Horwell temperaments: Difference between revisions

Latest revision as of 08:35, 9 November 2026

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 page collects rank-2 horwell temperaments, which temper out the horwell comma (monzo: [-16 1 5 1⟩, ratio: 65625/65536). For the family of rank-3 temperaments, see Horwell family.

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
Counterpara (+100442349/100000000) → Parakleismic 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

Considered below are fifthplus, mutt, oquatonic, emkay, kastro, soviet ferris wheel, 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 temperament. The name fifthplus means using a sharp fifth interval (such as a superpyth fifth) as a generator.

Fifthplus may be extended to the 2.3.5.7.17 subgroup, called prime archagall, derived from the fact that it is an extension of archagall to a prime subgroup. In either the 7-limit or the add-17 subgroup, 171edo is exceptionally efficient and accurate, but in the latter case 171edo is the tuning where 2401/2400 (S49), 2500/2499 (S50), and 1701/1700 (S18/S20) all vanish, which is natural because this temperament tempers out 1225/1224 (S35, S49⋅S50) and 5832/5831 ((S18/S20)/S49) while not tempering out any of above individually.

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, 105d, 127d, 149, 171, 1903c, 2074c, …, 3613ccd

Badness (Sintel): 0.654

Prime archagall

See also: MVP archagall

See also: 24576/24565

Subgroup: 2.3.5.7.17

Comma list: 1225/1224, 24576/24565, 57375/57344

Subgroup-val mapping: [⟨1 -12 10 -22 -3], ⟨0 23 -13 42 12]]

Optimal tunings:

WE: ~2 = 1200.0516 ¢, ~128/85 = 708.8057 ¢
CWE: ~2 = 1200.0000 ¢, ~128/85 = 708.7758 ¢

Optimal ET sequence: 22, 105d, 127d, 149, 171, 1219, 1390

Badness (Sintel): 0.421

Mutt

Main article: 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

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

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

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

Emkay

For the 5-limit version, see Miscellaneous 5-limit temperaments #Emka.

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.

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

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

Soviet ferris wheel

The soviet-ferris-wheel temperament may be described as 320 & 460. It was named by Eliora in 2022 because it has a period of 1/20 octave, and there are 20 cabins on a standard ferris wheel found throughout most of Eastern Europe and Central Asia (as in abandoned Pripyat wheel, for example).

Restricting the temperament to the 5-limit tempers out the comma which is produced by closing twenty 375/256's at 11 octaves, tempering this interval to 11\20.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 1977326743/1968300000

Mapping: [⟨20 0 57 35], ⟨0 3 -1 2]]

mapping generators: ~16807/16200, ~3456/2401

Optimal tunings:

WE: ~16807/16200 = 60.0061 ¢, ~3456/2401 = 634.0252 ¢ (~686/675 = 26.0415 ¢)

error map: ⟨+0.121 +0.121 +0.007 -0.563]

CWE: ~16807/16200 = 60.0000 ¢, ~3456/2401 = 633.9770 ¢ (~686/675 = 26.0230 ¢)

error map: ⟨0.000 -0.024 -0.291 -0.872]

Optimal ET sequence: 140, 320, 460, 600, 1060d

Badness (Sintel): 5.25

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 65625/65536, 184877/184320

Mapping: [⟨20 0 57 35 122], ⟨0 3 -1 2 5]]

Optimal tunings:

WE: ~512/495 = 60.0085 ¢, ~231/160 = 633.9990 ¢ (~625/616 = 26.0943 ¢)
CWE: ~512/495 = 60.0000 ¢, ~231/160 = 633.9127 ¢ (~625/616 = 26.0873 ¢)

Optimal ET sequence: 140, 320, 460

Badness (Sintel): 2.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 4225/4224, 4459/4455, 26411/26364

Mapping: [⟨20 0 57 35 122 74], ⟨0 3 -1 2 5 0]]

Optimal tunings:

