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
Error	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
Steps		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
Error	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
Steps		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].

Intervals in 186zpi
Degree	Cents	Ratios	Ups and Downs Notation	Step
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 %			⟨124 196] at every 3 steps Whole tone = 20 steps Limma = 12 steps Apotome = 8 steps
0	0.000		P1	0
1	29.025		^^^1	3
2	58.050	32/31, 31/30, 30/29, 29/28, 28/27, 27/26, 26/25, 25/24	vvA1, ^^d2	6
3	87.075	24/23, 23/22, 22/21, 21/20, 20/19, 19/18, 18/17	vvvm2	9
4	116.100	17/16, 16/15, 31/29, 15/14, 29/27, 14/13	m2	12
5	145.124	27/25, 13/12, 25/23, 12/11, 23/21	^^^m2	15
6	174.149	11/10, 32/29, 21/19, 31/28, 10/9	vvM2	18
7	203.174	29/26, 19/17, 28/25, 9/8, 26/23, 17/15	^M2	21
8	232.199	25/22, 8/7, 31/27, 23/20	^⁴M2	24
9	261.224	15/13, 22/19, 29/25, 7/6	^^^d3	27
10	290.249	27/23, 20/17, 13/11, 32/27, 19/16, 25/21, 31/26	vvm3	30
11	319.274	6/5, 29/24, 23/19	^m3	33
12	348.299	17/14, 28/23, 11/9, 27/22, 16/13	~3	36
13	377.323	21/17, 26/21, 31/25, 5/4	vM3	39
14	406.348	29/23, 24/19, 19/15, 14/11	^^M3	42
15	435.373	23/18, 32/25, 9/7, 31/24, 22/17	vvvA3	45
16	464.398	13/10, 30/23, 17/13, 21/16, 25/19, 29/22	v⁴4	48
17	493.423	4/3	v4	51
18	522.448	31/23, 27/20, 23/17, 19/14, 15/11	^^4	54
19	551.473	26/19, 11/8, 29/21, 18/13	vvvA4	57
20	580.498	25/18, 32/23, 7/5, 31/22	A4	60
21	609.523	24/17, 17/12, 27/19, 10/7	vd5	63
22	638.547	23/16, 13/9, 29/20, 16/11	^^d5	66
23	667.572	19/13, 22/15, 25/17, 28/19, 31/21	vvv5	69
24	696.597	3/2	P5	72
25	725.622	32/21, 29/19, 26/17, 23/15	^^^5	75
26	754.647	20/13, 17/11, 31/20, 14/9	vvA5, ^^d6	78
27	783.672	25/16, 11/7, 30/19, 19/12	vvvm6	81
28	812.697	27/17, 8/5, 29/18	m6	84
29	841.722	21/13, 13/8, 31/19, 18/11	^^^m6	87
30	870.746	23/14, 28/17, 5/3	vvM6	90
31	899.771	32/19, 27/16, 22/13	^M6	93
32	928.796	17/10, 29/17, 12/7, 31/18	^⁴M6	96
33	957.821	19/11, 26/15, 7/4	^^^d7	99
34	986.846	30/17, 23/13, 16/9	vvm7	102
35	1015.871	25/14, 9/5, 29/16	^m7	105
36	1044.896	20/11, 31/17, 11/6	~7	108
37	1073.921	24/13, 13/7, 28/15, 15/8	vM7	111
38	1102.946	32/17, 17/9, 19/10	^^M7	114
39	1131.970	21/11, 23/12, 25/13, 27/14, 29/15, 31/16	vvvA7	117
40	1160.995		v⁴1 +1 oct	120
41	1190.020	2/1	v1 +1 oct	123
42	1219.045		^^1 +1 oct	126
43	1248.070	31/15, 29/14	vvvA1 +1 oct	129
44	1277.095	27/13, 25/12, 23/11, 21/10	v⁴m2 +1 oct	132
45	1306.120	19/9, 17/8, 32/15, 15/7	vm2 +1 oct	135
46	1335.145	28/13, 13/6	^^m2 +1 oct	138
47	1364.170	24/11, 11/5, 31/14	vvvM2 +1 oct	141
48	1393.194	20/9, 29/13, 9/4	M2 +1 oct	144
49	1422.219	25/11, 16/7	^^^M2 +1 oct	147
50	1451.244	23/10, 30/13	vvA2 +1 oct, ^^d3 +1 oct	150
51	1480.269	7/3, 26/11	vvvm3 +1 oct	153
52	1509.294	19/8, 31/13, 12/5	m3 +1 oct	156
53	1538.319	29/12, 17/7, 22/9	^^^m3 +1 oct	159
54	1567.344	27/11, 32/13	vvM3 +1 oct	162
55	1596.369	5/2	^M3 +1 oct	165
