Meantone - Revision history

Vector at 01:54, 2 June 2026

2026-06-02T01:54:33Z

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	[[File:Meantone.png\|thumb\|Meantone equates four 3/2s to 5/1 and generates pentic (2L 3s) and mosdiatonic (5L 2s) scales.]]		[[File:Meantone.png\|thumb\|Meantone equates four 3/2s to 5/1 and generates pentic (2L 3s) and mosdiatonic (5L 2s) scales.]]
	'''Meantone''', or rarely '''Syntonic''' or '''Didymus,''' is a widespread historical [[temperament]] that forms the basis of Western music theory, where the [[Perfect fifth\|fifths]] are flattened to about 696[[Cent\|c]] to produce a [[diatonic major third]] tuned to roughly [[5/4]], enabling the use of 5-limit harmony in the diatonic scale. When all the fifths are tuned the same, Meantone is a [[regular temperament]], where the period is the [[octave]], the generator is 3/2, and four generators stack to reach the 5th harmonic, meaning that the '''syntonic comma''', which is the difference between Pythagorean intervals and nearby 5-limit intervals and has a ratio of 81/80, is tempered out.		'''Meantone''', or rarely '''Syntonic''' or '''Didymus,''' is a widespread historical [[temperament]] that forms the basis of Western music theory, where the [[Perfect fifth\|fifths]] are flattened to about 696[[Cent\|c]] to produce a [[diatonic major third]] tuned to roughly [[5/4]], enabling the use of 5-limit harmony in the diatonic scale. When all the fifths are tuned the same, Meantone is a [[regular temperament]], where the period is the [[octave]], the generator is 3/2, and four generators stack to reach the 5th harmonic, meaning that the '''syntonic comma''', which is the difference between Pythagorean intervals and nearby 5-limit intervals and has a ratio of 81/80, is tempered out.
			[[File:Meantone7.png\|thumb\|Meantone diatonic scale]]
	As a monocot temperament, Meantone can be notated with standard [[diatonic notation]], and in fact diatonic notation works the best for Meantone as the 5-limit 4:5:6 harmonic triad becomes simply C-E-G on C, and the chromatic semitone is usually smaller than the diatonic semitone. Meantone is a 7-form temperament, and is tuned well around the golden tuning of diatonic. Unsurprisingly, 7edo supports Meantone, and so does 12edo (which is the simplest ET to do so without exotempering the 5-limit), and so the best tunings of Meantone lie in between those two extremes. [7 & 12] is thus the ''edo join'' for meantone.		As a monocot temperament, Meantone can be notated with standard [[diatonic notation]], and in fact diatonic notation works the best for Meantone as the 5-limit 4:5:6 harmonic triad becomes simply C-E-G on C, and the chromatic semitone is usually smaller than the diatonic semitone. Meantone is a 7-form temperament, and is tuned well around the golden tuning of diatonic. Unsurprisingly, 7edo supports Meantone, and so does 12edo (which is the simplest ET to do so without exotempering the 5-limit), and so the best tunings of Meantone lie in between those two extremes. [7 & 12] is thus the ''edo join'' for meantone.

FloraC: /* Chords */ dominant seventh and half-diminished are 7-limit det chords too

2026-05-24T06:41:29Z

Chords: dominant seventh and half-diminished are 7-limit det chords too

← Older revision		Revision as of 06:41, 24 May 2026
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	== Chords ==		== Chords ==
	Meantone's main feature is its conflation of the standard harmonic triad 4:5:6 with the diatonic major triad ~~P1-M3-P5~~, thus equating the [[Diatonic#MOS diatonic\|MOS diatonic scale]] with the 5-limit tuning of [[Diatonic#Greek diatonic scales\|Ancient Greek diatonic]] and allowing for 5-limit consonances to be easily accessed within a continuous circle of fifths. Modern Western music theory, which is derived in large part from meantone practice, treats triadic harmony (chords made by stacking two thirds over a root) as the basis of concordance, as the only way to fit three [[5-odd-limit]] intervals in one octave is via some permutation of 4/3, 5/4, and 6/5, which will always make some rotation or retroversion of 4:5:6.		Meantone's main feature is its conflation of the standard harmonic triad 4:5:6 with the diatonic major triad P1–M3–P5, thus equating the [[Diatonic #MOS diatonic\|MOS diatonic scale]] with the 5-limit tuning of [[Diatonic #Greek diatonic scales\|Ancient Greek diatonic]] and allowing for 5-limit consonances to be easily accessed within a continuous circle of fifths. Modern Western music theory, which is derived in large part from meantone practice, treats triadic harmony (chords made by stacking two thirds over a root) as the basis of concordance, as the only way to fit three [[5-odd-limit]] intervals in one octave is via some permutation of 4/3, 5/4, and 6/5, which will always make some rotation or retroversion of 4:5:6.

