Horwell temperaments

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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.

Horwell temperaments temper out the horwell comma, [-16 1 5 1⟩ = 65625/65536.

Temperaments discussed elsewhere are

Mutt

Main article: Mutt temperament

Subgroup: 2.3.5

Comma list: [-44 -3 21⟩

Mapping: [⟨3 5 7], ⟨0 -7 -1]]

mapping generators: ~98304/78125, ~393216/390625

Optimal tuning (POTE): ~98304/78125 = 1\3, ~5/4 = 385.980 (~393216/390625 = 14.020)

Optimal ET sequence: 84, 87, 171, 771, 942, 1113, 1284, 1455

Badness: 0.162467

7-limit

Subgroup: 2.3.5.7

Comma list: 65625/65536, 250047/250000

Mapping: [⟨3 5 7 8], ⟨0 -7 -1 12]]

Optimal tuning (POTE): ~63/50 = 1\3, ~5/4 = 385.964 (~126/125 = 14.036)

Optimal ET sequence: 84, 87, 171

Badness: 0.028406

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 5 7 8 10], ⟨0 -7 -1 12 11]]

Optimal tuning (POTE): ~44/35 = 1\3, ~5/4 = 386.020 (~126/125 = 13.980)

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

Badness: 0.058344

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨3 5 7 8 10 11], ⟨0 -7 -1 12 11 3]]

Optimal tuning (POTE): ~44/35 = 1\3, ~5/4 = 386.022 (~126/125 = 13.978)

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

Badness: 0.029089

Fifthplus

Fifthplus (22 & 171) tempers out the sesesix comma, [-74 13 23⟩ in the 5-limit. The name "fifthplus" means using a sharp fifth interval (such as superpyth fifth) as a generator.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 420175/419904

Mapping: [⟨1 11 -3 20], ⟨0 -23 13 -42]]

Optimal tuning (POTE): ~2 = 1\1, ~5488/3645 = 708.774

Optimal ET sequence: 22, 149, 171, 1903c, 2074c, 2245cd, 2416cd, 2587cd, 2758cd, 2929cd, 3100cd, 3271ccd, 3442ccd, 3613ccd

Badness: 0.025840

Emkay

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

Subgroup: 2.3.5.7

Comma list: 65625/65536, 244140625/243045684

Mapping: [⟨1 14 6 -28], ⟨0 -27 -8 67]]

Optimal tuning (POTE): ~2 = 1\1, ~3125/2268 = 551.7745

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

Badness: 0.135696

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 = 1\1, ~11/8 = 551.7746

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

Badness: 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 = 1\1, ~11/8 = 551.7749

Optimal ET sequence: 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff

Badness: 0.017853

See also

Kastro

See also: 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 = 1\1, ~3375/3136 = 132.1845

Optimal ET sequence: 109, 118, 345d

Badness: 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 = 1\1, ~121/112 = 132.1864

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

Badness: 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 = 1\1, ~13/12 = 132.1789

Optimal ET sequence: 109, 118f, 227f

Badness: 0.046695

Oquatonic

For the 5-limit version of this temperament, 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): ~2 = 1\1, ~3/2 = 702.1137

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

Badness: 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): ~2 = 1\1, ~3/2 = 702.0186

Optimal ET sequence: 84, 140, 224, 364, 588, 1400cd, 1988cd, 2576ccdd

Badness: 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): ~2 = 1\1, ~3/2 = 702.0288

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

Badness: 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 = 1\32, ~3/2 = 701.610

Optimal ET sequence: 224, 544, 768, 1312

Badness: 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 = 1\32, ~3/2 = 701.601

Optimal ET sequence: 224, 544, 768

Badness: 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 = 1\32, ~3/2 = 701.593

Optimal ET sequence: 224, 544, 768, 1312

Badness: 0.0298

Notes

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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