User:BudjarnLambeth/42zpi

42 zeta peak index (abbreviated 42zpi), is the equal-step tuning system obtained from the 42nd peak of the Riemann zeta function.

Tuning			Strength			Closest edo			Integer limit
ZPI	Steps per 8ve	Step size (cents)	Height	Integral	Gap	Edo	Octave (cents)		Consistent	Distinct
							Size	Stretch
42zpi	13.900253	86.329367	4.592177	0.984037	14.097244	14edo	1208.611136	8.611136	7	5

Theory

42zpi is the closest zeta peak index to 14edo. It is effectively a stretched version of 14edo which improves on its tuning of most primes.

Harmonic series

Approximation of harmonics in 42zpi

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	Absolute (¢)	+8.6	-2.7	+17.2	-23.8	+5.9	-2.0	+25.8	-5.4	-15.2	-7.5	+14.5	-37.7	+6.6	-26.5	+34.4
	Relative (%)	+10.0	-3.1	+19.9	-27.5	+6.8	-2.3	+29.9	-6.3	-17.6	-8.7	+16.8	-43.7	+7.7	-30.7	+39.9
Step		14	22	28	32	36	39	42	44	46	48	50	51	53	54	56

Approximation of harmonics in 42zpi

Harmonic		17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
Error	Absolute (¢)	+15.8	+3.2	-4.1	-6.6	-4.7	+1.1	+10.5	+23.1	+38.8	-29.1	-8.1	+15.2	+40.8	-17.9	+11.7	+43.1
	Relative (%)	+18.3	+3.7	-4.7	-7.6	-5.4	+1.3	+12.1	+26.8	+44.9	-33.7	-9.4	+17.7	+47.3	-20.7	+13.5	+49.9
Step		57	58	59	60	61	62	63	64	65	65	66	67	68	68	69	70

Scala file

! 42zpi.scl
! Created using Scale Workshop 3.1.0
!
! https://scaleworkshop.plainsound.org/scale/g8qoINpGf
!
42nd zeta peak index
 14
!
 86.329
 172.659
 258.988
 345.317
 431.647
 517.976
 604.306
 690.635
 776.964
 863.294
 949.623
 1035.952
 1122.282
 1208.611

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