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 pions 7mus approximate ratios

in the 17-limit

ups and downs notation
0 do 0.000 1/1 perfect unison P1 D
1 di 26.087 27.652 33.391 (21.64316) up unison ^1 D^
2 ro 52.174 55.304 66.783 (42.C8616) downminor 2nd vm2 Ebv
3 rih 78.261 82.9565 100.174 (64.2C816) minor 2nd m2 Eb
4 ra 104.348 110.609 133.565 (85.90B16) 16/15, 17/16, 18/17 upminor 2nd ^m2 Eb^
5 ru (as in supraminor) 130.435 138.261 166.9565 (A6.F4E16) 13/12, 14/13, 15/14 downmid 2nd v~2 Eb^^
6 ruh (as in submajor) 156.522 165.913 200.348 (C8.59116) 12/11, 11/10 upmid 2nd ^~2 Evv
7 reh 182.609 193.565 233.739 (E9.BD3816) 10/9 downmajor 2nd vM2 Ev
8 re 208.696 221.217 267.13 (10B.21616) 9/8, 17/15 major 2nd M2 E
9 ri 234.783 248.87 300.522 (12C.85916) 8/7, 15/13* upmajor 2nd ^M2 E^
10 ma 260.87 276.522 333.913 (14D.E9C16) 7/6, 15/13* downminor 3rd vm3 Fv
11 meh 286.957 304.174 367.304 (16F.4DF16) 13/11, 20/17 minor 3rd m3 F
12 me 313.043 331.826 400.696 (190.B2116) 6/5 upminor 3rd ^m3 F^
13 mu 339.13 359.479 434.087 (1B2.16416) 11/9, 17/14 downmid 3rd v~3 F^^
14 muh 365.217 387.13 467.478 (1D3.7A716) 16/13 upmid 3rd ^~3 F#vv
15 mi 391.304 414.783 501.87 (1F5.DEA16) 5/4 downmajor 3rd vM3 F#v
16 maa 417.391 442.435 534.261 (216.42C816) 14/11 major 3rd M3 F#
17 mo 443.478 470.087 567.652 (237.A6F16) 9/7, 13/10, 22/17 upmajor 3rd ^M3 F#^
18 fe 469.565 497.739 601.0435 (259.0B216) 17/13 down 4th v4 Gv
19 fa 495.652 525.391 634.435 (27A.6F516) 4/3 perfect 4th P4 G
20 fih 521.739 553.0435 667.826 (29B.D3816) up 4th ^4 G^
21 fu 547.826 580.696 701.217 (2BD.37A16) 11/8, 15/11 double-up 4th ^^4 G^^
22 fi 573.913 608.348 734.609 (2DE.9BD16) 7/5, 18/13 double-down aug 4th,

dim 5th

vvA4, d5 G#vv, Ab
23 seh 600 636 768 (30016) 17/12, 24/17 downaug 4th, updim 5th vA4, ^d5 G#v, Ab^
24 se 626.087 663.652 801.391 (321.64316) 10/7, 13/9 aug 4th, double-up dim 5th A4, ^^d5 G#, Ab^^
25 su 652.174 691.304 834.783 (342.C8616) 16/11, 22/15 double-down 5th vv5 Avv
26 sih 678.261 718.9565 868.174 (364.2C816) down 5th v5 Av
27 sol 704.348 746.609 901.565 (385.90B16) 3/2 perfect 5th P5 A
28 si 730.435 774.261 934.9565 (3A6.F4E16) 26/17 up 5th ^5 A^
29 lo 756.522 801.913 968.348 (3C8.59116) 14/9, 20/13, 17/11 downminor 6th vm6 Bbv
30 leh 782.609 829.565 1001.739 (3E9.BD3816) 11/7 minor 6th m6 Bb
31 le 808.696 857.217 1035.13 (40B.21616) 8/5 upminor 6th ^m6 Bb^
32 lu 834.783 884.87 1068.522 (42C.85916) 13/8 downmid 6th v~6 Bb^^
33 luh 860.87 913.521 1101.913 (44D.E9C16 18/11, 28/17 upmid 6th ^~6 Bvv
34 la 886.957 940.174 1135.304 (46F.4DF16). 5/3 downmajor 6th vM6 Bv
35 laa 913.043 967.826 1168.696 (490.B2116) 22/13, 17/10 major 6th M6 B
36 li 939.13 995.478 1202.087 (4B2.16416) 12/7, 26/15* upmajor 6th ^M6 B^
37 ta 965.217 1023.13 1235.478 (4D3.7A716) 7/4, 26/15* downminor 7th vm7 Cv
38 teh 991.304 1050.783 1269.87 (4F5.DEA16) 16/9, 30/17 minor 7th m7 C
39 te 1017.391 1078.435 1302.261 (516.42C816)) 9/5 upminor 7th ^m7 C^
40 tu 1043.478 1106.087 1335.652 (537.A6F16) 11/6, 20/11 downmid 7th v~7 C^^
41 tuh 1069.565 1133.739 1369.0435 (559.0B216) 24/13, 13/7, 28/15 upmid 7th ^~7 C#vv
42 ti 1095.652 1161.391 1402.435 (57A.6F516) 15/8, 32/17, 17/9 downmajor 7th vM7 C#v
43 taa 1121.739 1189.0435 1435.826 (59B.D3816) major 7th M7 C#
44 to 1147.826 1216.696 1469.217 (5BD.37A16) upmajor 7th ^M7 C#^
45 da 1173.913 1244.348 1502.609 (5DE.9BD16) down 8ve v8 Dv
46 do 1200.000 1272 1536 (60016) 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 12.958

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

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