31 equal temperament

File:Syntonic tuning continuum.svg

Template:Nobr on the regular diatonic tuning continuum at Template:Nobr

In music, 31 equal temperament, Template:Nobr which can also be abbreviated Template:Nobr (31 tone Template:Sc) or Template:Nobr (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). {{errorTemplate:Main other|Audio file "31-tet scale on C.mid" not found}}Template:Category handler Each step represents a frequency ratio of Template:Radic, or 38.71 cents ({{errorTemplate:Main other|Audio file "1 step in 31-et on C.mid" not found}}Template:Category handler).

Template:Nobr 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 Template:Nobr is precisely the same as it is in any other syntonic tuning (such as Template:Nobr 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. 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 Template:Nobr, 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 Template:Nobr 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 Template:Nobr

Here are the sizes of some common intervals:

interval name	size (steps)	size (cents)	Template:Sc audio	just ratio	just (cents)	Template:Sc 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	{{errorTemplate:Main other|Audio file "25 steps in 31-et on C.mid" not found}}Template:Category handler	7:4	0968.83	{{errorTemplate:Main other|Audio file "Harmonic seventh on C.mid" not found}}Template:Category handler	−01.09
minor sixth	21	0812.90	{{errorTemplate:Main other|Audio file "21 steps in 31-et on C.mid" not found}}Template:Category handler	8:5	0813.69	{{errorTemplate:Main other|Audio file "Just minor sixth on C.mid" not found}}Template:Category handler	−00.78
perfect fifth	18	0696.77	{{errorTemplate:Main other|Audio file "18 steps in 31-et on C.mid" not found}}Template:Category handler	3:2	0701.96	{{errorTemplate:Main other|Audio file "Just perfect fifth on C.mid" not found}}Template:Category handler	−05.19
greater septimal tritone, diminished fifth	16	619.35		10:70	0617.49		+01.87
lesser septimal tritone, augmented fourth	15	0580.65	{{errorTemplate:Main other|Audio file "15 steps in 31-et on C.mid" not found}}Template:Category handler	7:5	0582.51	{{errorTemplate:Main other|Audio file "Lesser septimal tritone on C.mid" not found}}Template:Category handler	−01.86
undecimal tritone, half augmented fourth, 11th harmonic	14	0541.94	{{errorTemplate:Main other|Audio file "14 steps in 31-et on C.mid" not found}}Template:Category handler	11:80	0551.32	{{errorTemplate:Main other|Audio file "Eleventh harmonic on C.mid" not found}}Template:Category handler	−09.38
perfect fourth	13	0503.23	{{errorTemplate:Main other|Audio file "13 steps in 31-et on C.mid" not found}}Template:Category handler	4:3	0498.04	{{errorTemplate:Main other|Audio file "Just perfect fourth on C.mid" not found}}Template:Category handler	+05.19
septimal narrow fourth, half diminished fourth	12	0464.52	{{errorTemplate:Main other|Audio file "12 steps in 31-et on C.mid" not found}}Template:Category handler	21:16	0470.78	{{errorTemplate:Main other|Audio file "Twenty-first harmonic on C.mid" not found}}Template:Category handler	−06.26
tridecimal augmented third, and greater major third	12	0464.52	{{errorTemplate:Main other|Audio file "12 steps in 31-et on C.mid" not found}}Template:Category handler	13:10	0454.21	{{errorTemplate:Main other|Audio file "Tridecimal major third on C.mid" not found}}Template:Category handler	+10.31
