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6ed7/3

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6 equal divisions of 7/3 (abbreviated 6ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 6 equal parts of about 244 ¢ each. Each step represents a frequency ratio of (7/3)1/6, or the 6th root of 7/3.

← 5ed7/3 6ed7/3 7ed7/3 →
Prime factorization 2 × 3 (highly composite)
Step size 244.478 ¢ 
Octave 5\6ed7/3 (1222.39 ¢)
(semiconvergent)
Twelfth 8\6ed7/3 (1955.83 ¢) (→ 4\3ed7/3)
Consistency limit 4
Distinct consistency limit 4

Theory

Harmonics

Approximation of harmonics in 6ed7/3
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +22 +54 +45 -97 +76 +54 +67 +108 -75 +5 +99
Relative (%) +9.2 +22.0 +18.3 -39.7 +31.2 +22.0 +27.5 +44.1 -30.5 +2.0 +40.4
Steps
(reduced) 		5
(5) 		8
(2) 		10
(4) 		11
(5) 		13
(1) 		14
(2) 		15
(3) 		16
(4) 		16
(4) 		17
(5) 		18
(0)
Approximation of harmonics in 6ed7/3
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40 +76 -43 +90 -15 -114 +37 -52 +108 +27 -50 +121
Relative (%) -16.3 +31.2 -17.7 +36.6 -6.3 -46.8 +14.9 -21.4 +44.1 +11.1 -20.3 +49.5
Steps
(reduced) 		18
(0) 		19
(1) 		19
(1) 		20
(2) 		20
(2) 		20
(2) 		21
(3) 		21
(3) 		22
(4) 		22
(4) 		22
(4) 		23
(5)

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 244.5 7/6, 8/7, 9/8, 13/11, 15/13, 17/15, 19/16, 19/17, 20/17
2 489 4/3, 9/7, 13/10, 15/11, 17/13, 19/14, 21/16
3 733.4 3/2, 11/7, 14/9, 17/11, 20/13
4 977.9 7/4, 12/7, 16/9, 17/10, 19/11, 20/11
5 1222.4 2/1
6 1466.9 7/3, 16/7, 19/8

6ed7/3+7edo scale

On the Xenharmonic Alliance Discord in September 2025, Maeve Gutierrez noted that the notes of 3ed7/3 make for a nice chord when played simultaneously, and that 6ed7/3 is a good tuning for using said chord.

Gutierrez also noted that playing 6ed7/3 on one instrument/track simultaneously with 7edo on another (a polymicrotonal approach) makes for some useful effects: "6ed7/3+7edo together gives lots of shimmer to play with+2 different flavours of detuned perfect fifth and fourth".

Lériendil then noted that this 6ed7/3+7edo scale is very closely approximated by 49edo. Budjarn Lambeth expanded on this idea, mentioning that after going 3 octaves up or 3 octaves down from the root note, the discrepancy between the two tunings (6ed7/3 and a stack of 7/3 from 49edo) will be no more than 6 cents.

If one wished to use this 6ed7/3+7edo scale tempered to 49edo, then it would look as follows:

Within 49edo:

  • 6ed7/3 is the step pattern 10 10 10...
  • 7edo is the step pattern 7 7 7...

Which means that both scales sync up every 70 steps of 49edo, at the interval 1714.286c.

So (tempered to 49edo), the combined 6ed7/3 & 7edo scale is:

  • 7\49
  • 10\49
  • 14\49
  • 20\49
  • 21\49
  • 28\49
  • 30\49
  • 35\49
  • 40\49
  • 42\49
  • 49\49
  • 50\49
  • 56\49
  • 60\49
  • 63\49
  • 70\49 (period)

Lumatone mappings

Mapping the 6ed7/3+7edo scale onto a 2D isomorphic keyboard like the Lumatone, one can use 7\49 for the x-steps and 10\49 for the y-steps or vice versa.

6ed7/3 on the x-steps
 
0
10
17
27
37
47
8
24
34
44
5
15
25
35
45
41
2
12
22
32
42
3
13
23
33
43
48
9
19
29
39
0
10
20
30
40
1
11
21
31
16
26
36
46
7
17
27
37
47
8
18
28
38
48
9
19
29
23
33
43
4
14
24
34
44
5
15
25
35
45
6
16
26
36
46
7
17
40
1
11
21
31
41
2
12
22
32
42
3
13
23
33
43
4
14
24
34
44
5
15
47
8
18
28
38
48
9
19
29
39
0
10
20
30
40
1
11
21
31
41
2
12
22
32
42
3
25
35
45
6
16
26
36
46
7
17
27
37
47
8
18
28
38
48
9
19
29
39
0
10
20
30
40
1
13
23
33
43
4
14
24
34
44
5
15
25
35
45
6
16
26
36
46
7
17
27
37
47
8
18
11
21
31
41
2
12
22
32
42
3
13
23
33
43
4
14
24
34
44
5
15
25
35
48
9
19
29
39
0
10
20
30
40
1
11
21
31
41
2
12
22
32
42
46
7
17
27
37
47
8
18
28
38
48
9
19
29
39
0
10
34
44
5
15
25
35
45
6
16
26
36
46
7
17
32
42
3
13
23
33
43
4
14
24
34
20
30
40
1
11
21
31
41
18
28
38
48
9
6
16
6ed7/3 on the x-steps (alt.)
 
