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

31 equal divisions of the octave (abbreviated 31edo or 31ed2), also called 31-tone equal temperament (31tet) or 31 equal temperament (31et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 31 equal parts of about 38.7 ¢ each. Each step represents a frequency ratio of 2^1/31, or the 31st root of 2.

31edo is also referred to as the tricesimoprimal meantone temperament. The term tricesimoprimal was first used by Adriaan Fokker.

English Wikipedia has an article on:

31 equal temperament

Theory

31edo's perfect fifth is flat of just by 5.2 ¢, as befits a tuning of meantone. The major third is less than a cent sharp of just 5/4, making it slightly sharp of quarter-comma meantone. 31edo's approximation of 7/4, a cent flat, is also very close to just. Because of the near-just 5/4 and 7/4, 31edo is relatively quite accurate in the 7-limit. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course).

Prime 11 is somewhat less accurate, making intervals like 11/8 off by about 9 cents. However, intervals like 11/9 and 11/6 are approximated quite well because the errors cancel out. This makes 31edo a very tone-efficient melodic approximation of the 11-limit (and specifically the 11-odd-limit), although it conflates 9/7 with 14/11 and 11/8 with 15/11. It also maps most 15-odd-limit intervals consistently, the exceptions being 13/9, 13/11, and their octave complements.

Other ways 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. It is also a strict zeta edo, meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once.

One step of 31edo, measuring about 38.7 ¢, is called a diesis because it stands in for several intervals called dieses (most notably, 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.

In terms of interval categories, because 31edo is a meantone system, the major and minor seconds, thirds, sixth, and sevenths on the chain of fifths are equated to 5-limit intervals, those being 16/15, 10/9, 6/5, 5/4, and their octave complements. 31edo maps the chromatic semitone to two steps, meaning there are "neutral" intervals between minor and major ones, which are not found in 12edo. They can be represented by 11-limit intervals, with 11/10~12/11 being a neutral second, and 11/9~27/22 a neutral third. One step in the other direction from the classical intervals are the subminor and supermajor intervals, which can be seen as intervals of prime 7. The subminor second is 21/20~28/27, the supermajor second 8/7, the subminor third 7/6, and the supermajor third 9/7~14/11. 31edo thus has five varieties of seconds and thirds, which is much more than the two varieties in 12edo.

Prime harmonics

Approximation of prime harmonics in 31edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0	-5.2	+0.8	-1.1	-9.4	+11.1	+11.2	+12.2	-8.9	+15.6	+16.3
Error	Relative (%)	+0.0	-13.4	+2.0	-2.8	-24.2	+28.6	+28.9	+31.4	-23.0	+40.3	+42.0
Steps (reduced)		31 (0)	49 (18)	72 (10)	87 (25)	107 (14)	115 (22)	127 (3)	132 (8)	140 (16)	151 (27)	154 (30)

Approximation of prime harmonics in 31edo (continued)
Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	-19.1	-3.3	-8.3	-7.4	+16.8	-14.0	+5.7	-1.9	+13.9	+4.5	-16.1
Error	Relative (%)	-49.3	-8.4	-21.4	-19.2	+43.4	-36.2	+14.7	-4.9	+35.8	+11.5	-41.7
Steps (reduced)		161 (6)	166 (11)	168 (13)	172 (17)	178 (23)	182 (27)	184 (29)	188 (2)	191 (5)	192 (6)	195 (9)

As a tuning of other temperaments

Besides meantone, 31edo can be used as a tuning for mohajira, mothra or less optimally miracle and valentine. These temperaments split 31edo's fifth, at 18 steps, into two, three, six, and nine equal parts. In fact, 31edo can be defined as the unique temperament that tempers out 81/80, 99/98, 121/120, and 126/125.

If we split the meantone generator of ~3/2 into two neutral thirds, each representing 11/9~27/22, then we get the 2.3.5.11-subgroup temperament mohaha, tempering out 121/120 and 243/242. We can then map 7/4 to the semi-diminished seventh (-13 generators), tempering out 385/384, to get the full 11-limit mohajira temperament, which maps 7/6, 6/5, 11/9, 5/4, and 9/7 equidistant from each other. Alternatively, we can use the septimal meantone mapping of 7/4 (+20 generators) to get migration. Mohajira and migration merge in 31edo, and create a near-optimal 11-limit meantone structure in one unified system.

