71zpi

71 zeta peak index (abbreviated 71zpi), is the equal-step tuning system obtained from the 71st 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
71zpi	20.2248393119540	59.3329806724710	3.531097	0.613581	12.986080	20edo	1186.65961344942	6	6

Theory

71zpi marks the most prominent zeta peak index in the vicinity of 20edo. While 70zpi is the nearest peak to 20edo and closely competes with 71zpi in terms of strength, 71zpi remains superior across all measures of strength.

71zpi features a good 3:5:9:11:14:15:16:19:25:26:33 chord, which differs a lot from the harmonic characteristics of 20edo.

The Riemann zeta function around 71zpi

The nearest zeta peaks to 71zpi that surpass its strength are 65zpi and 75zpi.

Harmonic series

Approximation of harmonics in 71zpi

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	Absolute (¢)	-13.3	-3.3	-26.7	+2.3	-16.6	+13.2	+19.3	-6.6	-11.0	+2.0	+29.4	+9.4	-0.2	-1.0	+6.0
	Relative (%)	-22.5	-5.6	-45.0	+3.9	-28.0	+22.2	+32.5	-11.1	-18.5	+3.4	+49.5	+15.9	-0.3	-1.6	+10.1
Step		20	32	40	47	52	57	61	64	67	70	73	75	77	79	81

Approximation of harmonics in 71zpi

Harmonic		17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33
Error	Absolute (¢)	+19.7	-19.9	+5.1	-24.3	+9.9	-11.3	-29.0	+16.0	+4.7	-3.9	-9.9	-13.5	-14.9	-14.3	-11.7	-7.4	-1.3
	Relative (%)	+33.2	-33.6	+8.6	-41.0	+16.6	-19.1	-48.8	+27.0	+7.9	-6.6	-16.7	-22.8	-25.2	-24.1	-19.8	-12.4	-2.2
Step		83	84	86	87	89	90	91	93	94	95	96	97	98	99	100	101	102

Intervals

Step	Cents	Ratios
0	0.000	1/1
1	59.333	30/29, 29/28
2	118.666	15/14
3	177.999	10/9
4	237.332	8/7
5	296.665	13/11, 19/16, 6/5
6	355.998	11/9, 27/22, 16/13
7	415.331	5/4, 14/11
8	474.664	25/19, 4/3
9	533.997	15/11
10	593.330	7/5, 31/22
11	652.663	16/11, 19/13
12	711.996	3/2
13	771.329	14/9, 25/16, 11/7
14	830.662	8/5, 21/13, 13/8
15	889.995	5/3
16	949.328	19/11, 26/15, 7/4
17	1008.661	9/5
18	1067.994	13/7
19	1127.327	23/12
20	1186.660	2/1
22	1305.326	17/8
23	1364.659	11/5
25	1483.325	7/3
27	1601.990	5/2
28	1661.323	13/5
29	1720.656	8/3, 27/10
30	1779.989	14/5
32	1898.655	3/1
33	1957.988	31/10
34	2017.321	16/5
35	2076.654	10/3
36	2135.987	24/7
37	2195.320	7/2, 32/9
38	2254.653	11/3
39	2313.986	19/5
40	2373.319	4/1
44	2610.651	9/2
45	2669.984	14/3
46	2729.317	29/6
47	2788.650	5/1
51	3025.982	23/4
52	3085.315	6/1
57	3381.980	7/1
61	3619.312	8/1
63	3737.978	26/3
64	3797.311	9/1
67	3975.310	10/1
70	4153.309	11/1
75	4449.974	13/1
77	4568.640	14/1
78	4627.972	29/2
79	4687.305	15/1
80	4746.638	31/2
81	4805.971	16/1

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