4th-octave temperaments: Difference between revisions
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{{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. | {{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 [[ | 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. | [[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. | There are nonetheless other less common temperaments which divide the octave in four. | ||
Temperaments discussed elsewhere are: | |||
[[ | * [[Diminished family]] | ||
* [[Undim family]] | |||
[[ | * [[Very low accuracy temperaments #Quad|Quad]] | ||
[[ | |||
== Berylic == | == 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. | 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 | Subgroup: 2.11.37 | ||