Tolerant temperaments: Difference between revisions
Birth with hilarity |
+ minimal generators |
||
| Line 23: | Line 23: | ||
[[Comma list]]: 179200/177147, 390625/388962 | [[Comma list]]: 179200/177147, 390625/388962 | ||
{{Mapping|legend= | {{Mapping|legend=1| 16 0 -39 -82 | 0 1 3 5 }} | ||
: mapping generators: ~392/375, ~3 | : mapping generators: ~392/375, ~3 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~392/375 = 74.9713{{c}}, ~3/2 = 703.6843{{c}} | * [[WE]]: ~392/375 = 74.9713{{c}}, ~3/2 = 703.6843{{c}} (~81/80 = 28.9423{{c}}) | ||
: [[error map]]: {{val| -0.459 +1.271 -0.519 -0.347 }} | : [[error map]]: {{val| -0.459 +1.271 -0.519 -0.347 }} | ||
* [[CWE]]: ~392/375 = 75.0000{{c}}, ~3/2 = 703.8172{{c}} | * [[CWE]]: ~392/375 = 75.0000{{c}}, ~3/2 = 703.8172{{c}} (~81/80 = 28.8172{{c}}) | ||
: error map: {{val| 0.000 +1.862 +0.138 +0.260 }} | : error map: {{val| 0.000 +1.862 +0.138 +0.260 }} | ||
| Line 44: | Line 44: | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~392/375 = 74.9639{{c}}, ~3/2 = 703.6872{{c}} | * WE: ~392/375 = 74.9639{{c}}, ~3/2 = 703.6872{{c}} (~56/55 = 29.0120{{c}}) | ||
* CWE: ~392/375 = 75.0000{{c}}, ~3/2 = 703.8817{{c}} | * CWE: ~392/375 = 75.0000{{c}}, ~3/2 = 703.8817{{c}} (~56/55 = 28.8817{{c}}) | ||
{{Optimal ET sequence|legend=0| 80, 128, 208, 288be, 496bdee }} | {{Optimal ET sequence|legend=0| 80, 128, 208, 288be, 496bdee }} | ||
| Line 54: | Line 54: | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 325/324, 352/351, | Comma list: 325/324, 352/351, 364/363, 125000/124509 | ||
{{Mapping|legend=0| 16 0 -39 -82 30 110 | 0 1 3 5 1 -2 }} | {{Mapping|legend=0| 16 0 -39 -82 30 110 | 0 1 3 5 1 -2 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~392/375 = 74.9681{{c}}, ~3/2 = 703.6668{{c}} | * WE: ~392/375 = 74.9681{{c}}, ~3/2 = 703.6668{{c}} (~66/65 = 28.9543{{c}}) | ||
* CWE: ~392/375 = 75.0000{{c}}, ~3/2 = 703.9022{{c}} | * CWE: ~392/375 = 75.0000{{c}}, ~3/2 = 703.9022{{c}} (~66/65 = 28.9022{{c}}) | ||
{{Optimal ET sequence|legend=0| 80, 128, 208, 496bdeef }} | {{Optimal ET sequence|legend=0| 80, 128, 208, 496bdeef }} | ||
Revision as of 13:53, 13 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 tolerant temperaments, which temper out the tolerant comma (monzo: [10 -11 2 1⟩, ratio: 179200/177147). As the canonical extension of rank-3 tolerant to the 11-limit tempers out the commas 896/891 and 2200/2187, this provides a clean way to extend the various tolerant temperaments to the 11-limit as well.
Temperaments discussed elsewhere are:
- Sharptone (+21/20) → Meantone family
- Polypyth (+6144/6125) → Porwell temperaments
- Srutal (+2048/2025) → Diaschismic family
- Immunized (+64/63) → Immunity family
- Rodan (+245/243) → Gamelismic clan
- Bunya (+225/224) → Tetracot family
- Pentaorwell (+1728/1715) → Orwellismic temperaments
- Accord (+126/125) → Amity family
- Metakleismic (+15625/15552) → Kleismic family
- Pluto (+4000/3969) → Octagar temperaments
Considered below is hilarity.
Hilarity
Named by Tristan Bay in 2025, hilarity tempers out the dimcomp comma in addition to the tolerant comma, and may be described as the 80 & 128 temperament. It tempers 40/27 to 9\16.
Subgroup: 2.3.5.7
Comma list: 179200/177147, 390625/388962
Mapping: [⟨16 0 -39 -82], ⟨0 1 3 5]]
- mapping generators: ~392/375, ~3
- WE: ~392/375 = 74.9713 ¢, ~3/2 = 703.6843 ¢ (~81/80 = 28.9423 ¢)
- error map: ⟨-0.459 +1.271 -0.519 -0.347]
- CWE: ~392/375 = 75.0000 ¢, ~3/2 = 703.8172 ¢ (~81/80 = 28.8172 ¢)
- error map: ⟨0.000 +1.862 +0.138 +0.260]
Optimal ET sequence: 80, 128, 208, 496bd
Badness (Sintel): 7.65
11-limit
Subgroup: 2.3.5.7.11
Comma list: 896/891, 2200/2187, 125000/124509
Mapping: [⟨16 0 -39 -82 30], ⟨0 1 3 5 1]]
Optimal tunings:
- WE: ~392/375 = 74.9639 ¢, ~3/2 = 703.6872 ¢ (~56/55 = 29.0120 ¢)
- CWE: ~392/375 = 75.0000 ¢, ~3/2 = 703.8817 ¢ (~56/55 = 28.8817 ¢)
Optimal ET sequence: 80, 128, 208, 288be, 496bdee
Badness (Sintel): 2.93
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 352/351, 364/363, 125000/124509
Mapping: [⟨16 0 -39 -82 30 110], ⟨0 1 3 5 1 -2]]
Optimal tunings:
- WE: ~392/375 = 74.9681 ¢, ~3/2 = 703.6668 ¢ (~66/65 = 28.9543 ¢)
- CWE: ~392/375 = 75.0000 ¢, ~3/2 = 703.9022 ¢ (~66/65 = 28.9022 ¢)
Optimal ET sequence: 80, 128, 208, 496bdeef
Badness (Sintel): 2.52