186zpi

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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 previous record is 125zpi and the next one 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

Approximation of EDONOIs

Based on harmonics with less than 1 cent of error, 186zpi can be approximated by 96ed5, 124ed8 (or every 3 steps of 124edo), 143ed11, 153ed13, 169ed17, 187ed23, and 192ed25.

Intervals and notation

There are several ways to approach notation. The simplest method involves using the notations from 41edo. However, this method does not preserve octave compression when rendered by notation software. To address this issue, consider using the ups and downs notation from 124edo at every 3-degree step (i.e., the edonoi 124ed8).

It is important to note that 124edo provides two possible fifths (3/2). The closest one, from the val <124 197] (i.e. the patent val), is the fifth mapped to 73 steps of 124edo with a relative error of +46.465%. The second closest, from the val <124 196] (i.e. the val 124b), is mapped to 72 steps of 124edo with a relative error of -53.535%. This second fifth, which appears in 124ed8, also corresponds to the fifth of 31edo. Therefore, we choose to use the ups and downs notation of the 124b temperament, denoted as <124 196].

Todo: complete table

Incorporate 3 new columns for ups and downs notation from 124b at every 3-degree. column 1 = ups and downs notation in full, column 2 = ups and downs notation abbreviated, column 3 = octave

JI ratios are comprised of 32-integer limit ratios,
and are stylized as follows to indicate their accuracy:

