Meantone - Revision history

Unque: Greatly expanded chords section

2026-04-13T17:21:00Z

Greatly expanded chords section

← Older revision		Revision as of 17:21, 13 April 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 ~~chord~~ P1-M3-P5~~. This allows~~ 5-limit ~~harmony~~ to be ~~used in Meantone, and is the foundation~~ of ~~modern~~ Western music theory, which treats P1-M3-P5 as ~~stable~~ and P1-m3-P5 as ~~unstable (10~~:12:~~15)~~. 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).		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.

			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.

			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).

	== Tunings ==		== Tunings ==

Inthar: /* 11-limit */

2026-04-13T03:34:57Z

11-limit

← Older revision		Revision as of 03:34, 13 April 2026
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	However, one drawback of this temperament is the large degree of complexity required to get to the 11th and 13th harmonics. In fact, there are two main options. In both cases, the [[Tridecimal neutral thirds\|tridecimal neutral third]] 16/13 is conflated with the [[Undecimal neutral thirds\|undecimal neutral third]] 11/9, representing a characteristic tendency to make 11/9 the sharper of the two 11-limit neutral thirds. (As a result, one might find it useful to irregularly map 11/9.)		However, one drawback of this temperament is the large degree of complexity required to get to the 11th and 13th harmonics. In fact, there are two main options. In both cases, the [[Tridecimal neutral thirds\|tridecimal neutral third]] 16/13 is conflated with the [[Undecimal neutral thirds\|undecimal neutral third]] 11/9, representing a characteristic tendency to make 11/9 the sharper of the two 11-limit neutral thirds. (As a result, one might find it useful to irregularly map 11/9.)

	===== 11-limit =====		===== 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 (because both 12edo and 19edo do so). More accurate extensions are below.

Inthar: /* List of patent vals */

2026-04-10T15:20:29Z

List of patent vals

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	\|-		\|-
	\|5		\|5
	\|~~dominant~~		\|Dominant
	\|		\|
	\|720c		\|720c
	\|-		\|-
	\|12		\|12
	\|~~dominant~~, ~~septimal meantone~~		\|Dominant, Septimal Meantone
	\|[12 & 31], [12 & 19]		\|[12 & 31], [12 & 19]
	\|700c		\|700c
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	\|-		\|-
	\|43		\|43
	\|~~septimal meantone~~		\|Septimal Meantone
	\|[12 & 31]		\|[12 & 31]
	\|697.7c		\|697.7c
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	\|-		\|-
	\|74		\|74
	\|~~septimal meantone~~		\|Septimal Meantone
	\|[12 & 31]		\|[12 & 31]
	\|697.3c		\|697.3c
	\|-		\|-
	\|105		\|105
	\|~~septimal meantone~~		\|Septimal Meantone
	\|[12 & 31]		\|[12 & 31]
	\|697.1c		\|697.1c
	\|-		\|-
	\|31		\|31
	\|~~septimal meantone~~		\|Septimal Meantone
	\|[12 & 31], [19 & 31]		\|[12 & 31], [19 & 31]
	\|696.8c		\|696.8c
	\|-		\|-
	\|81		\|81
	\|~~septimal meantone~~		\|Septimal Meantone
	\|[19 & 31]		\|[19 & 31]
	\|696.3c		\|696.3c
	\|-		\|-
	\|50		\|50
	\|~~septimal meantone~~		\|Septimal Meantone
	\|[19 & 31]		\|[19 & 31]
	\|696c		\|696c
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	\|-		\|-
	\|19		\|19
	\|~~septimal meantone~~, ~~flattone~~		\|Septimal Meantone, Flattone
	\|[12 & 19], [19 & 31], (~~flattone~~)		\|[12 & 19], [19 & 31], (Flattone)
	\|694.7c		\|694.7c
	\|-		\|-
	\|45		\|45
	\|~~flattone~~		\|Flattone
	\|(~~flattone~~)		\|(Flattone)
	\|693.3c		\|693.3c
	\|-		\|-
	\|26		\|26
	\|~~flattone~~		\|Flattone
	\|(~~flattone~~)		\|(Flattone)
	\|692.3c		\|692.3c
	\|-		\|-
	\|7		\|7
	\|~~dominant~~, ~~flattone~~		\|Dominant, Flattone
	\|(~~flattone~~)		\|(Flattone)
	\|685.7c		\|685.7c
	\|}		\|}

