Meantone

From Xenharmonic Wiki
Jump to navigation Jump to search
English Wikipedia has an article on:

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 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. However, even today, the vast majority of common-practice Western music theory is based exclusively on meantone.

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¢
Eigenmonzo (unchanged-interval) basis: 2.1875 (4/17-comma tuning)
Skewed-equilateral CSEE: ~3/2 = 696.4534¢
Eigenmonzo (unchanged-interval) basis: 2.48828125/3 (11/43-comma tuning)
Benedetti/Wilson CBE: ~3/2 = 697.3738¢
Eigenmonzo (unchanged-interval) basis: 2.[0 25 36 (36/169-comma tuning)
Skewed-Benedetti/Wilson CSBE: ~3/2 = 696.7868¢
Eigenmonzo (unchanged-interval) basis: 2.[0 5 31 (31/129-comma tuning)
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¢
Eigenmonzo (unchanged-interval) basis: 2.[0 1 4 10
Skewed-equilateral CSEE: ~3/2 = 696.7248¢
Eigenmonzo (unchanged-interval) basis: 2.4117715/9
Benedetti/Wilson CBE: ~3/2 = 697.0147¢
Eigenmonzo (unchanged-interval) basis: 2.[0 1225 1764 2250
Skewed-Benedetti/Wilson CSBE: ~3/2 = 696.6306¢
Eigenmonzo (unchanged-interval) basis: 2.[0 -3290 3171 7215

Tuning spectrum

Edo
Generator
Eigenmonzo
(Unchanged-interval)
Generator
(¢)
Comments
27/20 680.449 Full comma
4\7 685.714 Lower bound of 5-odd-limit diamond monotone
567/512 688.323 1/2 septimal comma
[16 -10 690.225 1/2 Pythagorean comma, as M2.
51/38 690.603
19\33 690.909
[-19 9 0 2 691.049 2/5 septimal comma
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
15\26 692.308
[31 -19 692.571 2/5 Pythagorean comma, as m2
1701/1024 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, as m3
[-23 11 0 2 694.165 2/7 septimal 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
5103/4095 695.139 1/4 septimal comma
[27 -17 695.252 2/7 Pythagorean comma, as A1
35/27 695.389
51\88 695.455
1\2 + 1\(4π) 695.493 Lucy tuning
9/7 695.614
f4 - 2f - 2 = 0 695.630 Wilson fifth
40\69 695.652
25/24 695.810 2/7 comma
36/35 695.936
695.981 5/18 comma
49/27 695.987
29\50 696.000
8192/6561 696.090 1/4 Pythagorean comma, as M3
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
15309/8192 696.502 1/5 septimal comma
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
2187/2048 697.263 1/5 Pythagorean comma, as m2
43\74 697.297
697.347 3/14 comma
21/16 697.344
(sqrt (10) - 2)\2 697.367 Tungsten meantone
45927/32768 697.411 1/6 septimal comma
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
1024/729 698.045 1/6 Pythagorean comma, as A4
[-17 9 0 1 698.060 1/7 septimal comma
25/14 698.099
32\55 698.182
63/40 698.303
17/15 698.331 As d3
45/32 698.371 1/6 comma
39\67 698.507
698.514 4/25 comma
256/243 698.604 1/7 Pythagorean comma, as A1
45/34 698.661 As A3
46\79 698.734
135/128 698.883 1/7 comma
53\91 698.901
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, virtually 1/12 Pythagorean comma
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
64/63 702.272
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]

Music

See Quarter-comma meantone #Music.

External links