31edo

31edo supports Valentine and Miracle, alongside supporting Meantone.

31edo, or 31 equal divisions of the octave, is an equal tuning with a step size of approximately 39 cents. It is most commonly known as a tuning of Meantone, and for its accurate approximation of the 2.5.7 subgroup.

31edo as a whole contains a diverse palette of interval qualities and structures ranging from the very familiar to the quite exotic, and remarkably, almost all of these still have a reasonably simple harmonic interpretation. As a meantone system, 31edo's diatonic scale includes the basic qualities of the 5-limit, such as the perfect fourth and fifth, and the classical minor and major thirds (6/5 and 5/4). But 31edo also includes subminor and supermajor intervals, identifiable with septal ratios such as 7/6 and 9/7, and neutral intervals, identifiable with 11-limit ratios such as 11/9.

In terms of structures, or ways of organizing harmony, it should first be noted that 31edo's perfect fifth, of 18 steps, is quite divisible. The fifth can be split in two, giving us a neutral-third temperament, known in this case as Mohajira, which emphasizes heptatonic structure, the 11th harmonic, and 2.3.5.11. Splitting the fifth in three gives us Slendric (or in this case Mothra), formed by stacking 8/7, and which emphasizes pentatonic structure and 2.3.7. Combining these gives us Miracle, while splitting Slendric into three again gives us Valentine. In 31edo, each of these provides xenharmonic ways of accessing the 11-limit with more simplicity than Meantone. Yet another way to encompass the 11-limit is given by Orwell, generated by 31edo's subminor third; and of course, as a prime EDO, 31edo contains several more structures unique to itself.

Theory

Edostep interpretations

31edo's edostep has the following interpretations in the 11-limit:

• 128/125 (the residue between three stacked 5/4s and the octave)
• 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
• 49/48 (the difference between 8/7 and 7/6)
• 50/49 (the difference between 7/5 and 10/7)
• 64/63 (the difference between 8/7 and 9/8)
• 33/32 (the difference between 12/11 and 9/8)
• 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
• 55/54 (the difference between 6/5 and 11/9, or 12/11 and 10/9)
• 56/55 (the difference between 5/4 and 14/11)

JI approximation

31edo can be understood as a 7-limit system with a somewhat flat 3/2 but nearly-perfect 5th and 7th harmonics. In particular, the product of 5 and 7, 35/32, is approximated to within about 0.3¢. 31edo also has an approximation to the 11th harmonic that, while tuned flat, has the property that the flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral third. The harmonic 23 turns out to be flat in a very similar way to 11.

The intervening harmonics - 13, 17, and 19 - are tuned rather sharp, but by almost exactly the same amount; therefore the chord 13:17:19 is extremely well-approximated by 31edo, with the interval 17/13 tuned less than 0.1¢ off.

31edo's fifth generates a functional diatonic scale. Its whole tone, of 5 steps, is split into semitones of 2 and 3; as 31edo's fifth is flatter than that of 12edo, the chromatic semitone, comprised by 2 steps, is smaller than the diatonic semitone, which is 3 steps.

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
	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)

Thirds in 31edo

Quality (ADIN)	Subminor	Nearminor	Neutral	Nearmajor	Supermajor
Cents	271	310	348	387	426
Just interpretation	7/6 (+4.1¢)	6/5 (-6.0¢)	11/9 (+1.0¢)	5/4 (+0.8¢)	9/7 (-9.2¢)
Steps	7	8	9	10	11

Diatonic thirds are bolded.

Chords

Along with its diatonic major and minor chords which approximate 5-limit harmony, 31edo also has a narrow but functional supermajor triad, and a well-tuned subminor triad. It also supports arto and tendo chords, with its slendric chords of [0 6 18] and [0 12 18], and has a neutral triad [0 9 18] which represents both artoneutral and tendoneutral triads in the 11- and 13-limit.

Regular temperaments

Besides Meantone (for which it provides an excellent tuning and which is shared with 19edo), 31edo also supports variations of Rastmic temperament (like 24edo), Slendric (like 36edo), Miracle (like 41edo), and Orwell (like 22edo).

Scales

31edo does not temper out 64/63, meaning that it can be used to tune Diasem while representing some simpler 5-limit intervals. 31edo's step is called a diesis, and can function as an aberrisma. Due to being a prime number, 31edo has a large number of full-period MOS scales that exist in the edo. Orwell[9] (gramitonic) is one example, so is Mohajira[7] (mosh).

31edo also has a usable 12-note chromatic scale, approximating golden Meantone/monocot.

MOS scales

A key aspect of 31edo noted by several sources is the diversity of MOS scales represented. The 15 edo-distinct regular temperaments of 31edo are divided into three loops that are traversed by halving or doubling their generators.

