Horwell temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
→Bezique: where's your brackets and normalization?
 
(14 intermediate revisions by 7 users not shown)
Line 1: Line 1:
{{Technical data page}}
Horwell temperaments temper out the horwell comma, {{monzo|-16 1 5 1}} = 65625/65536.
Horwell temperaments temper out the horwell comma, {{monzo|-16 1 5 1}} = 65625/65536.


Discussed elsewhere are [[Hemimage temperaments|bisupermajor]], [[Kleismic family|countercata]], [[Mirkwai clan|eris]], [[Escapade family|escaped]], [[Hemimean clan|hemithirds]], [[Diaschismic family|keen]], [[Mabila family|mabila]], [[Maquila family|maquiloid]], [[Vishnuzmic family|narayana]], [[Semicomma family|orwell]], [[Amity family|paramity]], [[Schismatic family|pontiac]], [[Breedsmic temperaments|tertiaseptal]], and [[Würschmidt family|worschmidt]].
Temperaments discussed elsewhere are  
* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila|Mabila 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]]
* ''[[Keen]]'' (+875/864) → [[Diaschismic family #Keen|Diaschismic 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]]
* ''[[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]]


== Mutt ==
== Mutt ==
{{Main|Mutt temperament}}
{{Main| Mutt temperament }}


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


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


[[Mapping]]: [{{val|3 5 7}}, {{val|0 -7 -1}}]
{{Mapping|legend=1| 3 5 7 | 0 -7 -1 }}


[[POTE generator]]: ~5/4 = 385.980
: mapping generators: ~98304/78125, ~393216/390625


{{Val list|legend=1| 3, 84, 87, 171, 771, 942, 1113, 1284, 1455 }}
[[Optimal tuning]] ([[POTE]]): ~98304/78125 = 1\3, ~5/4 = 385.980 (~393216/390625 = 14.020)
 
{{Optimal ET sequence|legend=1| 84, 87, 171, 771, 942, 1113, 1284, 1455 }}


[[Badness]]: 0.162467
[[Badness]]: 0.162467


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 65625/65536, 250047/250000
[[Comma list]]: 65625/65536, 250047/250000


[[Mapping]]: [{{val|3 5 7 8}}, {{val|0 -7 -1 12}}]
{{Mapping|legend=1| 3 5 7 8 | 0 -7 -1 12 }}


{{Multival|legend=1|21 3 -36 -44 -116 -92}}
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~5/4 = 385.964 (~126/125 = 14.036)


[[POTE generator]]: ~5/4 = 385.964
{{Optimal ET sequence|legend=1| 84, 87, 171 }}
 
{{Val list|legend=1| 3, 84, 87, 171 }}


[[Badness]]: 0.028406
[[Badness]]: 0.028406
Line 38: Line 55:
Comma list: 441/440, 4375/4356, 16384/16335
Comma list: 441/440, 4375/4356, 16384/16335


Mapping: [{{val|3 5 7 8 10}}, {{val|0 -7 -1 12 11}}]
Mapping: {{mapping| 3 5 7 8 10 | 0 -7 -1 12 11 }}


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


Optimal GPV sequence: {{Val list| 3, 84, 87, 171, 258, 429e }}
{{Optimal ET sequence|legend=1| 84, 87, 171, 258, 429e }}


Badness: 0.058344
Badness: 0.058344
Line 51: Line 68:
Comma list: 364/363, 441/440, 625/624, 2200/2197
Comma list: 364/363, 441/440, 625/624, 2200/2197


Mapping: [{{val|3 5 7 8 10 11}}, {{val|0 -7 -1 12 11 3}}]
Mapping: {{mapping| 3 5 7 8 10 11 | 0 -7 -1 12 11 3 }}


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


Optimal GPV sequence: {{Val list| 3, 84, 87, 171, 258, 429ef }}
{{Optimal ET sequence|legend=1| 84, 87, 171, 258, 429ef }}


Badness: 0.029089
Badness: 0.029089


== 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.
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.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|1 11 -3 20}}, {{val|0 -23 13 -42}}]
{{Mapping|legend=1| 1 11 -3 20 | 0 -23 13 -42 }}
 
{{Multival|legend=1|23 -13 42 -74 2 134}}


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


{{Val list|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, 2245cd, 2416cd, 2587cd, 2758cd, 2929cd, 3100cd, 3271ccd, 3442ccd, 3613ccd }}


