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Meantone is a familar historical temperament based on a chain of fifths (or fourths), which is discussed here in the context of the associated family of temperaments, and here 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.

Common Meantone Temperaments (ie, tunings)


Spectrum of Meantone Tunings by Eigenmonzos

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

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