46edo

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46edo
Prime factorization 2 × 23
Step size 26.0870¢
Fifth 27\46 (704.3¢)
Major 2nd 8\46 (208.7¢)
Semitones (A1:m2) 5:3 (130.4¢ : 78.3¢)
Consistency limit 13
Monotonicity limit 17

The 46 equal divisions of the octave (46edo), or the 46(-tone) equal temperament (46tet, 46et) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 46 equally-sized steps. Each step has a size of about 26.1 cents, an interval close in size to 66/65, the interval from 13/11 to 6/5.

Theory

46et tempers out 507/500, 91/90, 686/675, 2048/2025, 121/120, 245/243, 126/125, 169/168, 176/175, 896/891, 196/195, 1029/1024, 5120/5103, 385/384, and 441/440 among other intervals, with various consequences. Rank two temperaments it supports include sensi, valentine, shrutar, rodan, leapday and unidec. The 11-limit minimax tuning for valentine temperament, (11/7)1/10, is only 0.01 cents flat of 3\46 octaves. In the opinion of some, 46et is the first equal division to deal adequately with the 13-limit, though others award that distinction to 41et. In fact, while 41 is a zeta integral EDO but not a zeta gap EDO, 46 is zeta gap but not zeta integral.

The fifth of 46 equal is 2.39 cents sharp, which some people (e.g. Margo Schulter) prefer, sometimes strongly, over both the just fifth and fifths of temperaments with flat fifths, such as meantone. It gives a characteristic bright sound to triads, distinct from the mellowness of a meantone triad.

46edo can be treated as two 23edo's separated by an interval of 26.087 cents.

Shrutar22 as srutis describes a possible use of 46edo for Indian music.

Prime harmonics

Approximation of prime harmonics in 46edo
Harmonic 2 3 5 7 11 13 17 19 23 29
Error absolute (¢) +0.0 +2.4 +5.0 -3.6 -3.5 -5.7 -0.6 -10.6 -2.2 -12.2
relative (%) +0 +9 +19 -14 -13 -22 -2 -40 -8 -47
Steps
(reduced) 		46
(0) 		73
(27) 		107
(15) 		129
(37) 		159
(21) 		170
(32) 		188
(4) 		195
(11) 		208
(24) 		223
(39)