56	1625.393	28/11, 23/9, 18/7	^⁴M3 +1 oct	168
57	1654.418	31/12, 13/5	^^^d4 +1 oct	171
58	1683.443	21/8, 29/11	vv4 +1 oct	174
59	1712.468	8/3, 27/10	^4 +1 oct	177
60	1741.493	19/7, 30/11, 11/4	~4 +1 oct	180
61	1770.518	25/9, 14/5	vA4 +1 oct	183
62	1799.543	31/11, 17/6	^^A4 +1 oct, vvd5 +1 oct	186
63	1828.568	20/7, 23/8, 26/9	^d5 +1 oct	189
64	1857.593	29/10, 32/11	~5 +1 oct	192
65	1886.617		v5 +1 oct	195
66	1915.642	3/1	^^5 +1 oct	198
67	1944.667	31/10	vvvA5 +1 oct	201
68	1973.692	28/9, 25/8, 22/7	v⁴m6 +1 oct	204
69	2002.717	19/6, 16/5	vm6 +1 oct	207
70	2031.742	29/9, 13/4	^^m6 +1 oct	210
71	2060.767	23/7	vvvM6 +1 oct	213
72	2089.792	10/3	M6 +1 oct	216
73	2118.816	27/8, 17/5, 24/7	^^^M6 +1 oct	219
74	2147.841	31/9	vvA6 +1 oct, ^^d7 +1 oct	222
75	2176.866	7/2	vvvm7 +1 oct	225
76	2205.891	32/9, 25/7, 18/5	m7 +1 oct	228
77	2234.916	29/8, 11/3	^^^m7 +1 oct	231
78	2263.941	26/7	vvM7 +1 oct	234
79	2292.966	15/4	^M7 +1 oct	237
80	2321.991	19/5, 23/6	^⁴M7 +1 oct	240
81	2351.016	27/7, 31/8	^^^d1 +2 oct	243
82	2380.040		vv1 +2 oct	246
83	2409.065	4/1	^1 +2 oct	249
84	2438.090		^⁴1 +2 oct	252
85	2467.115	29/7, 25/6	^^^d2 +2 oct	255
86	2496.140	21/5, 17/4	vvm2 +2 oct	258
87	2525.165	30/7, 13/3	^m2 +2 oct	261
88	2554.190	22/5	~2 +2 oct	264
89	2583.215	31/7	vM2 +2 oct	267
90	2612.239	9/2	^^M2 +2 oct	270
91	2641.264	32/7, 23/5	vvvA2 +2 oct	273
92	2670.289	14/3	v⁴m3 +2 oct	276
93	2699.314	19/4	vm3 +2 oct	279
94	2728.339	24/5, 29/6	^^m3 +2 oct	282
95	2757.364		vvvM3 +2 oct	285
96	2786.389	5/1	M3 +2 oct	288
97	2815.414		^^^M3 +2 oct	291
98	2844.439	31/6, 26/5	vvA3 +2 oct, ^^d4 +2 oct	294
99	2873.463	21/4	vvv4 +2 oct	297
100	2902.488	16/3	P4 +2 oct	300
101	2931.513	27/5	^^^4 +2 oct	303
102	2960.538	11/2	vvA4 +2 oct	306
103	2989.563	28/5, 17/3	^A4 +2 oct	309
104	3018.588	23/4	d5 +2 oct	312
105	3047.613	29/5	^^^d5 +2 oct	315
106	3076.638		vv5 +2 oct	318
107	3105.663	6/1	^5 +2 oct	321
108	3134.687		^⁴5 +2 oct	324
109	3163.712	31/5, 25/4	^^^d6 +2 oct	327
110	3192.737	19/3	vvm6 +2 oct	330
111	3221.762	32/5	^m6 +2 oct	333
112	3250.787	13/2	~6 +2 oct	336
113	3279.812	20/3	vM6 +2 oct	339
114	3308.837	27/4	^^M6 +2 oct	342
115	3337.862		vvvA6 +2 oct	345
116	3366.886	7/1	v⁴m7 +2 oct	348
117	3395.911		vm7 +2 oct	351
118	3424.936	29/4	^^m7 +2 oct	354
119	3453.961	22/3	vvvM7 +2 oct	357
120	3482.986	15/2	M7 +2 oct	360
121	3512.011	23/3	^^^M7 +2 oct	363
122	3541.036	31/4	vvA7 +2 oct, ^^d1 +3 oct	366
123	3570.061		vvv1 +3 oct	369
124	3599.086	8/1	P1 +3 oct	372
125	3628.110		^^^1 +3 oct	375
126	3657.135	25/3	vvA1 +3 oct, ^^d2 +3 oct	378
127	3686.160		vvvm2 +3 oct	381
128	3715.185	17/2	m2 +3 oct	384
129	3744.210	26/3	^^^m2 +3 oct	387
130	3773.235		vvM2 +3 oct	390
131	3802.260	9/1	^M2 +3 oct	393
132	3831.285		^⁴M2 +3 oct	396
133	3860.309	28/3	^^^d3 +3 oct	399
134	3889.334	19/2	vvm3 +3 oct	402
135	3918.359	29/3	^m3 +3 oct	405
136	3947.384		~3 +3 oct	408
137	3976.409	10/1	vM3 +3 oct	411
138	4005.434		^^M3 +3 oct	414
139	4034.459	31/3	vvvA3 +3 oct	417
140	4063.484	21/2	v⁴4 +3 oct	420
