Meantone

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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)

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

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