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.

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 the family of rank-3 temperaments tempering it out. There are nonetheless other less common temperaments which divide the octave in three.

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.

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:

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* [[Very low accuracy temperaments #Alteraugment|Alteraugment]]

* [[Very low accuracy temperaments #Ternary|Ternary]]

* [[No-fives subgroup temperaments #Deviaug|Deviaug]]

4th-octave temperaments:

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* [[Undim family]]

* [[Very low accuracy temperaments #Quad|Quad]]

* [[No-threes subgroup temperaments #Berylic|Berylic]]

[[Category:Fractional-octave temperaments|#]]

@@ Line 4: / Line 4: @@
 * 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 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 the family of rank-3 temperaments tempering it out. There are nonetheless other less common temperaments which divide the octave in three.
+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.
+nd-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]]
 rd-octave temperaments:
@@ Line 14: / Line 25: @@
 * [[Very low accuracy temperaments #Alteraugment|Alteraugment]]
 * [[Very low accuracy temperaments #Ternary|Ternary]]
+* [[No-fives subgroup temperaments #Deviaug|Deviaug]]
 th-octave temperaments:
@@ Line 19: / Line 31: @@
 * [[Undim family]]
 * [[Very low accuracy temperaments #Quad|Quad]]
+* [[No-threes subgroup temperaments #Berylic|Berylic]]
 [[Category:Fractional-octave temperaments|#]] <!-- 1-digit number -->