46edo
46 tone equal temperament[edit]
The 46 equal temperament, often abbreviated 46-tET, 46-EDO, or 46-ET, is the scale derived by dividing the octave into 46 equally-sized steps. Each step represents a frequency ratio of 26.087 cents, an interval close in size to 66/65, the interval from 13/11 to 6/5.
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 varied consequences it would take a very long article to describe. 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 41edo. 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 (eg, 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.
46edo srutis[edit]
Shrutar22 as srutis describes a possible use of 46edo for Indian music.
Intervals[edit]
| degrees of 46edo | solfege | cents value | approximate ratios
in the 17-limit |
ups and downs notation | ||
|---|---|---|---|---|---|---|
| 0 | do | 0.00 | 1/1 | perfect unison | P1 | D |
| 1 | di | 26.087 | up unison | ^1 | D^ | |
| 2 | ro | 52.174 | downminor 2nd | vm2 | Ebv | |
| 3 | rih | 78.261 | minor 2nd | m2 | Eb | |
| 4 | ra | 104.348 | 16/15, 17/16, 18/17 | upminor 2nd | ^m2 | Eb^ |
| 5 | ru (as in supraminor) | 130.435 | 13/12, 14/13, 15/14 | downmid 2nd | v~2 | Eb^^ |
| 6 | ruh (as in submajor) | 156.522 | 12/11, 11/10 | upmid 2nd | ^~2 | Evv |
| 7 | reh | 182.609 | 10/9 | downmajor 2nd | vM2 | Ev |
| 8 | re | 208.696 | 9/8, 17/15 | major 2nd | M2 | E |
| 9 | ri | 234.783 | 8/7, 15/13* | upmajor 2nd | ^M2 | E^ |
| 10 | ma | 260.87 | 7/6, 15/13* | downminor 3rd | vm3 | Fv |
| 11 | meh | 286.957 | 13/11, 20/17 | minor 3rd | m3 | F |
| 12 | me | 313.043 | 6/5 | upminor 3rd | ^m3 | F^ |
| 13 | mu | 339.13 | 11/9, 17/14 | downmid 3rd | v~3 | F^^ |
| 14 | muh | 365.217 | 16/13 | upmid 3rd | ^~3 | F#vv |
| 15 | mi | 391.304 | 5/4 | downmajor 3rd | vM3 | F#v |
| 16 | maa | 417.391 | 14/11 | major 3rd | M3 | F# |
| 17 | mo | 443.478 | 9/7, 13/10, 22/17 | upmajor 3rd | ^M3 | F#^ |
| 18 | fe | 469.565 | 17/13 | down 4th | v4 | Gv |
| 19 | fa | 495.652 | 4/3 | perfect 4th | P4 | G |
| 20 | fih | 521.739 | up 4th | ^4 | G^ | |
| 21 | fu | 547.826 | 11/8, 15/11 | double-up 4th | ^^4 | G^^ |
| 22 | fi | 573.913 | 7/5, 18/13 | double-down aug 4th,
dim 5th |
vvA4, d5 | G#vv, Ab |
| 23 | seh | 600 | 17/12, 24/17 | downaug 4th, updim 5th | vA4, ^d5 | G#v, Ab^ |
| 24 | se | 626.087 | 10/7, 13/9 | aug 4th, double-up dim 5th | A4, ^^d5 | G#, Ab^^ |
| 25 | su | 652.174 | 16/11, 22/15 | double-down 5th | vv5 | Avv |
| 26 | sih | 678.261 | down 5th | v5 | Av | |
| 27 | sol | 704.348 | 3/2 | perfect 5th | P5 | A |
| 28 | si | 730.435 | 26/17 | up 5th | ^5 | A^ |
| 29 | lo | 756.522 | 14/9, 20/13, 17/11 | downminor 6th | vm6 | Bbv |
| 30 | leh | 782.609 | 11/7 | minor 6th | m6 | Bb |
| 31 | le | 808.696 | 8/5 | upminor 6th | ^m6 | Bb^ |
| 32 | lu | 834.783 | 13/8 | downmid 6th | v~6 | Bb^^ |
| 33 | luh | 860.87 | 18/11, 28/17 | upmid 6th | ^~6 | Bvv |
| 34 | la | 886.957 | 5/3 | downmajor 6th | vM6 | Bv |
| 35 | laa | 913.043 | 22/13, 17/10 | major 6th | M6 | B |
| 36 | li | 939.13 | 12/7, 26/15* | upmajor 6th | ^M6 | B^ |
| 37 | ta | 965.217 | 7/4, 26/15* | downminor 7th | vm7 | Cv |
| 38 | teh | 991.304 | 16/9, 30/17 | minor 7th | m7 | C |
| 39 | te | 1017.391 | 9/5 | upminor 7th | ^m7 | C^ |
| 40 | tu | 1043.478 | 11/6, 20/11 | downmid 7th | v~7 | C^^ |
| 41 | tuh | 1069.565 | 24/13, 13/7, 28/15 | upmid 7th | ^~7 | C#vv |
| 42 | ti | 1095.652 | 15/8, 32/17, 17/9 | downmajor 7th | vM7 | C#v |
| 43 | taa | 1121.739 | major 7th | M7 | C# | |
| 44 | to | 1147.826 | upmajor 7th | ^M7 | C#^ | |
| 45 | da | 1173.913 | down 8ve | v8 | Dv | |
| 46 | do | 1200 | 2/1 | perfect 8ve | P8 | D |
- 15/13 (and its inversion 26/15) appears twice on the list. 9\46edo is closest 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. This discrepancy occurs because 46edo is not consistent in the 15-limit.