	Loop 1
Temperament	Didacus	Wurschmidt	Squares	Mohajira	Meantone
Complexity	15	8	4	2	1
Scale (albitonic)	5-5-5-5-5-6	8-1-1-8-1-1-8-1-1-1	2-2-7-2-2-7-2-7	5-4-5-4-5-4-4	5-5-3-5-5-5-3
Generator	5, 26	10, 21	20, 11	22, 9	18, 13
	Loop 2
Temperament	Miracle	Slendric	A-Team	Orwell	Casablanca
Complexity	6	3	14	7	12
Scale (albitonic)	3-3-3-3-3-3-3-3-3-4	6-6-6-6-7	2-5-2-5-5-2-5-5	4-3-4-3-4-3-4-3-3	3-3-3-3-5-3-3-3-5
Generator	3, 28	6, 25	12, 19	24, 7	17, 14
	Loop 3
Temperament	Slender	Valentine	Nusecond	Myna	Tritonic
Complexity	13	9	11	10	5
Scale (albitonic)	- (11-note scale has >10 steps interval)	- (11-note scale has >10 steps interval)	4-4-4-4-4-4-4-3	1-1-6-1-1-6-1-1-6-1-6	- (11-note scale has >10 steps interval)
Generator	1, 30	2, 29	4, 27	8, 23	15, 16

Notation

31edo, as one of the more popular edos, has a somewhat agreed-upon notation system. This notation is simply neutral diatonic notation applied to the edo, where a half-# or half-b represents an alteration by one diesis. In this manner, all notes can be spelled in a way that does not require multiple sharps or flats.

Step	Cents	ADIN	Neutral diatonic	Notation	Just intervals represented
0	0.00	unison	unison	A	1/1
1	38.71	superunison	semiaugmented unison	At	49/48, 50/49, 128/125
2	77.42	subminor second	semidiminished second	A#	25/24
3	116.13	nearminor second	minor second	Bb	16/15
4	154.84	neutral second	neutral second	Bd	11/10, 12/11
5	193.55	nearmajor second	major second	B	10/9, 9/8
6	232.26	supermajor second	semiaugmented second	Bt	8/7
7	270.97	subminor third	semidiminished third	Cd	7/6
8	309.68	nearminor third	minor third	C	6/5
9	348.39	neutral third	neutral third	Ct	11/9, 16/13
10	387.10	nearmajor third	major third	C#	5/4
11	425.81	supermajor third	semiaugmented third	Db	9/7
12	464.52	subfourth	semidiminished fourth	Dd	21/16, 13/10
13	503.23	perfect fourth	perfect fourth	D	4/3
14	541.94	neutral fourth	semiaugmented fourth	Dt	11/8, 15/11
15	580.65	nearaugmented fourth	augmented fourth	D#	7/5
16	619.35	neardiminished fifth	diminished fifth	Eb	10/7
17	658.06	neutral fifth	semidiminished fifth	Ed	16/11, 22/15
18	696.77	perfect fifth	perfect fifth	E	3/2
19	735.48	superfifth	semiaugmented fifth	Et	32/21, 20/13
20	774.19	subminor sixth	semidiminished sixth	Fd	14/9
21	812.90	nearminor sixth	minor sixth	F	8/5
22	851.61	neutral sixth	neutral sixth	Ft	13/8, 18/11
23	890.32	nearmajor sixth	major sixth	F#	5/3
24	929.03	supermajor sixth	semiaugmented sixth	Gb	12/7
25	967.74	subminor seventh	semidiminished seventh	Gd	7/4
26	1006.45	nearminor seventh	minor seventh	G	9/5, 16/9
27	1045.16	neutral seventh	neutral seventh	Gt	11/6, 20/11
28	1083.87	nearmajor seventh	major seventh	G#	15/8
29	1122.58	supermajor seventh	semiaugmented seventh	Ab	48/25
30	1161.29	suboctave	semidiminished octave	Ad	49/25, 125/64, 96/49
31	1200.00	octave	octave	A	2/1

Multiples

217edo

217edo is a theoretically strong system which keeps 31edo's tuning of 2.5.7.(13:17:19). 217edo is strong in the 19-limit and the smallest edo distinctly consistent in the 19-odd-limit.

Approximation of prime harmonics in 217edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	0.0	+0.3	+0.8	-1.1	+1.7	+0.0	+0.1	+1.1	+2.1	-1.0	-0.3	-2.5
	Relative (%)	0.0	+6.3	+14.2	-19.6	+30.3	+0.5	+2.1	+20.0	+38.7	-18.2	-6.1	-45.1
Steps (reduced)		217 (0)	344 (127)	504 (70)	609 (175)	751 (100)	803 (152)	887 (19)	922 (54)	982 (114)	1054 (186)	1075 (207)	1130 (45)