WE: ~512/495 = 60.0082 ¢, ~75/52 = 633.9971 ¢ (~65/64 = 26.2937 ¢)
CWE: ~512/495 = 60.0000 ¢, ~75/52 = 633.9127 ¢ (~65/64 = 26.0873 ¢)

Optimal ET sequence: 140, 320, 460

Badness (Sintel): 1.48

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 833/832, 1001/1000, 1225/1224, 4225/4224, 4459/4455

Mapping: [⟨20 0 57 35 122 74 124], ⟨0 3 -1 2 5 0 -4]]

Optimal tunings:

WE: ~88/85 = 60.0081 ¢, ~75/52 = 633.9982 ¢ (~65/64 = 26.0911 ¢)
CWE: ~88/85 = 60.0000 ¢, ~75/52 = 633.9117 ¢ (~65/64 = 26.0883 ¢)

Optimal ET sequence: 140, 320, 460

Badness (Sintel): 1.14

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 833/832, 1001/1000, 1225/1224, 1729/1728, 2926/2925, 4459/4455

Mapping: [⟨20 0 57 35 122 74 124 11], ⟨0 3 -1 2 5 0 -4 7]]

Optimal tunings:

WE: ~88/85 = 60.0075 ¢, ~75/52 = 633.9713 ¢ (~65/64 = 26.1108 ¢)
CWE: ~88/85 = 60.0000 ¢, ~75/52 = 633.8948 ¢ (~65/64 = 26.1052 ¢)

Optimal ET sequence: 140, 320, 460

Badness (Sintel): 1.16

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

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

References

↑ ^1.0 ^1.1 Yahoo! Tuning Group | Suggested names for the unclasified temperaments

[petr's_long_post-1] 1.0 ^1.1 Yahoo! Tuning Group | Suggested names for the unclasified temperaments

[1]

Horwell temperaments: Difference between revisions

FloraC (talk | contribs)

