31 equal temperament

In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 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 2^1/31, or 38.71 cents (Play^ⓘ).

History

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 and a third of a semitone. In 1666, Lemme Rossi first proposed an equal temperament of this order. Shortly thereafter, having discovered it independently, scientist Christiaan Huygens wrote about it also. Since the standard system of tuning at that time was quarter-comma meantone, in which the fifth is tuned to 5^1/4, 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.

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		154		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

Sometimes the five "double flat" notes and five "double sharp" notes are replaced by half sharps and half flats, similar to the quarter tone system.

Interval size

Here are the sizes of some common intervals:

interval name	size (steps)	size (cents)	midi	just ratio	just (cents)	midi	error
harmonic seventh	25	967.74	Playⓘ	7:4	968.83	Playⓘ	−1.09
perfect fifth	18	696.77	Playⓘ	3:2	701.96	Playⓘ	−5.19
greater septimal tritone	16	619.35		10:7	617.49		+1.87
lesser septimal tritone	15	580.65	Playⓘ	7:5	582.51	Playⓘ	−1.86
undecimal tritone, 11th harmonic	14	541.94	Playⓘ	11:8	551.32	Playⓘ	−9.38
perfect fourth	13	503.23	Playⓘ	4:3	498.04	Playⓘ	+5.19
septimal narrow fourth	12	464.52	Playⓘ	21:16	470.78		−6.26
tridecimal 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ⓘ	−9.27
undecimal major third	11	425.81	Playⓘ	14:11	417.51	Playⓘ	+8.30
major third	10	387.10	Playⓘ	5:4	386.31	Playⓘ	+0.79
tridecimal neutral third	9	348.39	Playⓘ	16:13	359.47	playⓘ	−11.09
undecimal neutral third	9	348.39	Playⓘ	11:9	347.41	Playⓘ	+0.98
minor third	8	309.68	Playⓘ	6:5	315.64	Playⓘ	−5.96
septimal minor third	7	270.97	Playⓘ	7:6	266.87	Playⓘ	+4.10
septimal whole tone	6	232.26	Playⓘ	8:7	231.17	Playⓘ	+1.09
whole tone, major tone	5	193.55	Playⓘ	9:8	203.91	Playⓘ	−10.36
whole tone, minor tone	5	193.55	Playⓘ	10:9	182.40	Playⓘ	+11.15
greater undecimal neutral second	4	154.84	Playⓘ	11:10	165.00		−10.16
lesser undecimal neutral second	4	154.84	Playⓘ	12:11	150.64	Playⓘ	+4.20
septimal diatonic semitone	3	116.13	Playⓘ	15:14	119.44	Playⓘ	−3.31
diatonic semitone, just	3	116.13	Playⓘ	16:15	111.73	Playⓘ	+4.40
septimal chromatic semitone	3	77.42	Playⓘ	21:20	84.47	Playⓘ	−7.05
chromatic semitone, just	2	77.42	Playⓘ	25:24	70.67	Playⓘ	+6.75
lesser diesis	1	38.71	Playⓘ	128:125	41.06	Playⓘ	−2.35
undecimal diesis	1	38.71	Playⓘ	45:44	38.91	Playⓘ	−0.20
septimal diesis	1	38.71	Playⓘ	49:48	35.70	Playⓘ	+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.

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.

Chords of 31 equal temperament

Circle of fifths in 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.

I-IV-V-I chord progression in 31 tone equal temperament. Play^ⓘ Whereas in 12TET B♮ is 11 steps, in 31-TET B♮ is 28 steps.

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.