Meantone
Meantone is a familiar historical temperament based on a chain of fifths (or fourths), possessing 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, 81/80, which means that stacking four fifths (such as C – G – D – A – E) results in a major third (C–E) that is close to the just interval 5/4 rather than the more complex Pythagorean interval 81/64; good tunings of meantone also lead to soft diatonic and chromatic scales, which are desirable for interval categorization.
| Meantone |
81/80, 126/125 (2.3.5.7)
9-odd-limit: 10.8 ¢
9-odd-limit: 12 notes
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.
Technical temperament data is discussed at Meantone family #Meantone in the context of the associated family of temperaments.
History
Meantone tunings 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.
Extensions
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 (+10 fifths, C–A♯), and is notably present in the augmented sixth chord; it can also be seen as a diesis-flat minor seventh, as the diesis represents 36/35~64/63. In septimal meantone, 7/5 is an augmented fourth, 7/6 is an augmented second, and 9/7 is a diminished fourth. Notably, septimal meantone equates the interval of a diminished fifth between the third and the seventh of a dominant seventh chord to 10/7, making it a 9-odd-limit essentially tempered chord. Septimal meantone is best tuned close to 31edo or 1/4-comma.
Extending meantone to the 11-limit is not as simple. For one, there is the factorization of 81/80 as (121/120)*(243/242), and tempering both out leads to mohaha in the 2.3.5.11 subgroup, which splits the perfect fifth into two 11/9~27/22 neutral thirds. Adding back the septimal meantone mapping of 7 (+20 neutral thirds) gives migration, but mohaha has an alternative mapping of 7/4 at the semi-diminished seventh (-13 neutral thirds), known as mohajira. Extensions to prime 11 generated by the perfect fifth are trickier. If 121/120 and 243/242 are not tempered out, then one of them must be mapped positively, and the other negatively. Since 121/120 is the difference between 11/10 and 12/11, it makes more sense to map it positively, and thus 243/242 negatively, leading 11/9 to be mapped wider than 27/22 and causing inconsistencies. Nonetheless, 31edo supports septimal meantone well while also having a neutral third, and there are two extensions generated by the fifth which map 11/9 to the neutral third. Undecimal meantone (also known as huygens) maps 11/9 to +16 fifths (C–D𝄪) and 11/8 to +18 fifths (C–E𝄪), tempering out 99/98, 176/175, and 441/440. Huygens works in the range from 31edo (696.8 ¢) to 12edo (700 ¢). The other extension is meanpop, which maps 11/9 to -15 fifths (C–F𝄫) and 11/8 to -13 fifths (C–G𝄫), tempering out 385/384 and 540/539. Tunings of meanpop range from 19edo (694.7 ¢) to 31edo (696.8 ¢).
Other septimal extensions
There are some alternative mappings of the 7-limit meantone, including flattone and dominant.
Flattone
Flattone is an alternative extension of meantone, which represents 7/4 as a diminished seventh rather than an 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. While less accurate than septimal meantone, flattone extends much more easily to the 11- and 13-limits, with 11/8 being an augmented fourth (+6 fifths, C–F♯) and 13/8 being a minor sixth (-4 fifths, C–A♭).
Dominant
Dominant is an alternative extension of meantone, which represents 7/4 as a minor seventh rather than an augmented sixth. This equates 6/5 with 7/6 and 5/4 with 9/7, tempering out 36/35 (septimal quarter tone) and 64/63 (Archytas' comma). Dominant was named because the dominant seventh chord of the diatonic scale represents 4:5:6:7 in it.
Intervals
In the following tables, odd harmonics 1–15 are labeled in bold.