Latest revision as of 08:35, 9 November 2026

Contents

Fifthplus

Prime archagall

Mutt

11-limit

13-limit

Oquatonic

11-limit

13-limit

Emkay

11-limit

13-limit

Kastro

11-limit

13-limit

Soviet ferris wheel

11-limit

13-limit

17-limit

19-limit

Bezique

11-limit

13-limit

References

Navigation menu

@@ Line 1: / Line 1: @@
-Horwell temperaments temper out the horwell comma, {{monzo|-16 1 5 1}} = 65625/65536.
+{{Technical data page}}
+This page collects [[rank-2 temperament|rank-2]] '''horwell temperaments''', which [[tempering out|temper out]] the [[horwell comma]] ({{monzo|legend=1| -16 1 5 1 }}, [[ratio]]: 65625/65536). For the family of [[rank-3 temperament]]s, see [[Horwell family]].
 Temperaments discussed elsewhere are
-* ''[[Mabila]]'' (+49/48) → [[Mabila family #Septimal mabila|Mabila family]]
+* [[Pontiac]] (+4375/4374) → [[Schismatic family #Pontiac|Schismatic family]]
+* ''[[Keen]]'' (+875/864) → [[Diaschismic family #Keen|Diaschismic family]]
+* ''[[Paramity]]'' (+1600000/1594323) → [[Amity family #Paramity|Amity family]]
+* ''[[Countercata]]'' (+5120/5103) → [[Kleismic family #Countercata|Kleismic family]]
+* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
 * ''[[Worschmidt]]'' (+126/125) → [[Würschmidt family #Worschmidt|Würschmidt family]]
 * ''[[Escaped]]'' (+245/243) → [[Escapade family #Escaped|Escapade family]]
-* ''[[Maquiloid]]'' (+686/675) → [[Maquila family #Maquiloid|Maquila family]]
+* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila|Mabila family]]
-* ''[[Keen]]'' (+875/864) → [[Diaschismic family #Keen|Diaschismic family]]
+* ''[[Counterpara]]'' (+100442349/100000000) → [[Parakleismic family #Counterpara|Parakleismic family]]
+* ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana|Vishnu family]]
 * [[Hemithirds]] (+1029/1024) → [[Hemimean clan #Hemithirds|Hemimean clan]]
-* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
-* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
-* [[Pontiac]] (+4375/4374) → [[Schismatic family #Pontiac|Schismatic family]]
-* ''[[Countercata]]'' (+5120/5103) → [[Kleismic family #Countercata|Kleismic family]]
 * ''[[Bisupermajor]]'' (+10976/10935) → [[Hemimage temperaments #Bisupermajor|Hemimage temperaments]]
-* ''[[Eris]]'' (+16875/16807) → [[Mirkwai clan #Eris|Mirkwai clan]]
+* ''[[Maquiloid]]'' (+686/675) → [[Maquila family #Maquiloid|Maquila family]]
-* ''[[Narayana]]'' (+321489/320000) → [[Vishnuzmic family #Narayana|Vishnuzmic family]]
-* ''[[Paramity]]'' (+1600000/1594323) → [[Amity family #Paramity|Amity family]]
 * ''[[Kaboom]]'' (+4802000/4782969) → [[Vavoom family #Kaboom|Vavoom family]]
-* ''[[Soviet ferris wheel]]'' (+{{monzo| -5 -9 -5 11 }}) → [[20th-octave temperaments #Soviet ferris wheel|20th-octave temperaments]]
+* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
+* ''[[Eris]]'' (+16875/16807) → [[Canopic clan #Eris|Canopic clan]]
+Considered below are fifthplus, mutt, oquatonic, emkay, kastro, soviet ferris wheel, 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 {{nowrap| 22 & 171 }} temperament. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator.
