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.
History
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
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)
- 19edo
- 1/3 syntonic comma meantone
- Golden Meantone
- 1/4 syntonic comma meantone
- 31edo
- 1/5 syntonic comma meantone
- 1/6 syntonic comma meantone
- 12edo
- Lucy Tuning
- 50edo
- 55edo
- Tungsten meantone
Spectrum of Meantone Tunings by Eigenmonzos
| Eigenmonzo | Fifth size | usual name |
|---|---|---|
| 567/512 | 688.323 | 1/2 septimal comma |
| | 16 -10 > | 690.225 | 1/2 Pythagorean comma, Pythagorean dilimma |
| 76/51 | 690.603 | |
| | -19 9 0 2 > | 691.049 | 2/5 septimal comma |
| 10/9 | 691.202 | 1/2 comma |
| 15\26 | 692.308 | |
| | 31 -19 > | 692.571 | 2/5 Pythagorean comma |
| 2048/1701 | 692.867 | 1/3 septimal comma |
| 26\45 | 693.333 | |
| 27/25 | 693.352 | 2/5 comma |
| 19683/16384 | 694.135 | 1/3 Pythagorean comma, Pythagorean augmented second |
| | -23 11 0 2 > | 694.165 | 2/7 septimal comma |
| 14/13 | 694.34 | Tridecimal Meantone |
| 56/45 | 694.651 | |
| 28/27 | 694.709 | |
| 81/70 | 694.732 | |
| 11\19 | 694.737 | |
| 6/5, 25/18 | 694.786 | 1/3 comma |
| 18/13 | 695.124 | Tridecimal Meantone |
| 5103/4095 | 695.139 | 1/4 septimal comma |
| 15/13 | 695.226 | Tridecimal Meantone |
| | 27 -17 > | 695.252 | 2/7 Pythagorean comma, 17-comma |
| 35/27 | 695.389 | |
| 51\88 | 695.455 | |
| 1\2 + 1\(4π) | 695.493 | Lucy Tuning |
| 13/12 | 695.612 | Tridecimal Meantone |
| 9/7 | 695.614 | |
| f^4 = 2f + 2 | 695.63 | Wilson fifth |
| 40\69 | 695.652 | |
| 25/24 | 695.81 | 2/7 comma |
| 13/10 | 695.838 | ratwolf fifth, Tridecimal Meantone and meanpop eigenmonzo |
| 81/80 | 695.869 | |
| 36/35 | 695.936 | |
| 54/49 | 695.987 | |
| 29\50 | 696 | |
| 16/13 | 696.035 | Tridecimal Meantone |
| 8192/6561 | 696.09 | 1/4 Pythagorean comma, Pythagorean diminished fourth |
| 15/14 | 696.111 | |
| 78125/73728 | 696.165 | 5-limit least squares |
| (8 - φ)\11 | 696.214 | Golden meantone |
| 49/45 | 696.245 | |
| 19/17 | 696.279 | Classical meantone |
| 47\81 | 696.296 | |
| 7/6 | 696.319 | |
| 48/35 | 696.399 | |
| 39/32 | 696.405 | |
| [19 9 -1 -11⟩ | 696.436 | 9-limit least squares |
| 16384/15309 | 696.502 | 1/5 septimal comma |
| 5/4 | 696.578 | 5-, 7-, 9- and 11- (Meanpop) limit minimax, 1/4 comma |
| 49/48 | 696.616 | |
| 60/49 | 696.626 | |
| [-55 -11 1 25⟩ | 696.648 | 7-limit least squares |
| 11/9 | 696.713 | 11-, 13- and 15- limit (Tridecimal Meantone) minimax |
| 18\31 | 696.774 | |
| 35/32 | 696.796 | |
| 8/7 | 696.883 | |
| 49/40 | 696.959 | |
| 12/11 | 697.021 | |
| 7/5 | 697.085 | |
| 15/11 | 697.158 | |
| 27/22 | 697.159 | |
| 75/64 | 697.176 | |
| 14/13 | 697.242 | 13, 15 limit minimax (Grosstone) |
| 2187/2048 | 697.263 | 1/5 Pythagorean comma, Pythagorean apotome |
| 13/10 | 697.289 | Grosstone |
| 11/8 | 697.295 | |
| 43\74 | 697.297 | |
| 21/16 | 697.344 | |
| 13/11 | 697.376 | Meridetone |
| 45927/32768 | 697.411 | 1/6 septimal comma |
| 18/13 | 697.465 | 13, 15 limit minimax (Meridetone) |
| 16/13 | 696.467 | Grosstone |
| 11/10 | 697.5 | |
| 15/13 | 697.511 | Grosstone |
| 13/12 | 697.637 | Meridetone |
| 16/15 | 697.654 | 1/5 comma |
| 25\43 | 697.674 | |
| 64/63 | 697.728 | |
| 21/20 | 697.781 | |
| 16/13 | 697.797 | Meridetone |
| 14/11 | 697.812 | |
| 15/13 | 697.83 | Meridetone |
| 18/13 | 697.966 | Grosstone |
| 13/10 | 698.009 | Meridetone |
| 1024/729 | 698.045 | 1/6 Pythagorean comma, lesser Pythagorean tritone |
| | - 17 9 0 1 > | 698.06 | 1/7 septimal comma |
| 28/25 | 698.099 | |
| 32\55 | 698.182 | |
| 33/28 | 698.272 | |
| 80/63 | 698.303 | |
| 17/15 | 698.331 | |
| 45/32 | 698.371 | 1/6 comma |
| 39\67 | 698.507 | |
| 256/243 | 698.604 | 1/7 Pythagorean comma, Pythagorean limma |
| 45/34 | 698.661 | |
| 46\79 | 698.734 | |
| 13/11 | 698.801 | Meridetone |
| 135/128 | 698.883 | 1/7 comma |
| 17/16 | 699.009 | |
| 25/21 | 699.384 | |
| 7\12 | 700 | |
| 18/17 | 700.209 | |
| 19/16 | 700.829 | |
| 81/80 | 701.792 | |
| 31\53 | 701.887 | |
| 3/2 | 701.955 | Pythagorean tuning |
| 64/63 | 702.272 | |
| 256/189 | 702.301 | |
| 33/26 | 703.186 | Tridecimal Meantone |
| 13/11 | 703.597 | Tridecimal Meantone |
| 88/81 | 710.4335 |
[5/4 7] eigenmonzos: meanwoo12, meanwoo19
Links
- http://www.kylegann.com/histune.html -- An Introduction to Historical Tunings, by Kyle Gann