Intervals

# Cents Approximate Ratios* Ups and Downs Notation Solfege
0 0.000 1/1 perfect unison P1 D do
1 26.087 81/80, 64/63, 49/48 up unison ^1 ^D di
2 52.174 28/27, 36/35, 33/32 downminor 2nd vm2 vEb ro
3 78.261 25/24, 21/20, 22/21, 24/23, 23/22 minor 2nd m2 Eb rih
4 104.348 16/15, 17/16, 18/17 upminor 2nd ^m2 ^Eb ra
5 130.435 13/12, 14/13, 15/14 downmid 2nd v~2 ^^Eb ru (as in supraminor)
6 156.522 12/11, 11/10, 23/21 upmid 2nd ^~2 vvE ruh (as in submajor)
7 182.609 10/9 downmajor 2nd vM2 vE reh
8 208.696 9/8 major 2nd M2 E re
9 234.783 8/7, 23/20 upmajor 2nd ^M2 ^E ri
10 260.870 7/6 downminor 3rd vm3 vF ma
11 286.957 13/11, 20/17 minor 3rd m3 F meh
12 313.043 6/5 upminor 3rd ^m3 ^F me
13 339.130 11/9, 17/14, 28/23 downmid 3rd v~3 ^^F mu
14 365.217 16/13, 26/21, 21/17 upmid 3rd ^~3 vvF# muh
15 391.304 5/4 downmajor 3rd vM3 vF# mi
16 417.391 14/11, 23/18 major 3rd M3 F# maa
17 443.478 9/7, 13/10, 22/17 upmajor 3rd ^M3 ^F# mo
18 469.565 21/16, 17/13 down 4th v4 vG fe
19 495.652 4/3 perfect 4th P4 G fa
20 521.739 27/20, 23/17 up 4th ^4 ^G fih
21 547.826 11/8 downmid 4th v~4 ^^G fu
22 573.913 7/5, 18/13, 32/23 upmid 4th, dim 5th ^~4, d5 vvG#, Ab fi
23 600.000 17/12, 24/17 downaug 4th, updim 5th vA4, ^d5 vG#, ^Ab seh
24 626.087 10/7, 13/9, 23/16 aug 4th, downmid 5th A4, v~5 G#, ^^Ab se
25 652.174 16/11 double-down 5th ^~5 vvA su
26 678.261 40/27, 34/23 down 5th v5 vA sih
27 704.348 3/2 perfect 5th P5 A sol
28 730.435 32/21, 26/17 up 5th ^5 ^A si
29 756.522 14/9, 20/13, 17/11 downminor 6th vm6 vBb lo
30 782.609 11/7 minor 6th m6 Bb leh
31 808.696 8/5 upminor 6th ^m6 ^Bb le
32 834.783 13/8, 21/13, 34/21 downmid 6th v~6 ^^Bb lu
33 860.870 18/11, 28/17, 23/14 upmid 6th ^~6 vvB luh
34 886.957 5/3 downmajor 6th vM6 vB la
35 913.043 22/13, 17/10 major 6th M6 B laa
36 939.130 12/7 upmajor 6th ^M6 ^B li
37 965.217 7/4, 40/23 downminor 7th vm7 vC ta
38 991.304 16/9, 23/13 minor 7th m7 C teh
39 1017.391 9/5 upminor 7th ^m7 ^C te
40 1043.478 11/6, 20/11, 42/23 downmid 7th v~7 ^^C tu
41 1069.565 24/13, 13/7, 28/15 upmid 7th ^~7 vvC# tuh
42 1095.652 15/8, 32/17, 17/9 downmajor 7th vM7 vC# ti
43 1121.739 48/25, 40/21, 21/11, 23/12, 44/23 major 7th M7 C# taa
44 1147.826 27/14, 35/18, 64/33 upmajor 7th ^M7 ^C# to
45 1173.913 160/81, 63/32, 96/49 down 8ve v8 vD da
46 1200.000 2/1 perfect 8ve P8 D do

* Based on treating 46edo as a 2.3.5.7.11.13.17.23 subgroup, without ratios of 15 (except the superparticulars). 46edo has the 15th harmonic poorly approximated in general, because, while both the 3rd and 5th harmonics are sharp by a fair amount and they add up, all the other primes are flat, making the difference even larger, to the extent that it is not consistent in the 15-odd-limit. This can be demonstrated with the discrepancy approximating 15/13 (and its inversion 26/15). 9\46edo is closer to 15/13 by a hair; 10\46edo represents the difference between, for instance, 46edo's 15/8 and 13/8, and is more likely to appear in chords actually functioning as 15/13.

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

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

All 46edo 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, lu, yo and ru triads:

color of the 3rd JI chord notes as edosteps notes of C chord written name spoken name
zo 6:7:9 0-10-27 C vEb G Cvm C downminor
gu 10:12:15 0-12-27 C ^Eb G C^m C upminor
ilo 18:22:27 0-13-27 C ^^Eb G Cv~ C downmid
lu 22:27:33 0-14-27 C vvE G C^~ C upmid
yo 4:5:6 0-15-27 C vE G Cv C downmajor or C down
ru 14:18:21 0-17-27 C ^E G C^ C upmajor or C up

For a more complete list, see Ups and Downs Notation #Chords and Chord Progressions.

Notation

Sagittal

The following table shows sagittal notation accidentals in one apotome for 46edo.

Steps 0 1 2 3 4 5
Symbol Sagittal natural.png Sagittal pai.png Sagittal pakai.png Sagittal jatai.png Sagittal sharp pao.png Sagittal sharp.png

JI approximation

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 46edo. Prime harmonics are in bold; inconsistent intervals are in italic.

