56ed5
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56 equal divisions of the 5th harmonic (abbreviated 56ed5) is a nonoctave tuning system that divides the interval of 5/1 into 56 equal parts of about 49.8 ¢ each. Each step represents a frequency ratio of 51/56, or the 56th root of 5.
Theory
56ed5 is related to 24edo, but with the 5th harmonic rather than the octave being just. The octave is compressed by about 5.8 ¢, a small but significant deviation. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning is also a hyperpyth, tempering out 135/133, 171/169, 225/221, and 1521/1445 in the patent val.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.9 | -11.2 | -11.7 | +0.0 | -17.1 | +14.6 | -17.6 | -22.5 | -5.9 | -21.6 | -23.0 |
| Relative (%) | -11.8 | -22.6 | -23.6 | +0.0 | -34.4 | +29.3 | -35.4 | -45.2 | -11.8 | -43.4 | -46.2 | |
| Steps (reduced) |
24 (24) |
38 (38) |
48 (48) |
56 (0) |
62 (6) |
68 (12) |
72 (16) |
76 (20) |
80 (24) |
83 (27) |
86 (30) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -12.3 | +8.7 | -11.2 | -23.5 | +20.8 | +21.4 | -22.4 | -11.7 | +3.3 | +22.3 | -4.9 | +20.9 |
| Relative (%) | -24.7 | +17.5 | -22.6 | -47.2 | +41.9 | +43.0 | -45.1 | -23.6 | +6.7 | +44.8 | -9.9 | +42.0 | |
| Steps (reduced) |
89 (33) |
92 (36) |
94 (38) |
96 (40) |
99 (43) |
101 (45) |
102 (46) |
104 (48) |
106 (50) |
108 (52) |
109 (53) |
111 (55) | |
Subsets and supersets
Since 56 factors into primes as 23 × 7, 56ed5 contains subset ed5's 2, 4, 7, 8, 14, and 28.
Intervals
| # | Cents | Approximated ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 49.8 | 35/34, 36/35 |
| 2 | 99.5 | 18/17 |
| 3 | 149.3 | 12/11 |
| 4 | 199.0 | 55/49 |
| 5 | 248.8 | 15/13 |
| 6 | 298.5 | 19/16 |
| 7 | 348.3 | 11/9 |
| 8 | 398.0 | 5/4 |
| 9 | 447.8 | 35/27 |
| 10 | 497.6 | 4/3 |
| 11 | 547.3 | 70/51 |
| 12 | 597.1 | 24/17 |
| 13 | 646.8 | |
| 14 | 696.6 | 3/2 |
| 15 | 746.3 | 20/13 |
| 16 | 796.1 | 19/12 |
| 17 | 845.8 | 44/27, 75/46 |
| 18 | 895.6 | 5/3 |
| 19 | 945.4 | 19/11 |
| 20 | 995.1 | 9/5, 16/9 |
| 21 | 1044.9 | 64/35 |
| 22 | 1094.6 | 32/17 |
| 23 | 1144.4 | |
| 24 | 1194.1 | 2/1 |
| 25 | 1243.9 | 39/19, 80/39 |
| 26 | 1293.6 | 19/9 |
| 27 | 1343.4 | 50/23 |
| 28 | 1393.2 | 38/17, 85/38 |
| 29 | 1442.9 | 23/10 |
| 30 | 1492.7 | 45/19 |
| 31 | 1542.4 | 39/16 |
| 32 | 1592.2 | 5/2 |
| 33 | 1641.9 | 13/5 |
| 34 | 1691.7 | 85/32 |
| 35 | 1741.4 | 175/64 |
| 36 | 1791.2 | 45/16 |
| 37 | 1841.0 | 55/19 |
| 38 | 1890.7 | 3/1 |
| 39 | 1940.5 | 46/15, 135/44 |
| 40 | 1990.2 | 60/19 |
| 41 | 2040.0 | 13/4 |
| 42 | 2089.7 | 10/3 |
| 43 | 2139.5 | 17/5 |
| 44 | 2189.2 | 85/24 |
| 45 | 2239.0 | 51/14 |
| 46 | 2288.8 | 15/4, 19/5 |
| 47 | 2338.5 | 27/7 |
| 48 | 2388.3 | 4/1 |
| 49 | 2438.0 | 45/11 |
| 50 | 2487.8 | 21/5 |
| 51 | 2537.5 | 13/3 |
| 52 | 2587.3 | 49/11 |
| 53 | 2637.0 | 55/12 |
| 54 | 2686.8 | 85/18 |
| 55 | 2736.6 | 34/7 |
| 56 | 2786.3 | 5/1 |