Vector at 19:48, 12 March 2026

2026-03-12T19:48:12Z

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 == List of patent vals ==
-todo: format this better
+{| class="wikitable"
-('''dominant''', septimal meantone, meanenneadecal, huygens)
+{{Navbox regtemp}}
 {{Cat|Temperaments}}

Vector at 19:22, 12 March 2026

2026-03-12T19:22:10Z

← Older revision		Revision as of 19:22, 12 March 2026
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	5 (dominant)		5 (dominant)

	12 ('''dominant''', septimal meantone, meanenneadecal)		12 ('''dominant''', septimal meantone, meanenneadecal, huygens)

	67		67

Vector at 19:21, 12 March 2026

2026-03-12T19:21:24Z

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 === [[12edo]] ===
 {{Navbox regtemp}}
 {{Cat|Temperaments}}

Vector at 21:53, 5 March 2026

2026-03-05T21:53:00Z

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	==== 7/4 as the diminished seventh (19 & 26) ====		==== 7/4 as the diminished seventh (19 & 26) ====
	This temperament, often called "Flattone", sets 7/4 equal to the diminished seventh, and is best tuned with the generator 3/2 around 693 cents, 5/4 at 372 cents, and 7/4 at 963 cents. It is a melodically intuitive extension, as it creates an [[equiheptatonic]] scale with a quartertone-sized chroma, and interval sizes tend to match with their corresponding interval categories. For example, it can be easily extended to map prime 11 to the augmented fourth (C-F#, +6 fifths) and 13 to the minor sixth (C-Ab, -4 fifths) tuned to around 558 and 828 cents respectively. 26edo is the most commonly used tuning, though it can be tuned more accurately with 45edo.		This temperament, often called "Flattone", sets 7/4 equal to the diminished seventh, and is best tuned with the generator 3/2 around 693 cents, 5/4 at 372 cents, and 7/4 at 963 cents. It is a melodically intuitive extension, as it creates an [[equiheptatonic]] scale with a quartertone-sized chroma, and interval sizes tend to match with their corresponding interval categories. For example, it can be easily extended to map prime 11 to the augmented fourth (C-F#, +6 fifths) and 13 to the minor sixth (C-Ab, -4 fifths) tuned to around 558 and 828 cents respectively. 26edo is the most commonly used tuning, though it can be tuned more accurately with 45edo. It is a 7-cluster temperament, as indicated by the edo join (26 - 19 = 7).

	== Chords ==		== Chords ==

Overthink: /* Tunings */ capitalize temperament names

2026-03-02T03:29:09Z

Tunings: capitalize temperament names

← Older revision		Revision as of 03:29, 2 March 2026
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	== Tunings ==		== Tunings ==
	As essentially the only temperament that is both [[regular]] and attested outside [[xenharmony]], ~~meantone~~ has a number of historical tunings that today correspond to various extensions and approximate edos. Here, "comma" refers to the syntonic comma.		As essentially the only temperament that is both [[regular]] and attested outside [[xenharmony]], Meantone has a number of historical tunings that today correspond to various extensions and approximate edos. Here, "comma" refers to the syntonic comma.

	=== Quarter-comma ~~meantone~~ ===		=== Quarter-comma Meantone ===
	This tunes the fifth 1/4-comma flat, to a size of 696.57 cents. It has a just 5/4, and a step ratio of 1.65 (quasisoft). It approximates [[31edo]], and is often (rather insultingly to the rest of 31edo) seen as the latter's primary feature. It extends to 11-limit 19 & 31.		This tunes the fifth 1/4-comma flat, to a size of 696.57 cents. It has a just 5/4, and a step ratio of 1.65 (quasisoft). It approximates [[31edo]], and is often (rather insultingly to the rest of 31edo) seen as the latter's primary feature. It extends to 11-limit 19 & 31.

