User:BudjarnLambeth/74zpi

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74 zeta peak index (abbreviated 74zpi), is the equal-step tuning system obtained from the 74th 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

74zpi is like 21edo but octave stretched by about 12 cents.

It sounds best with aperiodic or metallophone timbres (similar to gamelan instruments).

Compared to 21edo:

• It improves upon primes 3, 5, 7, 19 and 23
• It worsens primes 2, 11, 13 and 17

Harmonic series

Approximation of harmonics in 74zpi

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	Absolute (¢)	+12.3	+3.0	+24.5	-15.4	+15.3	-20.6	-20.9	+6.1	-3.1	+5.0	+27.6	+4.5	-8.4	-12.4	-8.6
	Relative (%)	+21.3	+5.3	+42.5	-26.7	+26.5	-35.8	-36.2	+10.5	-5.4	+8.7	+47.8	+7.7	-14.5	-21.4	-15.0
Step		21	33	42	48	54	58	62	66	69	72	75	77	79	81	83

Approximation of harmonics in 74zpi

Harmonic		17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
Error	Absolute (¢)	+1.9	+18.4	-17.5	+9.1	-17.6	+17.3	-1.9	-17.9	+26.9	+16.7	+9.1	+3.9	+0.9	-0.1	+0.9	+3.6
	Relative (%)	+3.2	+31.8	-30.3	+15.8	-30.5	+30.0	-3.3	-31.0	+46.6	+29.0	+15.8	+6.7	+1.5	-0.2	+1.5	+6.3
Step		85	87	88	90	91	93	94	95	97	98	99	100	101	102	103	104

21edo for comparison

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	Absolute (¢)	+0.0	-16.2	+0.0	+13.7	-16.2	+2.6	+0.0	+24.7	+13.7	+20.1	-16.2	+16.6	+2.6	-2.6	+0.0
	Relative (%)	+0.0	-28.4	+0.0	+24.0	-28.4	+4.6	+0.0	+43.2	+24.0	+35.2	-28.4	+29.1	+4.6	-4.5	+0.0
Step		21	33	42	49	54	59	63	67	70	73	75	78	80	82	84

21edo for comparison

Harmonic		17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
Error	Absolute (¢)	+9.3	+24.7	-11.8	+13.7	-13.6	+20.1	+0.3	-16.2	+27.4	+16.6	+8.4	+2.6	-1.0	-2.6	-2.2	+0.0
	Relative (%)	+16.3	+43.2	-20.6	+24.0	-23.9	+35.2	+0.5	-28.4	+47.9	+29.1	+14.7	+4.6	-1.8	-4.5	-3.8	+0.0
Step		86	88	89	91	92	94	95	96	98	99	100	101	102	103	104	105

Scala file

! 74zpi.scl
! Created using Scale Workshop 3.1.0
!
! https://scaleworkshop.plainsound.org/scale/wbuk2CLvA
!
74th zeta peak index
 21
!
 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

Scales

See also: 21edo#Scales

Values given in cents.

Pseudo-equiheptatonic

• 173.182
• 346.363
• 519.545
• 692.726
• 865.908
• 1039.09
• 1212.271

Pseudo-equipentatonic

• 230.909
• 519.545
• 692.726
• 981.362
• 1212.271

5-tone pseudo-pelog

• 115.454
• 288.636
• 692.726
• 808.181
• 1212.271

7-tone pseudo-pelog

• 115.454
• 288.636
• 519.545
• 692.726
• 808.181
• 1039.09
• 1212.271