Meantone

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Meantone is a familiar historical temperament based on a chain of fifths (or fourths). The more technical part is discussed in meantone family in the context of the associated family of temperaments.

History

Meantone with fifths flatter than 700 ¢ were 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. However, even today, the vast majority of common-practice Western music theory is based exclusively on meantone, as 12-tone equal temperament is itself a meantone tuning.

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.

Septimal meantone

English Wikipedia has an article on:

Septimal meantone or 7-limit meantone is a natural extension of meantone which also addresses septimal intervals including but not limited to 7/4, 7/5, and 7/6. By extending the circle of fifths, consonant septimal intervals start to appear. For example, 7/4 is represented by an augmented sixth and is notably present in the augmented sixth chord.

See meantone vs meanpop for a comparison of undecimal (11-limit) extensions.

Other septimal extensions

There are some alternative mappings of the 7-limit meantone, including flattone, dominant and sharptone.

Flattone

Flattone is an alternative extension of meantone, which 7/4 is represented by a diminished seventh rather than augmented sixth. The fifth interval is adjusted narrower, nine of which give 8/7 and thirteen of which give 10/7 with octave reduction. Alternatively, stacking three minor thirds results in a diminished seventh that is close to 7/4.

Dominant

Dominant is an alternative extension of meantone, which 7/4 is represented by a minor seventh rather than augmented sixth. This results equating 6/5 with 7/6 and 5/4 with 9/7, or tempering out 36/35 (septimal quarter tone) and 64/63 (Archytas' comma).

Sharptone

Sharptone is a low-accuracy temperament which 7/4 is represented by a major sixth. This results equating 7/6 with 9/8 and 4/3 with 7/5, or tempering out 21/20 and 28/27.

Chords

Meantone induces didymic chords, the essentially tempered chords and associated progressions which are not found in other temperaments. Notably, the roots of the common chord progression vi-ii-V-I make up such a tetrad. Moreover, the dominant seventh chord and the half-diminished seventh chord can be seen as essentially tempered by septimal meantone.

Scales

EDO tunings
Eigenmonzo (unchanged-interval) tunings
  • Meanwoo12 – chromatic scale in 5/4.7-eigenmonzo tuning
  • Meanwoo19 – enharmonic scale in 5/4.7-eigenmonzo tuning
  • Ratwolf – chromatic scale with 20/13 wolf fifth
Others
  • Meaneb471a – chromatic scale in one equal beating tuning of ~3/1 and ~5/1
  • Meaneb471 – chromatic scale in the other equal beating tuning of ~3/1 and ~5/1, also called "metameantone"

Tunings

Common meantone tunings can be classified into eigenmonzo (unchanged-interval) tunings, edo tunings, prime-optimized tunings and others. In eigenmonzo tunings such as the quarter-comma meantone, a certain interval is tuned pure and certain others are equally off. Edo tunings like 31edo have rational size relationship between steps, and happen to send an additional comma to unison. Prime-optimized tunings are optimized for all intervals. For a more complete list, see the table below. These different tunings are referred to as "temperaments" in traditional terms.

Notable eigenmonzo (unchanged-interval) tunings
Other optimized tunings

Prime-optimized tunings

5-limit Prime-Optimized Tunings
Weight-skew\Order Euclidean
Tenney CTE: ~3/2 = 697.2143¢
Weil CWE: ~3/2 = 696.6512¢
Equilateral CEE: ~3/2 = 696.8947¢ (4/17-comma)
Skewed-equilateral CSEE: ~3/2 = 696.4534¢ (11/43-comma)
Benedetti/Wilson CBE: ~3/2 = 697.3738¢ (36/169-comma)
Skewed-Benedetti/Wilson CSBE: ~3/2 = 696.7868¢ (31/129-comma)
7-limit Prime-Optimized Tunings
Weight-skew\Order Euclidean
Tenney CTE: ~3/2 = 696.9521¢
Weil CWE: ~3/2 = 696.6562¢
Equilateral CEE: ~3/2 = 696.8843¢
Skewed-equilateral CSEE: ~3/2 = 696.7248¢
Benedetti/Wilson CBE: ~3/2 = 697.0147¢
Skewed-Benedetti/Wilson CSBE: ~3/2 = 696.6306¢

Tuning spectrum

The below tuning chart assumes septimal meantone and is agnostic to higher-limit extensions.

