3rd-octave temperaments: Difference between revisions
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..." Tags: Mobile edit Mobile web edit |
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== Argument == | == Argument == | ||
Argument temperament tempers out | Argument temperament tempers out [[1372/1331]] in the 2.7.11 subgroup. | ||
[[Subgroup]]: 2.7.11 | |||
[[Comma list]]: 1372/1331 | [[Comma list]]: 1372/1331 | ||
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[[Badness]] (Sintel): 0.475 | [[Badness]] (Sintel): 0.475 | ||
== Deviaug == | |||
Deviaug temperament tempers out [[6912/6859]] in the 2.3.19 subgroup. | |||
[[Subgroup]]: 2.3.19 | |||
[[Comma list]]: 6912/6859 | |||
{{Mapping|legend=2| 3 0 8 | 0 1 1 }} | |||
: sval mapping generators: ~24/19, ~3 | |||
[[Optimal tuning]]s: | |||
* [[CTE]]: ~24/19 = 400.000{{c}} (1\3), ~3/2 = 701.412{{c}} | |||
* [[CWE]]: ~24/19 = 400.000{{c}} (1\3), ~3/2 = 701.881{{c}} | |||
{{Optimal ET sequence|legend=1| 9, 12, 75, 87, 99, 111, 123h }} | |||
[[Badness]] (Sintel): 0.237 | |||
{{Navbox fractional-octave}} | {{Navbox fractional-octave}} | ||
{{Todo| review }} | {{Todo| review }} | ||
Revision as of 05:58, 21 February 2026
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 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 1372/1331 in the 2.7.11 subgroup.
Subgroup: 2.7.11
Comma list: 1372/1331
Subgroup-val mapping: [⟨3 0 2], ⟨0 1 1]]
- sval mapping generators: ~14/11, ~7
Optimal ET sequence: 6, 9, 15
Badness (Sintel): 0.475
Deviaug
Deviaug temperament tempers out 6912/6859 in the 2.3.19 subgroup.
Subgroup: 2.3.19
Comma list: 6912/6859
Subgroup-val mapping: [⟨3 0 8], ⟨0 1 1]]
- sval mapping generators: ~24/19, ~3
Optimal ET sequence: 9, 12, 75, 87, 99, 111, 123h
Badness (Sintel): 0.237