Protolangwidge: Difference between revisions

'''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 {{monzo| -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 [[17/1|17th harmonic]], whereas [[722edo]] is the best tuning for the purest [[19/1|19th harmonic]] due to it being a convergent to log₂(19/16). Any tuning between them can be considered good compromises.  

Since the temperament is generated by the fifth, [[chain-of-fifths notation]] can be used. Note that -17 generator steps [[octave reduction|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.

[[Subgroup]]: 2.17.19

 

 

 

{{Optimal ET sequence|legend=1| 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.

Comma list: 24137569/24117248, 2.17.19.23 {{monzo| 69 3 -17 -2 }}

Sval mapping: {{mapping| 1 14 -1 64 | 0 -17 9 -102 }}

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