Nuwell temperaments

This is a collection of nuwell temperaments, which temper out the nuwell comma (monzo: [1 5 1 -4⟩, ratio: 2430/2401). Its fundamental equivalence entails that 7/6 is sharpened so that it stacks four times to reach 15/8. Most of these temperaments extend to the 11-limit by equating 11/8~15/11 with (7/6)², tempering out 99/98 and 121/120.

Temperaments discussed elsewhere are:

• Quasisuper (+64/63) → Archytas clan
• Oncle (+1029/1024) → Gamelismic clan
• Alphatrimot (+5120/5103) → Alphatricot family
• Squares (+81/80) → Meantone family
• Progression (+36/35) → Mint temperaments
• Orwell (+225/224) → Semicomma family
• Hamity (+4000/3969) → Amity family
• Nusecond (+126/125) → Starling temperaments
• Chromo (+875/864) → Keemic temperaments

Considered below is maja. For the rank-3 temperament, see Miscellaneous 7-limit temperaments #Nuwell.

Maja

For the 5-limit version, see Schismic–Mercator equivalence continuum #Maja (5-limit).

Maja tempers out 3125/3087, the gariboh comma, and 4375/4374, the ragisma, in addition to the nuwell comma, and may be described as the 8d & 53 temperament. It was named by Petr Pařízek in 2011 for ~13/10 is a generator for the 13-limit version and that M and J are the 13th letter and the 10th letter, respectively. It is unrelated to either the major quality or magic temperament. Note that in the data below, the generator is its octave complement, such that seventeen of them octave reduced reach the perfect fifth. Its ploidacot is iota-17-cot.

Subgroup: 2.3.5.7

Comma list: 2430/2401, 3125/3087

Mapping: [⟨1 -9 -12 -14], ⟨0 17 23 27]]

mapping generators: ~2, ~54/35

Optimal tunings:

• WE: ~2 = 1200.0230 ¢, ~54/35 = 747.0972 ¢

error map: ⟨-0.077 -0.612 -2.157 +3.873]

• CWE: ~2 = 1200.0000 ¢, ~54/35 = 747.1084 ¢

error map: ⟨0.000 -1.112 -2.820 +3.102]

Optimal ET sequence: 8d, …, 45, 53

Badness (Sintel): 2.26

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 3125/3087

Mapping: [⟨1 -9 -12 -14 -9], ⟨0 17 23 27 20]]

Optimal tunings:

• WE: ~2 = 1200.7417 ¢, ~54/35 = 747.5622 ¢
• CWE: ~2 = 1200.0000 ¢, ~54/35 = 747.2024 ¢

Optimal ET sequence: 8d, …, 45e, 53

Badness (Sintel): 1.65

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 169/168, 275/273

Mapping: [⟨1 -9 -12 -14 -9 -10], ⟨0 17 23 27 20 22]]

Optimal tunings:

• WE: ~2 = 1200.8497 ¢, ~20/13 = 747.6240 ¢
• CWE: ~2 = 1200.0000 ¢, ~20/13 = 747.2212 ¢

Optimal ET sequence: 8d, …, 45ef, 53

Badness (Sintel): 1.14