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]] 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.

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 is not a microtemperament, a lot of equal divisions support it.

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

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

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=== 2.3.35.11.19 subgroup ===

668edo does not map 36/35 consistently, with direct ~~mapping~~ being 27 steps and ~~consistent mapping being~~ 28 ~~steps~~.

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

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[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...

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-{{Fractional-octave navigation|4}}
+{{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]] 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.
-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 is not a microtemperament, a lot of equal divisions support it.
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 {{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
@@ Line 58: / Line 54: @@
 === 2.3.35.11.19 subgroup ===
-edo does not map 36/35 consistently, with direct mapping being 27 steps and consistent mapping being 28 steps.
+edo 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
@@ Line 69: / Line 65: @@
 [[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...
+{{Navbox fractional-octave}}
 {{Todo| review }}