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.

Tunings

Common meantone tunings include various eigenmonzo tunings such as the quarter-comma meantone, edo tunings like 31edo, and otherwise optimized tunings like the POTE tuning, shown below. For a more complete list, see #Spectrum of meantone tunings by eigenmonzos. These different tunings are referred to as "temperaments" in traditional terms.

Eigenmonzo tunings

• 1/2 syntonic comma meantone – with eigenmonzo 10/9
• 1/3 syntonic comma meantone – with eigenmonzo 5/3
• 2/7 syntonic comma meantone – with eigenmonzo 25/24
• 1/4 syntonic comma meantone – with eigenmonzo 5/4
• 1/5 syntonic comma meantone – with eigenmonzo 15/8
• 1/6 syntonic comma meantone – with eigenmonzo 45/32

Edo tunings

• 19\33
• 15\26
• 26\45
• 11\19
• 29\50
• 18\31
• 61\105
• 25\43
• 32\55
• 7\12

POTE tunings

• ~3/2 = 696.239¢ – 5-limit meantone
• ~3/2 = 696.495¢ – 7-limit meantone

Other optimized tunings

• Golden meantone
• Tungsten meantone
• Lucy tuning

Scales

• Meantone5 – pentatonic scale in 31edo
• Meantone7 – diatonic scale in 19edo and 31edo
• Meantone12 – chromatic scale in 31edo

[5/4 7] eigenmonzos:

• Meanwoo12
• Meanwoo19

Spectrum of meantone tunings by eigenmonzos

Eigenmonzo	Fifth size	Comments
567/512	688.323	1/2 septimal comma
[16 -10⟩	690.225	1/2 Pythagorean comma, Pythagorean dilimma
51/38	690.603
[-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, 25/18	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
f4 = 2f + 2	695.630	Wilson fifth
(40\69)	695.652
25/24	695.810	2/7 comma
81/80	695.869
36/35	695.936
54/49	695.987
(29\50)	696.000
8192/6561	696.090	1/4 Pythagorean comma, Pythagorean diminished fourth
15/14	696.111
78125/73728	696.165	5-odd-limit least squares
(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
48/35	696.399
[19 9 -1 -11⟩	696.436	9-odd-limit least squares
16384/15309	696.502	1/5 septimal comma
5/4	696.578	5-, 7-, and 9-odd-limit minimax, 1/4 comma
49/48	696.616
60/49	696.626
[-55 -11 1 25⟩	696.648	7-odd-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	2/9 comma
2187/2048	697.263	1/5 Pythagorean comma, Pythagorean apotome
(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, lesser Pythagorean tritone
[-17 9 0 1⟩	698.060	1/7 septimal comma
28/25	698.099
(32\55)	698.182
80/63	698.303
17/15	698.331	[clarification needed]
45/32	698.371	1/6 comma
(39\67)	698.507
256/243	698.604	1/7 Pythagorean comma, Pythagorean limma
45/34	698.661	[clarification needed]
(46\79)	698.734
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

Links

• An Introduction to Historical Tunings, by Kyle Gann
• Wikipedia: Meantone temperament
• Wikipedia: Septimal meantone temperament