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. | ||
[[4edo]] | 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 | |||
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. | 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. | 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 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 | ||
| Line 31: | Line 23: | ||
Comma list: 1874161/1874048 | Comma list: 1874161/1874048 | ||
{{Mapping|legend=2| 4 0 7 | 0 1 1 }} | |||
: sval mapping generators: ~44/37 | : sval mapping generators: ~44/37, ~11 | ||
Optimal tuning (CTE): ~11/8 = 551.326 | Optimal tuning (CTE): ~44/37 = 1\4, ~11/8 = 551.326 | ||
[[Support]]ing [[ET]]s: {{EDOs|24, 28, 148, 296, 320, 592, 616, 764}}, ... | [[Support]]ing [[ET]]s: {{EDOs|24, 28, 148, 296, 320, 592, 616, 764}}, ... | ||
== Darian calendar == | == Darian calendar == | ||
Darian calendar is described as 24 & 668 temperament and is named after a certain calendar layout by the same name. The generator is close to the [[36/35]] quartertone, | Darian calendar is described as 24 & 668 temperament in the 2.3.11.19 [[subgroup]] and is named after a certain calendar layout by the same name. The generator is close to the [[36/35]] quartertone, and this allows an extension to the 2.3.35.11.19 subgroup. 5 of them make [[11/8]], 8 of them make [[3/2]], and 6 of them make [[32/19]]. | ||
=== 2.3.11.19 subgroup === | === 2.3.11.19 subgroup === | ||
The temperament is simplest in this subgroup, although there is a tradeoff of breaking up the simplicity of the 36/35 quartertone. | The temperament is simplest in this subgroup, although there is a tradeoff of breaking up the simplicity of the 36/35 quartertone. | ||
Subgroup: 2.3.11.19 | [[Subgroup]]: 2.3.11.19 | ||
{{Mapping|legend=2| 4 5 13 18 | 0 8 5 -6 }} | |||
: sval mapping generators: ~6291456/5285401 | : sval mapping generators: ~6291456/5285401, ~25289/24576 | ||
Optimal tuning (CTE): 25289/24576 = 50.257 | [[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257 | ||
[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ... | [[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ... | ||
=== 2. | === 2.3.35.11.19 subgroup === | ||
668edo does not map 36/35 consistently, with direct | 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. | ||
Subgroup: 2.3.35.11.19 | |||
Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }} | |||
: sval mapping generators: ~2240/1881, ~36/35 | |||
Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288 | |||
: | [[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ... | ||
{{Navbox fractional-octave}} | |||
{{Todo| review }} | |||