Theory

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 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 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.

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

Intervals

degree	cents	solfege	ups and downs notation			approximate ratios in the 17-limit
0	0.000	do	perfect unison	P1	D	1/1
1	26.087	di	up unison	^1	^D
2	52.174	ro	downminor 2nd	vm2	vEb
3	78.261	rih	minor 2nd	m2	Eb
4	104.348	ra	upminor 2nd	^m2	^Eb	16/15, 17/16, 18/17
5	130.435	ru (as in supraminor)	downmid 2nd	v~2	^^Eb	13/12, 14/13, 15/14
6	156.522	ruh (as in submajor)	upmid 2nd	^~2	vvE	12/11, 11/10
7	182.609	reh	downmajor 2nd	vM2	vE	10/9
8	208.696	re	major 2nd	M2	E	9/8, 17/15
9	234.783	ri	upmajor 2nd	^M2	^E	8/7, 15/13*
10	260.87	ma	downminor 3rd	vm3	vF	7/6, 15/13*
11	286.957	meh	minor 3rd	m3	F	13/11, 20/17
12	313.043	me	upminor 3rd	^m3	^F	6/5
13	339.13	mu	downmid 3rd	v~3	^^F	11/9, 17/14
14	365.217	muh	upmid 3rd	^~3	vvF#	16/13
15	391.304	mi	downmajor 3rd	vM3	vF#	5/4
16	417.391	maa	major 3rd	M3	F#	14/11
17	443.478	mo	upmajor 3rd	^M3	^F#	9/7, 13/10, 22/17
18	469.565	fe	down 4th	v4	vG	17/13
19	495.652	fa	perfect 4th	P4	G	4/3
20	521.739	fih	up 4th	^4	^G
21	547.826	fu	downmid 4th	v~4	^^G	11/8, 15/11
22	573.913	fi	upmid 4th, dim 5th	^~4, d5	vvG#, Ab	7/5, 18/13
23	600	seh	downaug 4th, updim 5th	vA4, ^d5	vG#, ^Ab	17/12, 24/17
24	626.087	se	aug 4th, downmid 5th	A4, v~5	G#, ^^Ab	10/7, 13/9
25	652.174	su	double-down 5th	^~5	vvA	16/11, 22/15
26	678.261	sih	down 5th	v5	vA
27	704.348	sol	perfect 5th	P5	A	3/2
28	730.435	si	up 5th	^5	^A	26/17
29	756.522	lo	downminor 6th	vm6	vBb	14/9, 20/13, 17/11
30	782.609	leh	minor 6th	m6	Bb	11/7
31	808.696	le	upminor 6th	^m6	^Bb	8/5
32	834.783	lu	downmid 6th	v~6	^^Bb	13/8
33	860.87	luh	upmid 6th	^~6	vvB	18/11, 28/17
34	886.957	la	downmajor 6th	vM6	vB	5/3
35	913.043	laa	major 6th	M6	B	22/13, 17/10
36	939.13	li	upmajor 6th	^M6	^B	12/7, 26/15*
37	965.217	ta	downminor 7th	vm7	vC	7/4, 26/15*
38	991.304	teh	minor 7th	m7	C	16/9, 30/17
39	1017.391	te	upminor 7th	^m7	^C	9/5
40	1043.478	tu	downmid 7th	v~7	^^C	11/6, 20/11
41	1069.565	tuh	upmid 7th	^~7	vvC#	24/13, 13/7, 28/15
42	1095.652	ti	downmajor 7th	vM7	vC#	15/8, 32/17, 17/9
43	1121.739	taa	major 7th	M7	C#
44	1147.826	to	upmajor 7th	^M7	^C#
45	1173.913	da	down 8ve	v8	vD
46	1200.000	do	perfect 8ve	P8	D	2/1

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. 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 vEbv 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	0-43-27	C vvE G	C^~	C upmid
yo	4:5:6	0-15-27	C vE G	Cv	C downmajor or C down
ru	14:18:27	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.

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

Scales

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¢).