15-odd-limit intervals by direct mapping (even if inconsistent)
Interval, complement Error (abs, ¢) Error (rel, %)
11/7, 14/11 0.117 0.4
9/5, 10/9 0.205 0.8
13/7, 14/13 2.137 8.2
13/11, 22/13 2.253 8.6
3/2, 4/3 2.393 9.2
5/3, 6/5 2.598 10.0
11/8, 16/11 3.492 13.4
7/4, 8/7 3.609 13.8
9/8, 16/9 4.786 18.3
5/4, 8/5 4.991 19.1
13/8, 16/13 5.745 22.0
11/6, 12/11 5.885 22.6
7/6, 12/7 6.001 23.0
15/8, 16/15 7.383 28.3
13/12, 24/13 8.138 31.2
11/9, 18/11 8.278 31.7
9/7, 14/9 8.394 32.2
11/10, 20/11 8.482 32.5
7/5, 10/7 8.599 33.0
18/13, 13/9 10.531 40.4
13/10, 20/13 10.736 41.2
15/11, 22/15 10.875 41.7
15/14, 28/15 10.992 42.1
15/13, 26/15 12.958 49.7
15-odd-limit intervals by patent val mapping
Interval, complement Error (abs, ¢) Error (rel, %)
11/7, 14/11 0.117 0.4
9/5, 10/9 0.205 0.8
13/7, 14/13 2.137 8.2
13/11, 22/13 2.253 8.6
3/2, 4/3 2.393 9.2
5/3, 6/5 2.598 10.0
11/8, 16/11 3.492 13.4
7/4, 8/7 3.609 13.8
9/8, 16/9 4.786 18.3
5/4, 8/5 4.991 19.1
13/8, 16/13 5.745 22.0
11/6, 12/11 5.885 22.6
7/6, 12/7 6.001 23.0
15/8, 16/15 7.383 28.3
13/12, 24/13 8.138 31.2
11/9, 18/11 8.278 31.7
9/7, 14/9 8.394 32.2
11/10, 20/11 8.482 32.5
7/5, 10/7 8.599 33.0
13/9, 18/13 10.531 40.4
13/10, 20/13 10.736 41.2
15/11, 22/15 10.875 41.7
15/14, 28/15 10.992 42.1
15/13, 26/15 13.129 50.3

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [73 -46⟩ [⟨46 73]] -0.755 0.75 2.89
2.3.5 2048/2025, 78732/78125 [⟨46 73 107]] -1.219 0.90 3.45
2.3.5.7 126/125, 245/243, 1029/1024 [⟨46 73 107 129]] -0.595 1.34 5.12
2.3.5.7.11 121/120, 126/125, 176/175, 245/243 [⟨46 73 107 129 159]] -0.274 1.36 5.20
2.3.5.7.11.13 91/90, 121/120, 169/168, 176/175, 245/243 [⟨46 73 107 129 159 170]] +0.030 1.41 5.42
2.3.5.7.11.13.17 91/90, 121/120, 154/153, 169/168, 176/175, 245/243 [⟨41 65 95 115 142 152 174 188]] +0.047 1.31 5.02
2.3.5.7.11.13.17.23 91/90, 121/120, 154/153, 169/168, 176/175, 208/207, 231/230 [⟨41 65 95 115 142 152 174 188 208]] +0.101 1.23 4.72

46et is lower in relative error than any previous equal temperaments in the 17-, 19-, 23-limit, and others. The next ETs doing better in the aforementioned subgroups are 72, 72, 94, respectively. 46et is even more prominent in the no-19 23-limit, and the next ET doing better in this subgroup is 140.

