70ed6
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70 equal divisions of the 6th harmonic (abbreviated 70ed6) is a nonoctave tuning system that divides the interval of 6/1 into 70 equal parts of about 44.3 ¢ each. Each step represents a frequency ratio of 61/70, or the 70th root of 6.
Theory
70ed6 is closely related to 27edo, but with the 6th harmonic rather than the octave being just, which compresses the octave by about 3.53 ¢. The local zeta peak around 27 is located at 27.086614, which has a step size of 44.3023 ¢, making 70ed6 very close to optimal for 27edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.5 | +3.5 | -7.1 | +5.4 | +0.0 | -1.0 | -10.6 | +7.1 | +1.9 | +14.2 | -3.5 |
| Relative (%) | -8.0 | +8.0 | -15.9 | +12.3 | +0.0 | -2.2 | -23.9 | +15.9 | +4.3 | +32.0 | -8.0 | |
| Steps (reduced) |
27 (27) |
43 (43) |
54 (54) |
63 (63) |
70 (0) |
76 (6) |
81 (11) |
86 (16) |
90 (20) |
94 (24) |
97 (27) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -9.2 | -4.5 | +9.0 | -14.1 | +13.9 | +3.5 | -1.4 | -1.6 | +2.5 | +10.6 | -22.0 | -7.1 |
| Relative (%) | -20.7 | -10.2 | +20.3 | -31.9 | +31.3 | +8.0 | -3.3 | -3.7 | +5.7 | +24.0 | -49.7 | -15.9 | |
| Steps (reduced) |
100 (30) |
103 (33) |
106 (36) |
108 (38) |
111 (41) |
113 (43) |
115 (45) |
117 (47) |
119 (49) |
121 (51) |
122 (52) |
124 (54) | |
Subsets and supersets
Since 70 factors into primes as 2 × 5 × 7, 70ed6 has subset ed6's 2, 5, 7, 10, 14, and 35.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 44.3 | 37/36 |
| 2 | 88.6 | 20/19 |
| 3 | 132.9 | 27/25 |
| 4 | 177.3 | 31/28 |
| 5 | 221.6 | 25/22, 33/29 |
| 6 | 265.9 | 7/6 |
| 7 | 310.2 | |
| 8 | 354.5 | 27/22 |
| 9 | 398.8 | 34/27 |
| 10 | 443.1 | 22/17, 31/24 |
| 11 | 487.5 | |
| 12 | 531.8 | 19/14, 34/25 |
| 13 | 576.1 | |
| 14 | 620.4 | 10/7 |
| 15 | 664.7 | 22/15, 25/17 |
| 16 | 709 | |
| 17 | 753.3 | 17/11 |
| 18 | 797.6 | 19/12, 27/17 |
| 19 | 842 | 13/8 |
| 20 | 886.3 | 5/3 |
| 21 | 930.6 | 12/7 |
| 22 | 974.9 | |
| 23 | 1019.2 | 9/5 |
| 24 | 1063.5 | 24/13, 37/20 |
| 25 | 1107.8 | 19/10, 36/19 |
| 26 | 1152.2 | 35/18, 37/19 |
| 27 | 1196.5 | 2/1 |
| 28 | 1240.8 | |
| 29 | 1285.1 | 21/10 |
| 30 | 1329.4 | 28/13 |
| 31 | 1373.7 | 31/14 |
| 32 | 1418 | 25/11, 34/15 |
| 33 | 1462.4 | |
| 34 | 1506.7 | 31/13 |
| 35 | 1551 | 22/9, 27/11 |
| 36 | 1595.3 | |
| 37 | 1639.6 | 31/12 |
| 38 | 1683.9 | 37/14 |
| 39 | 1728.2 | 19/7 |
| 40 | 1772.5 | |
| 41 | 1816.9 | 20/7 |
| 42 | 1861.2 | |
| 43 | 1905.5 | 3/1 |
| 44 | 1949.8 | 37/12 |
| 45 | 1994.1 | 19/6 |
| 46 | 2038.4 | 13/4 |
| 47 | 2082.7 | 10/3 |
| 48 | 2127.1 | |
| 49 | 2171.4 | 7/2 |
| 50 | 2215.7 | 18/5 |
| 51 | 2260 | |
| 52 | 2304.3 | 34/9 |
| 53 | 2348.6 | 31/8, 35/9 |
| 54 | 2392.9 | |
| 55 | 2437.3 | |
| 56 | 2481.6 | 21/5 |
| 57 | 2525.9 | |
| 58 | 2570.2 | |
| 59 | 2614.5 | |
| 60 | 2658.8 | |
| 61 | 2703.1 | |
| 62 | 2747.4 | |
| 63 | 2791.8 | |
| 64 | 2836.1 | 36/7 |
| 65 | 2880.4 | 37/7 |
| 66 | 2924.7 | |
| 67 | 2969 | |
| 68 | 3013.3 | |
| 69 | 3057.6 | |
| 70 | 3102 | 6/1 |