septimal major third	11	0425.81	{{errorTemplate:Main other|Audio file "11 steps in 31-et on C.mid" not found}}Template:Category handler	9:7	0435.08	{{errorTemplate:Main other|Audio file "Septimal major third on C.mid" not found}}Template:Category handler	−09.27
diminished fourth	11	0425.81	{{errorTemplate:Main other|Audio file "11 steps in 31-et on C.mid" not found}}Template:Category handler	32:25	0427.37	{{errorTemplate:Main other|Audio file "Just diminished fourth on C.mid" not found}}Template:Category handler	−01.56
undecimal major third	11	0425.81	{{errorTemplate:Main other|Audio file "11 steps in 31-et on C.mid" not found}}Template:Category handler	14:11	0417.51	{{errorTemplate:Main other|Audio file "Undecimal major third on C.mid" not found}}Template:Category handler	+08.30
major third	10	0387.10	{{errorTemplate:Main other|Audio file "10 steps in 31-et on C.mid" not found}}Template:Category handler	5:4	0386.31	{{errorTemplate:Main other|Audio file "Just major third on C.mid" not found}}Template:Category handler	+00.79
tridecimal neutral third	09	0348.39	{{errorTemplate:Main other|Audio file "9 steps in 31-et on C.mid" not found}}Template:Category handler	16:13	0359.47	{{errorTemplate:Main other|Audio file "Tridecimal neutral third on C.mid" not found}}Template:Category handler	−11.09
undecimal neutral third	09	0348.39	{{errorTemplate:Main other|Audio file "9 steps in 31-et on C.mid" not found}}Template:Category handler	11:90	0347.41	{{errorTemplate:Main other|Audio file "Undecimal neutral third on C.mid" not found}}Template:Category handler	+00.98
minor third	08	0309.68	{{errorTemplate:Main other|Audio file "8 steps in 31-et on C.mid" not found}}Template:Category handler	6:5	0315.64	{{errorTemplate:Main other|Audio file "Just minor third on C.mid" not found}}Template:Category handler	−05.96
septimal minor third	07	0270.97	{{errorTemplate:Main other|Audio file "7 steps in 31-et on C.mid" not found}}Template:Category handler	7:6	0266.87	{{errorTemplate:Main other|Audio file "Septimal minor third on C.mid" not found}}Template:Category handler	+04.10
septimal whole tone	06	0232.26	{{errorTemplate:Main other|Audio file "6 steps in 31-et on C.mid" not found}}Template:Category handler	8:7	0231.17	{{errorTemplate:Main other|Audio file "Septimal major second on C.mid" not found}}Template:Category handler	+01.09
whole tone, major tone	05	0193.55	{{errorTemplate:Main other|Audio file "5 steps in 31-et on C.mid" not found}}Template:Category handler	9:8	0203.91	{{errorTemplate:Main other|Audio file "Major tone on C.mid" not found}}Template:Category handler	−10.36
whole tone, major second	05	0193.55	{{errorTemplate:Main other|Audio file "5 steps in 31-et on C.mid" not found}}Template:Category handler	28:25	0196.20		−02.65
mean tone, major second	05	0193.55		REDIRECT Template:Radic Template:Rcat shell:2	0193.16		+00.39
whole tone, minor tone	05	193.55	{{errorTemplate:Main other|Audio file "5 steps in 31-et on C.mid" not found}}Template:Category handler	10:90	0182.40	{{errorTemplate:Main other|Audio file "Minor tone on C.mid" not found}}Template:Category handler	+11.15
greater undecimal neutral second	04	0154.84	{{errorTemplate:Main other|Audio file "4 steps in 31-et on C.mid" not found}}Template:Category handler	11:10	0165.00		−10.16
lesser undecimal neutral second	04	0154.84	{{errorTemplate:Main other|Audio file "4 steps in 31-et on C.mid" not found}}Template:Category handler	12:11	0150.64	{{errorTemplate:Main other|Audio file "Lesser undecimal neutral second on C.mid" not found}}Template:Category handler	+04.20
