31 equal temperament

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File:Syntonic tuning continuum.svg
31 Template:Sc on the regular diatonic tuning continuum at p5 = 696.77 cents[1]

In music, 31 equal temperament, 31 Template:Sc, which can also be abbreviated 31 Template:Sc (31 tone Template:Sc) or 31 [[equal division of the octave|Template:Sc]] (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equally-proportioned steps (equal frequency ratios). Template:ErrorTemplate:Category handler Each step represents a frequency ratio of Template:Radic, or 38.71 cents (Template:ErrorTemplate:Category handler).

31 Template:Sc is a very good approximation of quarter-comma meantone temperament. More generally, it is a regular diatonic tuning in which the tempered perfect fifth is equal to 696.77 cents, as shown in Figure 1. On an isomorphic keyboard, the fingering of music composed in 31 Template:Sc is precisely the same as it is in any other syntonic tuning (such as [[12 equal temperament|12 Template:Sc]]), so long as the notes are spelled properly—that is, with no assumption of enharmonicity.

History and use

Division of the octave into 31 steps arose naturally out of Renaissance music theory; the lesser diesisTemplate:Px2 – the ratio of an octave to three major thirds, 128:125 or 41.06 cents – was approximately one-fifth of a tone or two-fifths of a semitone. In 1555, Nicola Vicentino proposed an extended-meantone tuning of 31 tones. In 1666, Lemme Rossi first proposed an equal temperament of this order. In 1691, having discovered it independently, scientist Christiaan Huygens wrote about it also.[2] Since the standard system of tuning at that time was quarter-comma meantone, in which the fifth is tuned to Template:Radic, the appeal of this method was immediate, as the fifth of 31 Template:Sc, at 696.77 cents, is only 0.19 cent wider than the fifth of quarter-comma meantone. Huygens not only realized this, he went farther and noted that 31 Template:Sc provides an excellent approximation of septimal, or 7 limit harmony.

In the twentieth century, physicist, music theorist, and composer Adriaan Fokker, after reading Huygens's work, led a revival of interest in this system of tuning which led to a number of compositions, particularly by Dutch composers. Fokker designed the Fokker organ, a 31 tone equal-tempered organ, which was installed in Teyler's Museum in Haarlem in 1951 and moved to Muziekgebouw aan 't IJ in 2010 where it has been frequently used in concerts since it moved.

Interval size

File:31ed2.svg
19 limit just intonation intervals approximated in 31 Template:Sc

Here are the sizes of some common intervals:

interval name size
(steps)Script error: No such module "Check for unknown parameters". 		size
(cents)Script error: No such module "Check for unknown parameters". 		Template:Sc
audio 		just
ratio 		just
(cents)Script error: No such module "Check for unknown parameters". 		Template:Sc
audio 		error
(cents)Script error: No such module "Check for unknown parameters".
octave 31 1200 2:1 1200 0
minor seventh 26 1006.45 9:5 1017.60 −11.15
grave just minor seventh 26 1006.45 16:9 996.09 +10.36
harmonic seventh, subminor seventh, augmented sixth 25 967.74 Template:ErrorTemplate:Category handler 7:4 968.83 Template:ErrorTemplate:Category handler −1.09
minor sixth 21 812.90 Template:ErrorTemplate:Category handler 8:5 813.69 Template:ErrorTemplate:Category handler −0.78
perfect fifth 18 696.77 Template:ErrorTemplate:Category handler 3:2 701.96 Template:ErrorTemplate:Category handler −5.19
greater septimal tritone, diminished fifth 16 619.35 10:7 617.49 +1.87
lesser septimal tritone, augmented fourth 15 580.65 Template:ErrorTemplate:Category handler 7:5 582.51 Template:ErrorTemplate:Category handler −1.86
undecimal tritone, half augmented fourth, 11th harmonic 14 541.94 Template:ErrorTemplate:Category handler 11:8 551.32 Template:ErrorTemplate:Category handler −9.38
perfect fourth 13 503.23 Template:ErrorTemplate:Category handler 4:3 498.04 Template:ErrorTemplate:Category handler +5.19
septimal narrow fourth, half diminished fourth 12 464.52 Template:ErrorTemplate:Category handler 21:16 470.78 Template:ErrorTemplate:Category handler −6.26
tridecimal augmented third, and greater major third 12 464.52 Template:ErrorTemplate:Category handler 13:10 454.21 Template:ErrorTemplate:Category handler +10.31
septimal major third 11 425.81 Template:ErrorTemplate:Category handler 9:7 435.08 Template:ErrorTemplate:Category handler −9.27
diminished fourth 11 425.81 Template:ErrorTemplate:Category handler 32:25 427.37 Template:ErrorTemplate:Category handler −1.56
undecimal major third 11 425.81 Template:ErrorTemplate:Category handler 14:11 417.51 Template:ErrorTemplate:Category handler +8.30
major third 10 387.10 Template:ErrorTemplate:Category handler 5:4 386.31 Template:ErrorTemplate:Category handler +0.79
tridecimal neutral third 9 348.39 Template:ErrorTemplate:Category handler 16:13 359.47 Template:ErrorTemplate:Category handler −11.09
undecimal neutral third 9 348.39 Template:ErrorTemplate:Category handler 11:9 347.41 Template:ErrorTemplate:Category handler +0.98
minor third 8 309.68 Template:ErrorTemplate:Category handler 6:5 315.64 Template:ErrorTemplate:Category handler −5.96
septimal minor third 7 270.97 Template:ErrorTemplate:Category handler 7:6 266.87 Template:ErrorTemplate:Category handler +4.10
septimal whole tone 6 232.26 Template:ErrorTemplate:Category handler 8:7 231.17 Template:ErrorTemplate:Category handler +1.09
whole tone, major tone 5 193.55 Template:ErrorTemplate:Category handler 9:8 203.91 Template:ErrorTemplate:Category handler −10.36
whole tone, major second 5 193.55 Template:ErrorTemplate:Category handler 28:25 196.20 −2.65
mean tone, major second 5 193.55
  1. REDIRECT Template:Radic