The supermajor second 8/7 is mapped to a third of the perfect fifth in 31edo, thus tempering out 1029/1024, supporting slendric in the 2.3.7-subgroup. Slendric is a cluster temperament with 5 clusters of notes in an octave, each with nearby intervals separated by the interval found at -5 generators, or 1 step of 31edo, representing 49/48~64/63. For example, 9/8, 8/7, and 7/6 are one step apart from each other, as well as 9/7, 21/16, and 4/3. 31edo supports the full 7-limit extension mothra, which tempers out 81/80, thus equating the 49/48~64/63 spacer with 36/35, so that 9/8~10/9, 8/7, 7/6, and 6/5 are all mapped equidistantly, as well as 5/4, 9/7, 21/16, and 4/3. Mothra splits into two 11-limit extensions: undecimal mothra (26 & 31) tempering out 99/98, and mosura (31 & 36) tempering out 176/175.

Miracle temperament splits the slendric generator in two parts and the perfect fifth in six, each representing 15/14~16/15, thus tempering out 225/224, so that 5/4 is found at -7 generators. The 11-limit version of miracle sets 11/9 to the neutral third, with prime 11 mapped at +15 generators. While 31edo supports miracle, a more accurate tuning is 72edo. Valentine temperament splits the slendric generator in three parts and the perfect fifth in nine, each representing 21/20, tempering out 126/125. Valentine can also be seen as Carlos Alpha but with octaves added. The canonical 11-limit extension equates the step with 22/21, thus tempering out 121/120, 176/175, and 441/440.

31edo also supports orwell, which splits the perfect twelfth into seven equal parts of ~7/6. Three of these reach 8/5, and two reach 11/8, with 1–7/6–11/8–8/5 being the orwell tetrad. Commas tempered out by orwell include 99/98, 121/120, 176/175, and 385/384, among others.

Another notable temperament it supports is myna, which is generated by the minor third, and sets the intervals 7/6, 6/5, 11/9~16/13, 5/4, and 9/7 being equidistant. Like mohajira, it creates five interval categories, but with 126/125 tempered out instead of 81/80.

31edo also supports squares, which splits the perfect eleventh into four equal parts, each representing 14/11~9/7, two of which make 18/11, and four of which make 8/3. The 2.3.7.11 subgroup version of this temperament is sometimes known as skwares, tempering out 99/98 and 243/242. Then, prime 5 is found by tempering out 81/80, completing the 11-limit.

Another temperament supported by 31edo is würschmidt, which is generated by 5/4, such that 8 intervals of 5/4 reach 6/1. Würschmidt extends to the 7- and 11-limit through the skwares mapping, also creating 5 interval categories, with the thirds being 7/6, 6/5, 11/9, 5/4, and 14/11~9/7, each equidistant from each other.

Subsets and supersets

31edo is the 11th prime edo, following 29edo and coming before 37edo. It does not contain any nontrivial subset edos, though it contains 31ed4. 62edo and 93edo, which double and triple it, respectively, provide alternative ways to extend the temperament to the 13-, 17-, and 19-limits, and in the case of 93edo, even to the 23-limit.

217edo, which slices the edostep in seven, provides a very good correction of primes 3, 13, 17 and 31, and is consistent in the 21-odd-limit.

