31edo

← 30edo

31edo

32edo →

Prime factorization	31 (prime)
Step size	38.7097 ¢
Fifth	18\31 (696.774 ¢)
Semitones (A1:m2)	2:3 (77.42 ¢ : 116.1 ¢)
Consistency limit	11
Distinct consistency limit	7

Thirty-one tone equal temperament, also called 31-tET, 31-EDO, 31-et, or tricesimoprimal meantone temperament, is the scale derived by dividing the octave into 31 equally large steps. The term Tricesimoprimal was first used by Adriaan Fokker.

Basic theory

		prime 2	prime 3	prime 5	prime 7	prime 11	prime 13	prime 17	prime 19	prime 23
Error	absolute (¢)	0	-5.18	+0.8	-1.1	-9.4	+11.1	+11.2	+12.2	-8.9
	relative (%)	0	-13	+2	-3	-24	+29	+29	+31	-23
nearest edomapping		31	18	10	25	14	22	3	8	16
fifthspan		0	+1	+4	+10	-13	+15	-5	-3	-6

Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 cents. 31's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5/4), making it slightly sharp of quarter-comma meantone. 31's approximation of 7/4, a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the 11-limit, although the fact that it equates 14/11 with 9/7, and 11/8 with 15/11, may be too off for some. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course).

Because of the near-just 5/4 and 7/4 and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31-et is relatively quite accurate and is the 6th zeta integral edo, the 7th zeta gap edo, a zeta peak edo and a zeta peak integer edo. (Another way in which 31edo is especially accurate is that it represents a record in Pepper ambiguity in the 7-, 9- and 11-odd-limit, which it is consistent through.)

One step of 31edo, measuring about 38.7¢, is called a diesis because it stands in for several intervals called "dieses" (such as 128/125 and 648/625) which are tempered out in 12edo. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in SpiralProgressions. Zhea Erose's 31edo music uses the interval frequently.

31edo is the 11th prime edo, following 29edo and coming before 37edo.

Intervals

Degree	Cents	Approximate Ratios	Ups and Downs Notation			Extended pythagorean notation
0	0.00	1/1	P1	perfect unison	D	P1	perfect unison	D
1	38.71	45/44, 49/48, 46/45, 128/125, 36/35	^1, d2	up-unison, dim 2nd	^D, Ebb	d2	dim 2nd	Ebb
2	77.42	25/24, 21/20, 22/21, 23/22	A1, vm2	aug 1sn, downminor 2nd	D#, vEb	A1	aug 1sn	D#
3	116.13	15/14, 16/15	m2	minor 2nd	Eb	m2	minor 2nd	Eb
4	154.84	12/11, 11/10, 35/32	~2	mid 2nd	vE	AA1, dd3	double-aug 1sn, double-dim 3rd	Dx, Fbb
5	193.55	9/8, 10/9, 19/17, 28/25	M2	major 2nd	E	M2	major 2nd	E
6	232.26	8/7, 144/125	^M2	upmajor 2nd	^E	d3	dim 3rd	Fb
7	270.97	7/6, 75/64	vm3	downminor 3rd	vF	A2	aug 2nd	E#
8	309.68	6/5, 25/21	m3	minor 3rd	F	m3	minor 3rd	F
9	348.39	11/9, 27/22, 16/13, 60/49, 49/40	~3	mid 3rd	^F	AA2, dd4	double-aug 2nd, double-dim 4th	Ex, Gbb
10	387.10	5/4	M3	major 3rd	F#	M3	major 3rd	F#
11	425.81	9/7, 14/11, 23/18, 32/25	^M3	upmajor 3rd	^F#	d4	dim 4th	Gb
12	464.52	21/16, 13/10, 17/13, 125/96	v4	down-4th	vG	A3	aug 3rd	Fx
13	503.23	4/3	P4	perfect 4th	G	P4	perfect 4th	G
14	541.94	11/8, 15/11, 26/19	^4, ~4	up-4th, mid 4th	^G	AA3, dd5	double-aug 3rd, double-dim 5th	Fx#, Abb
15	580.65	7/5, 45/32, 25/18	A4, vd5	aug 4th, downdim 5th	G#, vAb	A4	aug 4th	G#
16	619.35	10/7, 64/45, 36/25	^A4, d5	upaug 4th, dim 5th	^G#, Ab	d5	dim 5th	Ab
17	658.06	16/11, 22/15, 19/13	v5, ~5	down-5th, mid 5th	vA	AA4, dd6	double-aug 4th, double-dim 6th	Gx, Bbbb
18	696.77	3/2	P5	perfect 5th	A	P5	perfect 5th	A
19	735.48	32/21, 20/13, 26/17, 192/125	^5	up-5th	^A	d6	dim 6th	Bbb
20	774.19	14/9, 11/7, 25/16	vm6	downminor 6th	vBb	A5	aug 5th	A#
21	812.90	8/5	m6	minor 6th	Bb	m6	minor 6th	Bb
22	851.61	18/11, 44/27, 13/8, 49/30, 80/49	~6	mid 6th	vB	AA5, dd7	double-aug 5th, double-dim 7th	Ax, Cbb
23	890.32	5/3, 42/25	M6	major 6th	B	M6	major 6th	B
24	929.03	12/7, 128/75	^M6	upmajor 6th	^B	d7	dim 7th	Cb
25	967.74	7/4, 125/72	vm7	downminor 7th	vC	A6	aug 6th	B#
26	1006.45	16/9, 9/5, 34/19, 25/14	m7	minor 7th	C	m7	minor 7th	C
27	1045.16	11/6, 20/11, 64/35	~7	mid 7th	^C	AA6, dd8	double-aug 6th, double-dim 8ve	Bx, Dbb
28	1083.87	28/15, 15/8	M7	major 7th	C#	M7	major 7th	C#
29	1122.58	48/25, 40/21, 21/11, 44/23	^M7	upmajor 7th	^C#	d8	dim 8ve	Db
30	1161.29	88/45, 96/49, 45/23, 125/64, 35/18	v8	down-8ve	vD	A7	aug 7th	Cx
31	1200.00	2/1	P8	perfect 8ve	D	P8	perfect 8ve	D