+Fifthplus may be extended to the 2.3.5.7.17 subgroup, called prime archagall, derived from the fact that it is an extension of [[archagall]] to a prime subgroup. In either the 7-limit or the add-17 subgroup, [[171edo]] is exceptionally efficient and accurate, but in the latter case 171edo is the tuning where [[2401/2400]] (S49), [[2500/2499]] (S50), and [[1701/1700]] (S18/S20) all vanish, which is natural because this temperament tempers out [[1225/1224]] (S35, S49⋅S50) and [[5832/5831]] ((S18/S20)/S49) while not tempering out any of above individually.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 65625/65536, 420175/419904
+{{Mapping|legend=1| 1 -12 10 -22 | 0 23 -13 42 }}
+: mapping generators: ~2, ~5488/3645
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0934{{c}}, ~5488/3645 = 708.8291{{c}}
+: [[error map]]: {{val| +0.093 -0.007 -0.158 -0.059 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5488/3645 = 708.7752{{c}}
+: error map: {{val| 0.000 -0.126 -0.391 -0.268 }}
+{{Optimal ET sequence|legend=1| 22, 105d, 127d, 149, 171, 1903c, 2074c, …, 3613ccd }}
-== Mutt ==
+[[Badness]] (Sintel): 0.654
-{{Main| Mutt temperament }}
+=== Prime archagall ===
+{{ See also | MVP archagall }}
+{{ See also | 24576/24565 }}
+Subgroup: 2.3.5.7.17
-[[Subgroup]]: 2.3.5
+Comma list: 1225/1224, 24576/24565, 57375/57344
-[[Comma list]]: {{monzo| -44 -3 21 }}
+{{Mapping|legend=2| 1 -12 10 -22 -3 | 0 23 -13 42 12 }}
-{{Mapping|legend=1| 3 5 7 | 0 -7 -1 }}
+Optimal tunings:
+* WE: ~2 = 1200.0516{{c}}, ~128/85 = 708.8057{{cent}}
+* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 708.7758{{cent}}
-: mapping generators: ~98304/78125, ~393216/390625
+{{Optimal ET sequence|legend=0| 22, 105d, 127d, 149, 171, 1219, 1390 }}
-[[Optimal tuning]] ([[POTE]]): ~98304/78125 = 1\3, ~5/4 = 385.980 (~393216/390625 = 14.020)
+Badness (Sintel): 0.421
-{{Optimal ET sequence|legend=1| 84, 87, 171, 771, 942, 1113, 1284, 1455 }}
+== Mutt ==
+{{Main| Mutt }}
+: ''For the 5-limit version, see [[Father–3 equivalence continuum #Mutt (5-limit)]].''
-[[Badness]]: 0.162467
+Mutt tempers out the [[landscape comma]] in addition to the horwell comma, and may be described as the {{nowrap| 84 & 87 }} temperament.
-=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
 [[Comma list]]: 65625/65536, 250047/250000
-{{Mapping|legend=1| 3 5 7 8 | 0 -7 -1 12 }}
+{{Mapping|legend=1| 3 -2 6 20 | 0 7 1 -12 }}
+: mapping generators: ~63/50, ~5/4
-{{Multival|legend=1| 21 3 -36 -44 -116 -92 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~63/50 = 400.0351{{c}}, ~5/4 = 385.9974{{c}} (~126/125 = 14.0377{{c}})
-[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~5/4 = 385.964 (~126/125 = 14.036)
+: [[error map]]: {{val| +0.105 -0.043 -0.105 -0.092 }}
+* [[CWE]]: ~63/50 = 400.0000{{c}}, ~5/4 = 385.9638{{c}} (~126/125 = 14.0362{{c}})
+: error map: {{val| 0.000 -0.208 -0.350 -0.392 }}
 {{Optimal ET sequence|legend=1| 84, 87, 171 }}
-[[Badness]]: 0.028406
+[[Badness]] (Sintel): 0.719
 === 11-limit ===
@@ Line 56: / Line 92: @@
 Comma list: 441/440, 4375/4356, 16384/16335
-Mapping: {{mapping| 3 5 7 8 10 | 0 -7 -1 12 11 }}
+{{Mapping|legend=0| 3 -2 6 20 21 | 0 7 1 -12 -11 }}
-Optimal tuning (POTE): ~44/35 = 1\3, ~5/4 = 386.020 (~126/125 = 13.980)
+Optimal tunings:
+* WE: ~44/35 = 399.9783{{c}}, ~5/4 = 385.9993{{c}} (~126/125 = 13.9790{{c}})
+* CWE: ~44/35 = 400.0000{{c}}, ~5/4 = 386.0208{{c}} (~126/125 = 13.9792{{c}})
-{{Optimal ET sequence|legend=1| 84, 87, 171, 258, 429e }}
