# 31edo/Individual degrees

This article explains individual degrees of 31edo in depth.

## 1\31 – diesis or up-unison

A single step of 31-edo is about 38.71¢. Intervals around this size are called dieses (singular diesis). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the 11-limit, the diesis stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. 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. Demonstrated in SpiralProgressions.

## 2\31 – minor semitone, chromatic semitone, small minor 2nd or downminor 2nd

The difference between a major and minor third. The more 'expressive' of the 'half steps,' and the larger of 31's two "microtones". In meantone, it is the chromatic semitone, the interval that distinguishes major and minor intervals of the same generic interval class (e.g. thirds). 2\31 stands in for just ratios 28:27 (62.96¢); 25:24 (70.67¢); 22:21 (80.54¢); 21:20 (84.45¢) and others. Generates valentine temperament – aka semi-equalized Armodue.

MOS Scales generated by 2\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
15-tone (ME or quasi-equal) 1L 14s 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3
16-tone 15L 1s 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1

## 3\31 – major semitone, diatonic semitone, large minor 2nd or minor 2nd

The larger and clunkier of the 31edo semitones. In meantone, it is the diatonic semitone which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (119.44¢) and others. It is notable that two of these make an 8/7; this implies that the 3\31 is a secor and generates miracle temperament. It represents 343:320 (120.16¢) and 14:13 (128.29¢), and five of these give a 7/5, meaning it generates mercy temperament. The Pythagorean apotome 2187:2048 (113.69¢) is close to 3\31 in value, but is not consistent with the mapping of the primes 2 and 3 in 31edo (in fact the apotome of 31edo is the previous degree 2\31).

MOS Scales generated by 3\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
nonatonic 1L 8s 3 3 3 3 3 3 3 3 7
decatonic (quasi-equal) 1L 9s 3 3 3 3 3 3 3 3 3 4
11-tone 10L 1s 3 3 3 3 3 3 3 3 3 3 1
21-tone (Blackjack) 10L 11s 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1

## 4\31 – neutral tone, neutral 2nd or mid 2nd

Exactly one half of the minor third and twice the minor semitone. 4\31 stands in for 12:11 (150.64¢); 35:32 (155.14¢); 11:10 (165.00¢) and others. Although neutral seconds are typically associated with the 11-limit, 4\31 approximates the 7-limit interval 35/32 quite well, as the 5th harmonic of the 7th harmonic or vice versa, both of which are closely approximated in 31edo. And although 31 is not extremely accurate in the 11-limit, it is notable that since 11 and 3 are both flat, the interval that distinguishes them (12/11) is only about 4.5¢ off. Generates nusecond temperament.

MOS Scales generated by 4\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
heptatonic 1L 6s 4 4 4 4 4 4 7
octatonic (quasi-equal) 7L 1s 4 4 4 4 4 4 4 3
15-tone 8L 7s 1 3 1 3 1 3 1 3 1 3 1 3 1 3 3
23-tone 8L 15s 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 2

## 5\31 – whole tone or major 2nd

A rather smallish whole tone, 0.99¢ sharp of 19/17 (192.56¢). Sometimes called melodically dull. As it falls between (and functions as) just whole tones 9:8 and 10:9, 5\31 is considered a "meantone". Two meantones make a near-just major third. Perhaps it is worth noting that its relative narrowness (to JI 9/8) makes it easier to distinguish from the 8/7 approximation. And although it is over 10¢ flat of 9/8, 5\31 can function as a somewhat "active" (as opposed to perfectly stable) harmonic ninth, and it can be effective in combination with the also-narrow 11th harmonic. Indeed, the 11/9 approximation is excellent. Try, for instance 31's version of a 4:6:9:11 chord (steps 0-18-36-45). Generates hemithirds temperament and hemiwürschmidt temperament.

MOS Scales generated by 5\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
hexatonic (quasi-equal) 1L 5s 5 5 5 5 5 6
heptatonic 6L 1s 5 5 5 5 5 5 1
13-tone 6L 7s 4 1 4 1 4 1 4 1 4 1 4 1 1
19-tone 6L 13s 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 1
25-tone 6L 19s 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 1

## 6\31 – supermajor 2nd or upmajor 2nd

Exactly one half of a narrow fourth, twice a major semitone, or thrice a minor semitone. In 7-limit tonal music, 6\31 closely represents 8:7 (231.17¢). In meantone, it is a diminished third, e.g. C to Ebb. Generates mothra temperament.

