Langwidge

Langwidge 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.

The name "langwidge" originates from Adam Neely's video "Is Cb The Same Note as B?", where he mentions that there's "nothing technically incorrect about spelling the word language as "langwidge", but word structure-wise the information is different because it's not spelled right. In addition, he goes on to mention about how the "order of spelling in Western music theory is sacrosanct". Therefore, langwidge temperament is constructed with the purpose of defying the Western spelling.

In this case, the temperament's generator is a slightly flat fifth, 9 of which yield 38/1, meaning that when octave-reduced, this would require spelling root-3rd-P5 triad with 19:16 as C-D#-E and not as C-Eb-E, producing this peculiar violation of standard 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, and nine of them make 38/1, meaning 19/16 is mapped to C-D#. 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 langwidge

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