73edt

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73 equal divisions of the tritave[edit]

The 73 equal divisions of the tritave, often abbreviated 73edt, is the scale derived by dividing the tritave into 73 equally-sized steps. Each step represents a frequency ratio of 26.054178094046 cents, an interval close in size to 66/65, the interval from 13/11 to 6/5.

73edt tempers out {{#rreplace: 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|/(\d+)(\D+)?\/(\d+)(\D+)?/|\1/\3\2\4}} 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 0.086669307423804 cents sharp of 3\73 tritaves. In the opinion of some 73edt is the first equal division to deal adequately with the 13-limit, though others award that distinction to 65edt. In fact, while 65 is a zeta integral edt but not a zeta gap edt, 73 is zeta gap but not zeta integral.

The fifth of 73 equal is 1.5078076738655 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, a starling family sound if you will.

73edt can be treated as two 73ed9's separated by an interval of 26.054178094046 cents.

73edt srutis[edit]

Shrutar22 as srutis describes a possible use of 73edt for Indian music.

Intervals[edit]

degrees of 73edt solfege cents value approximate ratios in the 17-limit ups and downs notation with personalized naming
0 do 0 1/1 perfect unison P1 D
1 di 26.054178094046 up unison ^1 D^
2 ro 52.108356188093 downminor 2nd vm2 Ebv
3 rih 78.162534282139 minor 2nd m2 Eb
4 ra 104.21671237619 16/15, 17/16, 18/17 upminor 2nd ^m2 Eb^
5 ru (as in supraminor) 130.27089047023 13/12, 14/13, 15/14 downmid 2nd v~2 Eb^^
6 ruh (as in submajor) 156.32506856428 12/11, 11/10 upmid 2nd ^~2 Evv
7 reh 182.37924665832 10/9 downmajor 2nd vM2 Ev
8 re 208.43342475237 9/8, 17/15 major 2nd M2 E
9 ri 234.48760284642 8/7 upmajor 2nd ^M2 E^
10 ma 260.54178094046 7/6, 15/13 downminor 3rd vm3 Fv
11 meh 286.59595903451 13/11, 20/17 minor 3rd m3 F
12 me 312.65013712856 6/5 upminor 3rd ^m3 F^
13 mu 338.7043152226 11/9, 17/14 downmid 3rd v~3 F^^
14 muh 364.75849331665 16/13 upmid 3rd ^~3 F#vv
15 mi 390.8126714107 5/4 downmajor 3rd vM3 F#v
16 maa 416.86684950474 14/11 major 3rd M3 F#
17 mo 442.92102759879 9/7, 13/10, 22/17 upmajor 3rd ^M3 F#^
18 fe 468.97520569284 17/13 down 4th v4 Gv
19 fa 495.02938378688 4/3 perfect 4th P4 G
20 fih 521.08356188093 up 4th ^4 G^
21 fu 547.13773997497 11/8, 15/11 double-up 4th ^^4 G^^
22 fi 573.19191806902 7/5, 18/13 double-down aug 4th, dim 5th vvA4, d5 G#vv, Ab
23 seh 599.24609616307 17/12, 24/17 downaug 4th, updim 5th vA4, ^d5 G#v, Ab^
24 se 625.30027425711 10/7, 13/9 aug 4th, double-up dim 5th A4, ^^d5 G#, Ab^^
25 su 651.35445235116 16/11, 22/15 double-down 5th vv5 Avv
26 sih 677.40863044521 down 5th v5 Av
27 sol 703.46280853925 3/2 perfect 5th P5 A
28 si 729.5169866333 26/17 up 5th ^5 A^
29 lo 755.57116472735 14/9, 20/13, 17/11 downminor 6th vm6 Bbv
30 leh 781.62534282139 11/7 minor 6th m6 Bb
31 le 807.67952091544 8/5 upminor 6th ^m6 Bb^
32 lu 833.73369900949 13/8 downmid 6th v~6 Bb^^
33 luh 859.78787710353 18/11, 28/17 upmid 6th ^~6 Bvv
34 la 885.84205519758 5/3 downmajor 6th vM6 Bv
35 laa 911.89623329162 22/13, 17/10 major 6th M6 B
36 li 937.95041138567 12/7, 26/15* upmajor 6th ^M6 B^
37 ta 964.00458947972 7/4, 26/15* downminor 7th vm7 Cv
38 teh 990.05876757376 16/9, 30/17 minor 7th m7 C
39 te 1016.1129456678 9/5 upminor 7th ^m7 C^
40 tu 1042.1671237619 11/6, 20/11 downmid 7th v~7 C^^
41 tuh 1068.2213018559 24/13, 13/7, 28/15 upmid 7th ^~7 C#vv
42 ti 1094.2754799499 15/8, 32/17, 17/9 downmajor 7th vM7 C#v
43 taa 1120.329658044 major 7th M7 C#
44 to 1146.383836138 upmajor 7th ^M7 C#^
45 da 1172.4380142321 down 8ve v8 Dv
46 do 1198.4921923261 2/1 perfect 8ve P8 D