  • Bold Underlined: relative error < 8.333 %
  • Bold: relative error < 16.667 %
  • Normal: relative error < 25 %
  • Small: relative error < 33.333 %
  • Small Small: relative error < 41.667 %
  • Small Small Small: relative error < 50 %
Step Cents Ratios
0 0.000
1 29.025
2 58.050 32/31, 31/30, 30/29, 29/28, 28/27, 27/26, 26/25, 25/24
3 87.075 24/23, 23/22, 22/21, 21/20, 20/19, 19/18, 18/17
4 116.100 17/16, 16/15, 31/29, 15/14, 29/27, 14/13
5 145.124 27/25, 13/12, 25/23, 12/11, 23/21
6 174.149 11/10, 32/29, 21/19, 31/28, 10/9
7 203.174 29/26, 19/17, 28/25, 9/8, 26/23, 17/15
8 232.199 25/22, 8/7, 31/27, 23/20
9 261.224 15/13, 22/19, 29/25, 7/6
10 290.249 27/23, 20/17, 13/11, 32/27, 19/16, 25/21, 31/26
11 319.274 6/5, 29/24, 23/19
12 348.299 17/14, 28/23, 11/9, 27/22, 16/13
13 377.323 21/17, 26/21, 31/25, 5/4
14 406.348 29/23, 24/19, 19/15, 14/11
15 435.373 23/18, 32/25, 9/7, 31/24, 22/17
16 464.398 13/10, 30/23, 17/13, 21/16, 25/19, 29/22
17 493.423 4/3
18 522.448 31/23, 27/20, 23/17, 19/14, 15/11
19 551.473 26/19, 11/8, 29/21, 18/13
20 580.498 25/18, 32/23, 7/5, 31/22
21 609.523 24/17, 17/12, 27/19, 10/7
22 638.547 23/16, 13/9, 29/20, 16/11
23 667.572 19/13, 22/15, 25/17, 28/19, 31/21
24 696.597 3/2
25 725.622 32/21, 29/19, 26/17, 23/15
26 754.647 20/13, 17/11, 31/20, 14/9
27 783.672 25/16, 11/7, 30/19, 19/12
28 812.697 27/17, 8/5, 29/18
29 841.722 21/13, 13/8, 31/19, 18/11
30 870.746 23/14, 28/17, 5/3
31 899.771 32/19, 27/16, 22/13
32 928.796 17/10, 29/17, 12/7, 31/18
33 957.821 19/11, 26/15, 7/4
34 986.846 30/17, 23/13, 16/9
35 1015.871 25/14, 9/5, 29/16
36 1044.896 20/11, 31/17, 11/6
37 1073.921 24/13, 13/7, 28/15, 15/8
38 1102.946 32/17, 17/9, 19/10
39 1131.970 21/11, 23/12, 25/13, 27/14, 29/15, 31/16
40 1160.995
41 1190.020 2/1
42 1219.045
43 1248.070 31/15, 29/14
44 1277.095 27/13, 25/12, 23/11, 21/10
45 1306.120 19/9, 17/8, 32/15, 15/7
46 1335.145 28/13, 13/6
47 1364.170 24/11, 11/5, 31/14
48 1393.194 20/9, 29/13, 9/4
49 1422.219 25/11, 16/7
50 1451.244 23/10, 30/13
51 1480.269 7/3, 26/11
52 1509.294 19/8, 31/13, 12/5
53 1538.319 29/12, 17/7, 22/9
54 1567.344 27/11, 32/13
55 1596.369 5/2
56 1625.393 28/11, 23/9, 18/7
57 1654.418 31/12, 13/5
58 1683.443 21/8, 29/11
59 1712.468 8/3, 27/10
60 1741.493 19/7, 30/11, 11/4
61 1770.518 25/9, 14/5
62 1799.543 31/11, 17/6
63 1828.568 20/7, 23/8, 26/9
64 1857.593 29/10, 32/11
65 1886.617
66 1915.642 3/1
67 1944.667 31/10
68 1973.692 28/9, 25/8, 22/7
69 2002.717 19/6, 16/5
70 2031.742 29/9, 13/4
71 2060.767 23/7
72 2089.792 10/3
73 2118.816 27/8, 17/5, 24/7
74 2147.841 31/9
75 2176.866 7/2
76 2205.891 32/9, 25/7, 18/5
77 2234.916 29/8, 11/3
78 2263.941 26/7
79 2292.966 15/4
80 2321.991 19/5, 23/6
81 2351.016 27/7, 31/8
82 2380.040
83 2409.065 4/1
84 2438.090
85 2467.115 29/7, 25/6
86 2496.140 21/5, 17/4
87 2525.165 30/7, 13/3
88 2554.190 22/5
89 2583.215 31/7
90 2612.239 9/2
91 2641.264 32/7, 23/5
92 2670.289 14/3
93 2699.314 19/4
94 2728.339 24/5, 29/6
95 2757.364
96 2786.389 5/1
97 2815.414
98 2844.439 31/6, 26/5
99 2873.463 21/4
100 2902.488 16/3
101 2931.513 27/5
102 2960.538 11/2
103 2989.563 28/5, 17/3
104 3018.588 23/4
105 3047.613 29/5
106 3076.638
107 3105.663 6/1
108 3134.687
109 3163.712 31/5, 25/4
110 3192.737 19/3
111 3221.762 32/5
112 3250.787 13/2
113 3279.812 20/3
114 3308.837 27/4
115 3337.862
116 3366.886 7/1
117 3395.911
118 3424.936 29/4
119 3453.961 22/3
120 3482.986 15/2
121 3512.011 23/3
122 3541.036 31/4
123 3570.061
124 3599.086 8/1
125 3628.110
126 3657.135 25/3
127 3686.160
128 3715.185 17/2
129 3744.210 26/3
130 3773.235
131 3802.260 9/1
132 3831.285
133 3860.309 28/3
134 3889.334 19/2
135 3918.359 29/3
136 3947.384
137 3976.409 10/1
138 4005.434
139 4034.459 31/3
140 4063.484 21/2
141 4092.509 32/3
142 4121.533
143 4150.558 11/1
144 4179.583
145 4208.608
146 4237.633 23/2
147 4266.658
148 4295.683 12/1
149 4324.708
150 4353.732
151 4382.757 25/2
152 4411.782
153 4440.807 13/1
154 4469.832
155 4498.857 27/2
156 4527.882
157 4556.907 14/1
158 4585.932
159 4614.956
160 4643.981 29/2
161 4673.006
162 4702.031 15/1
163 4731.056 31/2
164 4760.081
165 4789.106 16/1
166 4818.131
167 4847.156
168 4876.180
169 4905.205 17/1
170 4934.230
171 4963.255
172 4992.280 18/1
173 5021.305
174 5050.330
175 5079.355
176 5108.379 19/1
177 5137.404
178 5166.429
179 5195.454 20/1
180 5224.479
181 5253.504
182 5282.529 21/1
183 5311.554
184 5340.579 22/1
185 5369.603
186 5398.628
187 5427.653 23/1
188 5456.678
189 5485.703
190 5514.728 24/1
191 5543.753
192 5572.778 25/1
193 5601.802
194 5630.827 26/1
195 5659.852
196 5688.877
197 5717.902 27/1
198 5746.927
199 5775.952 28/1
200 5804.977
201 5834.002 29/1
202 5863.026
203 5892.051 30/1
204 5921.076
205 5950.101 31/1
206 5979.126
207 6008.151 32/1

Approximation to JI

The following table illustrates the representation of the 32-integer limit intervals in 186zpi. Prime harmonics are in bold; inconsistent intervals are in italic.

