Meantone

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Meantone temperament

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

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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 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

• Meantone5 – pentic scale in 31edo
• Meantone7 – diatonic scale in 19edo and 31edo
• Meantone12 – chromatic scale in 31edo

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 include various eigenmonzo tunings such as the quarter-comma meantone, edo tunings like 31edo, and otherwise optimized tunings like the POTE tuning, shown below. For a more complete list, see the table below. These different tunings are referred to as "temperaments" in traditional terms.

Eigenmonzo tunings

• 1/2 syntonic comma meantone – with eigenmonzo 10/9
• 1/3 syntonic comma meantone – with eigenmonzo 5/3
• 2/7 syntonic comma meantone – with eigenmonzo 25/24
• 1/4 syntonic comma meantone – with eigenmonzo 5/4
• 1/5 syntonic comma meantone – with eigenmonzo 15/8
• 1/6 syntonic comma meantone – with eigenmonzo 45/32

Edo tunings

• 19\33
• 15\26
• 26\45
• 11\19
• 29\50
• 18\31
• 61\105
• 25\43
• 32\55
• 7\12

POTE tunings

• ~3/2 = 696.239¢ – 5-limit meantone
• ~3/2 = 696.495¢ – 7-limit meantone

Other optimized tunings

• Golden meantone
• Tungsten meantone
• Lucy tuning

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
	6/5, 25/18	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]\displaystyle{ 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
	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
	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
	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	[clarification needed]
	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	[clarification needed]
46\79		698.734
99\170		698.824
	135/128	698.883	1/7 comma
	17/16	699.009	[clarification needed]
	25/21	699.384
7\12		700.000
	18/17	700.209	[clarification needed]
	19/16	700.829	[clarification needed]
31\53		701.887
	3/2	701.955	Pythagorean tuning
	64/63	702.272
	256/189	702.301

External links

• An Introduction to Historical Tunings, by Kyle Gann