*26/15 appears twice on the list. 37\73edt is closest to 15/13 by a hair; 36\73edt represents the difference between, for instance 73edt's 26/9 and 5/3, and is more likely to appear in chords actually functioning as 26/15. This discrepancy occurs because 73edt is not consistent in the 26-3reduced-limit.

Combining ups and downs notation with Kite's color notation, qualities can be loosely associated with Kite's 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 73edt 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 73edt (ordered by absolute error).

Interval, complement Error (abs., in cents)
10/9, 27/10 0.024
14/11, 33/14 0.641
11/7, 21/11 0.867
5/3, 9/5 1.483
3/2, 2/1 1.508
21/16, 16/7 1.806
14/13, 39/14 1.973
13/11, 33/13 2.614
16/11, 33/16 2.672
6/5, 5/2 2.991
4/3, 9/4 3.016
8/7, 21/8 3.314
21/13, 13/7 3.48
11/8, 24/11 4.18
5/4, 12/5 4.499
9/8, 8/3 4.523
12/7, 7/4 4.821
16/13, 39/16 5.286
12/11, 11/4 5.688
8/5, 15/8 6.007
27/16, 16/9 6.031
7/6, 18/7 6.329
13/8, 24/13 6.794
18/11, 11/6 7.196
16/15, 45/16 7.515
10/7, 21/10 7.812
9/7, 7/3 7.837
13/12, 36/13 8.302
11/10, 30/11 8.679
11/9, 27/11 8.704
7/5, 15/7 9.32
14/9, 27/14 9.345
18/13, 13/6 9.81
15/11, 11/5 10.187
15/14, 14/5 10.828
13/10, 30/13 11.293
13/9, 27/13 11.317
15/13, 13/5 12.801

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 5 of the Harmonic Series[edit]

73edt represents overtones 5 through 15 (written as JI ratios 5:6:7:8:9:10:11:12:13:14:15) with degrees 0, 12, 22, 31, 39, 46, 52, 58, 63, 68, 73. In steps-in-between, that's 12, 10, 9, 8, 7, 6, 6, 5, 5, 5.

12\73edt (312.65¢) stands in for frequency ratio 6:5 (315.64¢).

10\73edt (260.54¢) stands in for frequency ratio 7:6 (266.87¢).

9\73edt (234.49¢) stands in for frequency ratio 8:7 (231.17¢).

8\73edt (208.43¢) stands in for frequency ratio 9:8 (203.91¢).

7\73edt (182.38¢) stands in for 10:9 (182.40¢).

6\73edt (156.33¢) stands in for 11:10 (165.00¢) and 12:11 (150.64¢).

5\73edt (130.27¢) stands in for 13:12 (138.57¢), 14:13 (128.30¢) and 15:14 (119.44¢).

Approximation to Mode 6 of the Harmonic Series[edit]

46edo represents overtones 6 through 18 (written as JI ratios 6:7:8:9:10:11:12:13:14:15:16:17:18) with degrees 0, 10, 19, 27, 34, 40, 46, 51, 56, 61, 65, 69, 73. In steps-in-between, that's 10, 9, 8, 7, 6, 6, 5, 5, 5, 4, 4, 4.