Edo
Generator
Eigenmonzo
(Unchanged-interval)
Generator
(¢)
Comments
27/20 680.449 Full comma (syntonic comma; from here onwards "comma" without an adjective refers to syntonic comma)
4\7 685.714 Lower bound of 5-odd-limit diamond monotone
51/38 690.603 As P4.
19\33 690.909 33cddd val
9/5 691.202 1/2 comma, tunings flatter than this do not fit the original sense of meantone, since their whole tones are no longer between 9/8 and 10/9, a.k.a. lower bound of 9-odd-limit diamond tradeoff
34\59 691.525 59bcddddd val
15\26 692.308
26\45 693.333 45dd val
27/25 693.352 2/5 comma
45/28 694.651
27/14 694.709
81/70 694.732
11\19 694.737 Lower bound of 7- and 9-odd-limit diamond monotone
5/3 694.786 1/3 comma
35/27 695.389
51\88 695.455 88dd val
1\2 + 1\(4π) 695.493 Lucy tuning
9/7 695.614
f4 - 2f - 2 = 0 695.630 1–3–5 equal-beating tuning, Wilson's "metameantone" (DR 4:5:6), virtually 5/17 comma
40\69 695.652 69d val
25/24 695.810 2/7 comma, virtually also DR 10:12:15
36/35 695.936
695.981 5/18 comma
49/27 695.987
29\50 696.000
15/14 696.111
78125/73728 696.165 7/26 comma, 5-odd-limit least squares
(8 - φ)\11 696.214 Golden meantone
49/45 696.245
19/17 696.279 Mediant of 9/8 and 10/9, known as classical meantone
47\81 696.296
7/6 696.319
19/16 696.340 As AAAA1
17/16 696.344 As AAA7
35/24 696.399
[19 9 -1 -11 696.436 9-odd-limit least squares
5/4 696.578 1/4 comma, 5-, 7-, and 9-odd-limit minimax
49/48 696.616
49/30 696.626
[-55 -11 1 25 696.648 7-odd-limit least squares
18\31 696.774
35/32 696.796
7/4 696.883
49/40 696.959
7/5 697.085
75/64 697.176 2/9 comma
f4 + 2f - 8 = 0 697.278 1–3–5 equal-beating tuning (DR 3:4:5), virtually 5/23 comma
43\74 697.297
697.347 3/14 comma
21/16 697.344
(sqrt (10) - 2)\2 697.367 Tungsten meantone
15/8 697.654 1/5 comma
25\43 697.674
64/63 697.728
21/20 697.781
17/10 697.929 As d7
25/14 698.099
32\55 698.182 55d val
63/40 698.303
17/15 698.331 As d3
45/32 698.371 1/6 comma
39\67 698.507 67d val
698.514 4/25 comma
45/34 698.661 As A3
46\79 698.734 79cdd val
135/128 698.883 1/7 comma
53\91 698.901 91cddd val
17/16 699.009 As m2
25/21 699.384
17/12 699.500 As d5
17/9 699.851 As d8
7\12 700.000 Upper bound of 7- and 9-odd-limit diamond monotone, 1/12 Pythagorean comma, virtually 1/11 syntonic comma (the difference is too small to appear in the digits provided here)
17/9 700.209 As M7
19/16 700.829 As m3
3/2 701.955 Pythagorean tuning, tunings sharper than this do not fit the original sense of meantone, since their whole tones are no longer between 9/8 and 10/9, a.k.a. upper bound of 5-, 7-, and 9-odd-limit diamond tradeoff
3\5 720.000 Upper bound of 5-odd-limit diamond monotone

Formula for n-comma meantone

The generator g of n-comma meantone, where n is a fraction (like 1/5, 2/9, etc.), can be found by

[math]\displaystyle g = g_J - ng_c[/math]

where gJ = 701.955001 cents is the size of the just perfect fifth, and gc = 21.506290 cents is the size of the syntonic comma.

Conversely, n can be found by

[math]\displaystyle n = (g_J - g)/g_c[/math]

Other tunings

  • DKW (2.3.5): ~2 = 1\1, ~3/2 = 696.353

Music

See Quarter-comma meantone #Music.

External links