	The major and minor seventh chords in meantone diatonic can be enumerated as 8:10:12:15 and 10:12:15:18 respectively, ~~and~~ the half-diminished seventh chord as 25:30:36:45, or ~~35:42:50:63~~ in septimal meantone. Additionally, the 5:6:7:9 chord is available as ~~P1-m3-A4-m7~~.		The major and minor seventh chords in meantone diatonic can be enumerated as 8:10:12:15 and 10:12:15:18 respectively. The dominant seventh chord is 20:25:30:36, or in septimal meantone, the 1–5/4–3/2–9/5 [[collection of chords #Essentially tempered chords\|essentially tempered chord]]; the half-diminished seventh chord is similarly 25:30:36:45, or in septimal meantone, the 1–6/5–10/7–9/5 essentially tempered chord. Additionally, the 5:6:7:9 chord is available as P1–m3–A4–m7.

	During the late Renaissance era, septimal meantone tunings were the basis of Augmented Sixth chords. The Italian Sixth chord can be enumerated as 4:5:7, with the intervals of a root, a major third, and an augmented sixth; the German Sixth chord adds an additional interval 3/2 above the root, providing a full 4:5:6:7, whereas the French Sixth chord adds the augmented fourth of 7/5, making a 20:25:28:35 chord.		During the late Renaissance era, septimal meantone tunings were the basis of Augmented Sixth chords. The Italian Sixth chord can be enumerated as 4:5:7, with the intervals of a root, a major third, and an augmented sixth; the German Sixth chord adds an additional interval 3/2 above the root, providing a full 4:5:6:7, whereas the French Sixth chord adds the augmented fourth of 7/5, making a 20:25:28:35 chord.
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	The septimal triads, 6:7:9 and 14:18:21, can additionally be found at P1-A2-P5 and P1-d4-P5 respectively. These can be further extended to the septimal seventh chords, 12:14:18:21 and 14:18:21:27, which are respectively P1-A2-P5-A6 and P1-d4-P5-d1 in septimal meantone.		The septimal triads, 6:7:9 and 14:18:21, can additionally be found at P1-A2-P5 and P1-d4-P5 respectively. These can be further extended to the septimal seventh chords, 12:14:18:21 and 14:18:21:27, which are respectively P1-A2-P5-A6 and P1-d4-P5-d1 in septimal meantone.

	Meantone also contains an ~~[[Collection of chords#Essentially tempered chords\|~~essentially tempered chord]], where 1-9/8-3/2-5/3-2 contains steps of 9/8, 4/3, 9/8, and 6/5. Note that in just intonation, the top interval would be 27/16, not 5/3, or the two whole tones would be different sizes (resulting in a 40/27 [[Wolf interval\|wolf]] fifth).		5-limit Meantone also contains an essentially tempered chord, where 1-9/8-3/2-5/3-2 contains steps of 9/8, 4/3, 9/8, and 6/5. Note that in just intonation, the top interval would be 27/16, not 5/3, or the two whole tones would be different sizes (resulting in a 40/27 [[Wolf interval\|wolf]] fifth).

	== Tunings ==		== Tunings ==

Overthink: /* 11-limit[12 & 19] */ add an equivalence

2026-05-23T23:22:19Z

11-limit[12 & 19]: add an equivalence

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	===== 11-limit[12 & 19] =====		===== 11-limit[12 & 19] =====
	The 11-limit form of 12 & 19 is an exotemperament called ''meanenneadecal'', which tunes 11/8 very sharp (because both 12edo and 19edo do so). More accurate extensions are below.		The 11-limit form of 12 & 19 is an exotemperament called ''meanenneadecal'', which tunes 11/8 very sharp and conflates 14/11 with 5/4 (because both 12edo and 19edo do so). More accurate extensions are below.

