186zpi

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

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