (~393216/390625 = 14.0210 ¢)
- error map: ⟨+0.068 +0.012 -0.176]
- CWE: ~98304/78125 = 400.0000 ¢, ~5/4 = 385.9858 ¢ (~393216/390625 = 14.0142 ¢)
- error map: ⟨0.000 -0.055 -0.328]
Optimal ET sequence: 84, 87, 171, 771, 942, 1113, 1284, 1455, 4194cc, 5649cc
Badness (Sintel): 3.81
7-limit
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
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
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 tuning (POTE): ~2 = 1200.0000 ¢, ~3125/2268 = 551.7745 ¢
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 14 6 -28 3], ⟨0 -27 -8 67 1]]
Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~11/8 = 551.7746 ¢
Optimal ET sequence: 87, 137, 224, 311, 535, 1381ce, 1916ce
Badness (Smith): 0.035586
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197
Mapping: [⟨1 14 6 -28 3 6], ⟨0 -27 -8 67 1 -5]]
Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~11/8 = 551.7749 ¢
Optimal ET sequence: 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff
Badness (Smith): 0.017853
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 5 1 6], ⟨0 -31 12 -29]]
Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3375/3136 = 132.1845 ¢
Optimal ET sequence: 109, 118, 345d
Badness (Smith): 0.183435
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 3388/3375, 12005/11979
Mapping: [⟨1 5 1 6 5], ⟨0 -31 12 -29 -14]]
Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~121/112 = 132.1864 ¢
Optimal ET sequence: 109, 118, 345de, 463de, 581dde
Badness (Smith): 0.052693
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 364/363, 385/384, 3388/3375
Mapping: [⟨1 5 1 6 5 7], ⟨0 -31 12 -29 -14 -30]]
Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~13/12 = 132.1789 ¢
Optimal ET sequence: 109, 118f, 227f
Badness (Smith): 0.046695
Oquatonic
- For the 5-limit version, see 28th-octave temperaments #Oquatonic (5-limit).
The oquatonic has a period of 1/28 octave and tempers out the horwell (65625/65536) and the dimcomp (390625/388962), as well as the hemfiness (4096000/4084101, saquinru-atriyo). In this temperament, major third of 5/4 is mapped into 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
Optimal tuning (POTE): ~128/125 = 42.8571 ¢, ~3/2 = 702.1137 ¢
Optimal ET sequence: 28, 56, 84, 140, 224, 364, 588, 952
Badness (Smith): 0.088286
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 tuning (POTE): ~128/125 = 42.8571 ¢, ~3/2 = 702.0186 ¢
Optimal ET sequence: 84, 140, 224, 364, 588, 1400cd, 1988cd, 2576ccdd
Badness (Smith): 0.047853
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 tuning (POTE): ~40/39 = 42.8571 ¢, ~3/2 = 702.0288 ¢
Optimal ET sequence: 84, 140, 224, 364, 588
Badness (Smith): 0.021968
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 tuning (CTE): ~100352/98415 = 37.500 ¢, ~3/2 = 701.610 ¢
Optimal ET sequence: 224, 544, 768, 1312
Badness (Smith): 0.270
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 tuning (CTE): ~45/44 = 37.500 ¢, ~3/2 = 701.601 ¢
Optimal ET sequence: 224, 544, 768
Badness (Smith): 0.0680
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 tuning (CTE): ~45/44 = 37.500 ¢, ~3/2 = 701.593 ¢
Optimal ET sequence: 224, 544, 768, 1312
Badness (Smith): 0.0298