Template:Rcat shell:2

193.16 +0.39
whole tone, minor tone 5 193.55 Template:ErrorTemplate:Category handler 10:9 182.40 Template:ErrorTemplate:Category handler +11.15
greater undecimal neutral second 4 154.84 Template:ErrorTemplate:Category handler 11:10 165.00 −10.16
lesser undecimal neutral second 4 154.84 Template:ErrorTemplate:Category handler 12:11 150.64 Template:ErrorTemplate:Category handler +4.20
septimal diatonic semitone 3 116.13 Template:ErrorTemplate:Category handler 15:14 119.44 Template:ErrorTemplate:Category handler −3.31
diatonic semitone, minor second 3 116.13 Template:ErrorTemplate:Category handler 16:15 111.73 Template:ErrorTemplate:Category handler +4.40
septimal chromatic semitone 2 77.42 Template:ErrorTemplate:Category handler 21:20 84.47 Template:ErrorTemplate:Category handler −7.05
chromatic semitone, augmented unison 2 77.42 Template:ErrorTemplate:Category handler 25:24 70.67 Template:ErrorTemplate:Category handler +6.75
lesser diesis 1 38.71 Template:ErrorTemplate:Category handler 128:125 41.06 Template:ErrorTemplate:Category handler −2.35
undecimal diesis 1 38.71 Template:ErrorTemplate:Category handler 45:44 38.91 Template:ErrorTemplate:Category handler −0.20
septimal diesis 1 38.71 Template:ErrorTemplate:Category handler 49:48 35.70 Template:ErrorTemplate:Category handler +3.01

The 31 equal temperament has a very close fit to the 7:6, 8:7, and 7:5 ratios, which have no approximate fits in 12 equal temperament and only poor fits in 19 equal temperament. The composer Joel Mandelbaum (born 1932) used this tuning system specifically because of its good matches to the 7th and 11th partials in the harmonic series.Template:Refn The tuning has poor matches to both the 9:8 and 10:9 intervals (major and minor tone in just intonation); however, it has a good match for the average of the two. Practically it is very close to quarter-comma meantone.

This tuning can be considered a meantone temperament. It has the necessary property that a chain of its four fifths is equivalent to its major third (the syntonic comma 81:80 is tempered out), which also means that it contains a "meantone" that falls between the sizes of 10:9 and 9:8 as the combination of one of each of its chromatic and diatonic semitones.

