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 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 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 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, subminor seventh	25	967.74	Play^ⓘ	7:4	968.83	Play^ⓘ	−01.09
minor sixth	21	812.90	Play^ⓘ	8:5	813.69	Play^ⓘ	−00.78
perfect fifth	18	696.77	Play^ⓘ	3:2	701.96	Play^ⓘ	−05.19
greater septimal tritone, diminished fifth	16	619.35		10:70	617.49		+01.87
lesser septimal tritone, augmented fourth	15	580.65	Play^ⓘ	7:5	582.51	Play^ⓘ	−01.86
undecimal tritone, half augmented fourth, 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, half diminished 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, major second	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, minor second	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, augmented unison	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		B		A♯		B♭		A		B		C♭		B♯		C		D		C♯		D♭		C		D		E		D♯		E♭		D		E		F♭		E♯		F		G		F♯		G♭		F		G		A		G♯		A♭		G		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								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–E–G, C–D–G or C–F–G, and the Orwell tetrad, which is C–E–F–B.

C subminor, C minor, C major, C supermajor (topped by A♭) in 31 equal temperament

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).

C seventh and G minor, twice in 31 equal temperament, then twice in 12 equal temperament

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.

References

^ ^a ^b 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.
^ Monzo, Joe (2005). "Equal-Temperament". Tonalsoft Encyclopedia of Microtonal Music Theory. Joe Monzo. Retrieved 28 February 2019.
^ 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

External links

The Huygens Fokker foundation for micro-tonal music, in Dutch and English
Fokker, Adriaan Daniël, Equal Temperament and the Thirty-one-keyed organ
Rapoport, Paul, About 31-tone Equal Temperament
Terpstra, Siemen, Toward a Theory of Meantone (and 31-et) Harmony
Barbieri, Patrizio. Enharmonic instruments and music, 1470-1900. (2008) Latina, Il Levante Libreria Editrice
M. Khramov, "Approximation to 7-Limit Just Intonation in a Scale of 31EDO", Proceedings of the FRSM-2009 International Symposium Frontiers of Research on Speech and Music, pp. 73–82, ABV IIITM, Gwalior, 2009.
31 Tone Equal Temperament

[Milne-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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