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 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 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.
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 | 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 | 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> |
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 by Mark Allan Barnes
- Ozone Layer play by Mark Allan Barnes
- Five canons by Claudi Meneghin:
- 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