	=== 1/3-comma ~~meantone~~ ===		=== 1/3-comma Meantone ===
	This is the tuning of the fifth to 694.78 cents, which has a just 6/5 and is extremely close to [[19edo]], having a step ratio of 1.503 (~monosoft). As a result, it does not cleanly extend to the 11-limit, although as it is slightly sharp of 19edo it does technically extend to 7-limit 12 & 19.		This is the tuning of the fifth to 694.78 cents, which has a just 6/5 and is extremely close to [[19edo]], having a step ratio of 1.503 (~monosoft). As a result, it does not cleanly extend to the 11-limit, although as it is slightly sharp of 19edo it does technically extend to 7-limit 12 & 19.

	=== 2/5-comma ~~meantone~~ ===		=== 2/5-comma Meantone ===
	This is very close to the [[45edo]] tuning of ~~meantone~~, tuning the fifth 693.35 cents and having a just 27/25 (note that 27/25 is tempered together with 16/15 in this system, resulting in a sharp minor second). As a result of the flat tuning, this extends to ~~flattone~~, rather than to ~~septimal meantone~~. Its step ratio is 1.401 (parasoft).		This is very close to the [[45edo]] tuning of Meantone, tuning the fifth 693.35 cents and having a just 27/25 (note that 27/25 is tempered together with 16/15 in this system, resulting in a sharp minor second). As a result of the flat tuning, this extends to Flattone, rather than to Septimal Meantone. Its step ratio is 1.401 (parasoft).

	=== 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).

	=== 2/7-comma ~~meantone~~ ===		=== 2/7-comma Meantone ===
	This is the tuning of ~~meantone~~ situated roughly between 50edo's and 69edo's tunings, with a fifth of 695.81 cents and a step ratio of 1.584 (quasisoft). Consequently, it extends to ~~septimal meantone~~, but with a rather poor approximation of 7/4. However, it tunes other septimal intervals like 9/7 and 7/6 more accurately. It tunes 25/24 purely, and can thus be considered a compromise between 1/4-comma's perfect 5/4 and 1/3-comma's perfect 6/5.		This is the tuning of Meantone situated roughly between 50edo's and 69edo's tunings, with a fifth of 695.81 cents and a step ratio of 1.584 (quasisoft). Consequently, it extends to Septimal Meantone, but with a rather poor approximation of 7/4. However, it tunes other septimal intervals like 9/7 and 7/6 more accurately. It tunes 25/24 purely, and can thus be considered a compromise between 1/4-comma's perfect 5/4 and 1/3-comma's perfect 6/5.

	=== 1/5-comma ~~meantone~~ ===		=== 1/5-comma Meantone ===
	This tuning of ~~meantone~~ equalizes the error on 3/2 and 5/4, tuning the former to 697.65 cents and the latter to 390.61 cents. Equivalently, it tunes 16/15 purely. Its step ratio is 1.748 (minisoft), and consequently it is well approximated by 43edo.		This tuning of Meantone equalizes the error on 3/2 and 5/4, tuning the former to 697.65 cents and the latter to 390.61 cents. Equivalently, it tunes 16/15 purely. Its step ratio is 1.748 (minisoft), and consequently it is well approximated by 43edo.

	=== 1/11-comma ~~meantone~~ ===		=== 1/11-comma Meantone ===
	This tuning of ~~meantone~~ is almost perfectly approximated by 12edo, having a fifth tuning of nearly exactly 700 cents and a step ratio of nearly exactly 2 (~basic). 12edo by definition lowers the fifth by 1/12 of a Pythagorean comma; setting 1/12 of a Pythagorean comma to 1/11 of a syntonic comma is done in edos such as [[34edo#612edo\|612edo]].		This tuning of Meantone is almost perfectly approximated by 12edo, having a fifth tuning of nearly exactly 700 cents and a step ratio of nearly exactly 2 (~basic). 12edo by definition lowers the fifth by 1/12 of a Pythagorean comma; setting 1/12 of a Pythagorean comma to 1/11 of a syntonic comma is done in edos such as [[34edo#612edo\|612edo]].

