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