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

quality	color name	monzo format	examples
downminor	zo	{a, b, 0, 1}	7/6, 7/4
minor	fourthward wa	{a, b}, b < -1	32/27, 16/9
"	gu	{a, b, -1}	6/5, 9/5
mid	ilo	{a, b, 0, 0, 1}	11/9, 11/6
"	lu	{a, b, 0, 0, -1}	12/11, 18/11
major	yo	{a, b, 1}	5/4, 5/3
"	fifthward wa	{a, b}, b > 1	9/8, 27/16
upmajor	ru	{a, b, 0, -1}	9/7, 12/7

All 31edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, yo and ru triads:

color of the 3rd	JI chord	edosteps	notes of C chord	written name	spoken name
zo	6:7:9	0-7-18	C vEb G	Cvm	C downminor
gu	10:12:15	0-8-18	C Eb G	Cm	C minor
ilo	18:22:27	0-9-18	C vE G	C~	C mid
yo	4:5:6	0-10-18	C E G	C	C major or C
ru	14:18:21	0-11-18	C ^E G	C^	C upmajor or C up

For a more complete list of chords, see Ups and Downs Notation - Chords and Chord Progressions.

Notations

From the appendix to The Sagittal Songbook by Jacob A. Barton, a diagram of how to notate 31-EDO in the Revo flavor of Sagittal:

Just approximation

Selected just intervals

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 31edo. Prime harmonics are in bold; inconsistent intervals are in italic.

Direct mapping (even if inconsistent)

Interval, complement	Error (abs, ¢)
5/4, 8/5	0.783
11/9, 18/11	0.979
8/7, 7/4	1.084
7/5, 10/7	1.867
15/14, 28/15	3.314
7/6, 12/7	4.097
12/11, 11/6	4.202
16/15, 15/8	4.398
15/11, 22/15	4.985
4/3, 3/2	5.181
6/5, 5/3	5.964
14/11, 11/7	8.298
9/7, 14/9	9.278
11/8, 16/11	9.382
11/10, 20/11	10.166
13/10, 20/13	10.302
9/8, 16/9	10.362
16/13, 13/8	11.085
10/9, 9/5	11.145
14/13, 13/7	12.169
15/13, 26/15	15.483
13/12, 24/13	16.266
18/13, 13/9	17.263
13/11, 22/13	18.242

Patent val mapping

Interval, complement	Error (abs, ¢)
5/4, 8/5	0.783
11/9, 18/11	0.979
8/7, 7/4	1.084
7/5, 10/7	1.867
15/14, 28/15	3.314
7/6, 12/7	4.097
12/11, 11/6	4.202
16/15, 15/8	4.398
15/11, 22/15	4.985
4/3, 3/2	5.181
6/5, 5/3	5.964
14/11, 11/7	8.298
9/7, 14/9	9.278
11/8, 16/11	9.382
11/10, 20/11	10.166
13/10, 20/13	10.302
9/8, 16/9	10.362
16/13, 13/8	11.085
10/9, 9/5	11.145
14/13, 13/7	12.169
15/13, 26/15	15.483
13/12, 24/13	16.266
13/11, 22/13	20.468
18/13, 13/9	21.447

Selected 19-limit intervals

Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 31et.