MOS Scales generated by 6\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
pentatonic (quasi-equal) 1L 4s 6 6 6 6 7
hexatonic 5L 1s 6 6 6 6 6 1
11-tone 5L 6s 5 1 5 1 5 1 5 1 5 1 1
16-tone 5L 11s 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 1
21-tone 5L 16s 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 1 1
26-tone 5L 21s 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1

## 7\31 – subminor 3rd or downminor 3rd

Exactly one half of a superfourth (11:8 approximation). In 7-limit tonal music, 7\31 stands in for 7:6 (266.87¢). In meantone temperament, it is an augmented 2nd, e.g. C to D#. Generates orwell temperament.

MOS Scales generated by 7\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
pentatonic 4L 1s 7 7 7 7 3
nonatonic (quasi-equal; Orwell[9]) 4L 5s 4 3 4 3 4 3 4 3 3
13-tone (Orwell[13]) 9L 4s 1 3 3 1 3 3 1 3 3 1 3 3 3
22-tone (Orwell[22]) 9L 13s 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 2

## 8\31 – minor third

A minor third, closer to the just 6:5 (315.64¢) than 12-edo, but still on the flat side. Exactly twice a neutral second, four times a minor semitone, and half of a large tritone. Generates myna temperament.

MOS Scales generated by 8\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tetratonic (quasi-equal) 3L 1s 8 8 8 7
heptatonic 4L 3s 1 7 1 7 1 7 7
11-tone 4L 7s 1 1 6 1 1 6 1 1 6 1 6
15-tone 4L 11s 1 1 1 5 1 1 1 5 1 1 1 5 1 1 5
19-tone 4L 15s 1 1 1 1 4 1 1 1 1 4 1 1 1 1 4 1 1 1 4
23-tone 4L 19s 1 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 3
27-tone 4L 23s 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 2

## 9\31 – neutral 3rd or mid 3rd

A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for mohajira temperament), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct. It is 11¢ shy of 16/13 (359.47¢), suggesting a 13-limit interpretation for 31edo. However, its close proximity to 11/9 makes it hard to hear it as 16/13, which in JI has a different quality (and, as a neutral third, is more "major-like" than "minor-like"). Also, its inversion, 22\31 (851.61¢) is wide of the 13th harmonic by about 11¢, which leaves the 143rd harmonic only about 2¢ wide after cancelling with the narrow 11th harmonic, while all the lower harmonics are either near-just or narrow. This means the errors can accumulate, for instance, with 13/9 (636.62¢) represented by 17\31 (658.06¢), a good 21.4¢ sharp.

MOS Scales generated by 9\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tetratonic 3L 1s 9 9 9 4
heptatonic (quasi-equal) 3L 4s 5 4 5 4 5 4 4
10-tone 7L 3s 1 4 4 1 4 4 1 4 4 4
17-tone 7L 10s 1 1 3 1 3 1 1 3 1 3 1 1 3 1 3 1 3
24-tone 7L 17s 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 2

## 10\31 – major 3rd

A near-just major 3rd (compare with 5:4 = 386.31¢). Has led to the characterization of 31-edo as "smooth". Generates würschmidt/worschmidt temperaments.

MOS Scales generated by 10\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tritonic (quasi-equal) 1L 2s 10 10 11
tetratonic 3L 1s 10 10 10 1
heptatonic 3L 4s 9 1 9 1 9 1 1
10-tone 3L 7s 8 1 1 8 1 1 8 1 1 1
13-tone 3L 10s 7 1 1 1 7 1 1 1 7 1 1 1 1
16-tone 3L 13s 6 1 1 1 1 6 1 1 1 1 6 1 1 1 1 1
19-tone 3L 16s 5 1 1 1 1 1 5 1 1 1 1 1 5 1 1 1 1 1 1
22-tone 3L 19s 4 1 1 1 1 1 1 4 1 1 1 1 1 1 4 1 1 1 1 1 1 1
25-tone 3L 22s 3 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1
28-tone 3L 25s 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1

## 11\31 – supermajor 3rd or upmajor 3rd

11\31 functions as 14:11 (417.51¢), 23:18 (424.36¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. In meantone temperament, it is a diminished fourth, e.g. C to Fb. It is notable as closely approximating an interval of the 23-limit, suggesting the possibility of treating 16\31 (619.35¢) as a flat version of 23/16 (628.27¢). It is perhaps also notable for being close to 6\17, the bright major third of the ever-popular 17edo. Generates squares temperament.