Intervals

See also: Table of 31edo intervals and 31edo/Individual degrees

#	Cents	Interval categories	Approximate ratios^{[note 1]}	Ups and downs notation
0	0.0	Unison	1/1	D
1	38.7	Super-unison	36/35, 45/44, 49/48, 50/49, 64/63, 128/125	^D, E♭♭
2	77.4	Subminor second	21/20, 22/21, 23/22, 25/24, 28/27	D♯, vE♭
3	116.1	Minor second	14/13, 15/14, 16/15	^D♯, E♭
4	154.8	Neutral second	11/10, 12/11, 13/12, 35/32	D𝄪, vE
5	193.5	Major second	9/8, 10/9, 19/17, 28/25	E
6	232.3	Supermajor second	8/7	^E, F♭
7	271.0	Subminor third	7/6	E♯, vF
8	309.7	Minor third	6/5, 25/21, 13/11	F
9	348.4	Neutral third	11/9, 16/13	^F, G♭♭
10	387.1	Major third	5/4	F♯, vG♭
11	425.8	Supermajor third	9/7, 14/11, 23/18, 32/25	^F♯, G♭
12	464.5	Subfourth	13/10, 17/13, 21/16	F𝄪, vG
13	503.2	Perfect fourth	4/3	G
14	541.9	Superfourth	11/8, 15/11, 26/19, 18/13, 48/35	^G, A♭♭
15	580.6	Augmented fourth	7/5, 25/18, 45/32	G♯, vA♭
16	619.4	Diminished fifth	10/7, 36/25, 64/45	^G♯, A♭
17	658.1	Subfifth	16/11, 19/13, 22/15, 13/9, 35/24	G𝄪, vA
18	696.8	Perfect fifth	3/2	A
19	735.5	Superfifth	20/13, 26/17, 32/21	^A, B♭♭
20	774.2	Subminor sixth	11/7, 14/9, 25/16	A♯, vB♭
21	812.9	Minor sixth	8/5	^A♯, B♭
22	851.6	Neutral sixth	13/8, 18/11	A𝄪, vB
23	890.3	Major sixth	5/3, 42/25, 22/13	B
24	929.0	Supermajor sixth	12/7	^B, C♭
25	967.7	Subminor seventh	7/4	B♯, vC
26	1006.5	Minor seventh	9/5, 16/9, 25/14, 34/19	C
27	1045.2	Neutral seventh	11/6, 20/11, 24/13, 64/35	^C, D♭♭
28	1083.9	Major seventh	13/7, 15/8, 28/15	C♯, vD♭
29	1122.6	Supermajor seventh	21/11, 27/14, 40/21, 44/23, 48/25	^C♯, D♭
30	1161.3	Sub-octave	35/18, 49/25, 63/32, 88/45, 96/49, 125/64	C𝄪, vD
31	1200.0	Octave	2/1	D

↑ As a 13-limit temperament, with additional ratios of 17, 19, and 23. Inconsistent intervals are in italics.

