46edo

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46 tone equal temperament

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

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

Intervals

degrees of 46edo solfege cents value approximate ratios

in the 17-limit

ups and downs notation
0 do 0.000 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 260.870 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 339.130 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 600.000 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 860.870 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 939.130 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 1200.000 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 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. Here are the zo, gu, ilo, 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 Ebv G C.vm 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 C.v~ C downmid
yo 4:5:6 0-15-27 C Ev G C.v C downmajor or C dot down
ru 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

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

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\46 339.130 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

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

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 (208.696¢) stands in for frequency ratio 9:8 (203.910¢).

7\46edo (182.609¢) stands in for 10:9 (182.404¢).

6\46edo (156.522¢) stands in for 11:10 (165.004¢) and 12:11 (150.637¢).

5\46edo (130.435¢) stands in for 13:12 (138.573¢), 14:13 (128.298¢) and 15:14 (119.443¢).

4\46edo (104.348¢) stands in for 16:15 (111.731¢).

Scales

Music

by Aaron Krister Johnson:

Satiesque

by Gene Ward Smith:

Chromosounds play

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