		3-limit	5-limit	7-limit	11-limit	13-limit
Octave stretch (¢)		+1.63	+0.98	+0.83	+1.21	+0.50
Error	absolute (¢)	1.64	1.63	1.43	1.49	2.07
	relative (%)	4.22	4.20	3.70	3.84	5.35

• 31et has a lower relative error than any previous ETs in all harmonic limits from 7 to 31.

Relationship to 12-edo

Whereas 12-edo has a circle of twelve 5ths, 31-edo has a spiral of twelve 5ths (since 18\31 is on the 7\12 kite in the scale tree). This spiral of 5th shows 31-edo in a 12-edo-friendly format. Excellent for introducing 31-edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.

31edo can be notated with a seperate semi/sesqui sharp/flat chain (like 17edo), with its own enharmonic circle of fifths.

MOS scales

The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other MOSes and MOS chains are also useful:

• 31edo's 9\31 neutral third generator generates ultrasoft mosh and superhard dicoid MOSes.
• Its 12\31 generator generates a semihard oneirotonic scale, similar to the 5L 3s scale in 13edo but with the 9/8, 5/4 and 7/6 better in tune and with the flat fifth closer to just.
• A chain of 5\31 whole tones is exceptionally rich in 4:5:7 chords, which are approximated very well in 31edo.
• If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0¢) subminor third generator. The ultrasoft 9-tone orwelloid (4L 5s) MOS could be treated as a 9-tone well temperament.
• It supports 6edf (miracle) and 9edf (Carlos alpha), fifth-equivalent equal temperaments that hit many good JI approximations.

Generator	Cents	MOSes
2\31	77.42	1L 14s 15L 1s
3\31	116.13	1L 8s 1L 9s 10L 1s 10L 11s
4\31	154.84	1L 6s 7L 1s 8L 7s 8L 15s
5\31	193.55	1L 5s 6L 7s 6L 13s 6L 19s
6\31	232.26	1L 4s 5L 6s 5L 11s 5L 16s 5L 21s
7\31	270.97	4L 1s 4L 5s 9L 4s 9L 13s
8\31	309.68	4L 3s 4L 7s 4L 11s 4L 15s
9\31	348.39	3L 4s 7L 3s 7L 10s 7L 17s
10\31	387.10	3L 7s 3L 10s 3L 13s 3L 16s 3L 19s 3L 22s 3L 25s
11\31	425.81	3L 2s 3L 5s 3L 8s 3L 11s 14L 3s
12\31	464.52	3L 2s 5L 3s 5L 8s 13L 5s
13\31	503.23	2L 3s 5L 2s 7L 5s 12L 7s
14\31	541.94	2L 5s 2L 7s 9L 2s 11L 9s

As a regular temperament

In the 13-limit 31edo doesn't do as well, but is the optimal patent val for the rank five temperament tempering out the 13-limit comma 66/65, which equates 6/5 and 13/11. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

Commas

31 EDO tempers out the following commas. (Note: This assumes the val ⟨31 49 72 87 107 115], comma values rounded to 5 significant digits.)

