102557edo

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.

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102557edo

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Prime factorization	73 × 13 × 23
Step size	0.0117008 ¢
Fifth	59992\102557 (701.955 ¢)
Semitones (A1:m2)	9716:7711 (113.7 ¢ : 90.22 ¢)
Consistency limit	39
Distinct consistency limit	39

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 2^1/102557, or the 102557th root of 2.

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

Approximation of prime harmonics in 102557edo

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)

Approximation of prime harmonics in 102557edo (continued)

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)

Approximation of prime harmonics in 102557edo (continued)

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)