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 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.

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.

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 tunings
  • Meanwoo12 – chromatic scale in {5/4, 7/1}-eigenmonzo tuning
  • Meanwoo19 – enharmonic scale in {5/4, 7/1}-eigenmonzo tuning
  • Ratwolf – chromatic scale with 20/13 wolf fifth
Others
  • Meaneb471a – chromatic scale in equal beating tuning

Tunings

Common meantone tunings can be classified into eigenmonzo 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 include TE, POTE, CTE, etc. For a more complete list, see the table below. These different tunings are referred to as "temperaments" in traditional terms.

Eigenmonzo tunings
Prime-optimized tunings
  • ~3/2 = 696.2387¢ – 5-limit POTE tuning
  • ~3/2 = 697.2143¢ – 5-limit CTE tuning
  • ~3/2 = 696.4949¢ – 7-limit POTE tuning
  • ~3/2 = 696.9521¢ – 7-limit CTE tuning
Other optimized tunings

Tuning spectrum

Edo
Generator
Eigenmonzo
(unchanged interval)
Generator
(¢)
Comments
567/512 688.323 1/2 septimal comma
[16 -10 690.225 1/2 Pythagorean comma, treating the eigenmonzo as a M2
51/38 690.603
[-19 9 0 2 691.049 2/5 septimal comma
10/9 691.202 1/2 comma
15\26 692.308
[31 -19 692.571 2/5 Pythagorean comma, treating the eigenmonzo as a m2
2048/1701 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, treating the eigenmonzo as a m3
[-23 11 0 2 694.165 2/7 septimal comma
56/45 694.651
28/27 694.709
81/70 694.732
11\19 694.737 Lower bound of 7- and 9-odd-limit diamond monotone
6/5 694.786 1/3 comma
5103/4095 695.139 1/4 septimal comma
[27 -17 695.252 2/7 Pythagorean comma, treating the eigenmonzo as an A1
35/27 695.389
51\88 695.455
1\2 + 1\(4π) 695.493 Lucy tuning
9/7 695.614
[math]f^4 = 2f + 2[/math] 695.630 Wilson fifth
40\69 695.652
25/24 695.810 2/7 comma
36/35 695.936
54/49 695.987
29\50 696.000
8192/6561 696.090 1/4 Pythagorean comma, treating the eigenmonzo as a M3
15/14 696.111
78125/73728 696.165 5-odd-limit least squares, 7/26 comma
(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
19/16 696.340 Meanpop mapping
17/16 696.344 Meanpop mapping
48/35 696.399
[19 9 -1 -11 696.436 9-odd-limit least squares
16384/15309 696.502 1/5 septimal comma
5/4 696.578 5-, 7-, and 9-odd-limit minimax, 1/4 comma
49/48 696.616
60/49 696.626
[-55 -11 1 25 696.648 7-odd-limit least squares
18\31 696.774
35/32 696.796
8/7 696.883
49/40 696.959
7/5 697.085
75/64 697.176 2/9 comma
2187/2048 697.263 1/5 Pythagorean comma, treating the eigenmonzo as a m2
43\74 697.297
21/16 697.344
(sqrt (10) - 2)\2 697.367 Tungsten meantone
45927/32768 697.411 1/6 septimal comma
16/15 697.654 1/5 comma
25\43 697.674
64/63 697.728
21/20 697.781
20/17 697.929 Treating the eigenmonzo as an A2
1024/729 698.045 1/6 Pythagorean comma, treating the eigenmonzo as an A4
[-17 9 0 1 698.060 1/7 septimal comma
28/25 698.099
32\55 698.182
80/63 698.303
17/15 698.331 Treating the eigenmonzo as a d3
45/32 698.371 1/6 comma
39\67 698.507
256/243 698.604 1/7 Pythagorean comma, treating the eigenmonzo as an A1
45/34 698.661 Treating the eigenmonzo as an A3
46\79 698.734
135/128 698.883 1/7 comma
17/16 699.009 Treating the eigenmonzo as a m2
25/21 699.384
24/17 699.500 Treating the eigenmonzo as an A4
18/17 699.851 Treating the eigenmonzo as an A1
7\12 700.000 Upper bound of 7- and 9-odd-limit diamond monotone
18/17 700.209 Treating the eigenmonzo as a m2
19/16 700.829 Treating the eigenmonzo as a m3
31\53 701.887
3/2 701.955 Pythagorean tuning
64/63 702.272
256/189 702.301

External links