324296edo

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.

324296 equal divisions of the octave (abbreviated 324296edo or 324296ed2), also called 324296-tone equal temperament (324296tet) or 324296 equal temperament (324296et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 324296 equal parts of about 0.0037 ¢ each. Each step represents a frequency ratio of 2^1/324296, or the 324296th root of 2.

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324296edo

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Prime factorization	23 × 7 × 5791
Step size	0.00370032 ¢
Fifth	189701\324296 (701.955 ¢)
Semitones (A1:m2)	30723:24383 (113.7 ¢ : 90.22 ¢)
Consistency limit	59
Distinct consistency limit	59

324296edo is notable for being an exceptionally good representation of the 47-limit, being the first edo with Dirichlet badness in this limit less than 1, and is distinctly consistent in the 59-odd-limit.

Prime harmonics

Approximation of prime harmonics in 324296edo

Harmonic		2	3	5	7	11	13	17	19	23
Error	Absolute (¢)	+0.00000	+0.00000	+0.00002	+0.00010	+0.00061	+0.00074	+0.00056	+0.00037	-0.00018
	Relative (%)	+0.0	+0.1	+0.7	+2.8	+16.4	+20.1	+15.0	+9.9	-4.8
Steps (reduced)		324296 (0)	513997 (189701)	752992 (104400)	910414 (261822)	1121880 (148992)	1200038 (227150)	1325548 (28364)	1377586 (80402)	1466973 (169789)

Approximation of prime harmonics in 324296edo (continued)

Harmonic		29	31	37	41	43	47	53	59	61
Error	Absolute (¢)	+0.00072	-0.00017	+0.00041	+0.00117	+0.00017	+0.00020	+0.00114	+0.00144	-0.00146
	Relative (%)	+19.5	-4.7	+11.1	+31.5	+4.5	+5.4	+30.8	+39.0	-39.6
Steps (reduced)		1575424 (278240)	1606626 (309442)	1689405 (67925)	1737433 (115953)	1759716 (138236)	1801331 (179851)	1857542 (236062)	1907718 (286238)	1923314 (301834)