[[Badness]]: 0.025840
[[Badness]]: 0.025840


== Emkay ==
== 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]] (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.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 65625/65536, 244140625/243045684
[[Comma list]]: 65625/65536, 244140625/243045684


[[Mapping]]: [{{val|1 14 6 -28}}, {{val|0 -27 -8 67}}]
{{Mapping|legend=1| 1 14 6 -28 | 0 -27 -8 67 }}


{{Multival|legend=1|27 8 -67 -50 -182 -178}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3125/2268 = 551.7745


[[POTE generator]]: ~3125/2268 = 551.7745
{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}
 
{{Val list|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}


[[Badness]]: 0.135696
[[Badness]]: 0.135696
Line 98: Line 111:
Comma list: 3025/3024, 4000/3993, 65625/65536
Comma list: 3025/3024, 4000/3993, 65625/65536


Mapping: [{{val|1 14 6 -28 3}}, {{val|0 -27 -8 67 1}}]
Mapping: {{mapping| 1 14 6 -28 3 | 0 -27 -8 67 1 }}


POTE generator: ~11/8 = 551.7746
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.7746


Optimal GPV sequence: {{Val list| 87, 137, 224, 311, 535, 1381ce, 1916ce }}
{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381ce, 1916ce }}


Badness: 0.035586
Badness: 0.035586
Line 111: Line 124:
Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197
Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197


Mapping: [{{val|1 14 6 -28 3 6}}, {{val|0 -27 -8 67 1 -5}}]
Mapping: {{mapping| 1 14 6 -28 3 6 | 0 -27 -8 67 1 -5 }}


POTE generator: ~11/8 = 551.7749
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.7749


Optimal GPV sequence: {{Val list| 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff }}
{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff }}


Badness: 0.017853
Badness: 0.017853
=== See also ===
* [[:File:Scale Tree Graph For Emkay.png]]


== Kastro ==
== Kastro ==
{{see also| Very high accuracy temperaments #Astro }}
{{See also| Very high accuracy temperaments #Astro }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 65625/65536, 117649/116640
[[Comma list]]: 65625/65536, 117649/116640


[[Mapping]]: [{{val| 1 5 1 6 }}, {{val| 0 -31 12 -29 }}]
{{Mapping|legend=1| 1 5 1 6 | 0 -31 12 -29 }}


[[POTE generator]]: ~3375/3136 = 132.1845
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3375/3136 = 132.1845


{{Val list|legend=1| 109, 118, 345d }}
{{Optimal ET sequence|legend=1| 109, 118, 345d }}


[[Badness]]: 0.183435
[[Badness]]: 0.183435
Line 139: Line 155:
Comma list: 385/384, 3388/3375, 12005/11979
Comma list: 385/384, 3388/3375, 12005/11979


Mapping: [{{val| 1 5 1 6 5 }}, {{val| 0 -31 12 -29 -14 }}]
Mapping: {{mapping| 1 5 1 6 5 | 0 -31 12 -29 -14 }}


POTE generator: ~121/112 = 132.1864
Optimal tuning (POTE): ~2 = 1\1, ~121/112 = 132.1864


Optimal GPV sequence: {{Val list| 109, 118, 345de, 463de, 581dde }}
{{Optimal ET sequence|legend=1| 109, 118, 345de, 463de, 581dde }}


Badness: 0.052693
Badness: 0.052693
Line 152: Line 168:
Comma list: 169/168, 364/363, 385/384, 3388/3375
Comma list: 169/168, 364/363, 385/384, 3388/3375


Mapping: [{{val| 1 5 1 6 5 7 }}, {{val| 0 -31 12 -29 -14 -30 }}]
Mapping: {{mapping| 1 5 1 6 5 7 | 0 -31 12 -29 -14 -30 }}


POTE generator: ~13/12 = 132.1789
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 132.1789


Optimal GPV sequence: {{Val list| 109, 118f, 227f }}
{{Optimal ET sequence|legend=1| 109, 118f, 227f }}


Badness: 0.046695
Badness: 0.046695


== Oquatonic ==
== Oquatonic ==
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.
: ''For the 5-limit version of this temperament, see [[28th-octave temperaments #Oquatonic (5-limit)]].''