Rank-2 temperaments

Periods
per octave 		Generator Cents Temperaments MOS Scales L:s
1 1\46 26.087 Sfourth
1 3\46 78.261 Valentine 1L 14s (15-tone)
15L 1s (16-tone)
16L 15s (31-tone) 		4:3 ~ quasi-equal
3:1
2:1 ~ QE
1 5\46 130.435 Twothirdtonic 1L 8s (9-tone)
9L 1s (10-tone)
9L 10s (19-tone)
9L 19s (28-tone)
9L 28s (37-tone) 		6:5 ~ QE
5:1
4:1
3:1
2:1 ~ QE
1 7\46 182.609 Minortone / mitonic 1L 5s (6-tone)
6L 1s (7-tone)
7L 6s (13-tone)
13L 7s (20-tone)
13L 20s (33-tone) 		11:7
7:4
4:3 ~ QE
3:1
2:1 ~ QE
1 9\46 234.783 Rodan 1L 4s (5-tone)
1L 5s (6-tone)
5L 6s (11-tone)
5L 11s (16-tone)
5L 16s (21-tone)
5L 21s (26-tone)
5L 26s (31-tone)
5L 31s (36-tone)
5L 36s (41-tone) 		10:9 ~QE
9:1
8:1
7:1
6:1
5:1
4:1
3:1
2:1 ~ QE, Pathological
1 11\46 286.957 4L 1s (5-tone)
4L 5s (9-tone)
4L 9s (13-tone)
4L 13s (17-tone)
4L 17s (21-tone)
21L 4s (25-tone) 		11:2
9:2
7:2
5:2
3:2 ~ QE, Golden
2:1 ~ QE
1 13\46 339.130 Amity / hitchcock 4L 3s (7-tone)
7L 4s (11-tone)
7L 11s (18-tone)
7L 18s (25-tone)
7L 25s (32-tone)
7L 32s (39-tone) 		7:6 ~ QE
6:1
5:1
4:1
3:1
2:1 ~ QE! Pathological
1 15\46 391.304 Magus / amigo 1L 2s (3-tone)
3L 1s (4-tone)
3L 4s (7-tone)
3L 7s (10-tone)
3L 10s (13-tone)
3L 13s (16-tone)
3L 16s (19-tone)
3L 19s (21-tone)
3L 21s (24-tone)
3L 24s (27-tone)
3L 27s (30-tone)
3L 30s (33-tone)
3L 33s (36-tone)
3L 36s (39-tone)
3L 39s (42-tone) 		16:15 ~ QE
15:1
14:1
13:1
12:1
11:1
10:1
9:1
8:1
7:1
6:1
5:1
4:1
3:1 ~ Pathological
2:1 ~ QE, Pathological
1 17\46 443.478 Sensi 3L 2s (5-tone)
3L 5s (8-tone)
8L 3s (11-tone)
8L 11s (19-tone)
19L 8s (27-tone) 		12:5
7:5
5:2
3:2 ~ QE, Golden
2:1
1 19\46 495.652 Leapday 2L 3s (5-tone)
5L 2s (7-tone)
5L 7s (12-tone)
12L 5s (17-tone)
17L 12s (29-tone) 		11:8
8:3
5:3 ~ Golden
3:2 ~ QE, Golden
2:1 ~ QE
1 21\46 547.826 Heinz 2L 3s (5-tone)
2L 5s (7-tone)
2L 7s (9-tone)
2L 9s (11-tone)
11L 2s (13-tone)
11L 13s (24-tone)
11L 24s (35-tone) 		17:4
13:4
9:4
5:4 ~ QE
4:1
3:1
2:1 ~ QE
2 1\46 26.087 Ketchup
2 2\46 52.174 Shrutar 2L 2s (4-tone)
2L 4s (6-tone)
2L 6s (8-tone)
2L 8s (10-tone)
2L 10s (12-tone)
2L 12s (14-tone)
2L 14s (16-tone)
2L 16s (18-tone)
2L 18s (20-tone)
2L 20s (22-tone)
22L 2s (24-tone) 		21:2
19:2
17:2
15:2
13:2
11:2
9:2
7:2
5:2
3:2 ~ QE, Golden
2:1 ~ QE
2 3\46 78.261 Semivalentine 2L 2s (4-tone)
2L 4s (6-tone)
2L 6s (8-tone)
2L 8s (10-tone)
2L 10s (12-tone)
2L 12s (14-tone)
14L 2s (16-tone)
16L 14s (30-tone) 		20:3
17:3
14:3
11:3
8:3
5:3 ~ Golden
3:2 ~ QE, Golden
2:1 ~ QE
2 4\46 104.348 Srutal / diaschismic 2L 2s (4-tone)
2L 4s (6-tone)
2L 6s (8-tone)
2L 8s (10-tone)
10L 2s (12-tone)
12L 10s (22-tone)
12L 22s (34-tone) 		19:4
15:4
11:4
7:4
4:3 ~ QE
3:1
2:1 ~ QE
2 5\46 130.435 2L 2s (4-tone)
2L 4s (6-tone)
2L 6s (8-tone)
8L 2s (10-tone)
8L 10s (18-tone)
18L 10s (28-tone) 		18:5
13:5
8:5 ~ Golden
5:3 ~ Golden
3:2 ~ QE, Golden
2:1 ~ QE
2 6\46 156.522 Bison 2L 2s (4-tone)
2L 4s (6-tone)
6L 2s (8-tone)
8L 6s (14-tone)
8L 14s (22-tone)
8L 22s (30-tone)
8L 30s (38-tone 		17:6
11:6
6:5 ~ QE
5:1
4:1
3:1
2:1 ~ QE, Pathological
2 7\46 182.609 Unidec / hendec 2L 2s (4-tone)
2L 4s (6-tone)
6L 2s (8-tone)
6L 8s (14-tone)
6L 14s (20-tone)
20L 6s (26-tone) 		16:7
9:7
7:2
5:2
3:2 ~ QE, Golden
2:1 ~ QE
2 8\46 208.696 Abigail 2L 2s (4-tone)
4L 2s (6-tone)
6L 2s (8-tone)
6L 8s (14-tone)
6L 14s (20-tone)
6L 20s (26-tone)
6L 26s (32-tone)
6L 32s (38-tone)
6L 38s (44-tone) 		15:8
8:7 ~ QE
8:1
7:1
6:1
5:1
4:1
3:1 ~ Pathological
2:1 ~ QE, Pathological
2 9\46 234.783 Echidnic 2L 2s (4-tone)
4L 2s (6-tone)
6L 4s (10-tone)
10L 6s (16-tone)
10L 16s (26-tone)
10L 26s (36-tone) 		14:9
9:5
5:4 ~ QE
4:1
3:1
2:1 ~ QE
2 10\46 260.87 Bamity 2L 2s (4-tone)
4L 2s (6-tone)
4L 6s (10-tone)
4L 10s (14-tone)
14L 4s (18-tone)
14L 18s (32-tone) 		13:10
10:3
7:3
4:3 ~ QE
3:1
2:1 ~ QE
2 11\46 286.957 Vines 2L 2s (4-tone)
4L 2s (6-tone)
4L 6s (10-tone)
4L 10s (14-tone)
4L 14s (18-tone)
4L 18s (22-tone)
4L 22s (26-tone)
4L 26s (30-tone)
4L 30s (34-tone)
4L 34s (38-tone)
4L 38s (42-tone) 		12:11 ~ QE
11:1
10:1
9:1
8:1
7:1
6:1
5:1
4:1
3:1 ~ Pathological
2:1 ~ QE, Pathological
23 1\46 26.087 Icositritonic

Scales

Harmonic scales

46edo represents overtones 8 through 16 (written as JI ratios 8:9:10:11:12:13:14:15:16) with degrees 0, 8, 15, 21, 27, 32, 37, 42, 46. In steps-in-between, that's 8, 7, 6, 6, 5, 5, 5, 4.

  • 8\46 (208.696¢) stands in for frequency ratio 9/8 (203.910¢).
  • 7\46 (182.609¢) stands in for 10/9 (182.404¢).
  • 6\46 (156.522¢) stands in for 11/10 (165.004¢) and 12/11 (150.637¢).
  • 5\46 (130.435¢) stands in for 13/12 (138.573¢), 14/13 (128.298¢) and 15/14 (119.443¢).
  • 4\46 (104.348¢) stands in for 16/15 (111.731¢).

Music

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