	===== 11/8 as the double-augmented third (12 & 31) =====		===== 11/8 as the double-augmented third (12 & 31) =====

Inthar: /* (Half Comma) Cleantone */

2026-05-05T23:08:38Z

(Half Comma) Cleantone

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	=== (Half Comma) Cleantone ===		=== (Half Comma) Cleantone ===
	Cleantone is Hans-Peter Deutsch's tuning of Meantone which tempers the octave to be sqrt(81/80) = 10.8c sharp and retains a just 4:5:6. It tunes 5/4, 6/5, 9/5, and 15/8 (~~generally~~, any interval in the JI group (3/2).(5/4)) justly, but 2/1 complements of these intervals are detuned.		Cleantone is Hans-Peter Deutsch's tuning of Meantone which tempers the octave to be sqrt(81/80) = 10.8c sharp and retains a just 4:5:6. It tunes 5/4, 6/5, 9/5, and 15/8 (in fact, any interval in the JI group (3/2).(5/4)) justly, but 2/1 complements of these intervals are detuned.

	=== Lucy Tuning ([[88edo\|88edo]]) ===		=== Lucy Tuning ([[88edo\|88edo]]) ===

Inthar: /* (Half Comma) Cleantone */

2026-05-05T23:00:14Z

(Half Comma) Cleantone

← Older revision		Revision as of 23:00, 5 May 2026
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	=== (Half Comma) Cleantone ===		=== (Half Comma) Cleantone ===
	Cleantone is Hans-Peter Deutsch's tuning of Meantone which tempers the octave to be sqrt(81/80) = 10.8c sharp and retains a just 4:5:6. It tunes 5/4, 6/5, 9/5, and 15/8 justly, but 2/1 complements of these intervals are detuned.		Cleantone is Hans-Peter Deutsch's tuning of Meantone which tempers the octave to be sqrt(81/80) = 10.8c sharp and retains a just 4:5:6. It tunes 5/4, 6/5, 9/5, and 15/8 (generally, any interval in the JI group (3/2).(5/4)) justly, but 2/1 complements of these intervals are detuned.

	=== Lucy Tuning ([[88edo\|88edo]]) ===		=== Lucy Tuning ([[88edo\|88edo]]) ===

Kili: /* (Half Comma) Cleantone */ credit inventior of cleantone

2026-05-05T18:39:43Z

(Half Comma) Cleantone: credit inventior of cleantone

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	=== (Half Comma) Cleantone ===		=== (Half Comma) Cleantone ===
	Cleantone is ~~Emikili~~'s tuning of Meantone which tempers the octave to be sqrt(81/80) = 10.8c sharp and retains a just 4:5:6. It tunes 5/4, 6/5, 9/5, and 15/8 justly, but 2/1 complements of these intervals are detuned.		Cleantone is Hans-Peter Deutsch's tuning of Meantone which tempers the octave to be sqrt(81/80) = 10.8c sharp and retains a just 4:5:6. It tunes 5/4, 6/5, 9/5, and 15/8 justly, but 2/1 complements of these intervals are detuned.

	=== Lucy Tuning ([[88edo\|88edo]]) ===		=== Lucy Tuning ([[88edo\|88edo]]) ===

Inthar: /* Lucy Tuning (88edo) */

2026-05-05T16:05:48Z

Lucy Tuning (88edo)

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	Cleantone is Emikili's tuning of Meantone which tempers the octave to be sqrt(81/80) = 10.8c sharp and retains a just 4:5:6. It tunes 5/4, 6/5, 9/5, and 15/8 justly, but 2/1 complements of these intervals are detuned.		Cleantone is Emikili's tuning of Meantone which tempers the octave to be sqrt(81/80) = 10.8c sharp and retains a just 4:5:6. It tunes 5/4, 6/5, 9/5, and 15/8 justly, but 2/1 complements of these intervals are detuned.