Scale diagram

File:31edo CoF semi and sesqui.png
Circle of fifths in 31 equal temperament

The following are the 31 notes in the scale:

Interval (cents) 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39
Note
name 		A G𝄪♯
B𝄫 		A♯ B♭ A𝄪
C𝄫 		B C♭ B♯ C B𝄪
D𝄫 		C♯ D♭ C𝄪
E♭𝄫 		D C𝄪♯
E𝄫 		D♯ E♭ D𝄪
F𝄫 		E F♭ E♯ F E𝄪
G𝄫 		F♯ G♭ F𝄪
A♭𝄫 		G F𝄪♯
A𝄫 		G♯ A♭ G𝄪
B♭𝄫 		A
Note (cents)  0   39 77 116 155 194 232 271 310 348 387 426 465 503 542 581 619 658 697 735 774 813 852 890 929 968 1006 1045 1084 1123 1161 1200

The five "double flat" notes and five "double sharp" notes may be replaced by half sharps and half flats, similar to the quarter tone system:

Interval (cents) 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39
Note name A Ahalf sharp A♯ B♭ Bhalf flat B Bhalf sharp Chalf flat C Chalf sharp C♯ D♭ Dhalf flat D Dhalf sharp D♯ E♭ Ehalf flat E Ehalf sharp Fhalf flat F Fhalf sharp F♯ G♭ Ghalf flat G Ghalf sharp G♯ A♭ Ahalf flat A
Note (cents) 0 39 77 116 155 194 232 271 310 348 387 426 465 503 542 581 619 658 697 735 774 813 852 890 929 968 1006 1045 1084 1123 1161 1200

Modes

Ionian mode (major scale)

Key signature Scale Number of
sharps 		Key signature Scale Number of
flats
C major C D E F G A B 0 (no sharps or flats)
G major G A B C D E F♯ 1
D major D E F♯ G A B C♯ 2
A major A B C♯ D E F♯ G♯ 3
E major E F♯ G♯ A B C♯ D♯ 4
B major B C♯ D♯ E F♯ G♯ A♯ 5
F♯ major F♯ G♯ A♯ B C♯ D♯ E♯ 6
C♯ major C♯ D♯ E♯ F♯ G♯ A♯ B♯ 7
G♯ major G♯ A♯ B♯ C♯ D♯ E♯ F𝄪 8
D♯ major D♯ E♯ F𝄪 G♯ A♯ B♯ C𝄪 9
A♯ major A♯ B♯ F𝄪 D♯ E♯ F𝄪 G𝄪 10 C♭𝄫 major C♭𝄫 D♭𝄫 E♭𝄫 F♭𝄫 G♭𝄫 A♭𝄫 B♭𝄫 21
E♯ major E♯ F𝄪 G𝄪 A♯ B♯ F𝄪 D𝄪 11 G♭𝄫 major G♭𝄫 A♭𝄫 B♭𝄫 C♭𝄫 D♭𝄫 E♭𝄫 F𝄫 20
B♯ major B♯ C𝄪 D𝄪 E♯ F𝄪 G𝄪 A𝄪 12 D♭𝄫 major D♭𝄫 E♭𝄫 F𝄫 G♭𝄫 A♭𝄫 B♭𝄫 C𝄫 19
F𝄪 major F𝄪 G𝄪 A𝄪 B♯ C𝄪 D𝄪 E𝄪 13 A♭𝄫 major A♭𝄫 B♭𝄫 C𝄫 D♭𝄫 E♭𝄫 F𝄫 G𝄫 18
C𝄪 major C𝄪 D𝄪 E𝄪 F𝄪 G𝄪 A𝄪 B𝄪 14 E♭𝄫 major E♭𝄫 F𝄫 G𝄫 A♭𝄫 B♭𝄫 C𝄫 D𝄫 17
G𝄪 major G𝄪 A𝄪 B𝄪 C𝄪 D𝄪 E𝄪 F♯𝄪 15 B♭𝄫 major B♭𝄫 C𝄫 D𝄫 E♭𝄫 F𝄫 G𝄫 A𝄫 16
D𝄪 major D𝄪 E𝄪 F♯𝄪 G𝄪 A𝄪 B𝄪 C♯𝄪 16 F𝄫 major F𝄫 G𝄫 A𝄫 B♭𝄫 C𝄫 D𝄫 E𝄫 15
A𝄪 major A𝄪 B𝄪 C♯𝄪 D𝄪 E𝄪 F♯𝄪 G♯𝄪 17 C𝄫 major C𝄫 D𝄫 E𝄫 F𝄫 G𝄫 A𝄫 B𝄫 14
E𝄪 major E𝄪 F♯𝄪 G♯𝄪 A𝄪 B𝄪 C♯𝄪 D♯𝄪 18 G𝄫 major G𝄫 A𝄫 B𝄫 C𝄫 D𝄫 E𝄫 F♭ 13
B𝄪 major B𝄪 C♯𝄪 D♯𝄪 E𝄪 F♯𝄪 G♯𝄪 A♯𝄪 19 D𝄫 major D𝄫 E𝄫 F♭ G𝄫 A𝄫 B𝄫 C♭ 12
F♯𝄪 major F♯𝄪 G♯𝄪 A♯𝄪 B𝄪 C♯𝄪 D♯𝄪 E♯𝄪 20 A𝄫 major A𝄫 B𝄫 C♭ D𝄫 E𝄫 F♭ G♭ 11
C♯𝄪 major C♯𝄪 D♯𝄪 E♯𝄪 F♯𝄪 G♯𝄪 A♯𝄪 B♯𝄪 21 E𝄫 major E𝄫 F♭ G♭ A𝄫 B𝄫 C♭ D♭ 10
B𝄫 major B𝄫 C♭ D♭ E𝄫 F♭ G♭ A♭ 9
F♭ major F♭ G♭ A♭ B𝄫 C♭ D♭ E♭ 8
C♭ major C♭ D♭ E♭ F♭ G♭ A♭ B♭ 7
G♭ major G♭ A♭ B♭ C♭ D♭ E♭ F 6
D♭ major D♭ E♭ F G♭ A♭ B♭ C 5
A♭ major A♭ B♭ C D♭ E♭ F G 4
E♭ major E♭ F G A♭ B♭ C D 3
B♭ major B♭ C D E♭ F G A 2
F major F G A B♭ C D E 1
C major C D E F G A B 0 (no flats or sharps)
Comparison between Template:SfracScript error: No such module "Check for unknown parameters". comma meantone and 31 Template:Sc (values in cents, rounded to 2 decimal places)
C C♯ D♭ D D♯ E♭ E E♯ F F♯ G♭ G G♯ A♭ A A♯ B♭ B C♭ C
Template:SfracScript error: No such module "Check for unknown parameters". comma: 0.00 76.05 117.11 193.16 269.21 310.26 386.31 462.36 503.42 579.47 620.53 696.58 772.63 813.69 889.74 965.78 1006.84 1082.89 1123.95 1200.00
31 Template:Sc: 0.00 77.42 116.13 193.55 270.97 309.68 387.10 464.52 503.23 580.65 619.35 696.77 774.19 812.90 890.32 967.74 1006.45 1083.87 1122.58 1200.00

