3rd-octave temperaments: Difference between revisions

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Created page with "{{Infobox fractional-octave|3}}3edo is much less used as a scale, rather as a chord. In many diatonic-based interval region schemes, one step of 3edo is known as a major third, and the stacking of them is the augmented triad. Usage of the 5/4 major third as one step of 3edo by tempering out 128/125, and therefore using 3edo as a augmented triad produced by stacking two major thirds is one of the features of standard Western music theory, and is..."
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{{Infobox fractional-octave|3}}[[3edo]] is much less used as a scale, rather as a chord. In many [[5L 2s|diatonic-based]] [[interval region]] schemes, one step of 3edo is known as a major third, and the stacking of them is the augmented triad.
#redirect [[2nd- to 4th-octave temperaments]]
 
Usage of the [[5/4]] major third as one step of 3edo by tempering out [[128/125]], and therefore using 3edo as a augmented triad produced by stacking two major thirds is one of the features of standard Western music theory, and is supported by [[12edo]]. See [[Augmented family]] for a collection of such temperaments.
 
[[24/19]] is much closer to 1/3-octave than 5/4, and while it is not a microtemperament, a lot of equal divisions support it.
 
An interval closer to 1\3 is [[63/50]], with the associated comma being the landscape comma. See [[Landscape family]] for a collection of rank-3 temperaments tempering it out.
 
There are nonetheless other less common temperaments which divide the octave in three.
 
== Argument ==
Argument temperament tempers out the [[1372/1331]] comma in the 2.7.11 subgroup. Subgroup: 2.7.11
 
[[Comma list]]: 1372/1331
 
{{Mapping|legend=2| 3 0 2 | 0 1 1 }}
 
: sval mapping generators: ~14/11, ~7
 
[[Optimal tuning]]s:
* [[CTE]]: ~14/11 = 400.000{{c}} (1\3), ~8/7 = 238.126{{c}}
* [[CWE]]: ~14/11 = 400.000{{c}} (1\3), ~8/7 = 237.253{{c}}
 
{{Optimal ET sequence|legend=1| 6, 9, 15 }}
 
[[Badness]] (Sintel): 0.475
 
{{Navbox fractional-octave}}
 
{{Todo| review }}

Latest revision as of 12:54, 2 July 2026

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