Proposed interval names and solfeges

See also: 31edo solfege

Table of proposed interval names and solfèges
#	Cents	Ups and downs notation (EUs: vvA1 and vd2)			Extended pythagorean notation			SKULO notation (S or U = 1)
0	0.0	P1	perfect unison	D	P1	perfect unison	D	P1	perfect unison	D
1	38.7	^1, d2	up-unison, dim 2nd	^D, Ebb	d2	dim 2nd	Ebb	S1/U1	super/uber unison	SD/UD
2	77.4	A1, vm2	aug 1sn, downminor 2nd	D#, vEb	A1	aug 1sn	D#	sm2	subminor 2nd	sEb
3	116.1	m2	minor 2nd	Eb	m2	minor 2nd	Eb	m2	minor 2nd	Eb
4	154.8	~2	mid 2nd	vE	AA1, dd3	double-aug 1sn, double-dim 3rd	Dx, Fbb	N2	neutral 2nd	UEb/uE
5	193.5	M2	major 2nd	E	M2	major 2nd	E	M2	major 2nd	E
6	232.3	^M2	upmajor 2nd	^E	d3	dim 3rd	Fb	SM2	supermajor 2nd	SE
7	271.0	vm3	downminor 3rd	vF	A2	aug 2nd	E#	sm3	subminor 3rd	sF
8	309.7	m3	minor 3rd	F	m3	minor 3rd	F	m3	minor 3rd	F
9	348.4	~3	mid 3rd	^F	AA2, dd4	double-aug 2nd, double-dim 4th	Ex, Gbb	N3	neutral 3rd	UF/uF#
10	387.1	M3	major 3rd	F#	M3	major 3rd	F#	M3	major 3rd	F#
11	425.8	^M3	upmajor 3rd	^F#	d4	dim 4th	Gb	SM3	supermajor 3rd	SF#
12	464.5	v4	down-4th	vG	A3	aug 3rd	Fx	s4	sub 4th	sG
13	503.2	P4	perfect 4th	G	P4	perfect 4th	G	P4	perfect 4th	G
14	541.9	^4, ~4	up-4th, mid 4th	^G	AA3, dd5	double-aug 3rd, double-dim 5th	Fx#, Abb	U4/N4	uber/neutral 4th	UG
15	580.6	A4, vd5	aug 4th, downdim 5th	G#, vAb	A4	aug 4th	G#	A4	aug 4th	G#
16	619.4	^A4, d5	upaug 4th, dim 5th	^G#, Ab	d5	dim 5th	Ab	d5	dim 5th	Ab
17	658.1	v5, ~5	down-5th, mid 5th	vA	AA4, dd6	double-aug 4th, double-dim 6th	Gx, Bbbb	u5/N5	unter/neutral 5th	uA
18	696.8	P5	perfect 5th	A	P5	perfect 5th	A	P5	perfect 5th	A
19	735.5	^5	up-5th	^A	d6	dim 6th	Bbb	S5	super 5th	SA
20	774.2	vm6	downminor 6th	vBb	A5	aug 5th	A#	sm6	subminor 6th	sBb
21	812.9	m6	minor 6th	Bb	m6	minor 6th	Bb	m6	minor 6th	Bb
22	851.6	~6	mid 6th	vB	AA5, dd7	double-aug 5th, double-dim 7th	Ax, Cbb	N6	neutral 6th	UBb/uB
23	890.3	M6	major 6th	B	M6	major 6th	B	M6	major 6th	B
24	929.0	^M6	upmajor 6th	^B	d7	dim 7th	Cb	SM6	supermajor 6th	SB
25	967.7	vm7	downminor 7th	vC	A6	aug 6th	B#	sm7	subminor 7th	sC
26	1006.5	m7	minor 7th	C	m7	minor 7th	C	m7	minor 7th	C
27	1045.2	~7	mid 7th	^C	AA6, dd8	double-aug 6th, double-dim 8ve	Bx, Dbb	N7	neutral 7th	UC/uC#
28	1083.9	M7	major 7th	C#	M7	major 7th	C#	M7	major 7th	C#
29	1122.6	^M7	upmajor 7th	^C#	d8	dim 8ve	Db	SM7	supermajor 7th	SC#
30	1161.3	v8	down-8ve	vD	A7	aug 7th	Cx	s8/u8	sub 8th, unter 8ve	sD/uD
31	1200.0	P8	perfect 8ve	D	P8	perfect 8ve	D	P8	perfect 8ve	D

Interval quality and chord names in color notation

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⟩ where b > −1	32/27, 16/9
minor	gu	[a b -1⟩	6/5, 9/5
mid	ilo	[a b 0 0 1⟩	11/9, 11/6
mid	lu	[a b 0 0 -1⟩	12/11, 18/11
major	yo	[a b 1⟩	5/4, 5/3
major	fifthward wa	[a b⟩ where 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 (7-over)	6:7:9	0 – 7 – 18	C – vE – G or C – E – G	Cvm	C downminor
gu (5-under)	10:12:15	0 – 8 – 18	C – E – G	Cm	C minor
ilo (11-over)	18:22:27	0 – 9 – 18	C – vE – G or C – E – G	C~	C mid
yo (5-over)	4:5:6	0 – 10 – 18	C – E – G	C, Cmaj	C, C major
ru (7-under)	14:18:21	0 – 11 – 18	C – ^E – G or C – E – G	C^	C up, C upmajor

For a more complete list of chords, see 31edo Chord Names and Ups and downs notation #Chords and chord progressions.

Notation

Ups and downs notation

Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp. The gamut runs D, ^D/vD#, D#, Eb, ^Eb/vE, E, ^E, vF, F etc.