+{{Optimal ET sequence|legend=0| 84, 87, 171, 258 }}
-Badness: 0.058344
+Badness (Sintel): 1.93
 === 13-limit ===
@@ Line 69: / Line 107: @@
 Comma list: 364/363, 441/440, 625/624, 2200/2197
-Mapping: {{mapping| 3 5 7 8 10 11 | 0 -7 -1 12 11 3 }}
+{{Mapping|legend=0| 3 -2 6 20 21 14 | 0 7 1 -12 -11 -3 }}
+Optimal tunings:
+* WE: ~44/35 = 399.9610{{c}}, ~5/4 = 385.9842{{c}} (~126/125 = 13.9768{{c}})
+* CWE: ~44/35 = 400.0000{{c}}, ~5/4 = 386.0231{{c}} (~126/125 = 13.9769{{c}})
+{{Optimal ET sequence|legend=0| 84, 87, 171, 258, 429ef }}
-Optimal tuning (POTE): ~44/35 = 1\3, ~5/4 = 386.022 (~126/125 = 13.978)
+Badness (Sintel): 1.20
-{{Optimal ET sequence|legend=1| 84, 87, 171, 258, 429ef }}
+== Oquatonic ==
+: ''For the 5-limit version, see [[28th-octave temperaments #Oquatonic (5-limit)]].''
-Badness: 0.029089
+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.
-== Fifthplus ==
+The name ''oquatonic'' was given by [[Petr Pařízek]] in 2011 as an abbreviation of the Italian [[wiktionary: ottantaquatro|''ottantaquatro'' ("eighty-four")]]<ref name="petr's long post"/>.
-Fifthplus (22 &amp; 171) tempers out the sesesix comma, {{monzo| -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
+[[Comma list]]: 65625/65536, 390625/388962
+{{Mapping|legend=1| 28 0 65 123 | 0 1 0 -1 }}
+: mapping generators: ~128/125, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~128/125 = 42.8570{{c}}, ~3/2 = 702.1112{{c}}
+: [[error map]]: {{val| -0.004 +0.152 -0.609 +0.477 }}
+* [[CWE]]: ~128/125 = 42.8571{{c}}, ~3/2 = 702.1132{{c}}
+: error map: {{val| 0.000 +0.158 -0.599 +0.489 }}
+{{Optimal ET sequence|legend=1| 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|legend=0| 28 0 65 123 230 | 0 1 0 -1 -3 }}
+Optimal tunings:
+* WE: ~128/125 = 42.8577{{c}}, ~3/2 = 702.0275{{c}}
+* CWE: ~128/125 = 42.8571{{c}}, ~3/2 = 702.0174{{c}}
+{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}
+Badness (Sintel): 1.58
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-{{Mapping|legend=1| 1 11 -3 20 | 0 -23 13 -42 }}
+Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197
-{{Multival|legend=1| 23 -13 42 -74 2 134 }}
+{{Mapping|legend=0| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5488/3645 = 708.774
+Optimal tunings:
+* WE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0289{{c}}
+* CWE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0288{{c}}
-{{Optimal ET sequence|legend=1| 22, 149, 171, 1903c, 2074c, 2245cd, 2416cd, 2587cd, 2758cd, 2929cd, 3100cd, 3271ccd, 3442ccd, 3613ccd }}
+{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}
-[[Badness]]: 0.025840
+Badness (Sintel): 0.908
 == Emkay ==
-[[Emkay]] (87 &amp; 224) tempers out the same 5-limit comma as the [[Hemimean clan #Emka|emka temperament]] (37 &amp; 50), but with the horwell (65625/65536) rather than the hemimean (3136/3125) tempered out.
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Emka]].''
+[[File:Scale Tree Graph For Emkay.png|thumb|Scale tree graph for emkay.]]
+Emkay may be described as the {{nowrap| 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
@@ Line 101: / Line 182: @@
 [[Comma list]]: 65625/65536, 244140625/243045684
-{{Mapping|legend=1| 1 14 6 -28 | 0 -27 -8 67 }}
+{{Mapping|legend=1| 1 -13 -2 39 | 0 27 8 -67 }}
