Meantone

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Revision as of 10:57, 2 February 2021 by FloraC (talk | contribs) (Remove meanwoo in the spectrum as it doesn't take pure octave, and add the section "scales")
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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

The common meantone temperaments (recte tunings) include:

Scales

[5/4 7] eigenmonzos:

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

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