45zpi

45 zeta peak index (abbreviated 45zpi), is the equal-step tuning system obtained from the 45th peak of the Riemann zeta function.

Tuning			Strength			Closest EDO		Integer limit
ZPI	Steps per octave	Step size (cents)	Height	Integral	Gap	EDO	Octave (cents)	Consistent	Distinct
45zpi	14.5944346577250	82.2231232756126	2.097730	0.344839	10.594800	15edo	1233.34684913419	2	2

Theory

45zpi is characterized by a very broad octave error, yet it maintains a quite decent zeta strength. This combination makes it an ideal candidate for no-octave tuning applications.

No other zeta peak indexes exhibit both a larger octave error and greater zeta height than 45zpi.

45zpi supports a complex chord structure with ratios of 1:3:4:5:7:9:13:15:18:19:20:21:22:23:24:25, which further exemplifies its capabilities.

The closest zeta peak indexes to 45zpi that exceed its strength are 42zpi and 47zpi, though 43zpi is nearly as strong as 45zpi.

Harmonic series

Approximation of harmonics in 45zpi

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	absolute (¢)	+33.3	-10.8	-15.5	+9.3	+22.5	+2.3	+17.8	-21.6	-39.6	-40.2	-26.4	-0.5	+35.7	-1.6	-31.1
	relative (%)	+41	-13	-19	+11	+27	+3	+22	-26	-48	-49	-32	-1	+43	-2	-38
Steps		15	23	29	34	38	41	44	46	48	50	52	54	56	57	58

Approximation of harmonics in 45zpi

Harmonic		17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
Error	absolute (¢)	+28.4	+11.7	+0.3	-6.3	-8.5	-6.8	-1.5	+7.0	+18.5	+32.9	-32.5	-13.2	+8.3	+31.8	-25.0	+2.3
	relative (%)	+35	+14	+0	-8	-10	-8	-2	+9	+23	+40	-39	-16	+10	+39	-30	+3
Steps		60	61	62	63	64	65	66	67	68	69	69	70	71	72	72	73

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