Quarter-comma meantone

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Quarter-comma meantone is the tuning of meantone temperament which makes the 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 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
0.00000 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 C## 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 D## Doubly augmented second AA2 [-9 0 4>
351.324 -15 Fbb Doubly diminished fourth dd2 [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 F## 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 G## 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 A## 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>

See Quarter-comma meantone

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