Step offset	0	1	2	3	4	5
Sharp symbol
Flat symbol

Neutral chain-of-fifths notation

Circle of fifths in 31edo showing equivalences and quartertone accidentals

Since a sharp raises by 2 steps, 31edo can be notated using quarter-tone accidentals. Between C and D (do and re) for example, we have the following notes:

Degree	Letter	Solfège	English full name
0	C	do	C
1	C⁠ ⁠	do ⁠ ⁠	C half-sharp
2	C♯	do ♯	C sharp
3	D♭	re ♭	D flat
4	D⁠ ⁠	re ⁠ ⁠	D half-flat
5	D	re	D

Stein–Zimmermann accidentals

Step offset	−4	−3	−2	−1	0	+1	+2	+3	+4
Symbol

Chain-of-fifths notation

Chain-of-fifths notation uses double sharps and double flats only:

Degree	Letter	Solfège	English full name
0	C	do	C
1	D𝄫	re 𝄫	D double flat
2	C♯	do ♯	C sharp
3	D♭	re ♭	D flat
4	C𝄪	do 𝄪	C double sharp
5	D	re	D

While using double sharps and double flats may seem confusing because it alternates between C and D, it provides a way of writing chords that is consistent with traditional notation. For example, the subminor7 chord 12:14:18:21 is written like so:

C / D♯ / G / A♯
C♯ / D𝄪 / G♯ / A𝄪
D♭ / E / A♭ / B
D / E♯ / A / B♯

In 12edo, the enharmonic equivalences include C♯ = D♭, E♯ = F, and E = F♭. But in 31edo we have:

C𝄪 = D⁠ ⁠
D𝄫 = C⁠ ⁠
E♯ = F⁠ ⁠
F♭ = E⁠ ⁠
E𝄪 = F⁠ ⁠
F𝄫 = E⁠ ⁠

Sagittal notation

This notation uses the same sagittal sequence as edos 17, 24, and 38, and is a subset of the notation for 62edo.

Evo flavor

Evo-SZ flavor

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

Revo flavor

We also have a diagram from the appendix to The Sagittal Songbook by Jacob A. Barton, which gives multiple spellings for each pitch, and up to the double-apotome:

Relationship to 12edo

31edo’s circle of 31 fifths can be bent into a 12-spoked "spiral of fifths". In Kite Giedraitis' theory, this is possible because going up 12 fifths in 31edo yields a difference (the absolute value of the dodeca-sharpness) of 1 edostep (which also implies that 18\31 is on the 7\12 kite in the scale tree).

This "spiral of fifths" can be a useful construct for introducing 31edo to musicians unfamiliar with microtonal music. It may help composers and musicians to make visual sense of the notation, and to understand what size of a jump is likely to land them where compared to 12edo.

The two innermost and two outermost intervals on the spiral are duplicates, reflecting the fact that it is a repeating circle at heart and the spiral shape is only a helpful illusion.

Approximation to JI

Selected 19-limit intervals approximated in 31edo

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 31edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 31edo (direct approximation, even if inconsistent)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
5/4, 8/5	0.783	2.0
11/9, 18/11	0.979	2.5
7/4, 8/7	1.084	2.8
7/5, 10/7	1.867	4.8
15/14, 28/15	3.314	8.6
7/6, 12/7	4.097	10.6
11/6, 12/11	4.202	10.9
15/8, 16/15	4.398	11.4
15/11, 22/15	4.985	12.9
3/2, 4/3	5.181	13.4
5/3, 6/5	5.964	15.4
11/7, 14/11	8.298	21.4
9/7, 14/9	9.278	24.0
11/8, 16/11	9.382	24.2
11/10, 20/11	10.166	26.3
13/10, 20/13	10.302	26.6
9/8, 16/9	10.362	26.8
13/8, 16/13	11.085	28.6
9/5, 10/9	11.145	28.8
13/7, 14/13	12.169	31.4
15/13, 26/15	15.483	40.0
13/12, 24/13	16.266	42.0
13/9, 18/13	17.263	44.6
13/11, 22/13	18.242	47.1