+: mapping generators: ~2, ~4536/3125
-{{Multival|legend=1| 27 8 -67 -50 -182 -178 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0279{{c}}, ~4536/3125 = 648.2405{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3125/2268 = 551.7745
+: [[error map]]: {{val| +0.028 +0.177 -0.445 +0.146 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~4536/3125 = 648.2254{{c}}
+: error map: {{val| 0.000 +0.133 -0.510 +0.069 }}
 {{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}
-[[Badness]]: 0.135696
+[[Badness]] (Sintel): 3.43
 === 11-limit ===
@@ Line 116: / Line 200: @@
 Comma list: 3025/3024, 4000/3993, 65625/65536
-Mapping: {{mapping| 1 14 6 -28 3 | 0 -27 -8 67 1 }}
+{{Mapping|legend=0| 1 -13 -2 39 4 | 0 27 8 -67 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.7746
+Optimal tunings:
+* WE: ~2 = 1199.9958{{c}}, ~16/11 = 648.2231{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2254{{c}}
-{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381ce, 1916ce }}
+{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535 }}
-Badness: 0.035586
+Badness (Sintel): 1.18
 === 13-limit ===
@@ Line 129: / Line 215: @@
 Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197
-Mapping: {{mapping| 1 14 6 -28 3 6 | 0 -27 -8 67 1 -5 }}
+{{Mapping|legend=0| 1 -13 -2 39 4 1 | 0 27 8 -67 -1 5 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.7749
+Optimal tunings:
+* WE: ~2 = 1199.9694{{c}}, ~16/11 = 648.2085{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2251{{c}}
-{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff }}
+{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535 }}
-Badness: 0.017853
+Badness (Sintel): 0.738
 == Kastro ==
-{{See also| Very high accuracy temperaments #Astro }}
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Astro]].''
+Kastro may be described as the {{nowrap| 109 & 118 }} temperament, named by [[Petr Pařízek]] in 2011 as a variation of ''astro''<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.
 [[Subgroup]]: 2.3.5.7
@@ Line 144: / Line 234: @@
 [[Comma list]]: 65625/65536, 117649/116640
-{{Mapping|legend=1| 1 5 1 6 | 0 -31 12 -29 }}
+{{Mapping|legend=1| 1 -26 13 -23 | 0 31 -12 29 }}
+: mapping generators: ~2, ~6272/3375
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3375/3136 = 132.1845
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1529{{c}}, ~6272/3375 = 1067.9515{{c}}
+: [[error map]]: {{val| +0.153 +0.567 +0.256 -1.749 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~6272/3375 = 1067.8174{{c}}
+: error map: {{val| 0.000 +0.384 -0.122 -2.122 }}
-{{Optimal ET sequence|legend=1| 109, 118, 345d }}
+{{Optimal ET sequence|legend=1| 109, 118, 345d, 463d, 581dd }}
-[[Badness]]: 0.183435
+[[Badness]] (Sintel): 4.64
 === 11-limit ===
@@ Line 157: / Line 252: @@
 Comma list: 385/384, 3388/3375, 12005/11979
-Mapping: {{mapping| 1 5 1 6 5 | 0 -31 12 -29 -14 }}
+{{Mapping|legend=0| 1 -26 13 -23 -9 | 0 31 -12 29 14 }}
-Optimal tuning (POTE): ~2 = 1\1, ~121/112 = 132.1864
+Optimal tunings:
+* WE: ~2 = 1200.2427{{c}}, ~224/121 = 1068.0296{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~224/121 = 1067.8166{{c}}
-{{Optimal ET sequence|legend=1| 109, 118, 345de, 463de, 581dde }}
+{{Optimal ET sequence|legend=0| 109, 118, 345de, 463de, 581dde }}
-Badness: 0.052693
+Badness (Sintel): 1.74
