4th-octave temperaments: Difference between revisions

{{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 [[Diminished family]] for a collection of such temperaments.

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

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.

668edo does not map 36/35 consistently, with its own [[direct approximation]] being 27 steps while the direct approximations of its constituent odd harmonics do not sum to that same amount: 3/2, 8/5, and 8/7 are 391, 453, and 129 steps, respectively, and 391 + 391 + 453 + 129 - 668 - 668 = 28, ≠ 27.