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
The common meantone temperaments (recte tunings) include:
- 1/2 syntonic comma meantone
- 33edo
- 26edo
- 2/5 syntonic comma meantone
- 45edo
- 1/3 syntonic comma meantone
- 19edo
- Golden meantone
- 2/7 syntonic comma meantone
- 50edo
- 1/4 syntonic comma meantone
- 31edo
- 1/5 syntonic comma meantone
- 43edo
- 105edo
- 1/6 syntonic comma meantone
- 55edo
- 12edo
- Lucy tuning
- Tungsten meantone
Spectrum of meantone tunings by eigenmonzos
| Eigenmonzo | Fifth size | Comments |
|---|---|---|
| 567/512 | 688.323 | 1/2 septimal comma |
| 16/13 | 689.868 | Meanplop |
| [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 |
| 13/12 | 692.285 | Meanplop |
| (15\26) | 692.308 | |
| [31 -19⟩ | 692.571 | 2/5 Pythagorean comma |
| 2048/1701 | 692.867 | 1/3 septimal comma |
| 33/26 | 693.178 | Grosstone |
| 13/10 | 693.223 | Meanplop |
| (26\45) | 693.333 | |
| 27/25 | 693.352 | 2/5 comma |
| 18/13 | 693.897 | Meanplop |
| 19683/16384 | 694.135 | 1/3 Pythagorean comma, Pythagorean augmented second |
| [-23 11 0 2⟩ | 694.165 | 2/7 septimal comma |
| 15/13 | 694.193 | Meanplop |
| 14/13 | 694.340 | 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 |
| 14/13 | 694.878 | Meanplop |
| 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 |
| 39/28 | 695.6095 | Tridecimal meantone, tridecimal meanpop |
| 13/12 | 695.612 | Tridecimal meantone |
| 9/7 | 695.614 | |
| f4 = 2f + 2 | 695.630 | Wilson fifth |
| (40\69) | 695.652 | |
| 25/24 | 695.810 | 2/7 comma |
| 40/33 | 695.815 | Meanpop |
| 13/10 | 695.838 | Ratwolf fifth, tridecimal meantone and meanpop eigenmonzo |
| 81/80 | 695.869 | |
| 112/99 | 695.886 | Meanpop |
| 36/35 | 695.936 | |
| 54/49 | 695.987 | |
| (29\50) | 696.000 | |
| 16/13 | 696.035 | Tridecimal meantone, tridecimal meanpop |
| 13/11 | 696.043 | 13- and 15-odd-limit minimax (tridecimal meanpop) |
| 11/8 | 696.052 | Meanpop |
| 8192/6561 | 696.09 | 1/4 Pythagorean comma, Pythagorean diminished fourth |
| 15/14 | 696.111 | |
| 78125/73728 | 696.165 | 5-odd-limit least squares |
| 11/10 | 696.176 | Meanpop |
| (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 | |
| 27/22 | 696.3635 | Meanpop |
| 48/35 | 696.399 | |
| 39/32 | 696.405 | Tridecimal meantone, tridecimal meanpop |
| 14/11 | 696.413 | Meanpop |
| [19 9 -1 -11⟩ | 696.436 | 9-odd-limit least squares |
| 12/11 | 696.474 | Meanpop |
| 16384/15309 | 696.502 | 1/5 septimal comma |
| 5/4 | 696.578 | 5-, 7-, 9- and 11-odd-limit (meanpop) minimax, 1/4 comma |
| 49/48 | 696.616 | |
| 60/49 | 696.626 | |
| [-55 -11 1 25⟩ | 696.648 | 7-odd-limit least squares |
| 11/9 | 696.713 | 11-, 13- and 15-odd-limit (tridecimal meantone) minimax |
| (18\31) | 696.774 | |
| 35/32 | 696.796 | |
| 11/9 | 696.839 | Meanpop |
| 8/7 | 696.883 | |
| 49/40 | 696.959 | |
| 12/11 | 697.021 | Undecimal meantone |
| 7/5 | 697.085 | |
| 15/11 | 697.158 | Undecimal meantone |
| 27/22 | 697.159 | |
| 39/32 | 697.168 | Grosstone |
| 75/64 | 697.176 | |
| 22/21 | 697.220 | Undecimal meantone |
| 14/13 | 697.242 | 13- and 15-odd-limit minimax (grosstone) |
| 2187/2048 | 697.263 | 1/5 Pythagorean comma, Pythagorean apotome |
| 13/10 | 697.289 | Grosstone |
| 11/8 | 697.295 | Undecimal meantone |
| (43\74) | 697.297 | |
| [5/4 7] | 697.339 | |
| 21/16 | 697.344 | |
| 13/11 | 697.376 | Meridetone |
| 45927/32768 | 697.411 | 1/6 septimal comma |
| 18/13 | 697.465 | 13-, 15-odd-limit minimax (meridetone) |
| 16/13 | 696.467 | Grosstone |
| 11/10 | 697.500 | Undecimal meantone |
| 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 | |
| 40/33, 16/13 | 697.797 | Undecimal meantone, meridetone |
| 14/11 | 697.812 | Undecimal meantone |
| 15/13 | 697.830 | 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⟩ | 698.060 | 1/7 septimal comma |
| 28/25 | 698.099 | |
| (32\55) | 698.182 | |
| 33/28 | 698.272 | Undecimal meantone |
| 80/63 | 698.303 | |
| 17/15 | 698.331 | |
| 45/32 | 698.371 | 1/6 comma |
| 33/26 | 698.407 | Meanplop, meridetone |
| (39\67) | 698.507 | |
| 256/243 | 698.604 | 1/7 Pythagorean comma, Pythagorean limma |
| 112/99 | 698.640 | Undecimal meantone |
| 45/34 | 698.661 | [clarification needed] |
| (46\79) | 698.734 | |
| 13/11 | 698.801 | Meridetone, meanplop |
| 135/128 | 698.883 | 1/7 comma |
| 17/16 | 699.009 | [clarification needed] |
| 25/21 | 699.384 | |
| (7\12) | 700.000 | |
| 18/17 | 700.209 | [clarification needed] |
| 19/16 | 700.829 | [clarification needed] |
| 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 |
| 22/21 | 703.356 | Meanpop |
| 13/11 | 703.597 | Tridecimal meantone |
| 88/81 | 707.946 | Meanpop |
| 88/81 | 710.4335 | Undecimal meantone |
[5/4 7] eigenmonzos: meanwoo12, meanwoo19