Meantone

Meantone

Subgroups	2.3.5, 2.3.5.7
Comma basis	81/80 (2.3.5); 81/80, 126/125 (2.3.5.7)
Reduced mapping	⟨1; 1 4 10]
ET join	12 & 19
Generators (CWE)	~3/2 = 696.7 ¢
MOS scales	2L 3s, 5L 2s, 7L 5s, 12L 7s
Ploidacot	monocot
Pergen	(P8, P5)
Color name	Guti
Minimax error	5-odd-limit: 5.4 ¢; 9-odd-limit: 10.8 ¢
Target scale size	5-odd-limit: 5 notes; 9-odd-limit: 12 notes

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.

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.

English Wikipedia has an article on:

Meantone temperament

History

See also: Historical temperaments

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

English Wikipedia has an article on:

Septimal meantone temperament

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

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. 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

Main article: Dominant (temperament)

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

Main article: Meantone intervals

In the following tables, odd harmonics 1–15 are labeled in bold.

Intervals fifthward

#	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

Intervals fourthward

#	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

Main article: 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

5-limit 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 ¢

7-limit norm-based tunings

	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 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 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

$$ n = (g_J - g)/g_c $$

Music

See Quarter-comma meantone #Music.

See also

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

External links

• An Introduction to Historical Tunings, by Kyle Gann