0
10
3
13
23
33
43
45
6
16
26
36
46
7
17
48
9
19
29
39
0
10
20
30
40
1
41
2
12
22
32
42
3
13
23
33
43
4
14
24
44
5
15
25
35
45
6
16
26
36
46
7
17
27
37
47
8
37
47
8
18
28
38
48
9
19
29
39
0
10
20
30
40
1
11
21
31
40
1
11
21
31
41
2
12
22
32
42
3
13
23
33
43
4
14
24
34
44
5
15
33
43
4
14
24
34
44
5
15
25
35
45
6
16
26
36
46
7
17
27
37
47
8
18
28
38
46
7
17
27
37
47
8
18
28
38
48
9
19
29
39
0
10
20
30
40
1
11
21
31
41
2
12
22
20
30
40
1
11
21
31
41
2
12
22
32
42
3
13
23
33
43
4
14
24
34
44
5
15
25
4
14
24
34
44
5
15
25
35
45
6
16
26
36
46
7
17
27
37
47
8
18
28
27
37
47
8
18
28
38
48
9
19
29
39
0
10
20
30
40
1
11
21
11
21
31
41
2
12
22
32
42
3
13
23
33
43
4
14
24
34
44
5
15
25
35
45
6
16
26
36
46
7
17
18
28
38
48
9
19
29
39
0
10
20
41
2
12
22
32
42
3
13
25
35
45
6
16
48
9
6ed7/3 on the y-steps
 
0
7
17
24
31
38
45
27
34
41
48
6
13
20
27
44
2
9
16
23
30
37
44
2
9
16
5
12
19
26
33
40
47
5
12
19
26
33
40
47
22
29
36
43
1
8
15
22
29
36
43
1
8
15
22
29
36
32
39
46
4
11
18
25
32
39
46
4
11
18
25
32
39
46
4
11
18
0
7
14
21
28
35
42
0
7
14
21
28
35
42
0
7
14
21
28
35
42
0
7
10
17
24
31
38
45
3
10
17
24
31
38
45
3
10
17
24
31
38
45
3
10
17
24
31
38
34
41
48
6
13
20
27
34
41
48
6
13
20
27
34
41
48
6
13
20
27
34
41
48
6
13
20
27
16
23
30
37
44
2
9
16
23
30
37
44
2
9
16
23
30
37
44
2
9
16
23
30
37
44
5
12
19
26
33
40
47
5
12
19
26
33
40
47
5
12
19
26
33
40
47
5
12
36
43
1
8
15
22
29
36
43
1
8
15
22
29
36
43
1
8
15
22
25
32
39
46
4
11
18
25
32
39
46
4
11
18
25
32
39
7
14
21
28
35
42
0
7
14
21
28
35
42
0
45
3
10
17
24
31
38
45
3
10
17
27
34
41
48
6
13
20
27
16
23
30
37
44
47
5
6ed7/3 on the y-steps (alt.)
 
0
7
46
4
11
18
25
36
43
1
8
15
22
29
36
33
40
47
5
12
19
26
33
40
47
5
23
30
37
44
2
9
16
23
30
37
44
2
9
16
20
27
34
41
48
6
13
20
27
34
41
48
6
13
20
27
34
10
17
24
31
38
45
3
10
17
24
31
38
45
3
10
17
24
31
38
45
7
14
21
28
35
42
0
7
14
21
28
35
42
0
7
14
21
28
35
42
0
7
14
46
4
11
18
25
32
39
46
4
11
18
25
32
39
46
4
11
18
25
32
39
46
4
11
18
25
1
8
15
22
29
36
43
1
8
15
22
29
36
43
1
8
15
22
29
36
43
1
8
15
22
29
36
43
12
19
26
33
40
47
5
12
19
26
33
40
47
5
12
19
26
33
40
47
5
12
19
26
33
40
30
37
44
2
9
16
23
30
37
44
2
9
16
23
30
37
44
2
9
16
23
30
37
41
48
6
13
20
27
34
41
48
6
13
20
27
34
41
48
6
13
20
27
10
17
24
31
38
45
3
10
17
24
31
38
45
3
10
17
24
21
28
35
42
0
7
14
21
28
35
42
0
7
14
39
46
4
11
18
25
32
39
46
4
11
1
8
15
22
29
36
43
1
19
26
33
40
47
30
37

See also

Retrieved from "https://en.xen.wiki/index.php?title=6ed7/3&oldid=210883"