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.
Contents
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 |
---|---|---|
10/9 | 691.202 | 1/2 comma |
15\26 | 692.308 | |
26\45 | 693.333 | |
27/25 | 693.352 | 2/5 comma |
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 | |
75/64 | 697.176 | |
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 | Pythagorean tuning |
[5/4 7] eigenmonzos: meanwoo12, meanwoo19
Links
- http://www.kylegann.com/histune.html -- An Introduction to Historical Tunings, by Kyle Gann