Tolerant temperaments

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

Optimal tunings:

• WE: ~392/375 = 74.9713 ¢, ~3/2 = 703.6843 ¢

error map: ⟨-0.459 +1.271 -0.519 -0.347]

• CWE: ~392/375 = 75.0000 ¢, ~3/2 = 703.8172 ¢

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 ¢
• CWE: ~392/375 = 75.0000 ¢, ~3/2 = 703.8817 ¢

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, 896/891, 540000/539539

Mapping: [⟨16 0 -39 -82 30 110], ⟨0 1 3 5 1 -2]]

Optimal tunings:

• WE: ~392/375 = 74.9681 ¢, ~3/2 = 703.6668 ¢
• CWE: ~392/375 = 75.0000 ¢, ~3/2 = 703.9022 ¢

Optimal ET sequence: 80, 128, 208, 496bdeef

Badness (Sintel): 2.52