Meantone
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
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
- Meantone5 – pentic scale in 31edo
- Meantone7 – diatonic scale in 19edo and 31edo
- Meantone12 – chromatic scale in 31edo
- 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
- 1/2-comma meantone – with eigenmonzo 10/9
- 1/3-comma meantone – with eigenmonzo 5/3
- 2/7-comma meantone – with eigenmonzo 25/24
- 1/4-comma meantone – with eigenmonzo 5/4
- 1/5-comma meantone – with eigenmonzo 15/8
- 1/6-comma meantone – with eigenmonzo 45/32
- Ratwolf tuning
- Other optimized tunings
- Golden meantone
- Tungsten meantone
- Mercury meantone
- Lucy tuning
- Equal beating tunings
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) |
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 | ||
f^{4} - 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 | |
f^{4} + 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 g_{J} = 701.955001 cents is the size of the just perfect fifth, and g_{c} = 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.