Meantone
Meantone is a familar historical temperament based on a chain of fifths (or fourths), which is discussed in meantone family in the context of the associated family of temperaments, and in meantone vs meanpop in terms of 11-limit extensions.
Meantone as mathematical approximation[edit]
Meantone are logarithmic approximations satisfying the equality b⁴÷a⁴÷c=1 '81/80', where a≈2, b≈3, and c≈5. A meantone system has pure octave when a=2. With pure octave, b becomes the remaining parameter as c=b⁴÷a⁴=b⁴÷16. In terms of those defined variables, octave is a '2/1', perfect fifth is b÷a '3/2', perfect fourth is a²÷b '4/3', major third is c÷a² '5/4', minor third is a×b÷c '6/5', major sixth is c÷b '5/3', and minor sixth is a³÷c '8/5'. Septimal meantone extends it by the approximation of next prime d=b²×c²÷a⁵≈7, resulting in equalities b²×c²÷a⁵÷d=1 '225/224' and c³÷a÷b²÷d=1 '126/125'.
History[edit]
Meantone was the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various Well Temperaments and eventually 12-tone Equal Temperament won in popularity.
Theory and Classification[edit]
Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a rank-2 temperament. The octave is typically pure or close to pure, and the fifth is a few cents narrower than pure. The rationale for narrowing the fifth is to temper out the syntonic comma. This means that stacking four fifths (such as C-G-D-A-E) results in a major third (C-E) that is close to just.
Intervals in meantone have standard names based on the number of steps of the diatonic scale they span (this corresponds to the val <7 11 16|), with a modifier {..."double diminished", "diminished", "minor", "major", "augmented", "double augmented"...} that tells you the specific interval in increments of a chromatic semitone. Note that in a general meantone system, all of these intervals are distinct. For example, a diminished fourth is a different interval from a major third.
Meantone Temperaments (ie, tunings)[edit]
- 19-edo
- 1/3 Syntonic Comma Meantone
- Golden Meantone
- 1/4 Syntonic Comma Meantone
- 31-edo
- 1/5 Syntonic Comma Meantone
- 1/6 Syntonic Comma Meantone
- 12-edo
- Lucy Tuning
- 50-edo
- 55-edo
- Tungsten meantone
Spectrum of Meantone Tunings by Eigenmonzos[edit]
| Eigenmonzo | Fifth size (usual name) |
|---|---|
| 10/9 | 691.202 (1/2 comma) |
| 15\26 | 692.308 |
| 56/45 | 694.651 |
| 28/27 | 694.709 |
| 81/70 | 694.732 |
| 11\19 | 694.737 |
| 6/5 | 694.786 (1/3 comma) |
| 35/27 | 695.389 |
| 51\88 | 695.455 |
| 1\2 + 1\(4π) | 695.493 (Lucy tuning) |
| 9/7 | 695.614 |
| f^4 = 2f + 2 | 695.630 (Wilson fifth) |
| 40\69 | 695.652 |
| 25/24 | 695.810 (2/7 comma) |
| 13/10 | 695.838 (ratwolf fifth, meanpop eigenmonzo) |
| 36/35 | 695.936 |
| 54/49 | 695.987 |
| 29\50 | 696.000 |
| 15/14 | 696.111 |
| 78125/73728 | 696.165 (5-limit least squares) |
| (8 - φ)\11 | 696.214 (Golden meantone) |
| 49/45 | 696.245 |
| 47\81 | 696.296 |
| 7/6 | 696.319 |
| 48/35 | 696.399 |
| | 19 9 -1 -11 > | 696.436 (9-limit least squares) |
| 5/4 | 696.578 (5- 7- and 9-limit minimax, 1/4 comma) |
| 49/48 | 696.616 |
| 60/49 | 696.626 |
| | -55 -11 1 25 > | 696.648 (7-limit least squares) |
| 18\31 | 696.774 |
| 35/32 | 696.796 |
| 8/7 | 696.883 |
| 49/40 | 696.959 |
| 7/5 | 697.085 |
| 43\74 | 697.297 |
| 21/16 | 697.344 |
| 16/15 | 697.654 (1/5 comma) |
| 25\43 | 697.674 |
| 64/63 | 697.728 |
| 21/20 | 697.781 |
| 28/25 | 698.099 |
| 32\55 | 698.182 |
| 80/63 | 698.303 |
| 45/32 | 698.371 (1/6 comma) |
| 39\67 | 698.507 |
| 46\79 | 698.734 |
| 25/21 | 699.384 |
| 7\12 | 700.000 |
| 31\53 | 701.887 |
| 3/2 | 701.955 |
[5/4 7] eigenmonos: meanwoo12, meanwoo19
Links[edit]
- http://www.kylegann.com/histune.html -- An Introduction to Historical Tunings, by Kyle Gann