10\73edt (260.54¢) stands in for frequency ratio 7:6 (266.87¢).

9\73edt (234.49¢) stands in for frequency ratio 8:7 (231.17¢).

8\73edt (208.43¢) stands in for frequency ratio 9:8 (203.91¢).

7\73edt (182.38¢) stands in for 10:9 (182.40¢).

6\73edt (156.33¢) stands in for 11:10 (165.00¢) and 12:11 (150.64¢).

5\73edt (130.27¢) stands in for 13:12 (138.57¢), 14:13 (128.30¢) and 15:14 (119.44¢).

4\73edt (104.22¢) stands in for 16:15 (111.73¢), 17:16 (104.96¢) and 18:17 (98.95¢).

Scales[edit]

Music[edit]

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

Table of 73edt intervals[edit]

Step Five limit Seven limit Eleven limit Thirteen limit
1 81/80 49/48 49/48 49/48
2 250/243 28/27 28/27 26/25
3 25/24 21/20 21/20 21/20
4 16/15 16/15 16/15 16/15
5 27/25 15/14 15/14 13/12
6 625/576 35/32 11/10 11/10
7 10/9 10/9 10/9 10/9
8 9/8 9/8 9/8 9/8
9 144/125 8/7 8/7 8/7
10 125/108 7/6 7/6 7/6
11 32/27 32/27 32/27 13/11
12 6/5 6/5 6/5 6/5
13 243/200 49/40 11/9 11/9
14 100/81 56/45 27/22 16/13
15 5/4 5/4 5/4 5/4
16 32/25 32/25 14/11 14/11
17 162/125 9/7 9/7 9/7
18 125/96 21/16 21/16 21/16
19 4/3 4/3 4/3 4/3
20 27/20 27/20 27/20 27/20
21 864/625 48/35 11/8 11/8
22 25/18 7/5 7/5 7/5
23 45/32 45/32 45/32 45/32
24 36/25 10/7 10/7 10/7
25 625/432 35/24 16/11 16/11
26 40/27 40/27 40/27 40/27
27 3/2 3/2 3/2 3/2
28 192/125 32/21 32/21 32/21
29 125/81 14/9 14/9 14/9
30 25/16 25/16 11/7 11/7
31 8/5 8/5 8/5 8/5
32 81/50 45/28 44/27 13/8
33 400/243 80/49 18/11 18/11
34 5/3 5/3 5/3 5/3
35 27/16 27/16 27/16 22/13
36 216/125 12/7 12/7 12/7
37 125/72 7/4 7/4 7/4
38 16/9 16/9 16/9 16/9
39 9/5 9/5 9/5 9/5
40 729/400 64/35 11/6 11/6
41 50/27 28/15 28/15 13/7
42 15/8 15/8 15/8 15/8
43 48/25 40/21 21/11 21/11
44 243/125 27/14 27/14 25/13
45 125/64 63/32 55/28 55/28
46 2/1 2/1 2/1 2/1
47 81/40 49/24 49/24 49/24
48 500/243 56/27 33/16 33/16
49 25/12 21/10 21/10 21/10
50 32/15 32/15 32/15 32/15
51 54/25 15/7 15/7 13/6
52 625/288 35/16 11/5 11/5
53 20/9 20/9 20/9 20/9
54 9/4 9/4 9/4 9/4
55 288/125 16/7 16/7 16/7
56 125/54 7/3 7/3 7/3
57 64/27 64/27 33/14 26/11
58 12/5 12/5 12/5 12/5
59 243/100 49/20 22/9 22/9
60 200/81 112/45 27/11 27/11
61 5/2 5/2 5/2 5/2
62 64/25 64/25 28/11 28/11
63 324/125 18/7 18/7 13/5
64 125/48 21/8 21/8 21/8
65 8/3 8/3 8/3 8/3
66 27/10 27/10 27/10 27/10
67 1728/625 96/35 11/4 11/4
68 25/9 14/5 14/5 14/5
69 45/16 45/16 45/16 45/16
70 72/25 20/7 20/7 20/7
71 625/216 35/12 32/11 32/11
72 80/27 80/27 80/27 65/22
73 3/1 3/1 3/1 3/1
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