| # | Category | Cents* | Approximate ratios |
|---|---|---|---|
| 0 | P1 | 0.0 | 1/1 |
| 1 | P5 | 696.7 | 3/2 |
| 2 | M2 | 193.3 | 9/8, 10/9, 28/25 |
| 3 | M6 | 890.0 | 5/3 |
| 4 | M3 | 386.6 | 5/4 |
| 5 | M7 | 1083.3 | 15/8, 28/15 |
| 6 | A4 | 579.9 | 7/5, 25/18 |
| 7 | A1 | 76.6 | 21/20, 25/24, 28/27 |
| 8 | A5 | 773.2 | 14/9, 25/16 |
| 9 | A2 | 269.9 | 7/6 |
| 10 | A6 | 966.6 | 7/4 |
| 11 | A3 | 463.2 | 21/16 |
| 12 | A7 | 1159.9 | 35/18, 49/25, 63/32 |
| # | Category | Cents* | Approximate ratios |
|---|---|---|---|
| 0 | P1 | 0.0 | 1/1 |
| −1 | P4 | 503.3 | 4/3 |
| −2 | m7 | 1006.7 | 9/5, 16/9, 25/14 |
| −3 | m3 | 310.0 | 6/5 |
| −4 | m6 | 813.4 | 8/5 |
| −5 | m2 | 116.7 | 15/14, 16/15 |
| −6 | d5 | 620.1 | 10/7, 36/25 |
| −7 | d8 | 1123.4 | 27/14, 40/21, 48/25 |
| −8 | d4 | 426.8 | 9/7, 32/25 |
| −9 | d7 | 930.1 | 12/7 |
| −10 | d3 | 233.4 | 8/7 |
| −11 | d6 | 736.8 | 32/21 |
| −12 | d2 | 40.1 | 36/35, 50/49, 64/63 |
* In 7-limit CWE tuning, octave reduced
Chords and harmony
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 31edo
- Meantone12 – chromatic scale in 31edo
- Unchanged-interval (eigenmonzo) 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, norm-based 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. Norm-based 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
- 5-limit DKW: ~2 = 1200.000 ¢, ~3/2 = 696.353 ¢
Norm-based tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~3/2 = 696.8947 ¢ (4/17 comma) |
CSEE: ~3/2 = 696.4534 ¢ (11/43 comma) |
POEE: ~3/2 = 695.2311 ¢ |
| Tenney | CTE: ~3/2 = 697.2143 ¢ | CWE: ~3/2 = 696.6512 ¢ | POTE: ~3/2 = 696.2387 ¢ |
| Benedetti, Wilson |
CBE: ~3/2 = 697.3738 ¢ (36/169 comma) |
CSBE: ~3/2 = 696.7868 ¢ (31/129 comma) |
POBE: ~3/2 = 696.2984 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~3/2 = 696.8843 ¢ | CSEE: ~3/2 = 696.7248 ¢ | POEE: ~3/2 = 696.4375 ¢ |
| Tenney | CTE: ~3/2 = 696.9521 ¢ | CWE: ~3/2 = 696.6562 ¢ | POTE: ~3/2 = 696.4949 ¢ |
| Benedetti, Wilson |
CBE: ~3/2 = 697.0147 ¢ | CSBE: ~3/2 = 696.6306 ¢ | POBE: ~3/2 = 696.4596 ¢ |
Target tunings
| Target | Minimax | Least squares | ||
|---|---|---|---|---|
| Generator | Eigenmonzo* | Generator | Eigenmonzo* | |
| 5-odd-limit | ~3/2 = 696.578 ¢ (1/4 comma) |
5/4 | ~3/2 = 696.165 ¢ (7/26 comma) |
[-13 -2 7⟩ |
| 7-odd-limit | ~3/2 = 696.578 ¢ | 5/4 | ~3/2 = 696.648 ¢ | [-55 -11 1 25⟩ |
| 9-odd-limit | ~3/2 = 696.578 ¢ | 5/4 | ~3/2 = 696.436 ¢ | [19 9 -1 -11⟩ |
Tuning spectrum
The below tuning chart assumes septimal meantone and is agnostic to higher-limit extensions.
| Edo generator |
Unchanged interval (eigenmonzo)* |
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 | 26d val | |
| 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, lower bound of 5- and 7-odd-limit diamond tradeoff | |
| 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 | ||
| 3125/2304 | 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 | ||
| 35/24 | 696.399 | ||
| 5/4 | 696.578 | 1/4 comma, 5-, 7-, and 9-odd-limit minimax | |
| 49/48 | 696.616 | ||
| 49/30 | 696.626 | ||
| 18\31 | 696.774 | ||
| 35/32 | 696.796 | ||
| 7/4 | 696.883 | ||
| 1875/1024 | 696.895 | 4/17-comma; 5-limit CEE tuning | |
| 49/40 | 696.959 | ||
| 7/5 | 697.085 | ||
| 61\105 | 697.143 | ||
| 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 | ||
| 21/16 | 697.344 | ||
| 1125/1024 | 697.347 | 3/14-comma | |
| (√(10) − 2)\2 | 697.367 | Tungsten meantone | |
| 68\117 | 697.436 | 117d val | |
| 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 | |
| 57\98 | 697.959 | 98d val | |
| 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 | |
| [-23 9 4⟩ | 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-comma† | |
| 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 |
* Besides the octave
† The difference is too small to appear in the digits provided here
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
$$ g = g_J - ng_c $$
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
$$ n = (g_J - g)/g_c $$
Music
See also
- Angel – fifth-equivalent or 5/1-equivalent meantone