Quarter-comma meantone

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Quarter-comma meantone

Quarter-comma meantone is the tuning of meantone temperament which makes the perfect fifth (3/2) the fourth root of 5, or in other words 696.578 cents. This means the fifth is flattened by 1/4 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 5.377 cents, hence the name quarter-comma or 1/4-comma meantone. It is the tuning where the major third (5/4) is tuned pure, and the minor third (6/5) and the fifth are equally flat by 1/4 syntonic comma. It is also the tuning where the whole tone is the exact geometric mean between the greater tone of 9/8 and the lesser tone of 10/9, and hence it can be argued it is the only tuning which is strictly "mean tone". It is the minimax tuning for 5-limit meantone, meaning the maximum error on the 5-limit tonality diamond is minimized. It is also the minimax tuning for septimal meantone in the 7- and 9-limit odd limits, and for meanpop, the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C-Gbb. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era.

Because of all of these features, it has a certain claim to be considered the canonical meantone tuning and often "meantone" is taken to mean quarter-comma specifically. The traditional interval names, listed below, may be considered to have their older traditional tuning in the quarter-comma meantone tunings listed below. However, other tunings (besides 12edo which is ubiquitous but not an especially good tuning for meantone) may be preferred. Note, for instance, that the doubly augmented second and the doubly diminished fourth are both neutral thirds, about six cents apart. If we want a tuning which will equate the two, we need to use 31edo instead.

Tuning profile

Projection map:

[⟨	1	1	0	-3	]
⟨	0	0	0	0	]
⟨	0	1/4	1	5/2	]
⟨	0	0	0	0	]]

Tuning map: ⟨1200 1896.5784 2786.3137 3365.7843]

Error map: ⟨0 -5.3766 0 -3.0416]

Intervals of quarter-comma meantone

Below is a table listing 36 of the notes of quarter-comma meantone, giving the size in cents, the number of fifths up or down on the chain of fifths, the letter notation for the interval relative to C as the tonic, the traditional name for the interval, a shorthand form of the name, and the size of the interval in fractional monzo notation.

Size in cents	Fifths dist.	Note C-based	Traditional interval name	Int. (short)	Fract. monzo
0.00000	0	C	Unison	P1	[0 0 0⟩
41.0590	−12	Dbb	Diminished second, diesis	d2	[7 0 -3⟩
76.0490	7	C#	Augmented unison, chromatic semitone	A1	[-4 0 7/4⟩
117.108	-5	Db	Minor second, diatonic semitone	m2	[3 0 -5/4⟩
152.098	14	Cx	Augmented chromatic semitone	AA1	[-8 0 7/2⟩
158.167	−17	Ebbb	Doubly diminished third	dd3	[10 0 -17/4⟩
193.157	2	D	Major second	M2	[-1 0 1/2⟩
234.216	−10	Ebb	Diminished third	d3	[6 0 -5/2⟩
269.206	9	D#	Augmented second	A2	[-5 0 9/4⟩
310.265	−3	Eb	Minor third	m3	[2 0 -3/4⟩
345.255	16	Dx	Doubly augmented second	AA2	[-9 0 4⟩
351.324	−15	Fbb	Doubly diminished fourth	dd4	[9 0 -15/4⟩
386.314	4	E	Major third	M3	[-2 0 1⟩
427.373	−8	Fb	Diminished fourth	d4	[5 0 -2⟩
462.363	11	E#	Augmented third	A3	[-6 0 11/4⟩
503.422	−1	F	"Perfect" fourth	P4	[1 0 -1/4⟩
544.480	−13	Gbb	Doubly diminished fifth	dd5	[8 0 -13/4⟩
579.471	6	F#	Augmented fourth	A4	[-3 0 3/2⟩
620.529	−6	Gb	Diminished fifth	d5	[4 0 -3/2⟩
655.520	13	Fx	Doubly augmented fourth	AA4	[-7 0 13/4⟩
696.578	1	G	"Perfect" fifth	P5	[0 0 1/4⟩
737.637	−11	Abb	Diminshed sixth	d6	[7 0 -11/4⟩
772.627	8	G#	Augmented fifth	A5	[-4 0 2⟩
813.686	−4	Ab	Minor sixth	m6	[3 0 -1⟩
848.676	15	Gx	Doubly augmented fifth	AA5	[-8 0 15/4⟩
854.745	−16	Bbbb	Doubly diminished seventh	dd7	[10 0 -4⟩
889.735	3	A	Major sixth	M6	[-1 0 3/4⟩
930.794	-9	Bbb	Diminished seventh	d7	[6 0 -9/4⟩
965.784	10	A#	Augmented sixth	A6	[-5 0 5/2⟩
1006.84	−2	Bb	Minor seventh	m7	[2 0 -1/2⟩
1041.83	17	Ax	Doubly augmented sixth	AA6	[-9 0 17/4⟩
1047.90	−14	C'bb	Doubly diminished octave	dd8	[9 0 -7/2⟩
1082.89	5	B	Major seventh	M7	[-2 0 5/4⟩
1123.95	−7	C'b	Diminished octave	d8	[5 0 -7/4⟩
1158.94	12	B#	Augmented seventh	A7	[-6 0 3⟩
1200.00	0	C'	Perfect octave	P8	[1 0 0⟩

Music

Modern renderings

Johann Sebastian Bach

Bach's Ricercar a 6 from A Musical Offering rendered by Claudi Meneghin

Chuck Berry

Reelin' and Rockin' (1957)

Giovanni Pietro Del Buono

Tre Sonate Sopra Ave Maris Stella (V, VI, VII) (c. 1641–1644)

François Couperin

Le Tic-Toc Choc (1722)

Sergei Rachmaninoff

Prelude in g minor Op. 23 #5 (1901)

Arthur Singer, John Medora and David White

At the Hop (1957)

Richard Wagner

"Prelude" to Die Meistersinger von Nürnberg (1868)

Nicola Vicentino

Laura che il verde lauro⁠ ⁠^[dead link] play

21st century

Mark Allan Barnes

Country Bumpkin (2010) play
Ozone Layer (2012) play

Francium

A Meantone Prelude (2023)

Adam Freese

Unnamed fugue (2023)

Claudi Meneghin

Five canons (2011)
- "Canon 2 in 1 in F at the octave" YouTube | play
- "Canon 2 in 1 in A- upon a ground" blog | play
- "Canon 2 in 1 in G- at the major seventh" YouTube | play
- "Canon at the minor second on an ancient Padanian folk theme" YouTube | play
- "Canon 2 in 1 on "Ah! vous dirai-je, maman" at the diminished fifth" YouTube | play