User:BudjarnLambeth/74zpi

This user page is editable by any wiki editor. As a general rule, most users expect their user space to be edited only by themselves, except for minor edits (e.g. maintenance), undoing obviously harmful edits such as vandalism or disruptive editing, and user talk pages. However, by including this message box, the author of this user page has indicated that this page is open to contributions from other users (e.g. content-related edits).

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

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
!
 57.727
 115.454
 173.182
 230.909
 288.636
 346.363
 404.090
 461.818
 519.545
 577.272
 634.999
 692.726
 750.454
 808.181
 865.908
 923.635
 981.362
 1039.090
 1096.817
 1154.544
 1212.271

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