10ed7/3
| ← 9ed7/3 | 10ed7/3 | 11ed7/3 → |
(convergent)
10 equal divisions of 7/3 (abbreviated 10ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 10 equal parts of about 147 ¢ each. Each step represents a frequency ratio of (7/3)1/10, or the 10th root of 7/3.
Theory
This tuning tempers out 16/15 in the 5-limit; 28/27, 50/49, and 36/35 in the 7-limit; 22/21, 56/55, 77/75, and 55/54 in the 11-limit; 26/25, 52/49, 65/63, 66/65, 78/77, and 40/39 in the 13-limit; 34/33, 51/49, 52/51, 75/68, 77/68, and 51/50 in the 17-limit; 20/19, 64/57, 76/75, 77/76, and 39/38 in the 19-limit; 24/23, 70/69, and 46/45 in the 23-limit; 30/29, 58/57, and 32/29 in the 29-limit; 33/31, 63/62, 65/62, and 34/31 in the 31-limit; 38/37, 39/37, 75/74, 77/74, and 40/37 in the 37-limit; 42/41 and 44/41 in the 41-limit; 43/42, 43/41, and 44/43 in the 43-limit; 47/46, 48/47, and 47/45 in the 47-limit; 54/53, 56/53, and 55/53 in the 53-limit; 59/58, 60/59, 64/59, and 59/57 in the 59-limit; 62/61, 65/61, 66/61, and 63/61 in the 61-limit; 69/67, 72/67, and 70/67 in the 67-limit; 71/67, 71/69, 72/71, and 71/70 in the 71-limit; 73/68, 75/73, 76/73, 77/73, 78/73; and 74/73 in the 73-limit; and 81/79 in the 79-limit.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 146.7 | 11/10, 12/11, 13/12, 14/13, 15/14, 21/19 |
| 2 | 293.4 | 6/5, 7/6, 13/11, 20/17 |
| 3 | 440.1 | 9/7, 13/10, 14/11, 17/13, 19/15, 22/17 |
| 4 | 586.7 | 7/5, 10/7, 17/12, 18/13 |
| 5 | 733.4 | 3/2, 14/9, 17/11, 20/13 |
| 6 | 880.1 | 5/3, 18/11, 22/13 |
| 7 | 1026.8 | 9/5, 11/6, 20/11 |
| 8 | 1173.5 | 2/1 |
| 9 | 1320.2 | 13/6, 15/7, 17/8, 19/9 |
| 10 | 1466.9 | 7/3 |
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -26.5 | +5.0 | +0.7 | +5.0 | -44.1 | -39.9 | -64.3 | +36.5 | -0.9 | +37.9 | +69.1 |
| Relative (%) | -18.1 | +3.4 | +0.5 | +3.4 | -30.0 | -27.2 | -43.8 | +24.9 | -0.6 | +25.8 | +47.1 | |
| Steps (reduced) |
8 (8) |
13 (3) |
19 (9) |
23 (3) |
28 (8) |
30 (0) |
33 (3) |
35 (5) |
37 (7) |
40 (0) |
41 (1) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +56.2 | +25.2 | -57.3 | -64.6 | +20.8 | -18.2 | +70.8 | +55.0 | -45.3 | +53.3 | +63.2 |
| Relative (%) | +38.3 | +17.2 | -39.1 | -44.0 | +14.2 | -12.4 | +48.3 | +37.5 | -30.9 | +36.3 | +43.1 | |
| Steps (reduced) |
43 (3) |
44 (4) |
44 (4) |
45 (5) |
47 (7) |
48 (8) |
49 (9) |
50 (0) |
50 (0) |
51 (1) |
52 (2) | |