Combining ups and downs notation with color notation, qualities can be loosely associated with colors:
| quality | color | monzo format | examples |
|---|---|---|---|
| downminor | blue | {a, b, 0, 1} | 7/6, 7/4 |
| minor | fourthward white | {a, b}, b < -1 | 32/27, 16/9 |
| upminor | green | {a, b, -1} | 6/5, 9/5 |
| downmid | jade | {a, b, 0, 0, 1} | 11/9, 11/6 |
| upmid | amber | {a, b, 0, 0, -1} | 12/11, 18/11 |
| downmajor | yellow | {a, b, 1} | 5/4, 5/3 |
| major | fifthward white | {a, b}, b > 1 | 9/8, 27/16 |
| upmajor | red | {a, b, 0, -1} | 9/7, 12/7 |
All 46edo chords can be named using ups and downs. Here are the blue, green, jade, yellow and red triads:
| color of the 3rd | JI chord | notes as edosteps | notes of C chord | written name | spoken name |
|---|---|---|---|---|---|
| blue | 6:7:9 | 0-10-27 | C Ebv G | C.vm | C downminor |
| green | 10:12:15 | 0-12-27 | C Eb^ G | C.^m | C upminor |
| jade | 18:22:27 | 0-13-27 | C Eb^^ G | C.v~ | C downmid |
| yellow | 4:5:6 | 0-15-27 | C Ev G | C.v | C downmajor or C dot down |
| red | 14:18:27 | 0-17-27 | C E^ G | C.^ | C upmajor or C dot up |
For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.
Selected just intervals by error[edit]
The following table shows how some prominent just intervals are represented in 46edo (ordered by absolute error).
| Interval, complement | Error (abs., in cents) |
|---|---|
| 14/11, 11/7 | 0.117 |
| 10/9, 9/5 | 0.205 |
| 14/13, 13/7 | 2.137 |
| 13/11, 22/13 | 2.253 |
| 4/3, 3/2 | 2.393 |
| 6/5, 5/3 | 2.598 |
| 11/8, 16/11 | 3.492 |
| 8/7, 7/4 | 3.609 |
| 9/8, 16/9 | 4.786 |
| 5/4, 8/5 | 4.991 |
| 16/13, 13/8 | 5.745 |
| 12/11, 11/6 | 5.885 |
| 7/6, 12/7 | 6.001 |
| 16/15, 15/8 | 7.383 |
| 13/12, 24/13 | 8.138 |
| 11/9, 18/11 | 8.278 |
| 9/7, 14/9 | 8.394 |
| 11/10, 20/11 | 8.482 |
| 7/5, 10/7 | 8.599 |
| 18/13, 13/9 | 10.531 |
| 13/10, 20/13 | 10.736 |
| 15/11, 22/15 | 10.875 |
| 15/14, 28/15 | 10.992 |
| 15/13, 26/15 | 13.129 |
Linear temperaments[edit]
| Periods
per octave |
Generator | Cents | Temperaments | MOS/DE Scales available | L:s |
|---|---|---|---|---|---|
| 1 | 1\46 | 26.087 | |||
| 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 | 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 |
| 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 | 339.13 | 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 |
| 1 | 15\46 | 391.304 | 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 2:1 ~ QE |
| 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 |
| 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 2:1 ~ QE |
| 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 2:1 ~ QE |
| 23 | 1\46 | 26.087 |
Approximation to Mode 8 of the Harmonic Series[edit]
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\46edo (208.70¢) stands in for frequency ratio 9:8 (203.91¢).
7\46edo (182.61¢) stands in for 10:9 (182.40¢).
6\46edo (156.52¢) stands in for 11:10 (165.00¢) and 12:11 (150.64¢).
5\46edo (130.43¢) stands in for 13:12 (138.57¢), 14:13 (128.30¢) and 15:14 (119.44¢).
4\46edo (104.35¢) stands in for 16:15 (111.73¢).
Approximation to Mode 9 of the Harmonic Series[edit]
46edo represents overtones 9 through 18 (written as JI ratios 9:10:11:12:13:14:15:16:17:18) with degrees 0, 7, 13, 19, 24, 29, 34, 38, 42, 46. In steps-in-between, that's 7, 6, 6, 5, 5, 5, 4, 4, 4.
7\46edo (182.61¢) stands in for 10:9 (182.40¢).
6\46edo (156.52¢) stands in for 11:10 (165.00¢) and 12:11 (150.64¢).
5\46edo (130.43¢) stands in for 13:12 (138.57¢), 14:13 (128.30¢) and 15:14 (119.44¢).
4\46edo (104.35¢) stands in for 16:15 (111.73¢), 17:16 (104.96¢) and 18:17 (98.95¢).
Scales[edit]
Music[edit]
Music For Your Ears play The central portion is in 27edo, the rest in 46edo.
by Andrew Heathwaite: Rats, play Tumbledown Stew play, Hypnocloudsmack 1 play, Hypnocloudsmack 2 play, Hypnocloudsmack 3 play
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