Intervals by direct approximation (even if inconsistent)
Ratio Error (abs, ¢) Error (rel,  %)
17/13 0.030 0.102
5/1 0.075 0.259
25/17 0.100 0.344
25/13 0.129 0.446
23/11 0.138 0.477
25/1 0.150 0.517
11/8 0.155 0.533
17/5 0.175 0.602
13/5 0.204 0.704
17/1 0.250 0.861
13/1 0.279 0.963
9/7 0.289 0.996
23/8 0.293 1.011
23/1 0.621 2.140
31/29 0.641 2.209
30/29 0.642 2.211
23/5 0.696 2.399
29/6 0.717 2.470
9/8 0.736 2.535
11/1 0.760 2.617
25/23 0.771 2.657
11/5 0.835 2.876
23/17 0.871 3.001
21/19 0.881 3.037
11/9 0.891 3.069
23/13 0.901 3.103
25/11 0.910 3.135
8/1 0.914 3.151
8/5 0.990 3.409
17/11 1.009 3.478
8/7 1.025 3.531
23/9 1.029 3.546
13/11 1.039 3.580
25/8 1.065 3.668
17/8 1.164 4.012
27/19 1.171 4.033
11/7 1.180 4.065
13/8 1.194 4.114
31/30 1.283 4.420
23/7 1.318 4.542
31/6 1.358 4.679
9/1 1.650 5.686
9/5 1.725 5.944
20/19 1.726 5.947
25/9 1.800 6.203
19/4 1.801 6.205
17/9 1.900 6.547
24/19 1.906 6.568
13/9 1.930 6.649
7/1 1.939 6.682
7/5 2.015 6.941
31/28 2.060 7.099
25/7 2.090 7.199
17/7 2.189 7.543
13/7 2.219 7.645
21/20 2.607 8.984
21/4 2.683 9.242
29/28 2.702 9.308
32/19 2.716 9.356
19/3 2.821 9.719
19/15 2.896 9.977
27/20 2.897 9.980
27/4 2.972 10.238
32/31 3.085 10.630
15/14 3.343 11.519
14/3 3.418 11.777
13/6 3.428 11.811
17/6 3.458 11.913
30/13 3.503 12.069
30/17 3.533 12.171
25/6 3.557 12.256
32/21 3.597 12.393
6/5 3.632 12.515
6/1 3.708 12.774
32/29 3.726 12.839
28/19 3.741 12.887
30/1 3.783 13.032
32/27 3.886 13.389
31/4 4.000 13.781
31/20 4.075 14.039
29/13 4.145 14.280
29/17 4.174 14.382
29/25 4.274 14.726
23/6 4.329 14.914
12/7 4.333 14.928
29/5 4.349 14.985
16/15 4.368 15.050
30/23 4.404 15.172
29/1 4.424 15.243
16/3 4.443 15.309
11/6 4.467 15.391
22/15 4.523 15.583
30/11 4.542 15.649
20/3 4.547 15.666
22/3 4.598 15.842
4/3 4.622 15.924
29/4 4.641 15.990
15/4 4.697 16.183
29/20 4.716 16.248
31/13 4.786 16.489
31/17 4.816 16.591
28/27 4.911 16.920
31/25 4.915 16.935
31/5 4.990 17.194
29/23 5.046 17.383
31/1 5.066 17.452
27/14 5.069 17.463
29/11 5.184 17.860
15/2 5.283 18.201
29/8 5.339 18.394
3/2 5.358 18.459
10/3 5.433 18.718
12/11 5.513 18.993
32/3 5.536 19.075
26/15 5.562 19.164
32/15 5.612 19.334
26/3 5.637 19.422
7/6 5.647 19.456
23/12 5.651 19.470
31/23 5.687 19.592
30/7 5.722 19.714
31/19 5.801 19.986
31/11 5.825 20.069
31/8 5.980 20.603
29/9 6.075 20.929
27/16 6.094 20.994
19/14 6.239 21.496
27/22 6.248 21.528
12/1 6.272 21.610
12/5 6.347 21.869
29/7 6.364 21.925
21/16 6.383 21.991
25/12 6.422 22.127
29/19 6.442 22.195
17/12 6.522 22.471
19/18 6.528 22.492
22/21 6.538 22.524
13/12 6.552 22.573
28/3 6.561 22.606
28/15 6.637 22.865
31/21 6.682 23.023
31/9 6.716 23.138
28/13 6.846 23.588
28/17 6.876 23.690
31/27 6.972 24.019
28/25 6.976 24.034
31/7 7.005 24.134
27/2 7.008 24.145
28/5 7.051 24.292
27/10 7.083 24.404
30/19 7.084 24.406
28/1 7.126 24.551
19/6 7.159 24.665
19/16 7.264 25.027
27/26 7.288 25.108
21/2 7.297 25.141
29/21 7.324 25.232
21/10 7.372 25.400
22/19 7.419 25.561
26/21 7.577 26.104
29/27 7.613 26.228
31/24 7.707 26.554
28/23 7.747 26.691
26/7 7.761 26.739
32/13 7.871 27.119
