User:Fredg999/99693edo

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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99693edo

99694edo →

Prime factorization	32 × 11 × 19 × 53
Step size	0.012037 ¢
Fifth	58317\99693 (701.959 ¢) (→ 19439\33231)
Semitones (A1:m2)	9447:7494 (113.7 ¢ : 90.2 ¢)
Dual sharp fifth	58317\99693 (701.959 ¢) (→ 19439\33231)
Dual flat fifth	58316\99693 (701.947 ¢)
Dual major 2nd	16940\99693 (203.906 ¢) (→ 1540\9063)
Consistency limit	7
Distinct consistency limit	7

99693 equal divisions of the octave (abbreviated 99693edo or 99693ed2), also called 99693-tone equal temperament (99693tet) or 99693 equal temperament (99693et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 99693 equal parts of about 0.012 ¢ each. Each step represents a frequency ratio of 2^1/99693, or the 99693rd root of 2.

99693edo is notable for being the edo below 100000 with the lowest maximum error, using direct approximation for each harmonic, for the first 547 harmonics (547 being the 100th prime number).

Prime harmonics

Approximation of odd harmonics in 99693edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31
Error	Absolute (¢)	+0.004	+0.000	+0.004	-0.004	-0.001	+0.001	+0.004	-0.005	+0.004	-0.004	-0.006	+0.001	+0.000	+0.004	+0.004
	Relative (%)	+33.3	+2.2	+36.6	-33.3	-11.6	+6.3	+35.6	-43.3	+36.2	-30.1	-46.2	+4.5	+0.0	+30.1	+30.7
Steps (reduced)		158010 (58317)	231480 (32094)	279874 (80488)	316019 (16940)	344881 (45802)	368908 (69829)	389490 (90411)	407491 (8719)	423489 (24717)	437883 (39111)	450967 (52195)	462960 (64188)	474029 (75257)	484307 (85535)	493899 (95127)