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Quarter comma meantone is the tuning of meantone temperament which makes the fifth 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 also the tuning where the major tone is the exact logarithmic average (e.g. cents average) 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 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.

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
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
C##
Augmented chromatic semitone

[-8 0 7/2>
158.167
-17
Ebbb
Doubly diminished third

[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
D##
Doubly augmented second

[-9 0 4>
351.324
-15
Fbb
Doubly diminished fourth

[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

[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
F##
Doubly augmented fourth

[-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
G##
Doubly augmented fifth

[-8 0 15/4>
854.745
-16
Bbbb
Doubly diminished seventh

[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
A##
Doubly augmented sixth

[-9 0 17/4>
1047.90
-14
C'bb
Doubly diminished octave

[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
0
C'
Perfect octave
P8
[1 0 0>

See Quarter-comma meantone

Music

Reelin' and Rockin' by Chuck Berry
Le Tic-Toc Choc by François Couperin
Prelude in g minor Op 23 #5 by Sergei Rachmaninoff
At the Hop by Singer, Medora and White
Prelude to Die Meistersinger von Nürnberg by Richard Wagner
Laura che il verde lauro play by Nicola Vicentino
Country Bumpkinplay
Ozone Layer play
both by Mark Allan Barnes
Five canons by Claudi Meneghin:
Canon 2 in 1 in F at the octave play
Canon 2 in 1 in A- upon a ground play
Canon 2 in 1 in G- at the major seventh play
Canon at the minor second on an ancient Padanian folk theme play
Canon 2 in 1 on "Ah! vous dirai-je, maman" at the diminished fifth play
Bach's Ricercar a 6 from A Musical Offering rendered by Claudi Meneghin
A Quarter of a Mean Tone by Chris Vaisvil
No Voice Leading by Chris Vaisvil
Tre Sonate Sopra Ave Maris Stella (V, VI, VII) by Giovanni Pietro Del Buono