Chords of 31 equal temperament

Many chords of 31 Template:Sc are discussed in the article on septimal meantone temperament. Chords not discussed there include the neutral thirds triad (Template:ErrorTemplate:Category handler), which might be written C–Ehalf flat–G, C–D𝄪–G or C–F𝄫–G, and the Orwell tetrad, which is C–E–F𝄪–B𝄫.

File:Simple I-IV-V-I isomorphic 31-TET.png
I–IV–V–I chord progression in 31 tone equal temperament.[1]File:Simple I-IV-V-I isomorphic 31-TET.mid Whereas in 12 Template:Sc B♮ is 11 steps, in 31 Template:Sc B♮ is 28 steps.
File:Csub Cmin Cmaj Csup.ogg
C subminor, C minor, C major, C supermajor (topped by A♭) in 31 Template:Sc

Usual chords like the major chord are rendered nicely in 31 Template:Sc because the third and the fifth are very well approximated. Also, it is possible to play subminor chords (where the first third is subminor) and supermajor chords (where the first third is supermajor).

File:Cmaj7 Gmin 31ET 12ET.ogg
C seventh and G minor, twice in 31 Template:Sc, then twice in 12 Template:Sc

It is also possible to render nicely the harmonic seventh chord. For example on tonic C, with C–E–G–A♯. The seventh here is different from stacking a fifth and a minor third, which instead yields B♭ to make a dominant seventh. This difference cannot be made in [[12 equal temperament|12 Template:Sc]].

Footnotes

Template:Notelist

See also

References

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External links

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