4th-octave temperaments: Difference between revisions

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[[4edo]] is much less used as a scale, rather as a chord. In many [[5L 2s|diatonic-based]] [[interval region]] schemes, one step of 4edo is known as a minor third, and the stacking of them is the diminished seventh chord.

{{Infobox fractional-octave|4}}[[4edo]] is much less used as a scale, rather as a chord. In many [[5L 2s|diatonic-based]] [[interval region]] schemes, one step of 4edo is 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.

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.

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

~~== Quad ==~~

Temperaments discussed elsewhere are:

[[~~Subgroup~~]]~~: 2.3.5.7~~

* [[Diminished family]]

* [[Undim family]]

[[~~Comma list~~]]~~: 9/8, 25/24~~

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

~~{{Mapping|legend=1| 4 6 9 0 | 0 0 0 1 }}~~

~~{{Multival|legend=1| 0 0 4 0 6 9 }}~~

[[~~Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~8/7 = 324.482~~

~~{{Optimal ET sequence~~|~~legend=1| 4 }}~~

~~[[Badness~~]]~~: 0.045911~~

== Berylic ==

Berylic 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. Beryllic is a rare example of a temperament which has an astronomically low [[badness]] by all metrics (generally several thousands of times lower than most temperaments), being a very high-accuracy [[microtemperament]] with low-to-average [[complexity]] for the harmonics in its [[subgroup]]. This also makes it simultaneously supported by EDO systems as low as [[16edo]] and up into the tens of thousands. The tradeoff with this temperament, not captured within the metric of badness, is that it is defined within the obscure subgroup 2.11.37.

If one wishes to explore harmony in this temperament, a great way is to use the 8-note [[4L 4s]] [[mos]], and use the [[32:37:44]] triad and its inversion [[296:352:407|1/(44:37:32)]] as the root chords. However, the consonance of the 37th harmonic is questionable.

Subgroup: 2.11.37

@@ Line 1: / Line 1: @@
-[[4edo]] is much less used as a scale, rather as a chord. In many [[5L 2s|diatonic-based]] [[interval region]] schemes, one step of 4edo is known as a minor third, and the stacking of them is the diminished seventh chord.
+{{Infobox fractional-octave|4}}[[4edo]] is much less used as a scale, rather as a chord. In many [[5L 2s|diatonic-based]] [[interval region]] schemes, one step of 4edo is 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.
+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.
@@ Line 9: / Line 9: @@
 There are nonetheless other less common temperaments which divide the octave in four.
-== Quad ==
+Temperaments discussed elsewhere are:
-[[Subgroup]]: 2.3.5.7
+* [[Diminished family]]
+* [[Undim family]]
-[[Comma list]]: 9/8, 25/24
+* [[Very low accuracy temperaments #Quad|Quad]]
-{{Mapping|legend=1| 4 6 9 0 | 0 0 0 1 }}
-{{Multival|legend=1| 0 0 4 0 6 9 }}
-[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~8/7 = 324.482
-{{Optimal ET sequence|legend=1| 4 }}
-[[Badness]]: 0.045911
 == Berylic ==
 Berylic 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. Beryllic is a rare example of a temperament which has an astronomically low [[badness]] by all metrics (generally several thousands of times lower than most temperaments), being a very high-accuracy [[microtemperament]] with low-to-average [[complexity]] for the harmonics in its [[subgroup]]. This also makes it simultaneously supported by EDO systems as low as [[16edo]] and up into the tens of thousands. The tradeoff with this temperament, not captured within the metric of badness, is that it is defined within the obscure subgroup 2.11.37.
+If one wishes to explore harmony in this temperament, a great way is to use the 8-note [[4L 4s]] [[mos]], and use the [[32:37:44]] triad and its inversion [[296:352:407|1/(44:37:32)]] as the root chords. However, the consonance of the 37th harmonic is questionable.
 Subgroup: 2.11.37