Prime limit	Ratio	Monzo	Cents	Color name	Name(s)
5	(16 digits)	[-25 7 6⟩	31.567	Lala-tribiyo	Ampersand
5	81/80	[-4 4 -1⟩	21.506	Gu	Syntonic comma, Didymos comma, meantone comma
5	(12 digits)	[17 1 -8⟩	11.445	Saquadbigu	Würschmidt comma
5	(14 digits)	[-21 3 7⟩	10.061	Lasepyo	Semicomma, Fokker comma
5	(24 digits)	[38 -2 -15⟩	1.3843	Sasa-quintrigu	Hemithirds comma
7	(18 digits)	[-10 7 8 -7⟩	22.413	Lasepru-aquadbiyo	Blackjackisma
7	64827/64000	[-9 3 -3 4⟩	22.227	Laquadzo-atrigu	Squalentine
7	2430/2401	[1 5 1 -4⟩	20.785	Quadru-ayo	Nuwell
7	50421/50000	[-4 1 -5 5⟩	14.516	Quinzogu	Trimyna
7	126/125	[1 2 -3 1⟩	13.795	Zotrigu	Septimal semicomma, Starling comma
7	1728/1715	[6 3 -1 -3⟩	13.074	Trizo-agu	Orwellisma, Orwell comma
7	1029/1024	[-10 1 0 3⟩	8.4327	Latrizo	Gamelisma
7	225/224	[-5 2 2 -1⟩	7.7115	Ruyoyo	Septimal kleisma, Marvel comma
7	16875/16807	[0 3 4 -5⟩	6.9903	Quinru-aquadyo	Mirkwai
7	3136/3125	[6 0 -5 2⟩	6.0832	Zozoquingu	Hemimean
7	6144/6125	[11 1 -3 -2⟩	5.3621	Sarurutrigu	Porwell
7	(18 digits)	[-26 -1 1 9⟩	3.7919	Latritrizo-ayo	Wadisma
7	65625/65536	[-16 1 5 1⟩	2.3495	Lazoquinyo	Horwell
7	(12 digits)	[-11 2 7 -3⟩	1.6283	Latriru-asepyo	Meter
7	2401/2400	[-5 -1 -2 4⟩	0.72120	Bizozogu	Breedsma
11	99/98	[-1 2 0 -2 1⟩	17.576	Loruru	Mothwellsma
11	121/120	[-3 -1 -1 0 2⟩	14.367	Lologu	Biyatisma
11	176/175	[4 0 -2 -1 1⟩	9.8646	Lorugugu	Valinorsma
11	243/242	[-1 5 0 0 -2⟩	7.1391	Lulu	Rastma
11	385/384	[-7 -1 1 1 1⟩	4.5026	Lozoyo	Keenanisma
11	441/440	[-3 2 -1 2 -1⟩	3.9302	Luzozogu	Werckisma
11	540/539	[2 3 1 -2 -1⟩	3.2090	Lururuyo	Swetisma
11	3025/3024	[-4 -3 2 -1 2⟩	0.57240	Loloruyoyo	Lehmerisma

Rank-2 temperaments

List of 31et rank two temperaments by badness

List of edo-distinct 31et rank two temperaments

Generator	Cents	Temperaments	Pergen
1\31	38.71	Slender	(P8, P4/13)
2\31	77.42	Valentine/Lupercalia	(P8, P5/9)
3\31	116.13	Miracle	(P8, P5/6)
4\31	154.84	Nusecond	(P8, P11/11)
5\31	193.55	Luna/Hemithirds/Hemiwürschmidt	(P8, WWP4/15)
6\31	232.26	Mothra/Mosura	(P8, P5/3)
7\31	270.97	Orwell/Orson/Winston	(P8, P12/7)
8\31	309.68	Myna	(P8, WWP5/10)
9\31	348.39	Vicentino/Mohajira/Migration	(P8, P5/2)
10\31	387.10	Würschmidt/Worschmidt	(P8, WWP5/8)
11\31	425.81	Squares/Sentinel	(P8, P11/4)
12\31	464.52	A-Team/Semisept	(P8, M9/3)
13\31	503.23	Meantone/Meanpop	(P8, P5)
14\31	541.94	Casablanca/Cypress/Oracle	(P8, W5P4/12)
15\31	580.65	Tritonic/Tritoni	(P8, WWP4/5)

Scales

Harmonic Scale

31edo approximates Mode 8 of the harmonic series O.K., but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated O.K., but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 31's version of 13/8 is quite wide and only vaguely suggests the 13-limit.

Overtones in "Mode 8":	8	9	10	11	12	13	14	15	16
...as JI Ratio from 1/1:	1/1	9/8	5/4	11/8	3/2	13/8	7/4	15/8	2/1
...in cents:	0	203.9	386.3	551.3	702.0	840.5	968.8	1088.3	1200.0
Nearest degree of 31edo:	0	5	10	14	18	22	25	28	31
...in cents:	0	193.5	387.1	541.9	696.8	851.6	967.7	1083.9	1200.0

In mode 16, the most closely-matched harmonics are the composite ones, 21 and 25. Of the other harmonics:

• 17 is sharp, like 13. In fact, the 17:13 ratio is matched within a tenth of a cent.
• 19 is also sharp, like 13 and 17. The 19:17 ratio is about one cent sharp. 31edo could be considered a tuning of the 2.5.7.13.17.19 subgroup, on which it is consistent.
• 23 is about as flat as 11. The chromatic semitone is about half a cent off from 23:22. 31edo could be considered a tuning of the 2.3.5.7.11.23 subgroup, on which it is consistent.
• 27 is quite flat, as it's 3^3 and the error from the meantone fifths accumulates.
• 29 and 31 are both very sharp, and intervals involving them are unlikely to play any major role.

