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
| -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
56/45	694.651
28/27	694.709
81/70	694.732
11\19	694.737
6/5	694.786	1/3 comma
5103/4095	695.139	1/4 septimal comma
| 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
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
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
47\81	696.296
7/6	696.319
48/35	696.399
[19 9 -1 -11⟩	696.436	9-limit least squares
16384/15309	696.502	1/5 septimal comma
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
2187/2048	697.263	1/5 Pythagorean comma, Pythagorean aptome
43\74	697.297
21/16	697.344
45927/32768	697.411	1/6 septimal comma
16/15	697.654	1/5 comma
25\43	697.674
64/63	697.728
21/20	697.781
1024/729	698.045	1/6 Pythagorean comma, Pythagorean tritone
| - 17 9 0 1 >	698.06	1/7 septimal comma
28/25	698.099
32\55	698.182
80/63	698.303
45/32	698.371	1/6 comma
39\67	698.507
256/243	698.604	1/7 Pythagorean comma, Pythagorean limma
46\79	698.734
135/128	698.883	1/7 comma
17/16	699.009
25/21	699.384
7\12	700
18/17	700.209
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