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 sequence12, 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

See also