Subgroup: 2.3.5.7
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 name ''oquatonic'' was given by [[Petr Pařízek]] in 2011 as an abbreviation of the Italian [[wiktionary: ottantaquatro|''ottantaquatro'' ("eighty-four")]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 65625/65536, 390625/388962
[[Comma list]]: 65625/65536, 390625/388962


[[Mapping]]: [{{val| 28 0 65 123 }}, {{val| 0 1 0 -1 }}]
{{Mapping|legend=1| 28 0 65 123 | 0 1 0 -1 }}


Mapping generators: ~128/125, ~3
: mapping generators: ~128/125, ~3


{{Multival|legend=1| 28 0 -28 -65 -123 -65 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.1137


[[POTE generator]]: ~3/2 = 702.1137
{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}
 
{{Val list|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}


[[Badness]]: 0.088286
[[Badness]]: 0.088286
Line 184: Line 202:
Comma list: 1375/1372, 6250/6237, 65625/65536
Comma list: 1375/1372, 6250/6237, 65625/65536


Mapping: [{{val| 28 0 65 123 230 }}, {{val| 0 1 0 -1 -3 }}]
Mapping: {{mapping| 28 0 65 123 230 | 0 1 0 -1 -3 }}


POTE generator: ~3/2 = 702.0186
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0186


Optimal GPV sequence: {{Val list| 84, 140, 224, 364, 588, 1400cd, 1988cd, 2576ccdd }}
{{Optimal ET sequence|legend=1| 84, 140, 224, 364, 588, 1400cd, 1988cd, 2576ccdd }}


Badness: 0.047853
Badness: 0.047853
Line 197: Line 215:
Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197
Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197


Mapping: [{{val| 28 0 65 123 230 148 }}, {{val| 0 1 0 -1 -3 -1 }}]
Mapping: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}


POTE generator: ~3/2 = 702.0288
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0288


Optimal GPV sequence: {{Val list| 84, 140, 224, 364, 588 }}
{{Optimal ET sequence|legend=1| 84, 140, 224, 364, 588 }}


Badness: 0.021968
Badness: 0.021968
Line 212: Line 230:
[[Comma list]]: 65625/65536, 847288609443/843308032000
[[Comma list]]: 65625/65536, 847288609443/843308032000


[[Mapping]]: [{{val| 32 0 125 -113 }}, {{val| 0 1 -1 4 }}]
{{Mapping|legend=1| 32 0 125 -113 | 0 1 -1 4 }}


Mapping generators: ~100352/98415, ~3
: mapping generators: ~100352/98415, ~3


[[Optimal tuning]] ([[CTE]]): ~100352/98415 = 1\32, ~3/2 = 701.610
[[Optimal tuning]] ([[CTE]]): ~100352/98415 = 1\32, ~3/2 = 701.610
{{Optimal ET sequence|legend=1| 224, 544, 768, 1312 }}
[[Badness]]: 0.270


=== 11-limit ===
=== 11-limit ===
Line 223: Line 245:
Comma list: 9801/9800, 46656/46585, 65625/65536
Comma list: 9801/9800, 46656/46585, 65625/65536


Mapping: [{{val| 32 0 125 -113 60 }}, {{val| 0 1 -1 4 1 }}]
Mapping: {{mapping| 32 0 125 -113 60 | 0 1 -1 4 1 }}


Optimal tuning (CTE): ~45/44 = 1\32, ~3/2 = 701.601
Optimal tuning (CTE): ~45/44 = 1\32, ~3/2 = 701.601
{{Optimal ET sequence|legend=1| 224, 544, 768 }}
Badness: 0.0680


=== 13-limit ===
=== 13-limit ===
Line 232: Line 258:
Comma list: 729/728, 1575/1573, 4225/4224, 6656/6655
Comma list: 729/728, 1575/1573, 4225/4224, 6656/6655


Mapping: [{{val| 32 0 125 -113 60 17 }}, {{val| 0 1 -1 4 1 2 }}]
Mapping: {{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 tuning (CTE): ~45/44 = 1\32, ~3/2 = 701.593


Vals: 224, 320, 544, 768, 992, 1216, 1312, ...
{{Optimal ET sequence|legend=1| 224, 544, 768, 1312 }}
 
Badness: 0.0298
 
== Notes ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Horwell temperaments| ]] <!-- main article -->
[[Category:Horwell temperaments| ]] <!-- main article -->
[[Category:Horwell]]
[[Category:Horwell| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:38, 24 June 2025

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
Retrieved from "https://en.xen.wiki/index.php?title=Horwell_temperaments&oldid=201147"