186 zeta peak index (abbreviated 186zpi), is the equal-step tuning system obtained from the 186st 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
|
| 186zpi
|
41.3438354846780
|
29.0248832971658
|
1.876590
|
0.241233
|
11.567493
|
41edo
|
1190.02021518380
|
2
|
2
|
Theory
Record on the Riemann zeta function with primes 2 and 3 removed
186zpi sets a height record on the Riemann zeta function with primes 2 and 3 removed. The last record is 125zpi and the next is 565zpi. It is important to highlight that the optimal equal tunings obtained by excluding the prime numbers 2 and 3 from the Riemann zeta function differs very slightly from the optimal equal tuning corresponding to the same peaks on the unmodified Riemann zeta function.
| Unmodified Riemann zeta function
|
Riemann zeta function with primes 2 and 3 removed
|
| Tuning
|
Strength
|
Closest EDO
|
Tuning
|
Strength
|
Closest EDO
|
| ZPI
|
Steps per octave
|
Step size (cents)
|
Height
|
EDO
|
Octave (cents)
|
Steps per octave
|
Step size (cents)
|
Height
|
EDO
|
Octave (cents)
|
| 125zpi
|
30.6006474885974
|
39.2148564976330
|
1.468164
|
31edo
|
1215.66055142662
|
30.5974484926723
|
39.2189564527704
|
3.769318
|
31edo
|
1215.78765003588
|
| 186zpi
|
41.3438354846780
|
29.0248832971658
|
1.876590
|
41edo
|
1190.02021518380
|
41.3477989230936
|
29.0221010852836
|
4.469823
|
41edo
|
1189.90614449663
|
| 565zpi
|
98.6209462564991
|
12.1678005084130
|
2.305330
|
99edo
|
1204.61225033289
|
98.6257548378926
|
12.1672072570942
|
4.883729
|
99edo
|
1204.55351845233
|
Harmonic series
As a non-octave, non-tritave scale, 186zpi features a well-balanced harmonic series segment from 5 to 9, and performs exceptionally well across all prime harmonics from 5 to 23, with the exception of 19.
Approximation of harmonics in 186zpi
| Harmonic
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
15
|
16
|
| Error
|
Absolute (¢)
|
-10.0
|
+13.7
|
+9.1
|
+0.1
|
+3.7
|
-1.9
|
-0.9
|
-1.7
|
-9.9
|
-0.8
|
-6.3
|
+0.3
|
-11.9
|
+13.8
|
-10.9
|
| Relative (%)
|
-34.4
|
+47.2
|
+31.2
|
+0.3
|
+12.8
|
-6.7
|
-3.2
|
-5.7
|
-34.1
|
-2.6
|
-21.6
|
+1.0
|
-41.1
|
+47.4
|
-37.5
|
| Step
|
41
|
66
|
83
|
96
|
107
|
116
|
124
|
131
|
137
|
143
|
148
|
153
|
157
|
162
|
165
|
Approximation of harmonics in 186zpi
| Harmonic
|
17
|
18
|
19
|
20
|
21
|
22
|
23
|
24
|
25
|
26
|
27
|
28
|
29
|
30
|
31
|
32
|
| Error
|
Absolute (¢)
|
+0.2
|
-11.6
|
+10.9
|
+9.1
|
+11.7
|
-10.7
|
-0.6
|
+12.8
|
+0.2
|
-9.7
|
+12.0
|
+7.1
|
+4.4
|
+3.8
|
+5.1
|
+8.2
|
| Relative (%)
|
+0.9
|
-40.1
|
+37.4
|
+31.5
|
+40.5
|
-37.0
|
-2.1
|
+44.0
|
+0.5
|
-33.4
|
+41.5
|
+24.6
|
+15.2
|
+13.0
|
+17.5
|
+28.1
|
| Step
|
169
|
172
|
176
|
179
|
182
|
184
|
187
|
190
|
192
|
194
|
197
|
199
|
201
|
203
|
205
|
207
|