15-odd-limit intervals in 31edo (patent val mapping)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
5/4, 8/5	0.783	2.0
11/9, 18/11	0.979	2.5
7/4, 8/7	1.084	2.8
7/5, 10/7	1.867	4.8
15/14, 28/15	3.314	8.6
7/6, 12/7	4.097	10.6
11/6, 12/11	4.202	10.9
15/8, 16/15	4.398	11.4
15/11, 22/15	4.985	12.9
3/2, 4/3	5.181	13.4
5/3, 6/5	5.964	15.4
11/7, 14/11	8.298	21.4
9/7, 14/9	9.278	24.0
11/8, 16/11	9.382	24.2
11/10, 20/11	10.166	26.3
13/10, 20/13	10.302	26.6
9/8, 16/9	10.362	26.8
13/8, 16/13	11.085	28.6
9/5, 10/9	11.145	28.8
13/7, 14/13	12.169	31.4
15/13, 26/15	15.483	40.0
13/12, 24/13	16.266	42.0
13/11, 22/13	20.468	52.9
13/9, 18/13	21.447	55.4

Consistent circles

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%). Remarkably, 31edo tempers out 83521/83486, the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's oneirotonic (5L 3s) mos accurate 13:17:19 chords.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[-49 31⟩	[⟨31 49]]	+1.637	1.637	4.228
2.3.5	81/80, 393216/390625	[⟨31 49 72]]	+0.976	1.628	4.204
2.3.5.7	81/80, 126/125, 1029/1024	[⟨31 49 72 87]]	+0.828	1.432	3.700
2.3.5.7.11	81/80, 99/98, 121/120, 126/125	[⟨31 49 72 87 107]]	+1.205	1.487	3.841
2.3.5.7.11.13	66/65, 81/80, 99/98, 105/104, 121/120	[⟨31 49 72 87 107 115]]	+0.502	2.072	5.353
2.3.5.7.11.23	81/80, 99/98, 126/125, 161/160, 231/230	[⟨31 49 72 87 107 140]]	+1.333	1.387	3.584

31et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next equal temperaments doing better in those subgroups are 72, 72, 41, and 46, respectively.
31et excels in the 2.5.7 subgroup (the JI chord 4:5:7 is represented highly consistently: to distance 10.36). In 2.5.7 it tempers out the didacus comma 3136/3125 and the quince comma 823543/819200, thus also tempering out the very small rainy comma, the simplest 2.5.7 comma tempered out by the 7-limit microtemperament 171edo.
In the 17-limit it tempers out 120/119, equating the otonal tetrad of 4:5:6:7 and the inversion of the 10:12:15:17 minor tetrad.

Uniform maps

13-limit uniform maps between 30.8 and 31.2
Min. size	Max. size	Wart notation	Map
30.7934	30.8119	31ddf	⟨31 49 72 86 107 114]
30.8119	30.9423	31f	⟨31 49 72 87 107 114]
30.9423	31.0745	31	⟨31 49 72 87 107 115]
31.0745	31.1681	31e	⟨31 49 72 87 108 115]
31.1681	31.2125	31de	⟨31 49 72 88 108 115]

