31 equal temperament

In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET (31 tone ET) or 31-EDO (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps (equal frequency ratios). Playⓘ Each step represents a frequency ratio of 31√2, or 38.71 cents (Playⓘ).

31-ET on the regular diatonic tuning continuum at P5= 696.77 cents[1]

31-ET 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-ET is precisely the same as it is in any other syntonic tuning (such as 12-ET), 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 diesis—the ratio of an octave to three major thirds, 128:125 or 41.06 cents—was approximately a fifth of a tone or a third 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 4√5, the appeal of this method was immediate, as the fifth of 31-ET, 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-ET 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
21-Limit just intonation intervals approximated in 31-ET

Here are the sizes of some common intervals:

interval name size (steps) size (cents) midi just ratio just (cents) midi error
octave 31 1200 2:1 1200 0
minor seventh 26 1006.45 9:5 1017.60 −11.15
small just minor seventh 26 1006.45 16:9 996.09 +10.36
harmonic seventh 25 967.74 Playⓘ 7:4 968.83 Playⓘ −01.09
perfect fifth 18 696.77 Playⓘ 3:2 701.96 Playⓘ −05.19
greater septimal tritone 16 619.35 10:70 617.49 +01.87
lesser septimal tritone 15 580.65 Playⓘ 7:5 582.51 Playⓘ −01.86
undecimal tritone, 11th harmonic 14 541.94 Playⓘ 11:80 551.32 Playⓘ −09.38
perfect fourth 13 503.23 Playⓘ 4:3 498.04 Playⓘ +05.19
septimal narrow fourth 12 464.52 Playⓘ 21:16 470.78 Playⓘ −06.26
tridecimal augmented third, and greater major third 12 464.52 Playⓘ 13:10 454.21 Playⓘ +10.31
septimal major third 11 425.81 Playⓘ 9:7 435.08 Playⓘ −09.27
diminished fourth 11 425.81 Playⓘ 32:25 427.37 Playⓘ −01.56
undecimal major third 11 425.81 Playⓘ 14:11 417.51 Playⓘ +08.30
major third 10 387.10 Playⓘ 5:4 386.31 Playⓘ +00.79
tridecimal neutral third 09 348.39 Playⓘ 16:13 359.47 Playⓘ −11.09
undecimal neutral third 09 348.39 Playⓘ 11:90 347.41 Playⓘ +00.98
minor third 08 309.68 Playⓘ 6:5 315.64 Playⓘ −05.96
septimal minor third 07 270.97 Playⓘ 7:6 266.87 Playⓘ +04.10
septimal whole tone 06 232.26 Playⓘ 8:7 231.17 Playⓘ +01.09
whole tone, major tone 05 193.55 Playⓘ 9:8 203.91 Playⓘ −10.36
whole tone, middle 05 193.55 Playⓘ 28:25 196.20 −02.65
whole tone, minor tone 05 193.55 Playⓘ 10:90 182.40 Playⓘ +11.15
greater undecimal neutral second 04 154.84 Playⓘ 11:10 165.00 −10.16
lesser undecimal neutral second 04 154.84 Playⓘ 12:11 150.64 Playⓘ +04.20
septimal diatonic semitone 03 116.13 Playⓘ 15:14 119.44 Playⓘ −03.31
diatonic semitone, just 03 116.13 Playⓘ 16:15 111.73 Playⓘ +04.40
septimal chromatic semitone 02 077.42 Playⓘ 21:20 084.47 Playⓘ −07.05
chromatic semitone, Just 02 077.42 Playⓘ 25:24 070.67 Playⓘ +06.75
lesser diesis 01 038.71 Playⓘ 128:125 041.06 Playⓘ −02.35
undecimal diesis 01 038.71 Playⓘ 45:44 038.91 Playⓘ −00.20
septimal diesis 01 038.71 Playⓘ 49:48 035.70 Playⓘ +03.01

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.[3] 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

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 Bdouble flat A♯ B♭ Adouble sharp B C♭ B♯ C Ddouble flat C♯ D♭ Cdouble sharp D Edouble flat D♯ E♭ Ddouble sharp E F♭ E♯ F Gdouble flat F♯ G♭ Fdouble sharp G Adouble flat G♯ A♭ Gdouble sharp 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
Circle of fifths in 31 equal temperament
Key signature Number of
sharps		 Key signature Number of
flats
C major C D E F G A B 0
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♯ C𝄪 D♯ E♯ F𝄪 G𝄪 10 C𝄫♭ major C𝄫♭ D𝄫♭ E𝄫♭ F𝄫♭ G𝄫♭ A𝄫♭ B𝄫♭ 21
E♯ major E♯ F𝄪 G𝄪 A♯ B♯ C𝄪 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
Comparison between 1/4-comma meantone and 31-ET (values in cents, rounded to 2 decimals)
  C C♯ D♭ D D♯ E♭ E E♯ F F♯ G♭ G G♯ A♭ A A♯ B♭ B C♭ C
1/4 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-ET: 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-ET are discussed in the article on septimal meantone temperament. Chords not discussed there include the neutral thirds triad (Playⓘ), which might be written C–Ehalf flat–G, C–Ddouble sharp–G or C–Fdouble flat–G, and the Orwell tetrad, which is C–E–Fdouble sharp–Bdouble flat.

I–IV–V–I chord progression in 31 tone equal temperament.[1]Whereas in 12TET B♮ is 11 steps, in 31-TET B♮ is 28 steps.

Usual chords like the major chord are rendered nicely in 31-ET 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).

It is also possible to render nicely the harmonic seventh chord. For example on 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-ET.

See also
  • Archicembalo, alternate keyboard instrument with 36 keys per octave that was sometimes tuned as 31TET.

References
  1. Milne, A., Sethares, W. A. and Plamondon, J., "Isomorphic Controllers and Dynamic Tuning: Invariant Fingerings Across a Tuning Continuum", Computer Music Journal, Winter 2007, vol. 31, no. 4, pp. 15–32.
  2. Monzo, Joe (2005). "Equal-Temperament". Tonalsoft Encyclopedia of Microtonal Music Theory. Joe Monzo. Retrieved 28 February 2019.
  3. Keislar, Douglas. "Six American Composers on Nonstandard Tunnings: Easley Blackwood; John Eaton; Lou Harrison; Ben Johnston; Joel Mandelbaum; William Schottstaedt", Perspectives of New Music, vol. 29, no. 1. (Winter 1991), pp. 176–211. JSTOR 833076
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