| 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
|
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.
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
