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

Approximation errors

Ratio	Error (abs, ¢)	Error (rel, %)
14/1	0.186	0.314
11/5	0.346	0.583
17/8	0.370	0.624
31/22	0.388	0.654
21/13	0.408	0.688
25/19	0.451	0.759
26/3	0.595	1.003
30/29	0.641	1.081
31/10	0.733	1.236
32/9	0.770	1.297
15/14	0.777	1.309
19/16	0.848	1.429
15/1	0.963	1.623
23/12	1.007	1.698
27/10	1.105	1.863
25/16	1.299	2.189
29/28	1.418	2.390
27/22	1.451	2.445
31/2	1.603	2.702
29/2	1.605	2.705
29/6	1.695	2.857
11/1	1.991	3.355
14/11	2.177	3.669
23/4	2.292	3.864
5/1	2.336	3.938
14/5	2.523	4.252
19/5	2.787	4.697
24/7	2.858	4.817
26/15	2.931	4.940
15/11	2.954	4.979
14/3	3.113	5.247
19/11	3.133	5.280
3/1	3.300	5.561
16/13	3.474	5.856
16/5	3.635	6.127
13/7	3.708	6.250
16/11	3.981	6.709
19/13	4.323	7.285
10/9	4.405	7.424
11/3	5.290	8.916
5/3	5.636	9.499
16/1	5.971	10.064
8/7	6.158	10.378
14/9	6.413	10.808
9/1	6.599	11.122
9/2	6.741	11.362
13/5	7.110	11.982
13/11	7.455	12.565
10/3	7.704	12.985
11/9	8.590	14.478
9/5	8.936	15.060
13/1	9.446	15.920
13/8	9.866	16.628
3/2	10.041	16.923
7/5	10.818	18.232
10/1	11.004	18.546
11/7	11.163	18.815
7/1	13.154	22.170
2/1	13.340	22.484
5/2	15.677	26.422
7/3	16.454	27.731
6/1	16.640	28.045
8/5	16.975	28.610
6/5	18.976	31.983
8/1	19.312	32.548
7/4	19.498	32.862
8/3	22.611	38.109
4/3	23.381	39.407
7/2	26.494	44.654
4/1	26.681	44.968
5/4	29.017	48.906

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