28/11 7.886 27.168
32/17 7.901 27.221
10/7 7.965 27.443
32/25 8.001 27.565
7/2 8.040 27.702
26/9 8.050 27.735
32/5 8.076 27.824
32/1 8.151 28.082
19/2 8.179 28.178
19/10 8.254 28.437
10/9 8.254 28.439
9/2 8.329 28.698
29/24 8.348 28.763
26/19 8.458 29.141
31/3 8.622 29.705
31/15 8.697 29.964
32/23 8.772 30.222
28/9 8.776 30.237
13/4 8.786 30.270
22/7 8.800 30.319
17/4 8.815 30.372
20/13 8.861 30.529
20/17 8.891 30.631
32/11 8.910 30.699
25/4 8.915 30.716
26/11 8.941 30.803
16/7 8.955 30.852
5/4 8.990 30.974
4/1 9.065 31.233
26/23 9.079 31.281
22/9 9.089 31.315
20/1 9.140 31.492
11/10 9.145 31.508
11/2 9.220 31.766
16/9 9.244 31.848
29/3 9.263 31.914
23/10 9.284 31.985
29/15 9.338 32.173
23/2 9.359 32.243
23/4 9.686 33.373
18/7 9.691 33.387
26/1 9.700 33.421
23/20 9.762 33.632
26/5 9.775 33.679
32/9 9.801 33.768
11/4 9.825 33.850
26/25 9.850 33.938
20/11 9.900 34.109
10/1 9.905 34.125
26/17 9.950 34.282
2/1 9.980 34.384
5/2 10.055 34.642
32/7 10.090 34.764
23/22 10.118 34.861
25/2 10.130 34.901
16/11 10.135 34.917
17/10 10.155 34.986
13/10 10.184 35.088
17/2 10.230 35.244
13/2 10.259 35.346
14/9 10.269 35.380
23/16 10.273 35.394
19/13 10.587 36.475
19/17 10.617 36.577
29/12 10.697 36.853
9/4 10.716 36.919
25/19 10.716 36.921
22/1 10.739 37.001
20/9 10.791 37.177
19/5 10.791 37.180
22/5 10.814 37.259
19/1 10.866 37.438
18/11 10.870 37.452
25/22 10.890 37.518
16/1 10.894 37.534
16/5 10.969 37.793
22/17 10.989 37.862
7/4 11.005 37.915
23/18 11.009 37.929
22/13 11.019 37.964
25/16 11.044 38.052
20/7 11.080 38.174
17/16 11.144 38.395
14/11 11.160 38.448
16/13 11.174 38.497
23/14 11.298 38.925
31/12 11.338 39.062
21/13 11.468 39.512
23/19 11.488 39.579
21/17 11.498 39.614
25/21 11.598 39.958
19/11 11.626 40.056
18/1 11.630 40.069
21/5 11.673 40.216
18/5 11.705 40.328
21/1 11.748 40.475
27/13 11.758 40.508
25/18 11.780 40.587
19/8 11.781 40.589
27/17 11.787 40.610
18/17 11.880 40.930
19/12 11.886 40.952
27/25 11.887 40.954
18/13 11.910 41.032
14/1 11.919 41.066
27/5 11.962 41.213
14/5 11.994 41.324
27/1 12.037 41.471
31/14 12.040 41.482
25/14 12.069 41.583
17/14 12.169 41.926
14/13 12.199 42.028
31/18 12.329 42.478
23/21 12.369 42.615
24/13 12.493 43.044
21/11 12.507 43.092
19/9 12.517 43.124
24/17 12.523 43.146
25/24 12.623 43.489
27/23 12.658 43.611
21/8 12.662 43.626
29/14 12.681 43.691
24/5 12.698 43.748
24/1 12.773 44.006
27/11 12.797 44.089
19/7 12.806 44.120
27/8 12.951 44.622
29/18 12.970 44.687
31/16 13.065 45.014
31/22 13.220 45.547
15/7 13.323 45.902
24/23 13.394 46.147
7/3 13.398 46.161
13/3 13.408 46.194
17/3 13.437 46.296
15/13 13.483 46.453
17/15 13.513 46.555
24/11 13.532 46.624
25/3 13.537 46.640
5/3 13.612 46.898
3/1 13.687 47.157
29/16 13.706 47.223
15/1 13.762 47.416
29/22 13.861 47.756
27/7 13.976 48.153
31/2 13.980 48.164
31/10 14.055 48.423
29/26 14.125 48.664
31/26 14.259 49.127
23/3 14.308 49.297
24/7 14.313 49.312
29/10 14.329 49.368
15/8 14.348 49.434
23/15 14.384 49.556
29/2 14.404 49.627
8/3 14.423 49.692
11/3 14.447 49.774
15/11 14.503 49.967
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