Werckismic temperaments: Difference between revisions
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* ''[[Festival]]'' (+45/44 or 50/49) → [[Jubilismic family #Festival|Jubilismic family]] | * ''[[Festival]]'' (+45/44 or 50/49) → [[Jubilismic family #Festival|Jubilismic family]] | ||
* ''[[Vulcan]]'' (+56/55 or 64/63) → [[Archytas family #Vulcan|Archytas family]] | * ''[[Vulcan]]'' (+56/55 or 64/63) → [[Archytas family #Vulcan|Archytas family]] | ||
* ''[[Euterpe]]'' (+81/80 or 99/98) → [[Didymus rank | * ''[[Euterpe]]'' (+81/80 or 99/98) → [[Didymus rank-3 family #Euterpe|Didymus rank-3 family]] | ||
* ''[[Nakika]]'' (+100/99 or 245/242) → [[Octagar family #Nakika|Octagar family]] | * ''[[Nakika]]'' (+100/99 or 245/242) → [[Octagar family #Nakika|Octagar family]] | ||
* ''[[Aphrodite]]'' (+121/120) → [[Biyatismic clan #Aphrodite|Biyatismic clan]] | * ''[[Aphrodite]]'' (+121/120) → [[Biyatismic clan #Aphrodite|Biyatismic clan]] | ||
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* ''[[Demeter]]'' (+686/675) → [[Sengic family #Demeter|Sengic family]] | * ''[[Demeter]]'' (+686/675) → [[Sengic family #Demeter|Sengic family]] | ||
* ''[[Pele]]'' (+896/891) → [[Hemifamity family #Pele|Hemifamity family]] | * ''[[Pele]]'' (+896/891) → [[Hemifamity family #Pele|Hemifamity family]] | ||
* ''[[Tiberius]]'' (+2048/2025) → [[Diaschismic rank | * ''[[Tiberius]]'' (+2048/2025) → [[Diaschismic rank-3 family #Tiberius|Diaschismic rank-3 family]] | ||
* ''[[Belobog]]'' (+3136/3125) → [[Hemimean family #Belobog|Hemimean family]] | * ''[[Belobog]]'' (+3136/3125) → [[Hemimean family #Belobog|Hemimean family]] | ||
* ''[[Turan]]'' (+10976/10935) → [[Hemimage family #Turan|Hemimage family]] | * ''[[Turan]]'' (+10976/10935) → [[Hemimage family #Turan|Hemimage family]] | ||
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* [[11-odd-limit]] | * [[11-odd-limit]] | ||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/17 12/17 0 -6/17 6/17 }}, {{monzo| -27/17 4/17 0 -2/17 19/17 }}, {{monzo| 8/17 -10/17 0 5/17 12/17 }}, {{monzo| 0 0 0 0 1 }}] | : [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/17 12/17 0 -6/17 6/17 }}, {{monzo| -27/17 4/17 0 -2/17 19/17 }}, {{monzo| 8/17 -10/17 0 5/17 12/17 }}, {{monzo| 0 0 0 0 1 }}] | ||
: [[Eigenmonzo basis| | : [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11 | ||
{{Optimal ET sequence|legend=1| 14c, 15, 29, 43, 57, 58, 72, 130, 202, 217, 289, 491 }} | {{Optimal ET sequence|legend=1| 14c, 15, 29, 43, 57, 58, 72, 130, 202, 217, 289, 491 }} | ||
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* 13-odd-limit | * 13-odd-limit | ||
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/4 0 0 -3/8 0 3/8 }}, {{monzo| 0 0 1 0 0 0 }}, {{monzo| 7/8 0 1 7/16 0 -7/16 }}, {{monzo| 5/4 0 1 1/8 0 -1/8 }}, {{monzo| 7/8 0 1 -9/16 0 9/16 }}] | : [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/4 0 0 -3/8 0 3/8 }}, {{monzo| 0 0 1 0 0 0 }}, {{monzo| 7/8 0 1 7/16 0 -7/16 }}, {{monzo| 5/4 0 1 1/8 0 -1/8 }}, {{monzo| 7/8 0 1 -9/16 0 9/16 }}] | ||
: | : unchanged-interval (eigenmonzo) basis: 2.5.13/7 | ||
{{Optimal ET sequence|legend=1| 14cf, 15, 29, 43, 58, 72, 87, 130, 217, 289 }} | {{Optimal ET sequence|legend=1| 14cf, 15, 29, 43, 58, 72, 87, 130, 217, 289 }} | ||
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[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category:Pages with mostly numerical content]] | |||
[[Category:Werckismic temperaments| ]] <!-- main article --> | [[Category:Werckismic temperaments| ]] <!-- main article --> | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||
[[Category:Varuna]] | [[Category:Varuna]] | ||
[[Category:Trimyna]] | [[Category:Trimyna]] | ||
Latest revision as of 13:04, 27 August 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.
This is a collection of rank-3 temperaments tempering out the werckisma, 441/440 = [-3 2 -1 2 -1⟩. For the rank-4 werckismic temperament, see Catalog of rank-4 temperaments #Werckismic (441/440).
