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 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
| [⟨ | 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]
Mistuning 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 | 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
Modern renderings
- Bach's Ricercar a 6 from A Musical Offering rendered by Claudi Meneghin
- Reelin' and Rockin' (1957)
- Tre Sonate Sopra Ave Maris Stella (V, VI, VII) (c. 1641–1644)
- Le Tic-Toc Choc (1722)
- Prelude in g minor Op. 23 #5 (1901)
- At the Hop (1957)
21st century
- Country Bumpkin (2010) play
- Ozone Layer (2012) play
- A Meantone Prelude (2023)
- Unnamed fugue (2023)
- 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
- A Quarter of a Mean Tone
- No Voice Leading (2012) blog | play