4th-octave temperaments - Revision history

Xenllium at 21:52, 2 March 2026

2026-03-02T21:52:10Z

← Older revision		Revision as of 21:52, 2 March 2026
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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 ==

Overthink: /* Berylic */ + note on harmony

2025-12-27T10:09:00Z

Berylic: + note on harmony

← Older revision		Revision as of 10:09, 27 December 2025
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	== 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

FloraC: - quad (addressed in very low-accuracy temps)

2025-12-25T10:01:35Z

- quad (addressed in very low-accuracy temps)

← Older revision		Revision as of 10:01, 25 December 2025
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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.

	~~== Quad ==~~
	~~[[Subgroup]]: 2.3.5.7~~

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

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

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

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

	~~[[Badness]]: 0.045911~~

	== Berylic ==		== Berylic ==

FloraC at 12:20, 21 August 2025

2025-08-21T12:20:06Z

← Older revision		Revision as of 12:20, 21 August 2025
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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 [[~~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.		[[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.

VectorGraphics at 05:18, 12 June 2025

2025-06-12T05:18:59Z

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	{{Mapping\|legend=1\| 4 6 9 0 \| 0 0 0 1 }}		{{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 tuning]] ([[POTE]]): ~6/5 = 1\4, ~8/7 = 324.482

ArrowHead294 at 03:56, 14 February 2025

2025-02-14T03:56:13Z

← Older revision		Revision as of 03:56, 14 February 2025
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	{{~~Fractional~~-octave ~~navigation~~\|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 [[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 [[Dimipent family]] for a collection of such temperaments.

BudjarnLambeth at 05:24, 4 December 2024

2024-12-04T05:24:25Z

← Older revision		Revision as of 05:24, 4 December 2024
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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.		{{Fractional-octave navigation\|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 [[Dimipent family]] for a collection of such temperaments.

ArrowHead294 at 01:38, 4 December 2024

2024-12-04T01:38:32Z

← Older revision		Revision as of 01:38, 4 December 2024
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	~~{{Fractional-octave navigation\|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.

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

			{{Navbox fractional-octave}}

	{{Todo\| review }}		{{Todo\| review }}

Cmloegcmluin: /* 2.3.35.11.19 subgroup */

2024-05-25T01:51:16Z

2.3.35.11.19 subgroup

← Older revision		Revision as of 01:51, 25 May 2024
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	=== 2.3.35.11.19 subgroup ===		=== 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		Subgroup: 2.3.35.11.19

CompactStar: /* Berylic */

2024-03-05T08:18:23Z

Berylic

← Older revision		Revision as of 08:18, 5 March 2024
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	== 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 ~~basis elements of~~ 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.

	Subgroup: 2.11.37		Subgroup: 2.11.37