Horwell temperaments: Difference between revisions
→Oquatonic: normalize mappings and generators |
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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]]: [{{val|3 5 7 8}}, {{val|0 -7 -1 12}}] | ||
{{Multival|legend=1|21 3 -36 -44 -116 -92}} | {{Multival|legend=1|21 3 -36 -44 -116 -92}} | ||
POTE generator: ~5/4 = 385.964 | [[POTE generator]]: ~5/4 = 385.964 | ||
{{Val list|legend=1| 3, 84, 87, 171 }} | |||
Badness: 0.028406 | [[Badness]]: 0.028406 | ||
=== 11-limit === | === 11-limit === | ||
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POTE generator: ~5/4 = 386.020 | POTE generator: ~5/4 = 386.020 | ||
Optimal GPV sequence: {{Val list| 3, 84, 87, 171, 258, 429e }} | |||
Badness: 0.058344 | Badness: 0.058344 | ||
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POTE generator: ~5/4 = 386.022 | POTE generator: ~5/4 = 386.022 | ||
Optimal GPV sequence: {{Val list| 3, 84, 87, 171, 258, 429ef }} | |||
Badness: 0.029089 | Badness: 0.029089 | ||
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== 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 | ||
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== 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 | ||
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POTE generator: ~11/8 = 551.7746 | POTE generator: ~11/8 = 551.7746 | ||
Optimal GPV sequence: {{Val list| 87, 137, 224, 311, 535, 1381ce, 1916ce }} | |||
Badness: 0.035586 | Badness: 0.035586 | ||
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POTE generator: ~11/8 = 551.7749 | POTE generator: ~11/8 = 551.7749 | ||
Optimal GPV sequence: {{Val list| 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff }} | |||
Badness: 0.017853 | Badness: 0.017853 | ||
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POTE generator: ~121/112 = 132.1864 | POTE generator: ~121/112 = 132.1864 | ||
Optimal GPV sequence: {{Val list| 109, 118, 345de, 463de, 581dde }} | |||
Badness: 0.052693 | Badness: 0.052693 | ||
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POTE generator: ~13/12 = 132.1789 | POTE generator: ~13/12 = 132.1789 | ||
Optimal GPV sequence: {{Val list| 109, 118f, 227f }} | |||
Badness: 0.046695 | Badness: 0.046695 | ||
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== 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. | 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. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
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POTE generator: ~3/2 = 702.0186 | POTE generator: ~3/2 = 702.0186 | ||
Optimal GPV sequence: {{Val list| 84, 140, 224, 364, 588, 1400cd, 1988cd, 2576ccdd }} | |||
Badness: 0.047853 | Badness: 0.047853 | ||
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POTE generator: ~3/2 = 702.0288 | POTE generator: ~3/2 = 702.0288 | ||
Optimal GPV sequence: {{Val list| 84, 140, 224, 364, 588 }} | |||
Badness: 0.021968 | Badness: 0.021968 | ||
Revision as of 22:51, 8 January 2022
Horwell temperaments temper out the horwell comma, [-16 1 5 1⟩ = 65625/65536.
Discussed elsewhere are bisupermajor, countercata, eris, escaped, hemithirds, keen, maquiloid, narayana, orwell, paramity, pontiac, tertiaseptal, and worschmidt.
Mutt
Subgroup: 2.3.5
Comma list: [-44 -3 21⟩
Mapping: [⟨3 5 7], ⟨0 -7 -1]]
POTE generator: ~5/4 = 385.980
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 ]]
POTE generator: ~5/4 = 385.964
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]]
POTE generator: ~5/4 = 386.020
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~5/4 = 386.022
Optimal GPV sequence: Template:Val list
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 ]]
POTE generator: ~5488/3645 = 708.774
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 ]]
POTE generator: ~3125/2268 = 551.7745
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]]
POTE generator: ~11/8 = 551.7746
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~11/8 = 551.7749
Optimal GPV sequence: Template:Val list
Badness: 0.017853
Kastro
Subgroup: 2.3.5.7
Comma list: 65625/65536, 117649/116640
Mapping: [⟨1 5 1 6], ⟨0 -31 12 -29]]
POTE generator: ~3375/3136 = 132.1845
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]]
POTE generator: ~121/112 = 132.1864
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~13/12 = 132.1789
Optimal GPV sequence: Template:Val list
Badness: 0.046695
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 (4096000/4084101, saquinru-atriyo). In this temperament, major third of 5/4 is mapped into 9\28.
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 ]]
POTE generator: ~3/2 = 702.1137
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]]
POTE generator: ~3/2 = 702.0186
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 702.0288
Optimal GPV sequence: Template:Val list
Badness: 0.021968