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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.

English Wikipedia has an article on:

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:
See also: Meantone family § 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

Main article: 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.

Dominant

Main article: Dominant (temperament)

Dominant is an alternative extension of meantone, which represents 7/4 isas 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).

Sharptone

See also: Meantone family § Sharptone

Sharptone is a low-accuracy temperament (exotemperament) which represents 7/4 as a major sixth. This equates 7/6 with 9/8 and 4/3 with 7/5, 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
Euclidean
Unskewed Skewed
Equilateral CEE: ~3/2 = 696.895¢
(4/17-comma) 		CSEE: ~3/2 = 696.453¢
(11/43-comma)
Tenney CTE: ~3/2 = 697.214¢ CWE: ~3/2 = 696.651¢
Benedetti,
Wilson 		CBE: ~3/2 = 697.374¢
(36/169-comma) 		CSBE: ~3/2 = 696.787¢
(31/129-comma)
7-limit Prime-Optimized Tunings
Euclidean
Unskewed Skewed
Equilateral CEE: ~3/2 = 696.884¢ CSEE: ~3/2 = 696.725¢
Tenney CTE: ~3/2 = 696.952¢ CWE: ~3/2 = 696.656¢
Benedetti,
Wilson 		CBE: ~3/2 = 697.015¢ CSBE: ~3/2 = 696.631¢

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
1875/1024 696.895 4/17 comma; 2.3.5 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
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
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 meantone (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

* Besides the octave

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

Other tunings

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

Scales

5-note Meantone scales

! meantone5.scl
!
Meantone[5] in 31edo tuning
5
!
193.54839
387.09677
696.77419
890.32258
2/1

Mode: 4|0

Suggested keyboard mappings:

  • Central C, providing C-G-D-A-E

7-note Meantone scales

! meantone7.scl
!
Meantone[7] in 31edo tuning
7
!
193.54839
387.09677
503.22581
696.77419
890.32258
1083.87097
2/1

! meantone7-19.scl
!
Meantone[7] in 19edo tuning
7
!
189.47368
378.94737
505.26316
694.73684
884.21053
1073.68421
2/1

Mode: 5|1 (Ionian, LLsLLLs)

Suggested keyboard mappings:

  • Central C, providing notes from F through B.

Fifth-equivalent

! meantone-fifths7.scl
!
Meantone-fifths[7] fifths-repetition MOS, pure 2 and 5 (1/4 comma)
7
!
117.10786
193.15686
310.26471
386.31371
503.42157
620.52943
696.57843

12-note Meantone scales

! meantone12.scl
!
Meantone[12] in 31edo tuning
12
!
116.12903
193.54839
309.67742
387.09677
503.22581
580.64516
696.77419
812.90323
890.32258
1006.45161
1083.87097
2/1

Mode: 5|6 (LsLsLsLLsLsL)

Suggested keyboard mappings:

  • Central G, providing notes from A♭ through C♯
  • Central D, providing notes from E♭ through G♯
  • Central A, providing notes from B♭ through D♯

11-note Meantone scales (fifth-equivalent)

! meantone-fifths11.scl
!
Meantone-fifths[11] fifths-repetition MOS, pure 2 and 5 (1/4 comma)
11
!
76.04900
117.10786
193.15686
269.20586
310.26471
386.31371
427.37257
503.42157
579.47057
620.52943
696.57843

91edo Meantone scales

Meantone[43]

Is also Domineering[43]. 

!
Rank 2 scale (698.9010989010989, 1200.)
 43
!
 13.186812
 79.120881
 92.307693
 105.494505
 118.681317
 184.615386
 197.802198
 210.989010
 276.923079
 290.109891
 303.296703
 316.483515
 382.417584
 395.604396
 408.791208
 421.978020
 487.912089
 501.098901
 514.285713
 580.219782
 593.406594
 606.593406
 619.780218
 685.714287
 698.901099
 712.087911
 778.021980
 791.208792
 804.395604
 817.582416
 883.516485
 896.703297
 909.890109
 923.076921
 989.010990
 1002.197802
 1015.384614
 1081.318683
 1094.505495
 1107.692307
 1120.879119
 1186.813188
 1200.

Meantone[55]

Is also Domineering[55].

!
Rank 2 scale (698.9010989010989, 1200.)
 55
!
 13.186812
 26.373624
 79.120881
 92.307693
 105.494505
 118.681317
 171.428574
 184.615386
 197.802198
 210.989010
 224.175822
 276.923079
 290.109891
 303.296703
 316.483515
 329.670327
 382.417584
 395.604396
 408.791208
 421.978020
 474.725277
 487.912089
 501.098901
 514.285713
 527.472525
 580.219782
 593.406594
 606.593406
 619.780218
 672.527475
 685.714287
 698.901099
 712.087911
 725.274723
 778.021980
 791.208792
 804.395604
 817.582416
 870.329673
 883.516485
 896.703297
 909.890109
 923.076921
 975.824178
 989.010990
 1002.197802
 1015.384614
 1028.571426
 1081.318683
 1094.505495
 1107.692307
 1120.879119
 1173.626376
 1186.813188
 1200.

Subsets

Tempered-octave scales

12 notes

! meanwoo12.scl
!
[5/4 7] eigenmonzo meantone, Eb to G# gamut
12
!
75.28859
193.15686
311.02512
386.31371
504.18198
579.47057
697.33883
772.62743
890.49569
1008.36395
1083.65255
1201.52081

Mode: 3|8 (sLLsLsLsLLsL)

Suggested keyboard mappings:

  • Central C, providing notes from Eb to G#.

19 notes

! meanwoo19.scl
!
[5/4 7] eigenmonzo meantone, Gb to B# gamut
19
!
75.28859
117.86826
193.15686
268.44545
311.02512
386.31371
461.60231
504.18198
579.47057
622.05024
697.33883
772.62743
815.20710
890.49569
965.78428
1008.36395
1083.65255
1158.94114
1201.52081

Music

See Quarter-comma meantone #Music.

See also

  • Angel – fifth-equivalent or 5/1-equivalent meantone

External links

Retrieved from "https://en.xen.wiki/index.php?title=Meantone&oldid=186739"