	=== Lucy Tuning ([[88edo\|88edo)]] ===		=== Lucy Tuning ([[88edo\|88edo]]) ===
	This is nearly indistinguishable from the 88edo tuning of Meantone, with a fifth (600 + 300/π cents) just 0.038 cents higher than 88edo's 695.55-ish fifth. Its major third (1200/π = 381.97 cents) is flat 4.3 cents, but closer than 1/3 Meantone's. The proper extension to the 7-limit (and possible 11-limit) is Mothra (8/7 = 200+100/π = 231.83 cents), which yields a 7/4 flat by only 0.659 cents.		This is nearly indistinguishable from the 88edo tuning of Meantone, with a fifth (600 + 300/π cents) just 0.038 cents higher than 88edo's 695.55-ish fifth. Its major third (1200/π = 381.97 cents) is flat 4.3 cents, but closer than 1/3 Meantone's. The proper extension to the 7-limit (and possible 11-limit) is Mothra (8/7 = 200+100/π = 231.83 cents), which yields a 7/4 flat by only 0.659 cents.

Inthar: /* (Half Comma) Cleantone */

2026-05-05T16:05:30Z

(Half Comma) Cleantone

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	=== (Half Comma) Cleantone ===		=== (Half Comma) Cleantone ===
	Cleantone is ~~a unique~~ tuning of ~~Pythagorean~~ Meantone ~~as it~~ tempers the octave to be ~~1 half of a Syntonsma~~ (81/80) ~~sharp, approximately~~ 10.~~75 cents. Because of that small change, it is able to retain~~ a ~~true-to-~~just ~~consistency in the~~ 5~~-limit with intervals such as~~ 5/4, 6/5, 9/5, 15/8, ~~and others, all with a pure 3~~/2.		Cleantone is Emikili's tuning of Meantone which tempers the octave to be sqrt(81/80) = 10.8c sharp and retains a just 4:5:6. It tunes 5/4, 6/5, 9/5, and 15/8 justly, but 2/1 complements of these intervals are detuned.

	=== Lucy Tuning ([[88edo\|88edo)]] ===		=== Lucy Tuning ([[88edo\|88edo)]] ===

Inthar: /* List of patent vals */ We only consider strong extensions in these tables because weak extensions no longer have the same generator

2026-05-05T16:00:33Z

List of patent vals: We only consider strong extensions in these tables because weak extensions no longer have the same generator

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	\|+		\|+
	!EDO		!EDO
	!7-limit extensions		!7-limit strong extensions
	!11-limit extensions		!11-limit strong extensions
	!Generator tuning		!Generator tuning
	\|-		\|-
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	\|-		\|-
	\|88		\|88
	\|~~Mothra~~		\|
	\|[26 & 31]		\|[26 & 31]
	\|695.5c		\|695.5c

Kili: added cleantone

2026-05-05T15:09:56Z

added cleantone

← Older revision		Revision as of 15:09, 5 May 2026
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	=== 1/2-comma Meantone ===		=== 1/2-comma Meantone ===
	This is close to 33edo's diatonic tuning, which is not Meantone. As a result, it can be considered the lower bound of Meantone's tuning, where the tone is tuned to a just 10/9. It tunes the fifth to 691.2 cents. Its step ratio is 1.26 (ultrasoft).		This is close to 33edo's diatonic tuning, which is not Meantone. As a result, it can be considered the lower bound of Meantone's tuning, where the tone is tuned to a just 10/9. It tunes the fifth to 691.2 cents. Its step ratio is 1.26 (ultrasoft).

			=== (Half Comma) Cleantone ===
			Cleantone is a unique tuning of Pythagorean Meantone as it tempers the octave to be 1 half of a Syntonsma (81/80) sharp, approximately 10.75 cents. Because of that small change, it is able to retain a true-to-just consistency in the 5-limit with intervals such as 5/4, 6/5, 9/5, 15/8, and others, all with a pure 3/2.

	=== Lucy Tuning ([[88edo\|88edo)]] ===		=== Lucy Tuning ([[88edo\|88edo)]] ===