31 equal temperament

(Redirected from 31 tone 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 equally-proportioned steps (equal frequency ratios). Play Each step represents a frequency ratio of 312 , or 38.71 cents (Play).

31 EDO on the regular diatonic tuning continuum at p5 = 696.77 cents[1]

31 EDO 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 EDO is precisely the same as it is in any other syntonic tuning (such as 12 EDO), so long as the notes are spelled properly—that is, with no assumption of enharmonicity.

History and use

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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 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 45 , the appeal of this method was immediate, as the fifth of 31 EDO, 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 EDO 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

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21 limit just intonation intervals approximated in 31 EDO

Here are the sizes of some common intervals:

interval name size
(steps)
size
(cents)
MIDI
audio
just
ratio
just
(cents)
MIDI
audio
error
(cents)
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 16:90 996.09 +10.36
harmonic seventh, subminor seventh, augmented sixth 25 967.74 Play 7:4 0968.83 Play 01.09
minor sixth 21 0812.90 Play 8:5 0813.69 Play 00.78
perfect fifth 18 0696.77 Play 3:2 0701.96 Play 05.19
greater septimal tritone, diminished fifth 16 619.35 10:70 0617.49 +01.87
lesser septimal tritone, augmented fourth 15 0580.65 Play 7:5 0582.51 Play 01.86
undecimal tritone, half augmented fourth, 11th harmonic 14 0541.94 Play 11:80 0551.32 Play 09.38
perfect fourth 13 0503.23 Play 4:3 0498.04 Play +05.19
septimal narrow fourth, half diminished fourth 12 0464.52 Play 21:16 0470.78 Play 06.26
tridecimal augmented third, and greater major third 12 0464.52 Play 13:10 0454.21 Play +10.31
septimal major third 11 0425.81 Play 9:7 0435.08 Play 09.27
diminished fourth 11 0425.81 Play 32:25 0427.37 Play 01.56
undecimal major third 11 0425.81 Play 14:11 0417.51 Play +08.30
major third 10 0387.10 Play 5:4 0386.31 Play +00.79
tridecimal neutral third 09 0348.39 Play 16:13 0359.47 Play −11.09
undecimal neutral third 09 0348.39 Play 11:90 0347.41 Play +00.98
minor third 08 0309.68 Play 6:5 0315.64 Play 05.96
septimal minor third 07 0270.97 Play 7:6 0266.87 Play +04.10
septimal whole tone 06 0232.26 Play 8:7 0231.17 Play +01.09
whole tone, major tone 05 0193.55 Play 9:8 0203.91 Play −10.36
whole tone, major second 05 0193.55 Play 28:25 0196.20 02.65
mean tone, major second 05 0193.55  1 / 2 5  0193.16 +00.39
whole tone, minor tone 05 193.55 Play 10:90 0182.40 Play +11.15
greater undecimal neutral second 04 0154.84 Play 11:10 0165.00 −10.16
lesser undecimal neutral second 04 0154.84 Play 12:11 0150.64 Play +04.20
septimal diatonic semitone 03 0116.13 Play 15:14 0119.44 Play 03.31
diatonic semitone, minor second 03 0116.13 Play 16:15 0111.73 Play +04.40
septimal chromatic semitone 02 0077.42 Play 21:20 0084.47 Play 07.05
chromatic semitone, augmented unison 02 0077.42 Play 25:24 0070.67 Play +06.75
lesser diesis 01 0038.71 Play 128:125 0041.06 Play 02.35
undecimal diesis 01 0038.71 Play 45:44 0038.91 Play 00.20
septimal diesis 01 0038.71 Play 49:48 0035.70 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

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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 A  A B B  B B  C  C C  C D D  D D  D E E  E E  F  F F  F G G  G G  G A A  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
Key signature Scale Number of
sharps
Key signature Scale 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 EDO (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
 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 EDO: 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

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Many chords of 31 EDO are discussed in the article on septimal meantone temperament. Chords not discussed there include the neutral thirds triad (Play), which might be written C–E –G, C–D –G or C–F –G, and the Orwell tetrad, which is C–E–F –B .

 
I–IV–V–I chord progression in 31 tone equal temperament.[1] Whereas in 12 EDO B is 11 steps, in 31 EDO B is 28 steps.
C subminor, C minor, C major, C supermajor (topped by A) in 31 EDO

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

C seventh and G minor, twice in 31 EDO, then twice in 12 EDO

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 EDO.

Footnotes

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  1. ^ The following composers mentioned in the title of Keislar (1991)'s journal article[3] have Wikipedia articles:

See also

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

References

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  1. ^ a b Milne, A.; Sethares, W.A.; Plamondon, J. (Winter 2007). "Isomorphic controllers and dynamic tuning: Invariant fingerings across a tuning continuum". Computer Music Journal. 31 (4): 15–32 – via mitpressjournals.org.
  2. ^ Monzo, Joe (2005). "Equal temperament". Tonalsoft Encyclopedia of Microtonal Music Theory. Joe Monzo / Tonalsoft. Retrieved 28 February 2019.
  3. ^ a b Keislar, Douglas (Winter 1991). "Six American composers on nonstandard tunnings: Easley Blackwood; John Eaton; Lou Harrison; Ben Johnston; Joel Mandelbaum; William Schottstaedt". Perspectives of New Music. 29 (1): 176–211. JSTOR 833076.[a]
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