Horwell temperaments: Difference between revisions

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[[Comma list]]: {{monzo| -44 -3 21 }}
[[Comma list]]: {{monzo| -44 -3 21 }}


{{Mapping|legend=1| 3 5 7 | 0 -7 -1 }}
{{Mapping|legend=1| 3 -2 6 | 0 7 1 }}
: mapping generators: ~98304/78125, ~393216/390625
: mapping generators: ~98304/78125, ~5/4


[[Optimal tuning]] ([[POTE]]): ~98304/78125 = 400.000{{c}}, ~5/4 = 385.980{{c}} (~393216/390625 = 14.020{{c}})
[[Optimal tuning]]s:
* [[WE]]: ~98304/78125 = 400.0227{{c}}, ~5/4 = 386.0017{{c}} (~393216/390625 = 14.0210{{c}})
: [[error map]]: {{val| +0.068 +0.012 -0.176 }}
* [[CWE]]: ~98304/78125 = 400.0000{{c}}, ~5/4 = 385.9858{{c}} (~393216/390625 = 14.0142{{c}})
: error map: {{val| 0.000 -0.055 -0.328 }}


{{Optimal ET sequence|legend=1| 84, 87, 171, 771, 942, 1113, 1284, 1455 }}
{{Optimal ET sequence|legend=1| 84, 87, 171, 771, 942, 1113, 1284, 1455, 4194cc, 5649cc }}


[[Badness]] (Smith): 0.162467
[[Badness]] (Sintel): 3.81


=== 7-limit ===
=== 7-limit ===
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[[Comma list]]: 65625/65536, 250047/250000
[[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, ~126/125
: mapping generators: ~63/50, ~5/4


[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~5/4 = 385.964{{c}} (~126/125 = 14.036{{c}})
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0351{{c}}, ~5/4 = 385.9974{{c}} (~126/125 = 14.0377{{c}})
: [[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 }}
{{Optimal ET sequence|legend=1| 84, 87, 171 }}


[[Badness]] (Smith): 0.028406
[[Badness]] (Sintel): 0.719


=== 11-limit ===
=== 11-limit ===
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Comma list: 441/440, 4375/4356, 16384/16335
Comma list: 441/440, 4375/4356, 16384/16335


Mapping: {{mapping| 3 5 7 8 10 | 0 -7 -1 12 11 }}
Mapping: {{mapping| 3 -2 6 20 21 | 0 7 1 -12 -11 }}


Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~5/4 = 386.020{{c}} (~126/125 = 13.980{{c}})
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=0| 84, 87, 171, 258, 429e }}
{{Optimal ET sequence|legend=0| 84, 87, 171, 258 }}


Badness (Smith): 0.058344
Badness (Sintel): 1.93


=== 13-limit ===
=== 13-limit ===
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Comma list: 364/363, 441/440, 625/624, 2200/2197
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: {{mapping| 3 -2 6 20 21 14 | 0 7 1 -12 -11 -3 }}


Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~5/4 = 386.022{{c}} (~126/125 = 13.978{{c}})
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 ET sequence|legend=0| 84, 87, 171, 258, 429ef }}


Badness (Smith): 0.029089
Badness (Sintel): 1.20


== Fifthplus ==
== Fifthplus ==
Fifthplus (22 & 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. It is a restriction of [[24576/24565 #2.3.5.7.17 subgroup (prime archagall)|prime archagall]].
: ''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 }}. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator. It is a restriction of [[24576/24565 #2.3.5.7.17 subgroup (prime archagall)|prime archagall]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 65625/65536, 420175/419904
[[Comma list]]: 65625/65536, 420175/419904


{{Mapping|legend=1| 1 11 -3 20 | 0 -23 13 -42 }}
{{Mapping|legend=1| 1 -12 10 -22 | 0 23 -13 42 }}
: mapping generators: ~2, ~5488/3645


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5488/3645 = 708.774{{c}}
[[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, 149, 171, 1903c, 2074c, 2245cd, 2416cd, 2587cd, 2758cd, 2929cd, 3100cd, 3271ccd, 3442ccd, 3613ccd }}
{{Optimal ET sequence|legend=1| 22, 149, 171, 1903c, 2074c, …, 3613ccd }}


[[Badness]] (Smith): 0.025840
[[Badness]] (Sintel): 0.654


== Emkay ==
== Emkay ==
[[File:Scale Tree Graph For Emkay.png|thumb|Scale tree graph for emkay.]]
[[File:Scale Tree Graph For Emkay.png|thumb|Scale tree graph for emkay.]]


[[Emkay]] (87 & 224) tempers out the same 5-limit comma as the [[Hemimean clan #Emka|emka temperament]] (37 & 50), but with the horwell (65625/65536) rather than the hemimean (3136/3125) tempered out.
[[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
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 65625/65536, 244140625/243045684
[[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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3125/2268 = 551.7745{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3125/2268 = 551.7745{{c}}
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{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}
{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}


[[Badness]] (Smith): 0.135696
[[Badness]] (Sintel): 3.43


=== 11-limit ===
=== 11-limit ===

Revision as of 09:56, 22 June 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 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

Mutt

Main article: Mutt

Subgroup: 2.3.5

Comma list: [-44 -3 21⟩

Mapping: [⟨3 -2 6], ⟨0 7 1]]

mapping generators: ~98304/78125, ~5/4

Optimal tunings:

  • 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

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

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

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, 149, 171, 1903c, 2074c, …, 3613ccd

Badness (Sintel): 0.654

Emkay

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 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

References

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