Odd overtones in "Mode 16":	17	19	21	23	25	27	29	31
...as JI Ratio from 1/1:	17/16	19/16	21/16	23/16	25/16	27/16	29/16	31/16
...in cents:	105.0	297.5	470.8	628.3	772.6	905.9	1029.6	1145.0
Nearest degree of 31edo:	3	8	12	16	20	23	27	30
...in cents:	116.1	309.7	464.5	619.4	774.2	890.3	1045.1	1161.3

Various subsets

A large open list of subsets from 31edo that people have named:

• 31edo modes.
• Strictly proper 7-note 31edo scales.
• Interesting (to somebody) 9-note 31edo scales.
• See also 31edo MOS scales.

Some of the popular ones:

• 31-tone major: 5 5 3 5 5 5 3
• Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3
• Harmonic scale 8: 5 5 4 4 4 3 3 3
• the Euler-Fokker genera (technically JI but representable in 31)

Trivia

31edo is close to a circle made by stacking 31 pure 17/13 subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents (relative error 7.1%).

Music

• An alphabetical list of Tricesimoprimal Compositions.
• Aire #2 in 31-equal temperament by Jon Lyle Smith ^{[dead link]}
• I Stand Hopeless Before the Gray Sea by Chuckles McGee
• Chaconna en G=, La Padana, ou la septimala (‘The Padanian, or the septimal’) by Claudi Meneghin
• earwig by Andrew Heathwaite
• Fanfare and Toccata by Juhani Nuorvala
• Orphanage of the Dutch Music IX: Sonate no. II in the 31-tone temperament - YouTube
• "Lively Up Yourself" cover featuring Paul Erlich on 31edo guitar
• Q: IS ART REVOLUTIONARY? A: LOL NO by Diamond Doll (xen-pop)
• Cool Guitar Girl by Diamond Doll (xen-pop)
• Some songs by Benyamin Linus

By Cam Taylor

• Enharmonic melody for guitar
• What use is a boy
• Back to 31: Hyperchromatic progression on C^

By Johann alias Circular17

• Mystère et tolérance
• Wave from the past
• Curieuse planète
• Deep but not too much
• Heal
• Désir mimétique (virtually 31edo)

By Zhea Erose

• Glitterdance
• Divinate

See Also

Pedagogy

The MicroPedagogyCollective is currently at work producing demonstrative material which will encourage and enable more people to learn this system. There have been two ThirtyOneToneSinginCamps as well.

See also: 31edo solfege, Tricesimoprimal Tetrachordal Tesseract, Pentachords of 31edo.

Books

External image: http://ronsword.com/images/TSG_sm.jpg ^{[dead link]}

WARNING: MediaWiki doesn't have very good support for external images.

Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

Sword, Ronald. "Tricesimoprimal Scales for Guitar." IAAA Press, UK-USA. First Ed: March 2009. ^{[dead link]} - A comprehensive approach to 31-EDO and all the families associated for Guitar. Features over 300 scale charts / scale examples.

Articles

• Wikipedia: 31 equal temperament
• de Beer, Anton, The Development of 31-tone Music Permalink
• Fokker, Adriaan Daniël, Equal Temperament and the Thirty-one-keyed organ Permalink
• Fokker, A.D., "New Music with 31 Notes" translated by Leigh Gerdine
• Rapoport, Paul, About 31-tone Equal Temperament Permalink
• Terpstra, Siemen, Toward a Theory of Meantone (and 31-et) Harmony Permalink
• Tonalsoft Encyclopedia article Permalink

Videos

• Quarter sharps and flats in the same diatonic key signature – Youtube by Stephen Weigel – a list of diatonic key signatures and major scales in 31edo (including semi- and sesqui-sharps); and docs in its description.

Software

• Virtual Piano Keyboard in 31-Tone Equal Temperament
• 31EDO Piano — Mini synthesizer in Pixilang