Temperaments discussed elsewhere are:
- Festival (+45/44 or 50/49) → Jubilismic family
- Vulcan (+56/55 or 64/63) → Archytas family
- Euterpe (+81/80 or 99/98) → Didymus rank-3 family
- Nakika (+100/99 or 245/242) → Octagar family
- Aphrodite (+121/120) → Biyatismic clan
- Thrush (+126/125) → Starling family
- Prodigy (+225/224) → Marvel family
- Jove (+243/242) → Breed family
- Sensawer (+245/243) → Sensamagic family
- Portent (+385/384) → Gamelismic family
- Demeter (+686/675) → Sengic family
- Pele (+896/891) → Hemifamity family
- Tiberius (+2048/2025) → Diaschismic rank-3 family
- Belobog (+3136/3125) → Hemimean family
- Turan (+10976/10935) → Hemimage family
- Snape (+19683/19600) → Cataharry family
Considered below are history, varuna, and trimyna.
History
History splits the perfect fourth into six, each for 21/20~22/21. A recommendable tuning is given by 289edo.
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4000/3993
Mapping: [⟨1 2 0 0 1], ⟨0 -6 0 7 2], ⟨0 0 1 1 1]]
- mapping generators: ~2, ~21/20, ~5
- [[1 0 0 0 0⟩, [4/17 12/17 0 -6/17 6/17⟩, [-27/17 4/17 0 -2/17 19/17⟩, [8/17 -10/17 0 5/17 12/17⟩, [0 0 0 0 1⟩]
- unchanged-interval (eigenmonzo) basis: 2.9/7.11
Optimal ET sequence: 14c, 15, 29, 43, 57, 58, 72, 130, 202, 217, 289, 491
Badness: 0.604 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 676/675
Mapping: [⟨1 2 0 0 1 2], ⟨0 -6 0 7 2 -9], ⟨0 0 1 1 1 1]]
- mapping generators: ~2, ~21/20, ~5
Minimax tuning:
- 13-odd-limit
- [[1 0 0 0 0 0⟩, [5/4 0 0 -3/8 0 3/8⟩, [0 0 1 0 0 0⟩, [7/8 0 1 7/16 0 -7/16⟩, [5/4 0 1 1/8 0 -1/8⟩, [7/8 0 1 -9/16 0 9/16⟩]
- unchanged-interval (eigenmonzo) basis: 2.5.13/7
Optimal ET sequence: 14cf, 15, 29, 43, 58, 72, 87, 130, 217, 289
Badness: 0.540 × 10-3
Complexity spectrum: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7
Varuna
7-limit
Subgroup: 2.3.5.7
Comma list: 321489/320000
Mapping: [⟨2 0 0 9], ⟨0 1 0 -4], ⟨0 0 1 2]]
- mapping generators: ~567/400, ~3, ~5
Optimal ET sequence: 12, 26, 34, 46, 58, 72, 118, 130, 202, 320, 450, 522, 972bd, 1174bd
Badness: 0.501 × 10-3
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 8019/8000
Mapping: [⟨2 0 0 9 12], ⟨0 1 0 -4 -6], ⟨0 0 1 2 3]]
Optimal ET sequence: 12, 26, 44ce, 46, 58, 72, 118, 130, 190, 248, 320
Badness: 0.419 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 364/363, 441/440
Mapping: [⟨2 0 0 9 12 11], ⟨0 1 0 -4 -6 -7], ⟨0 0 1 2 3 4]]
Optimal ET sequence: 12f, 14cf, 26, 46, 58, 72, 130
Badness: 0.636 × 10-3
Trimyna
Trimyna can be described as the 27e & 29 & 31 temperament. All three equal temperaments have a fairly accurate tritone for ~7/5 (or ~10/7), and trimyna takes it as one of its generators. The name trimyna is a contraction of tritonic and myna.
7-limit
Subgroup: 2.3.5.7
Comma list: 50421/50000
Mapping: [⟨1 4 0 0], ⟨0 -5 0 1], ⟨0 0 1 1]]
- mapping generators: ~2, ~7/5, ~5
Optimal ET sequence: 27, 31, 58, 87, 118, 267d, 385d, 412d *
Badness: 0.870 × 10-3
Projection pair: 3 50000/16807 to 2.5.7
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3388/3375
Mapping: [⟨1 4 0 0 5], ⟨0 -5 0 1 -8], ⟨0 0 1 1 1]]
Optimal ET sequence: 27e, 31, 58, 87, 118, 294, 412d
Badness: 0.512 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 1001/1000
Mapping: [⟨1 4 0 0 5 -2], ⟨0 -5 0 1 -8 7], ⟨0 0 1 1 1 1]]
Optimal ET sequence: 27e, 29, 31, 56, 58, 87, 118, 145, 232
Badness: 0.814 × 10-3
Trimy
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 676/675, 847/845
Mapping: [⟨1 4 0 0 5 5], ⟨0 -10 0 2 -16 -15], ⟨0 0 1 1 1 1]]
- mapping generators: ~2, ~13/11, ~5
Optimal ET sequence: 29, 54ceeff, 58, 87, 145, 149, 207, 236d, 294
Badness: 1.478 × 10-3