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Langwidge 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.
'''Langwidge''' is a [[rank-2 temperament]] in the 2.3.17.19 [[subgroup]] [[generator|generated]] by a [[perfect fifth]]. It was found in a search for a temperament that would defy the tradition of tertian harmony (→ [[#Notation]]). It is an [[extension]] of [[protolangwidge]].  


The name "langwidge" originates from Adam Neely's video "''Is Cb The Same Note as B?''", where he mentions that there's "nothing technically incorrect about spelling the word language as "langwidge", but word structure-wise the information is different<ref>[https://www.youtube.com/watch?v=SZftrA-aCa4&t=210s&pp=ygUYSXMgQyMgdGhlIHNhbWUgbm90ZSBhcyBC Is Cb the same note as B?] by Adam Neely</ref> because it's not spelled right. In addition, he goes on to mention about how the "order of spelling in Western music theory is sacrosanct". Therefore, langwidge temperament is constructed with the purpose of defying the Western spelling.
The name ''langwidge'' was given by [[Eliora]] in 2023, originating from Adam Neely's video "''Is Cb The Same Note as B?''", where he mentions that there's "nothing technically incorrect about spelling the word language as "langwidge", but word structure-wise the information is different because it is not spelled right. In addition, he goes on to mention about how the "order of spelling in Western music theory is sacrosanct".<ref>[https://www.youtube.com/watch?v=SZftrA-aCa4&t=210s&pp=ygUYSXMgQyMgdGhlIHNhbWUgbm90ZSBhcyBC ''Is Cb the same note as B?''] by Adam Neely</ref>


In this case, the temperament's generator is a slightly flat fifth, 9 of which yield [[38/1]], meaning that when octave-reduced, this would require spelling root-3rd-P5 triad with 19:16 as C-D#-E and not as C-Eb-E, producing this peculiar violation of standard Western music theory.  
== 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, it is considered to present a challenge to the tradition of tertian harmony, since the simplest harmonic building block, the 1-19/16-3/2 triad, is C-D#-G and not C-Eb-G.  


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]], and nine of them make [[38/1]], meaning [[19/16]] is mapped to C-D#. 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]].
This temperament is, however, neither the first nor the most successful to raise the notational issue, and there are a number of ways to address it. First, whether 19/16 must be notated as a minor third is debatable. Western harmony mainly dealt with the [[5-limit]], and only the mapping of [[5/1|5]] is fully established. Most conceptualization systems of [[just intonation]] ([[FJS]], [[HEJI]], [[Color notation]],etc.) indeed treat 19/16 as a minor third, but [[Sagittal notation|Sagittal]] is a notable exception in that it is equipped with an accidental of ratio 19683/19456 besides the more common [[513/512]], so 19/16 can be an augmented second there. Otherwise, if one wants to notate 19/16 as a minor third, one can adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C-^Eb-G. (Color notation can notate 19/16 as either an ino 3rd or a noqu 2nd.) There are also other temperaments known to raise the notational issue in much simpler chords, such as [[schismatic]] temperament which represents the 5-limit 10:12:15 triad as C-D#-G.
 
In the 17-limit, 17th harmonic is reached, coincidentally, 17 generators down, meaning [[17/16]] is mapped to C-Ebbb.


== Temperament data ==
== Temperament data ==
[[Subgroup]]: 2.3.17.19


Subgroup: 2.17.19
[[Comma list]]: 6144/6137, 19683/19456


Comma list: 2.17.19 {{monzo|-109 9 17}}
{{Mapping|legend=2| 1 0 31 -10 | 0 1 -17 9 }}


Sval mapping: [{{val|1 14 -1}}, {{val|0 -17 9}}]
: sval mapping generators: ~2, ~3


Optimal tuning (CTE): ~6137/4096 = 699.712
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 699.7519


{{Optimal ET sequence|legend=1|12, 271, 283, 295, 307, 319, 331, 343, 355, 367, 379 ,391, 403, 415, 722}}, ...
{{Optimal ET sequence|legend=1| 12, 235, 247, 259b, 271b, …, 355b, 367b }}


=== 23-limit langwidge ===
== See also ==
* [[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.
== References ==
 
<references/>
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}}


== References ==
[[Category:Langwidge| ]] <!-- main article -->
[[Category:Temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:19-limit]]
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]
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