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46 tone equal temperament | 46edo srutis | Intervals | Linear temperaments | Approximation to Mode 8 of the Harmonic Series | Scales | Music

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
0.00
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.

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
339.13
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


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¢).

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