Protolangwidge - Revision history

FloraC: - parent category

2025-04-28T14:35:01Z

- parent category

← Older revision		Revision as of 14:35, 28 April 2025
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	* [[Langwidge]]		* [[Langwidge]]

	[[Category:~~Temperaments~~]]		[[Category:Protolangwidge\| ]] <!-- main article -->
	[[Category:Rank-2 temperaments]]		[[Category:Rank-2 temperaments]]
			[[Category:Subgroup temperaments]]

Lériendil at 23:38, 19 April 2025

2025-04-19T23:38:51Z

← Older revision		Revision as of 23:38, 19 April 2025
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	[[Category:Temperaments]]		[[Category:Temperaments]]
			[[Category:Rank-2 temperaments]]

Fredg999: See also

2024-01-14T05:33:34Z

FloraC: Review (finally!)

2024-01-12T15:31:14Z

Review (finally!)

← Older revision		Revision as of 15:31, 12 January 2024
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	{{Novelty}}		{{Novelty}}

	Protolangwidge 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~~.		'''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]]).

	~~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 {{monzo\|-109 0 0 0 0 0 9 17}} comma. This means that the generator fifth in ~~question is mapped to [[6137/4096]]. 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]]~~.		The name ''protolangwidge'' was given by [[Eliora]] in 2023.

	~~In the~~ 17~~-limit, 17th harmonic is reached, coincidentally~~, 17 ~~generators down~~, ~~meaning~~ [[17/16]] ~~is mapped to C-Ebbb.~~		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]].
	~~==Temperament data==~~
	~~Subgroup: 2~~.~~17.19~~

	~~Comma list:~~ 2.17.~~19 {{monzo\|-109 9 17}}~~		[[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.

	~~Sval mapping:~~ [~~{{val~~\|1 14 -~~1}}~~, ~~{{val\|0~~ -17 ~~9}}]~~		== 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.

	~~Optimal tuning (CTE)~~: ~~~6137/4096 = 699~~.~~712~~		== Temperament data ==
			[[Subgroup]]: 2.17.19

	{{Optimal ET sequence\|legend=1\|12, ~~271~~, ~~283~~, ~~295~~, ~~307~~, 319, 331, 343, ~~355~~, ~~367~~, ~~379~~ ,~~391, 403, 415, 722~~}}~~, ...~~		[[Comma list]]: 2.17.19 {{monzo\| -109 9 17 }}
	===23~~-limit protolangwidge~~===
	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.		{{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: [{{~~val~~\|1 14 -1 64~~}}, {{val~~\|0 -17 9 -102}}]		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~~\|legend=1~~\|12, 343, 355, 367, 379, 722, 1077, 1089, 1432}}		Optimal ET sequence: {{Optimal ET sequence\| 12, 343, 355, 367, 379, 722, 1077, 1089, 1432 }}

			[[Category:Temperaments]]

Eliora: per FloraC's and Kite's recommendation: langwidge - generator mapped to 3/2, this is something else

2023-10-29T13:10:15Z

per FloraC's and Kite's recommendation: langwidge - generator mapped to 3/2, this is something else

New page

{{Novelty}}

Protolangwidge 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.

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 {{monzo|-109 0 0 0 0 0 9 17}} comma. This means that the generator fifth in question is mapped to [[6137/4096]]. 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 {{monzo|-109 9 17}}

Sval mapping: [{{val|1 14 -1}}, {{val|0 -17 9}}]

Optimal tuning (CTE): ~6137/4096 = 699.712

{{Optimal ET sequence|legend=1|12, 271, 283, 295, 307, 319, 331, 343, 355, 367, 379 ,391, 403, 415, 722}}, ...
===23-limit protolangwidge===
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 {{monzo|69 3 -17 -2}}

Sval mapping: [{{val|1 14 -1 64}}, {{val|0 -17 9 -102}}]

Optimal tuning (CTE): ~6137/4096 = 699.722

{{Optimal ET sequence|legend=1|12, 343, 355, 367, 379, 722, 1077, 1089, 1432}}

← Older revision		Revision as of 05:33, 14 January 2024
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	Optimal ET sequence: {{Optimal ET sequence\| 12, 343, 355, 367, 379, 722, 1077, 1089, 1432 }}		Optimal ET sequence: {{Optimal ET sequence\| 12, 343, 355, 367, 379, 722, 1077, 1089, 1432 }}

			== See also ==
			* [[Langwidge]]

	[[Category:Temperaments]]		[[Category:Temperaments]]