Protolangwidge: Difference between revisions
per FloraC's and Kite's recommendation: langwidge - generator mapped to 3/2, this is something else |
Review (finally!) |
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{{Novelty}} | {{Novelty}} | ||
Protolangwidge is a rank-2 temperament whose generator is an interval close to the perfect fifth | '''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 log<sub>2</sub>(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 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. | |||
== Temperament data == | |||
[[Subgroup]]: 2.17.19 | |||
{{Optimal ET sequence|legend=1|12, | [[Comma list]]: 2.17.19 {{monzo| -109 9 17 }} | ||
===23 | |||
Since 355edo and 722edo are good at 2.17.19.23 subgroup, it | {{Mapping|legend=2| 1 14 -1 | 0 -17 9 }} | ||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6137/4096 = 699.712 | |||
{{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. | |||
Subgroup: 2.17.19.23 | Subgroup: 2.17.19.23 | ||
Comma list: 24137569/24117248, 2.17.19.23 {{monzo|69 3 -17 -2}} | Comma list: 24137569/24117248, 2.17.19.23 {{monzo| 69 3 -17 -2 }} | ||
Sval mapping: | Sval mapping: {{mapping| 1 14 -1 64 | 0 -17 9 -102 }} | ||
Optimal tuning (CTE): ~6137/4096 = 699.722 | Optimal tuning (CTE): ~6137/4096 = 699.722 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 12, 343, 355, 367, 379, 722, 1077, 1089, 1432 }} | ||
[[Category:Temperaments]] | |||
Revision as of 15:31, 12 January 2024
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This page presents a novelty topic.
It may contain ideas which are less likely to find practical applications in music, or numbers or structures that are arbitrary or exceedingly small, large, or complex. Novelty topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks. |
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⟩
Subgroup-val 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