Commas

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

Prime limit	Ratio^{[note 1]}	Monzo	Cents	Color name	Name
3	(30 digits)	[-49 31⟩	160.605	Quadlawa	31-comma
5	(16 digits)	[-25 7 6⟩	31.567	Lala-tribiyo	Ampersand comma
5	81/80	[-4 4 -1⟩	21.506	Gu	Syntonic comma
5	(12 digits)	[17 1 -8⟩	11.445	Saquadbigu	Würschmidt comma
5	(14 digits)	[-21 3 7⟩	10.061	Lasepyo	Semicomma
5	(24 digits)	[38 -2 -15⟩	1.3843	Sasa-quintrigu	Hemithirds comma
7	59049/57344	[-13 10 0 -1⟩	50.72	Laru	Harrison's comma
7	3645/3584	[-9 6 1 -1⟩	29.22	Laruyo	Schismean 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 comma
7	2430/2401	[1 5 1 -4⟩	20.785	Quadru-ayo	Nuwell comma
7	50421/50000	[-4 1 -5 5⟩	14.516	Quinzogu	Trimyna comma
7	126/125	[1 2 -3 1⟩	13.795	Zotrigu	Starling comma, septimal semicomma
7	1728/1715	[6 3 -1 -3⟩	13.074	Trizo-agu	Orwellisma
7	1029/1024	[-10 1 0 3⟩	8.4327	Latrizo	Gamelisma
7	225/224	[-5 2 2 -1⟩	7.7115	Ruyoyo	Marvel comma, septimal kleisma
7	16875/16807	[0 3 4 -5⟩	6.9903	Quinru-aquadyo	Mirkwai comma
7	3136/3125	[6 0 -5 2⟩	6.0832	Zozoquingu	Hemimean comma
7	6144/6125	[11 1 -3 -2⟩	5.3621	Sarurutrigu	Porwell comma
7	(18 digits)	[-26 -1 1 9⟩	3.7919	Latritrizo-ayo	Wadisma
7	65625/65536	[-16 1 5 1⟩	2.3495	Lazoquinyo	Horwell comma
7	(12 digits)	[-11 2 7 -3⟩	1.6283	Latriru-asepyo	Metric comma
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
13	105/104	[-3 1 1 1 0 -1⟩	16.567	Thuzoyo	Animist comma
13	144/143	[4 2 0 0 -1 -1⟩	12.064	Thulu	Grossma
13	196/195	[2 -1 -1 2 0 -1⟩	8.8554	Thuzozogu	Mynucuma
13	351/350	[-1 3 -2 -1 0 1⟩	4.94	Thorugugu	Ratwolfsma
13	352/351	[5 -3 0 0 1 -1⟩	4.93	Thulo	Minor minthma
13	625/624	[-4 -1 4 0 0 -1⟩	2.77	Thuquadyo	Tunbarsma
13	1001/1000	[-3 0 -3 1 1 1⟩	1.73	Tholozotrigu	Fairytale comma, sinbadma
13	4096/4095	[12 -2 -1 -1 0 -1⟩	0.42	Sathurugu	Minisma

↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

Rank-2 temperaments

31edo provides the optimal patent val for the rank-5 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.

Rank-2 temperaments by generators
Generator*	Cents*	Mos scales	Temperaments	Pergen
1\31	38.71		Slender	(P8, P4/13)
2\31	77.42	1L 14s, 15L 1s	Valentine / lupercalia	(P8, P5/9)
3\31	116.13	1L 9s, 10L 1s, 10L 11s	Mercy / miracle	(P8, P5/6)
4\31	154.84	1L 6s, 7L 1s, 8L 7s, 8L 15s	Greeley / nusecond	(P8, P11/11)
5\31	193.55	1L 5s, 6L 1s, 6L 7s, 6L 13s, 6L 19s	Luna / didacus / hemithirds / hemiwürschmidt / tutone	(P8, ccP4/15)
6\31	232.26	1L 4s, 5L 1s, 5L 6s, 5L 11s, 5L 16s, 5L 21s	Mothra / mosura Quadrawell	(P8, P5/3)
7\31	270.97	1L 3s, 4L 1s, 4L 5s, 9L 4s, 9L 13s	Orson / orwell / winston	(P8, P12/7)
8\31	309.68	3L 1s, 4L 3s, 4L 7s, 4L 11s, 4L 15s, 4L 19s, 4L 23s	Myna Triwell	(P8, ccP5/10)
9\31	348.39	3L 1s, 3L 4s, 7L 3s, 7L 10s, 7L 17s	Mohaha / vicentino / mohajira / migration	(P8, P5/2)
10\31	387.10	3L 1s, 3L 4s, 3L 7s, 3L 10s, 3L 13s, 3L 16s, 3L 19s, 3L 22s, 3L 25s	Würschmidt / worschmidt	(P8, ccP5/8)
11\31	425.81	3L 2s, 3L 5s, 3L 8s, 3L 11s, 14L 3s	Squares / sentinel	(P8, P11/4)
12\31	464.52	3L 2s, 5L 3s, 5L 8s, 13L 5s	A-Team Semisept	(P8, c⁵P4/14)
13\31	503.23	2L 3s, 5L 2s, 7L 5s, 12L 7s	Meantone / meanpop	(P8, P5)
14\31	541.94	2L 3s, 2L 5s, 2L 7s, 9L 2s, 11L 9s	Casablanca Cypress Oracle	(P8, c⁵P4/12)
15\31	580.65	2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s, 2L 13s, 2L 15s, 2L 17s, 2L 19s, 2L 21s, 2L 23s, 2L 25s, 2L 27s	Tritonic / tritoni	(P8, ccP4/5)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Octave stretch or compression

