2nd- to 4th-octave temperaments: Difference between revisions
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* One step of 4edo is known as a minor third, and the stacking of them is the diminished seventh chord. | * One step of 4edo is known as a minor third, and the stacking of them is the diminished seventh chord. | ||
Usage of the [[5/4]] major third as one step of 3edo by tempering out [[128/125]], and therefore using 3edo as | Usage of [[7/5]] or [[10/7]] as one step of 2edo by tempering out [[50/49]], and therefore using 2edo as a tritone is one of the features of standard Western music theory, and is supported by [[12edo]]. See [[Jubilismic clan]] or [[Jubilismic family]] for a collection of such temperaments. Other interpretations as one step of 2edo include [[Diaschismic family|diaschismic]] (tempering out [[2048/2025]]), [[semitonismic]] (tempering out [[289/288]]), and [[kalismic]] (tempering out [[9801/9800]]). | ||
Usage of the [[5/4]] major third as one step of 3edo by tempering out [[128/125]], and therefore using 3edo as an 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 the family of rank-3 temperaments tempering it out. There are nonetheless other less common temperaments which divide the octave in three. | |||
Usage of the [[6/5]] minor third as one step of 4edo by tempering out [[648/625]], and therefore using 4edo as a diminished seventh chord produced by stacking three minor thirds is one of the features of standard Western music theory, and is supported by [[12edo]]. See [[Diminished family]] for a collection of such temperaments. [[19/16]], the 19th harmonic octave-reduced, is much closer to quarter-octave than 6/5, and while it is not a microtemperament, a lot of equal divisions support it. An interval closer to 1\4 is [[25/21]], with the associated comma being the dimcomp comma. See [[Dimcomp family]] for the family of rank-3 temperaments tempering it out. There are nonetheless other less common temperaments which divide the octave in four. | Usage of the [[6/5]] minor third as one step of 4edo by tempering out [[648/625]], and therefore using 4edo as a diminished seventh chord produced by stacking three minor thirds is one of the features of standard Western music theory, and is supported by [[12edo]]. See [[Diminished family]] for a collection of such temperaments. [[19/16]], the 19th harmonic octave-reduced, is much closer to quarter-octave than 6/5, and while it is not a microtemperament, a lot of equal divisions support it. An interval closer to 1\4 is [[25/21]], with the associated comma being the dimcomp comma. See [[Dimcomp family]] for the family of rank-3 temperaments tempering it out. There are nonetheless other less common temperaments which divide the octave in four. | ||
2nd-octave temperaments: | |||
* [[Diaschismic family|Diaschismic temperaments]] | |||
* [[Vishnuzmic family|Vishnuzmic temperaments]] | |||
* [[Jubilismic clan|Jubilismic temperaments]] | |||
* [[Varunismic temperaments]] | |||
* [[Lokismic temperaments]] | |||
* [[Nimona|Nimona temperament]] | |||
* [[Very low accuracy temperaments #Antitonic|Antitonic]] | |||
3rd-octave temperaments: | 3rd-octave temperaments: | ||
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* [[Very low accuracy temperaments #Alteraugment|Alteraugment]] | * [[Very low accuracy temperaments #Alteraugment|Alteraugment]] | ||
* [[Very low accuracy temperaments #Ternary|Ternary]] | * [[Very low accuracy temperaments #Ternary|Ternary]] | ||
* [[No-fives subgroup temperaments #Deviaug|Deviaug]] | |||
4th-octave temperaments: | 4th-octave temperaments: | ||
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* [[Undim family]] | * [[Undim family]] | ||
* [[Very low accuracy temperaments #Quad|Quad]] | * [[Very low accuracy temperaments #Quad|Quad]] | ||
* [[No-threes subgroup temperaments #Berylic|Berylic]] | |||
[[Category:Fractional-octave temperaments|#]] <!-- 1-digit number --> | [[Category:Fractional-octave temperaments|#]] <!-- 1-digit number --> | ||
Latest revision as of 02:13, 4 July 2026
2-, 3- and 4edo are much less used as scales, rather as chords. In many diatonic-based interval region schemes:
- One step of 2edo is known as a tritone.
- One step of 3edo is known as a major third, and the stacking of them is the augmented triad.
- One step of 4edo is known as a minor third, and the stacking of them is the diminished seventh chord.
Usage of 7/5 or 10/7 as one step of 2edo by tempering out 50/49, and therefore using 2edo as a tritone is one of the features of standard Western music theory, and is supported by 12edo. See Jubilismic clan or Jubilismic family for a collection of such temperaments. Other interpretations as one step of 2edo include diaschismic (tempering out 2048/2025), semitonismic (tempering out 289/288), and kalismic (tempering out 9801/9800).
Usage of the 5/4 major third as one step of 3edo by tempering out 128/125, and therefore using 3edo as an 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 the family of rank-3 temperaments tempering it out. There are nonetheless other less common temperaments which divide the octave in three.
Usage of the 6/5 minor third as one step of 4edo by tempering out 648/625, and therefore using 4edo as a diminished seventh chord produced by stacking three minor thirds is one of the features of standard Western music theory, and is supported by 12edo. See Diminished family for a collection of such temperaments. 19/16, the 19th harmonic octave-reduced, is much closer to quarter-octave than 6/5, and while it is not a microtemperament, a lot of equal divisions support it. An interval closer to 1\4 is 25/21, with the associated comma being the dimcomp comma. See Dimcomp family for the family of rank-3 temperaments tempering it out. There are nonetheless other less common temperaments which divide the octave in four.
2nd-octave temperaments:
- Diaschismic temperaments
- Vishnuzmic temperaments
- Jubilismic temperaments
- Varunismic temperaments
- Lokismic temperaments
- Nimona temperament
- Antitonic
3rd-octave temperaments:
4th-octave temperaments: