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
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 | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
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.
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 |
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
- Satiesque by Aaron Krister Johnson.
- Chromosounds play by Gene Ward Smith.
- Music For Your Ears play by Gene Ward Smith. The central portion is in 27edo, the rest is in 46edo.
- Rats play by Andrew Heathwaite.
- Tumbledown Stew play by Andrew Heathwaite.
- Hypnocloudsmack 1 play by Andrew Heathwaite.
- Hypnocloudsmack 2 play by Andrew Heathwaite.
- Hypnocloudsmack 3 play by Andrew Heathwaite.
- Bach BWV 1029 in 46 equal Claudi Meneghin version
- Bach Contrapunctus 4 Claudi Meneghin version
- A Seed Planted - (Yet another version: 46 EDO) by Jake Freivald
- Locrian Suite Gavotte by Inthar lus Lăneaf