4th-octave temperaments: Difference between revisions

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[[4edo]] is much less used as a scale, rather as a chord.

[[4edo]] is much less used as a scale, rather as a chord. In the Western theory, one step of 4edo is usually known as a minor third and the stacking of them is the diminished seventh chord.

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 [[Dimipent 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's 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 a collection of rank-3 temperaments tempering it out.

There are nonetheless other less common temperaments which divide the octave in four.

== Hunt 19-cycle or Saquadnu (rank-1) ==

Associates [[19/16]] to one step of 4edo, and hence [[19/1|19th harmonic]] by octavation to 17 steps of 4edo.

Subgroup: 2.19

Comma list: 131072/130321

: mapping generator: ~19/16 = 1\4

Pure [[TE tuning]]: 299.927¢ = 1/4.00097 of the octave

[[Support]]ing [[ET]]s: 4N, N = 1 to 60, largest: [[240edo|240]]

== Quadthisolu ==

Quadthisolu temperament tempers out the [[1874161/1874048]] comma in the 2.11.37 subgroup, representing [[44/37]] ~~as quarter-octave~~.

Quadthisolu temperament tempers out the [[1874161/1874048]] comma in the 2.11.37 subgroup, representing the fact that [[44/37]] is a [[wikipedia:continued fraction|continued fraction]] convergent to the fourth root of 2.

Subgroup: 2.11.37

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 {{Fractional-octave navigation|4}}
-[[4edo]] is much less used as a scale, rather as a chord.
+[[4edo]] is much less used as a scale, rather as a chord. In the Western theory, one step of 4edo is usually known as a minor third and the stacking of them is the diminished seventh chord.
 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 [[Dimipent 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's 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 a collection of rank-3 temperaments tempering it out.
 There are nonetheless other less common temperaments which divide the octave in four.
+== Hunt 19-cycle or Saquadnu (rank-1) ==
+Associates [[19/16]] to one step of 4edo, and hence [[19/1|19th harmonic]] by octavation to 17 steps of 4edo.
+Subgroup: 2.19
+Comma list: 131072/130321
+{{Mapping|legend=2|4 17}}
+: mapping generator: ~19/16 = 1\4
+Pure [[TE tuning]]: 299.927¢ = 1/4.00097 of the octave
+[[Support]]ing [[ET]]s: 4N, N = 1 to 60, largest: [[240edo|240]]
 == Quadthisolu ==
-Quadthisolu temperament tempers out the [[1874161/1874048]] comma in the 2.11.37 subgroup, representing [[44/37]] as quarter-octave.
+Quadthisolu temperament tempers out the [[1874161/1874048]] comma in the 2.11.37 subgroup, representing the fact that [[44/37]] is a [[wikipedia:continued fraction|continued fraction]] convergent to the fourth root of 2.
 Subgroup: 2.11.37