31edo can benefit from slightly stretching the octave, especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 11th harmonic at the expense of somewhat less accurate approximations of 5, 7, and 13.

Good options include:

127zpi: Good 13-limit option
80ed6: Great 11-limit option but bad harmonic 13
49edt: Good 13-limit option for the opposite mapping of 13

Scales

MOS scales

Main article: List of MOS scales in 31edo

The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other MOSes and MOS chains^{[clarification needed]} are also useful:

9\31, the neutral third, generates ultrasoft mosh and superhard dicotonic MOSes.
11\31, the supermajor third or diminished fourth, generates a parahard sensoid scale with resolution from neutral thirds, sixths, and sevenths to perfect fourths, fifths, and octaves, and a semihard 3L 8s scale with a jagged-but-chromatic feel.
12\31 generator generates a semihard oneirotonic (5L 3s) 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 close to 19/13.
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 has close approximations to 6edf (→ miracle) and 9edf (→ Carlos Alpha), fifth-equivalent equal divisions that hit many good JI approximations.

See #Rank-2 temperaments for a table of MOSes and their temperament interpretations.

Harmonic scales

31edo approximates Mode 8 of the harmonic series okay, 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 okay, but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated much better. 31edo's closest approximation of 13/8, the neutral sixth, is significantly sharper than just and only vaguely suggests the 13-limit.

The steps are: 5 5 4 4 4 3 3 3.

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 (see mercy temperament).
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 3³ and the error from the meantone fifths accumulates.
29 and 31 are both almost critically 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

Lists of scales

31edo modes
Strictly proper 7-tone 31edo scales
Interesting (to somebody) 9-tone 31edo scales
the Erose–McClain double modes, which are nonoctave

Individual scales

the Euler–Fokker genus (technically JI but representable in 31)
the altered pentad
diasem (2.3.7 subgroup scale; 5 2 5 1 5 2 5 1 5 or 5 1 5 2 5 1 5 2 5 in 31edo)
the moon dust scale^{[idiosyncratic term]} (technically JI but representable in 31)

Instruments

Keyboard Instruments

String Instruments

Guitar

Other Instruments

31edo array kalimba built by Tristan Bay; 3 octaves, 94 keys, and laid out in circle-of-fourths meantone tuning

Lumatone

Lumatone mapping for 31edo

Skip fretting

Skip fretting system 31 2 9 is a skip fretting system for 31edo.

Skip fretting system 31 3 7 is another skip fretting system for 31edo.

Skip fretting system 31 2 5 is another skip fretting system for 31edo. All examples on this page are for 7-string guitar.

Prime harmonics

1/1: string 2 open

2/1: string 7 fret 3

3/2: string 4 fret 4

5/4: string 4 open

7/4: string 7 open

11/8: string 4 fret 2

13/8: string 6 fret 1

17/16: string 1 fret 4

19/16: string 2 fret 4

23/16: string 4 fret 3

29/16: string 7 fret 1

31/16: string 1 fret 2

Music

Main article: 31edo/Music

See also: Category:31edo tracks

External links

Websites

31edo.com by Alex Racz

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.
Playlist of 31edo music theory videos on YouTube by Zhea Erose

Software

Diagrams

Keys and Modes of 31Et by Juhan Puhm (2016)
Keyboard Mapping for 31Et by Juhan Puhm (2017)
Mapping Range for 31Et by Juhan Puhm (2017)

[1] As a 13-limit temperament, with additional ratios of 17, 19, and 23. Inconsistent intervals are in italics.

[2] Ratios longer than 10 digits are presented by placeholders with informative hints.

[note 1]

[note 1]