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

The common meantone temperaments (recte tunings) include:

• 1/2 syntonic comma meantone
• 33edo
• 26edo
• 2/5 syntonic comma meantone
• 45edo
• 1/3 syntonic comma meantone
• 19edo
• Golden meantone
• 2/7 syntonic comma meantone
• 50edo
• 1/4 syntonic comma meantone
• 31edo
• 1/5 syntonic comma meantone
• 43edo
• 105edo
• 1/6 syntonic comma meantone
• 55edo
• 12edo
• Lucy tuning
• Tungsten meantone

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
16/13	689.868	Meanplop
[16 -10⟩	690.225	1/2 Pythagorean comma, Pythagorean dilimma
76/51	690.603
[-19 9 0 2⟩	691.049	2/5 septimal comma
10/9	691.202	1/2 comma
13/12	692.285	Meanplop
(15\26)	692.308
[31 -19⟩	692.571	2/5 Pythagorean comma
2048/1701	692.867	1/3 septimal comma
33/26	693.178	Grosstone
13/10	693.223	Meanplop
(26\45)	693.333
27/25	693.352	2/5 comma
18/13	693.897	Meanplop
19683/16384	694.135	1/3 Pythagorean comma, Pythagorean augmented second
[-23 11 0 2⟩	694.165	2/7 septimal comma
15/13	694.193	Meanplop
14/13	694.340	Tridecimal meantone
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
14/13	694.878	Meanplop
18/13	695.124	Tridecimal meantone
5103/4095	695.139	1/4 septimal comma
15/13	695.226	Tridecimal Meantone
[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
39/28	695.6095	Tridecimal meantone, tridecimal meanpop
13/12	695.612	Tridecimal meantone
9/7	695.614
f4 = 2f + 2	695.630	Wilson fifth
(40\69)	695.652
25/24	695.810	2/7 comma
40/33	695.815	Meanpop
13/10	695.838	Ratwolf fifth, tridecimal meantone and meanpop eigenmonzo
81/80	695.869
112/99	695.886	Meanpop
36/35	695.936
54/49	695.987
(29\50)	696.000
16/13	696.035	Tridecimal meantone, tridecimal meanpop
13/11	696.043	13- and 15-odd-limit minimax (tridecimal meanpop)
11/8	696.052	Meanpop
8192/6561	696.09	1/4 Pythagorean comma, Pythagorean diminished fourth
15/14	696.111
78125/73728	696.165	5-odd-limit least squares
11/10	696.176	Meanpop
(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
27/22	696.3635	Meanpop
48/35	696.399
39/32	696.405	Tridecimal meantone, tridecimal meanpop
14/11	696.413	Meanpop
[19 9 -1 -11⟩	696.436	9-odd-limit least squares
12/11	696.474	Meanpop
16384/15309	696.502	1/5 septimal comma
5/4	696.578	5-, 7-, 9- and 11-odd-limit (meanpop) minimax, 1/4 comma
49/48	696.616
60/49	696.626
[-55 -11 1 25⟩	696.648	7-odd-limit least squares
11/9	696.713	11-, 13- and 15-odd-limit (tridecimal meantone) minimax
(18\31)	696.774
35/32	696.796
11/9	696.839	Meanpop
8/7	696.883
49/40	696.959
12/11	697.021	Undecimal meantone
7/5	697.085
15/11	697.158	Undecimal meantone
27/22	697.159
39/32	697.168	Grosstone
75/64	697.176
22/21	697.220	Undecimal meantone
14/13	697.242	13- and 15-odd-limit minimax (grosstone)
2187/2048	697.263	1/5 Pythagorean comma, Pythagorean apotome
13/10	697.289	Grosstone
11/8	697.295	Undecimal meantone
(43\74)	697.297
21/16	697.344
13/11	697.376	Meridetone
45927/32768	697.411	1/6 septimal comma
18/13	697.465	13-, 15-odd-limit minimax (meridetone)
16/13	696.467	Grosstone
11/10	697.500	Undecimal meantone
15/13	697.511	Grosstone
13/12	697.637	Meridetone
16/15	697.654	1/5 comma
(25\43)	697.674
64/63	697.728
21/20	697.781
40/33, 16/13	697.797	Undecimal meantone, meridetone
14/11	697.812	Undecimal meantone
15/13	697.830	Meridetone
18/13	697.966	Grosstone
13/10	698.009	Meridetone
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
33/28	698.272	Undecimal meantone
80/63	698.303
17/15	698.331
45/32	698.371	1/6 comma
33/26	698.407	Meanplop, meridetone
(39\67)	698.507
256/243	698.604	1/7 Pythagorean comma, Pythagorean limma
112/99	698.640	Undecimal meantone
45/34	698.661	[clarification needed]
(46\79)	698.734
13/11	698.801	Meridetone, meanplop
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
33/26	703.186	Tridecimal meantone
22/21	703.356	Meanpop
13/11	703.597	Tridecimal meantone
88/81	707.946	Meanpop
88/81	710.4335	Undecimal meantone

Links

• An Introduction to Historical Tunings, by Kyle Gann