MOS Scales generated by 11\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tritonic 2L 1s 11 11 9
pentatonic 3L 2s 2 9 2 9 9
octatonic 3L 5s 2 2 7 2 2 7 2 7
11-tone 3L 8s 2 2 2 5 2 2 2 5 2 2 5
14-tone (quasi-equal) 3L 11s 2 2 2 2 3 2 2 2 2 3 2 2 2 3
17-tone 14L 3s 2 2 2 2 2 1 2 2 2 2 2 2 1 2 2 2 1

## 12\31 – narrow 4th, subfourth or down 4th

Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). Although 31edo does not offer reasonable approximations of the 17th or 13th harmonics, 12\31 is less than 0.09¢ flat of the 17-limit interval 17/13 (464.43¢); combining this with 17\31's 1.08-cent-sharp approximation to 19/13 yields a good 13:17:19, which helps make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-odd-limit consonances, and the only intervals larger than 2\31 and smaller than 29\31 that are not 15-odd-limit consonances. Generates A-Team and semisept temperaments.

MOS Scales generated by 12\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tritonic 2L 1s 12 12 7
pentatonic 3L 2s 5 7 5 7 7
octatonic 5L 3s 5 5 2 5 5 2 5 2
13-tone (quasi-equal) 5L 8s 3 2 3 2 2 3 2 3 2 2 3 2 2
18-tone 13L 5s 1 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2 2

## 13\31 – perfect 4th

A slightly wide perfect fourth (compare to 4:3 = 498.04¢). As such, it functions marvelously as a generator for meantone temperament.

MOS Scales generated by 13\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tritonic 2L 1s 13 13 5
pentatonic 2L 3s 8 5 8 5 5
heptatonic 5L 2s 3 5 5 3 5 5 5
12-tone (quasi-equal) 7L 5s 3 3 2 3 2 3 3 2 3 2 3 2
19-tone 12L 7s 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 1 2 2

## 14\31 – superfourth or up 4th

Exactly twice a subminor third, and 1.08c flat of 26/19 (543.02¢). When viewing 31edo as a 11-limit temperament, the interval functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths (121/120 is tempered out). Thus it makes possible a symmetrical tempered version of an 8:11:15 triad. As 11/8, 14\31 is about 9¢ flat; however, it fits nicely with the also-flat 9/8, allowing a near-just 11/9. Nonetheless, most 11-limit chords in 31edo have a somewhat unstable quality which distinguishes them from their just counterparts. Generates casablanca and joan temperaments.

MOS Scales generated by 14\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tritonic 2L 1s 14 14 3
pentatonic 2L 3s 11 3 11 3 3
heptatonic 2L 5s 8 3 3 8 3 3 3
nonatonic 2L 7s 5 3 3 3 5 3 3 3 3
11-tone (quasi-equal) 9L 2s 2 3 3 3 3 2 3 3 3 3 3
20-tone 11L 9s 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1

## 15\31 – small tritone, augmented 4th, subdiminished 5th or downdim 5th

In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates tritonic temperament.

MOS Scales generated by 15\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tritonic 2L 1s 15 15 1
pentatonic 2L 3s 14 1 14 1 1
heptatonic 2L 5s 13 1 1 13 1 1 1
nonatonic 2L 7s 12 1 1 1 12 1 1 1 1
11-tone 2L 9s 11 1 1 1 1 11 1 1 1 1 1
13-tone 2L 11s 10 1 1 1 1 1 10 1 1 1 1 1 1
15-tone 2L 13s 9 1 1 1 1 1 1 9 1 1 1 1 1 1 1
17-tone 2L 15s 8 1 1 1 1 1 1 1 8 1 1 1 1 1 1 1 1
19-tone 2L 17s 7 1 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 1 1
21-tone 2L 19s 6 1 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 1 1 1
23-tone 2L 21s 5 1 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 1
25-tone 2L 23s 4 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1
27-tone 2L 25s 3 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1
29-tone 2L 27s 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1