56ed5: Difference between revisions
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== See also == | |||
* [[14edf]] – relative edf | |||
* [[24edo]] – relative edo | |||
* [[38edt]] – relative edt | |||
* [[62ed6]] – relative ed6 | |||
* [[86ed12]] – relative ed12 | |||
Revision as of 10:24, 27 May 2025
| ← 55ed5 | 56ed5 | 57ed5 → |
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 about 5.87 cents compressed. 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 |