Horwell temperaments

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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]]

Wedgie: ⟨⟨ 21 3 -36 -44 -116 -92 ]]

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]]

Wedgie: ⟨⟨ 23 -13 42 -74 2 134 ]]

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]]

Wedgie: ⟨⟨ 27 8 -67 -50 -182 -178 ]]

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

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

Wedgie: ⟨⟨ 28 0 -28 -65 -123 -65 ]]

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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