102557edo
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
102557 equal divisions of the octave (abbreviated 102557edo or 102557ed2), also called 102557-tone equal temperament (102557tet) or 102557 equal temperament (102557et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 102557 equal parts of about 0.0117 ¢ each. Each step represents a frequency ratio of 21/102557, or the 102557th root of 2.
| ← 102556edo | 102557edo | 102558edo → |
102557edo is notable for being a good high-limit system, and specializes in the 17-limit with a lower relative error than any smaller equal temperaments. It is consistent to the 39-odd-limit and is the first non-trivial edo to be consistent in the 32-odd-prime-sum-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00000 | +0.00001 | +0.00024 | +0.00118 | +0.00084 | +0.00004 | +0.00086 | +0.00349 | +0.00066 |
| Relative (%) | +0.0 | +0.1 | +2.0 | +10.1 | +7.1 | +0.4 | +7.3 | +29.8 | +5.6 | |
| Steps (reduced) |
102557 (0) |
162549 (59992) |
238130 (33016) |
287914 (82800) |
354789 (47118) |
379506 (71835) |
419198 (8970) |
435655 (25427) |
463923 (53695) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00050 | +0.00572 | +0.00107 | -0.00539 | -0.00508 | +0.00036 | -0.00396 | -0.00261 | +0.00434 |
| Relative (%) | +4.3 | +48.9 | +9.1 | -46.1 | -43.4 | +3.1 | -33.8 | -22.3 | +37.1 | |
| Steps (reduced) |
498220 (87992) |
508088 (97860) |
534266 (21481) |
549454 (36669) |
556501 (43716) |
569662 (56877) |
587438 (74653) |
603306 (90521) |
608239 (95454) | |
| Harmonic | 67 | 71 | 73 | 79 | 83 | 89 | 97 | 101 | 103 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00106 | +0.00450 | +0.00190 | +0.00185 | +0.00095 | +0.00242 | -0.00306 | -0.00228 | -0.00508 |
| Relative (%) | +9.1 | +38.5 | +16.3 | +15.8 | +8.1 | +20.7 | -26.1 | -19.5 | -43.5 | |
| Steps (reduced) |
622120 (6778) |
630700 (15358) |
634810 (19468) |
646497 (31155) |
653805 (38463) |
664132 (48790) |
676867 (61525) |
682846 (67504) |
685747 (70405) | |