 === 13-limit ===
@@ Line 170: / Line 267: @@
 Comma list: 169/168, 364/363, 385/384, 3388/3375
-Mapping: {{mapping| 1 5 1 6 5 7 | 0 -31 12 -29 -14 -30 }}
+{{Mapping|legend=0| 1 -26 13 -23 -9 -23 | 0 31 -12 29 14 30 }}
-Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 132.1789
+Optimal tunings:
+* WE: ~2 = 1200.4303{{c}}, ~13/7 = 1068.2040{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1067.8267{{c}}
-{{Optimal ET sequence|legend=1| 109, 118f, 227f }}
+{{Optimal ET sequence|legend=0| 109, 118f, 227f }}
-Badness: 0.046695
+Badness (Sintel): 1.93
-== Oquatonic ==
+== Soviet ferris wheel ==
-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 temperaments|hemfiness]] (4096000/4084101, saquinru-atriyo). In this temperament, major third of [[5/4]] is mapped into 9\28.
+The soviet-ferris-wheel temperament may be described as 320 & 460. It was named by [[Eliora]] in 2022 because it has a period of 1/20 octave, and there are 20 cabins on a standard ferris wheel found throughout most of Eastern Europe and Central Asia (as in abandoned Pripyat wheel, for example).
+Restricting the temperament to the 5-limit tempers out the comma which is produced by closing twenty [[375/256]]'s at 11 octaves, tempering this interval to 11\20.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 65625/65536, 390625/388962
+[[Comma list]]: 65625/65536, 1977326743/1968300000
-{{Mapping|legend=1| 28 0 65 123 | 0 1 0 -1 }}
+{{Mapping|legend=1| 20 0 57 35 | 0 3 -1 2 }}
+: mapping generators: ~16807/16200, ~3456/2401
-: mapping generators: ~128/125, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~16807/16200 = 60.0061{{c}}, ~3456/2401 = 634.0252{{c}} (~686/675 = 26.0415{{c}})
+: [[error map]]: {{val| +0.121 +0.121 +0.007 -0.563 }}
+* [[CWE]]: ~16807/16200 = 60.0000{{c}}, ~3456/2401 = 633.9770{{c}} (~686/675 = 26.0230{{c}})
+: error map: {{val| 0.000 -0.024 -0.291 -0.872 }}
-{{Multival|legend=1| 28 0 -28 -65 -123 -65 }}
+{{Optimal ET sequence|legend=1| 140, 320, 460, 600, 1060d }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.1137
+[[Badness]] (Sintel): 5.25
-{{Optimal ET sequence|legend=1| 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
+Comma list: 9801/9800, 65625/65536, 184877/184320
-Mapping: {{mapping| 28 0 65 123 230 | 0 1 0 -1 -3 }}
+{{Mapping|legend=0| 20 0 57 35 122 | 0 3 -1 2 5 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0186
+Optimal tunings:
+* WE: ~512/495 = 60.0085{{c}}, ~231/160 = 633.9990{{c}} (~625/616 = 26.0943{{c}})
+* CWE: ~512/495 = 60.0000{{c}}, ~231/160 = 633.9127{{c}} (~625/616 = 26.0873{{c}})
-{{Optimal ET sequence|legend=1| 84, 140, 224, 364, 588, 1400cd, 1988cd, 2576ccdd }}
+{{Optimal ET sequence|legend=0| 140, 320, 460 }}
-Badness: 0.047853
+Badness (Sintel): 2.92
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197
+Comma list: 1001/1000, 4225/4224, 4459/4455, 26411/26364
+{{Mapping|legend=0| 20 0 57 35 122 74 | 0 3 -1 2 5 0 }}
+Optimal tunings:
+* WE: ~512/495 = 60.0082{{c}}, ~75/52 = 633.9971{{c}} (~65/64 = 26.2937{{c}})
+* CWE: ~512/495 = 60.0000{{c}}, ~75/52 = 633.9127{{c}} (~65/64 = 26.0873{{c}})
+{{Optimal ET sequence|legend=0| 140, 320, 460 }}
+Badness (Sintel): 1.48
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 833/832, 1001/1000, 1225/1224, 4225/4224, 4459/4455
+{{Mapping|legend=0| 20 0 57 35 122 74 124 | 0 3 -1 2 5 0 -4 }}