septimal diatonic semitone	03	0116.13	{{errorTemplate:Main other|Audio file "3 steps in 31-et on C.mid" not found}}Template:Category handler	15:14	0119.44	{{errorTemplate:Main other|Audio file "Septimal diatonic semitone on C.mid" not found}}Template:Category handler	−03.31
diatonic semitone, minor second	03	0116.13	{{errorTemplate:Main other|Audio file "3 steps in 31-et on C.mid" not found}}Template:Category handler	16:15	0111.73	{{errorTemplate:Main other|Audio file "Just diatonic semitone on C.mid" not found}}Template:Category handler	+04.40
septimal chromatic semitone	02	0077.42	{{errorTemplate:Main other|Audio file "2 steps in 31-et on C.mid" not found}}Template:Category handler	21:20	0084.47	{{errorTemplate:Main other|Audio file "Septimal chromatic semitone on C.mid" not found}}Template:Category handler	−07.05
chromatic semitone, augmented unison	02	0077.42	{{errorTemplate:Main other|Audio file "2 steps in 31-et on C.mid" not found}}Template:Category handler	25:24	0070.67	{{errorTemplate:Main other|Audio file "Just chromatic semitone on C.mid" not found}}Template:Category handler	+06.75
lesser diesis	01	0038.71	{{errorTemplate:Main other|Audio file "1 step in 31-et on C.mid" not found}}Template:Category handler	128:125	0041.06	{{errorTemplate:Main other|Audio file "Septimal diesis on C.mid" not found}}Template:Category handler	−02.35
undecimal diesis	01	0038.71	{{errorTemplate:Main other|Audio file "1 step in 31-et on C.mid" not found}}Template:Category handler	45:44	0038.91	{{errorTemplate:Main other|Audio file "Undecimal diesis on C.mid" not found}}Template:Category handler	−00.20
septimal diesis	01	0038.71	{{errorTemplate:Main other|Audio file "1 step in 31-et on C.mid" not found}}Template:Category handler	49:48	0035.70	{{errorTemplate:Main other|Audio file "Septimal diesis on C.mid" not found}}Template:Category handler	+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.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		Template:Nobr B𝄫		A♯		B♭		A𝄪 C𝄫		B		C♭		B♯		C		B𝄪 D𝄫		C♯		D♭		C𝄪 Template:Nobr		D		Template:Nobr E𝄫		D♯		E♭		D𝄪 F𝄫		E		F♭		E♯		F		E𝄪 G𝄫		F♯		G♭		F𝄪 A♭𝄫		G		Template:Nobr 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		ATemplate:Music		A♯		B♭		BTemplate:Music		B		BTemplate:Music		CTemplate:Music		C		CTemplate:Music		C♯		D♭		DTemplate:Music		D		DTemplate:Music		D♯		E♭		ETemplate:Music		E		ETemplate:Music		FTemplate:Music		F		FTemplate:Music		F♯		G♭		GTemplate:Music		G		GTemplate:Music		G♯		A♭		ATemplate:Music		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)

Template:Left

	C	C♯	D♭	D	D♯	E♭	E	E♯	F	F♯	G♭	G	G♯	A♭	A	A♯	B♭	B	C♭	C
Template:Nobr:	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
Template:Nobr:	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 Template:Nobr are discussed in the article on septimal meantone temperament. Chords not discussed there include the neutral thirds triad ({{errorTemplate:Main other|Audio file "Neutral thirds triad in 31-tet.mid" not found}}Template:Category handler), which might be written C–ETemplate:Music–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.File:Simple I-IV-V-I isomorphic 31-TET.mid Whereas in Template:Nobr BTemplate:Music is 11 steps, in Template:Nobr BTemplate:Music is 28 steps.

File:Csub Cmin Cmaj Csup.ogg

C subminor, C minor, C major, C supermajor (topped by ATemplate:Music) in Template:Nobr

Usual chords like the major chord are rendered nicely in Template:Nobr 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 Template:Nobr, then twice in Template:Nobr

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

See also

• Archicembalo, alternate keyboard instrument with 36 keys per octave that was sometimes tuned as Template:Nobr