Protolangwidge

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Protolangwidge is a rank-2 temperament whose generator is an interval close to the perfect fifth, and it is constructed with purpose of exploiting a loophole involving enharmonicity in Western music theory.

Since 355edo and 722edo are good at supporting this kind of mapping, and they're also good at approximating 17/16, this makes 355 & 722 2.7.19 subgroup the most natural and simplest way to tune this temperament, producing a rank-2 temperament associated with the [-109 0 0 0 0 0 9 17⟩ comma. This means that the generator fifth in question is mapped to 6137/4096. For the purest 19th harmonic, 722edo is the best due to it being a convergent to log2(19/16). The generator fifth is flat of pure 3/2 by 6144/6137.

In the 17-limit, 17th harmonic is reached, coincidentally, 17 generators down, meaning 17/16 is mapped to C-Ebbb.

Temperament data

Subgroup: 2.17.19

Comma list: 2.17.19 [-109 9 17⟩

Sval mapping: [⟨1 14 -1], ⟨0 -17 9]]

Optimal tuning (CTE): ~6137/4096 = 699.712

Optimal ET sequence: 12, 271, 283, 295, 307, 319, 331, 343, 355, 367, 379 ,391, 403, 415, 722, ...

23-limit protolangwidge

Since 355edo and 722edo are good at 2.17.19.23 subgroup, it's possible to extend this temperament into the 23-limit, although it is quite complex.

Subgroup: 2.17.19.23

Comma list: 24137569/24117248, 2.17.19.23 [69 3 -17 -2⟩

Sval mapping: [⟨1 14 -1 64], ⟨0 -17 9 -102]]

Optimal tuning (CTE): ~6137/4096 = 699.722

Optimal ET sequence: 12, 343, 355, 367, 379, 722, 1077, 1089, 1432