+Optimal tunings:
+* WE: ~88/85 = 60.0081{{c}}, ~75/52 = 633.9982{{c}} (~65/64 = 26.0911{{c}})
+* CWE: ~88/85 = 60.0000{{c}}, ~75/52 = 633.9117{{c}} (~65/64 = 26.0883{{c}})
+{{Optimal ET sequence|legend=0| 140, 320, 460 }}
+Badness (Sintel): 1.14
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 833/832, 1001/1000, 1225/1224, 1729/1728, 2926/2925, 4459/4455
-Mapping: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}
+{{Mapping|legend=0| 20 0 57 35 122 74 124 11 | 0 3 -1 2 5 0 -4 7 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0288
+Optimal tunings:
+* WE: ~88/85 = 60.0075{{c}}, ~75/52 = 633.9713{{c}} (~65/64 = 26.1108{{c}})
+* CWE: ~88/85 = 60.0000{{c}}, ~75/52 = 633.8948{{c}} (~65/64 = 26.1052{{c}})
-{{Optimal ET sequence|legend=1| 84, 140, 224, 364, 588 }}
+{{Optimal ET sequence|legend=0| 140, 320, 460 }}
-Badness: 0.021968
+Badness (Sintel): 1.16
 == 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.
+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
@@ Line 231: / Line 367: @@
 {{Mapping|legend=1| 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 tuning]]s:
+* [[WE]]: ~100352/98415 = 37.5038{{c}}, ~3/2 = 701.6058{{c}}
+: [[error map]]: {{val| +0.120 -0.229 -0.071 +0.154 }}
+* [[CWE]]: ~100352/98415 = 37.5000{{c}}, ~3/2 = 701.5544{{c}}
+: error map: {{val| 0.000 -0.401 -0.368 -0.108 }}
-{{Optimal ET sequence|legend=1| 224, 544, 768, 1312 }}
+{{Optimal ET sequence|legend=1| 96d, 224, 544, 768, 1312, 2080bc }}
-[[Badness]]: 0.270
+[[Badness]] (Sintel): 6.82
 === 11-limit ===
@@ Line 245: / Line 384: @@
 Comma list: 9801/9800, 46656/46585, 65625/65536
-Mapping: {{mapping| 32 0 125 -113 60 | 0 1 -1 4 1 }}
+{{Mapping|legend=0| 32 0 125 -113 60 | 0 1 -1 4 1 }}
-Optimal tuning (CTE): ~45/44 = 1\32, ~3/2 = 701.601
+Optimal tunings:
+* WE: ~45/44 = 37.5025{{c}}, ~3/2 = 701.5912{{c}}
+* CWE: ~45/44 = 37.5000{{c}}, ~3/2 = 701.5566{{c}}
-{{Optimal ET sequence|legend=1| 224, 544, 768 }}
+{{Optimal ET sequence|legend=0| 96d, 224, 544, 768 }}
-Badness: 0.0680
+Badness (Sintel): 2.25
 === 13-limit ===
@@ Line 258: / Line 399: @@
 Comma list: 729/728, 1575/1573, 4225/4224, 6656/6655
-Mapping: {{mapping| 32 0 125 -113 60 17 | 0 1 -1 4 1 2 }}
+{{Mapping|legend=0| 32 0 125 -113 60 17 | 0 1 -1 4 1 2 }}
-Optimal tuning (CTE): ~45/44 = 1\32, ~3/2 = 701.593
+Optimal tunings:
+* WE: ~45/44 = 37.5021{{c}}, ~3/2 = 701.5769{{c}}
+* CWE: ~45/44 = 37.5000{{c}}, ~3/2 = 701.5490{{c}}
-{{Optimal ET sequence|legend=1| 224, 544, 768, 1312 }}
+{{Optimal ET sequence|legend=0| 96d, 224, 544, 768, 1312 }}
-Badness: 0.0298
+Badness (Sintel): 1.23
+== References ==
-[[Category:Temperament collections]]
 [[Category:Horwell temperaments| ]] <!-- main article -->
 [[Category:Horwell| ]] <!-- key article -->
-[[Category:Rank 2]]
+[[Category:Temperament collections]]
+[[Category:Catalogs of rank-2 temperaments]]

Horwell temperaments: Difference between revisions

FloraC (talk | contribs)

Latest revision as of 08:35, 9 November 2026

Fifthplus

Prime archagall

Mutt

11-limit

13-limit

Oquatonic

11-limit

13-limit

Emkay

11-limit

13-limit

Kastro

11-limit

13-limit

Soviet ferris wheel

11-limit

13-limit

17-limit

19-limit

Bezique

11-limit

13-limit

References

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