	=== [[31edo]] ===		=== [[31edo]] ===
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	=== [[12edo]] ===		=== [[12edo]] ===
	{{~~Cat\|Temperaments~~}}{{~~Navbox regtemp~~}}
			{{Navbox regtemp}}

			{{Cat\|Temperaments}}

Overthink: /* 7/4 as the diminished seventh (19 & 26) */ Caps in accordance with XR:G; better wording

2026-03-02T03:21:40Z

7/4 as the diminished seventh (19 & 26): Caps in accordance with XR:G; better wording

← Older revision		Revision as of 03:21, 2 March 2026
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	==== 7/4 as the diminished seventh (19 & 26) ====		==== 7/4 as the diminished seventh (19 & 26) ====
	This temperament, often called "~~flattone~~", sets 7/4 equal to the diminished seventh, and is best tuned with the generator 3/2 around 693 cents, 5/4 at 372 cents, and 7/4 at 963 cents. It is a melodically intuitive extension, as it creates an [[equiheptatonic]] scale with a quartertone-sized chroma, and interval sizes tend to match with their corresponding interval categories. For example, it can be easily extended to map prime 11 to the augmented fourth (C-F#, +6 fifths) and 13 to the minor sixth (C-Ab, -4 fifths) tuned to around 558 and 828 cents respectively. 26edo is the ~~usual~~ tuning, though it can be tuned more accurately with 45edo.		This temperament, often called "Flattone", sets 7/4 equal to the diminished seventh, and is best tuned with the generator 3/2 around 693 cents, 5/4 at 372 cents, and 7/4 at 963 cents. It is a melodically intuitive extension, as it creates an [[equiheptatonic]] scale with a quartertone-sized chroma, and interval sizes tend to match with their corresponding interval categories. For example, it can be easily extended to map prime 11 to the augmented fourth (C-F#, +6 fifths) and 13 to the minor sixth (C-Ab, -4 fifths) tuned to around 558 and 828 cents respectively. 26edo is the most commonly used tuning, though it can be tuned more accurately with 45edo.

	== Chords ==		== Chords ==

Vector at 22:06, 1 March 2026

2026-03-01T22:06:25Z

← Older revision		Revision as of 22:06, 1 March 2026
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	'''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.

	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.		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.

	== Other primes ==		== Other primes ==
	If an unmapped (not equated to a stack of anything else, as [[Classical major third\|5/4]] is in blackwood) prime 7 is introduced as a second generator, then the result can be called Didymus.7.		If an unmapped (not equated to a stack of anything else, as [[Classical major third\|5/4]] is in blackwood) prime 7 is introduced as a second generator, then the result can be called Didymus.7.

	~~Extensions~~ of Meantone's diatonic structure to include other primes follow.		The edo join [7 & 12] results in an exotempered extension called ''dominant'' where 7/4 and 9/5 are equated, and is tuned best around Pythagorean tuning.

			More accurate extensions of Meantone's diatonic structure to include other primes follow.

	==== 7/4 as the augmented sixth (12 & 19) ====		==== 7/4 as the augmented sixth (12 & 19) ====
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	However, one drawback of this temperament is the large degree of complexity required to get to the 11th and 13th harmonics. In fact, there are two main options. In both cases, the [[Tridecimal neutral thirds\|tridecimal neutral third]] 16/13 is conflated with the [[Undecimal neutral thirds\|undecimal neutral third]] 11/9, representing a characteristic tendency to make 11/9 the sharper of the two 11-limit neutral thirds. (As a result, one might find it useful to irregularly map 11/9.)		However, one drawback of this temperament is the large degree of complexity required to get to the 11th and 13th harmonics. In fact, there are two main options. In both cases, the [[Tridecimal neutral thirds\|tridecimal neutral third]] 16/13 is conflated with the [[Undecimal neutral thirds\|undecimal neutral third]] 11/9, representing a characteristic tendency to make 11/9 the sharper of the two 11-limit neutral thirds. (As a result, one might find it useful to irregularly map 11/9.)

			===== 11-limit =====
			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.

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