Protolangwidge
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Protolangwidge is a rank-2 temperament in the 2.17.19 subgroup whose generator is an interval close to the perfect fifth. It was found in a search for a temperament that would defy the tradition of tertian harmony (→ #Notation).
The name protolangwidge was given by Eliora in 2023.
This temperament tempers out [-109 0 0 0 0 0 9 17⟩, and can be described as the 343 & 355 temperament in the 2.17.19 subgroup. The generator fifth in question represents a ratio of 6137/4096, flat of pure 3/2 by 6144/6137.
343edo offers a tuning with a near-pure 17th harmonic, whereas 722edo is the best tuning for the purest 19th harmonic due to it being a convergent to log2(19/16). Any tuning between them can be considered good compromises.
Notation
Since the temperament is generated by the fifth, chain-of-fifths notation can be used. Note that -17 generator steps octave-reduced yield 17/16, so that 17/16 is C–Ebbb. +9 generator steps octave-reduced yield 19/16, so that 19/16 is C-D#. As such, the simplest harmonic building block, the 1-17/16-19/16 triad, is C-Ebbb-D#. If one wants to notate the 17/16 as an augmented unison or minor second, or the 19/16 as a minor third, they can achieve it by adopting an additional module of accidentals such as arrows to represent the comma step.
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): ~2 = 1\1, ~6137/4096 = 699.712
Optimal ET sequence: 12, 199g, 211g, 223, 235, …, 319, 331, 343, 698, 1739, 2437, 3135
2.17.19.23 subgroup
Since 355edo and 722edo are good at 2.17.19.23 subgroup, it is 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