31edt

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31edt divides the tritave into equal parts of 61.353 cents, corresponding to the non-octave third tone scale of 39edo where each degree gets ~.185 cents flatter than the corresponding degree of 39edo. It supports the same higher-limit meantone temperament as 12 edt with better intonation of triads. It also contains a flat version of the BP nonatonic scale and the fair Sigma and false Father scales.

Intervals

1\31 tritave- approx. 61.35¢ - Third tone

A single step of 31-edt is about 61.35¢. Intervals around this size are called third tones. In 31 it is equivalent to the difference between one tritave 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). The third tone is a defining sound of 31edt; 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.

2\31 tritave - approx. 122.71¢ - Two-third tone or Small Minor Second

The difference between a major and minor third and the closest thing to a 'half step'; in macromeantone, it is exactly analogus to the chromatic semitone, the interval that distinguishes major and minor intervals of the same generic interval class (eg. thirds).

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 tritave - approx. 184.06¢- Whole tone or Large Minor Second

A small whole tone only ~1.6 cents wide of 10:9 which is an interval sometimes called melodically dull; in macromeantone, it is exactly analogus to the diatonic semitone which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave.

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) 9L 1s 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 (silimlar to Blackjack) 11L 10s 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1

4\31 tritave - approx. 245.41¢ - Classical hemifourth or Neutral Second

Exactly one half of the minor third and twice the minor semitone, 4\31 stands in for 15:13 (247.74¢). Although 31 is not extremely accurate with 5 or 13, it is notable that the inaccuracies of these harmonics cancel out so much, leaving the interval that distinguishes them (15/13) only about 2.3¢ off.

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 tritave - approx. 306.77¢ - Sesquitone or Major Second

At ~8.8 cents flat of a just 6:5, 5\31 is considered a "sesquitone". Two of this sesquitone make a near-just 10:7 tritone. Because it is fairly close to no intelligibly small integer ratio but 6/5, 5\31 can function as a semi-stabilized harmonic ninth.

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 tritave - approx. 368.12¢ - Supermajor Second

Exactly one half of a narrow fourth, twice a tone, or thrice a two-third tone. In 17-limit tonal music, 6\31 closely represents 21:17 (365.825¢). In macromeantone, it is a diminished third, eg. C to Ebb.

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 tritave - approx. 429.47¢ - Subminor Third

Exactly one half of a superfourth. In 7-limit tonal music, 7\31 stands in for 9:7 (435.08¢). In macromeantone temperament, it is an augmented 2nd, eg. C to D#.

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; similar to Orwell[9]) 4L 5s 4 3 4 3 4 3 4 3 3
13-tone (similar to Orwell[13]) 9L 4s 1 3 3 1 3 3 1 3 3 1 3 3 3
22-tone (similar to 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 octave - approx. 490.83¢ - Minor Third

A 4:3 ~1/3 syntonic comma flat. Exactly twice a neutral second, four times a minor semitone, and half of a large tritone. Generates a Fair Sigma scale

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 tritave - approx. 552.18¢ - Neutral Third

A neutral 3rd, about 1¢ away from 11:8 (551.32¢). 9\31 is half a perfect fifth (making it a suitable generator for macromohajira temperament), and also a very small tritone. It is closer in quality to a minor third than a major third, but indeed, it is distinct. It is 11¢ shy of 18/13 (563.38¢), suggesting a 13-limit interpretation for 31edt. However, its close proximity to 11/8 makes it hard to hear it as 18/13, which in JI has a different quality (and, as a neutral third, is more "major-like" than "minor-like")..

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 tritave - approx. 613.53¢ - Major Third

A near-enough-just greater septimal tritone (compare with 10:7 = 617.49¢). Generates wurshmidt/worshmidt 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 tritave - approx. 674.89¢ - Supermajor Third

11\31 functions as 126:85 (681.47¢). In macromeantone temperament, it is a diminished fourth, eg. C to Fb. It is notable as closely approximating the 9/16edo Armodue sixth. Generates the Unfair Mu scale.

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 3L 14s 2 2 2 2 2 1 2 2 2 2 2 2 1 2 2 2 1

12\31 tritave - approx. 736.52¢ - Narrow Fourth or Subfourth

Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 32:21 (729.22¢). It is also quite close to the 17-limit interval 26/17 (735.57¢) and 19\31edo (735.48¢). However, although 31edt offers up a reasonable approximation of the 17th harmonic (18\31), no such approximation of the 13th comes with it to help make this identity clear. Generates false Father temperament.

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 tritave - approx. 797.54¢ - Perfect Fourth

A slightly narrow perfect fourth (compare to 27:17 = 800.91¢). As such, it functions marvelously as a generator for macromeantone 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 tritave - approx. 858.95¢ - Superfourth

Exactly twice a subminor third, this interval functions as both the 28:17 (863.86¢) septendecimal and 23:14 (859.44¢) vicesmotertial superfourths (392/391 is tempered out). Thus it makes possible a symmetrical tempered version of a 17:28:46 triad. As either, 14\31 is flat (about 5¢ or about .5¢); however, it fits nicely with the sharp 17, allowing a even-nearer-just 28/27.

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 tritave - approx. 920.3¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth

In 23-limit tonal music, functions quite well as 46:27 (922.41¢). Exactly thrice a whole tone. Generates Trans-Arcturus 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