11358058edo
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11358058edo, or 11358058 equal divisions of the octave, is an equal tuning system with a step size of only about 0.00010565 cents, far beyond the human melodic just-noticeable difference. It has been noted for its highly accurate approximation of the 31-limit.
While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, like all other equal temperaments, the consistency of this tuning is limited, and the sequence of intervals may eventually start to deviate from their true JI counterparts.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.0000000 | -0.0000012 | +0.0000028 | -0.0000033 | +0.0000030 | +0.0000059 | -0.0000025 | -0.0000082 | -0.0000067 | +0.0000100 | -0.0000039 |
| relative (%) | +0 | -1 | +3 | -3 | +3 | +6 | -2 | -8 | -6 | +9 | -4 | |
| Steps (reduced) |
11358058 (0) |
18002096 (6644038) |
26372594 (3656478) |
31886100 (9169984) |
39292425 (5218251) |
42029809 (7955635) |
46425640 (993408) |
48248207 (2815975) |
51378879 (5946647) |
55177230 (9744998) |
56270049 (10837817) | |