141	4092.509	32/3	v4 +3 oct	423
142	4121.533		^^4 +3 oct	426
143	4150.558	11/1	vvvA4 +3 oct	429
144	4179.583		A4 +3 oct	432
145	4208.608		vd5 +3 oct	435
146	4237.633	23/2	^^d5 +3 oct	438
147	4266.658		vvv5 +3 oct	441
148	4295.683	12/1	P5 +3 oct	444
149	4324.708		^^^5 +3 oct	447
150	4353.732		vvA5 +3 oct, ^^d6 +3 oct	450
151	4382.757	25/2	vvvm6 +3 oct	453
152	4411.782		m6 +3 oct	456
153	4440.807	13/1	^^^m6 +3 oct	459
154	4469.832		vvM6 +3 oct	462
155	4498.857	27/2	^M6 +3 oct	465
156	4527.882		^⁴M6 +3 oct	468
157	4556.907	14/1	^^^d7 +3 oct	471
158	4585.932		vvm7 +3 oct	474
159	4614.956		^m7 +3 oct	477
160	4643.981	29/2	~7 +3 oct	480
161	4673.006		vM7 +3 oct	483
162	4702.031	15/1	^^M7 +3 oct	486
163	4731.056	31/2	vvvA7 +3 oct	489
164	4760.081		v⁴1 +4 oct	492
165	4789.106	16/1	v1 +4 oct	495
166	4818.131		^^1 +4 oct	498
167	4847.156		vvvA1 +4 oct	501
168	4876.180		v⁴m2 +4 oct	504
169	4905.205	17/1	vm2 +4 oct	507
170	4934.230		^^m2 +4 oct	510
171	4963.255		vvvM2 +4 oct	513
172	4992.280	18/1	M2 +4 oct	516
173	5021.305		^^^M2 +4 oct	519
174	5050.330		vvA2 +4 oct, ^^d3 +4 oct	522
175	5079.355		vvvm3 +4 oct	525
176	5108.379	19/1	m3 +4 oct	528
177	5137.404		^^^m3 +4 oct	531
178	5166.429		vvM3 +4 oct	534
179	5195.454	20/1	^M3 +4 oct	537
180	5224.479		^⁴M3 +4 oct	540
181	5253.504		^^^d4 +4 oct	543
182	5282.529	21/1	vv4 +4 oct	546
183	5311.554		^4 +4 oct	549
184	5340.579	22/1	~4 +4 oct	552
185	5369.603		vA4 +4 oct	555
186	5398.628		^^A4 +4 oct, vvd5 +4 oct	558
187	5427.653	23/1	^d5 +4 oct	561
188	5456.678		~5 +4 oct	564
189	5485.703		v5 +4 oct	567
190	5514.728	24/1	^^5 +4 oct	570
191	5543.753		vvvA5 +4 oct	573
192	5572.778	25/1	v⁴m6 +4 oct	576
193	5601.802		vm6 +4 oct	579
194	5630.827	26/1	^^m6 +4 oct	582
195	5659.852		vvvM6 +4 oct	585
196	5688.877		M6 +4 oct	588
197	5717.902	27/1	^^^M6 +4 oct	591
198	5746.927		vvA6 +4 oct, ^^d7 +4 oct	594
199	5775.952	28/1	vvvm7 +4 oct	597
200	5804.977		m7 +4 oct	600
201	5834.002	29/1	^^^m7 +4 oct	603
202	5863.026		vvM7 +4 oct	606
203	5892.051	30/1	^M7 +4 oct	609
204	5921.076		^⁴M7 +4 oct	612
205	5950.101	31/1	^^^d1 +5 oct	615
206	5979.126		vv1 +5 oct	618
207	6008.151	32/1	^1 +5 oct	621

Approximation to JI

Interval mappings

The following tables show how 32-integer-limit intervals are represented in 186zpi. Prime harmonics are in bold; inconsistent intervals are in italics.

32-integer-limit intervals in 186zpi (by direct approximation)
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

32-integer-limit intervals in 186zpi (by patent val mapping)
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
17/5	+0.175	+0.602
13/5	+0.204	+0.704
17/1	+0.250	+0.861
13/1	+0.279	+0.963
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
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
23/13	-0.901	-3.103
25/11	+0.910	+3.135
17/11	+1.009	+3.478
13/11	+1.039	+3.580
11/7	+1.180	+4.065
31/30	+1.283	+4.420
23/7	+1.318	+4.542
31/6	+1.358	+4.679
7/1	-1.939	-6.682
7/5	-2.015	-6.941
25/7	+2.090	+7.199
17/7	+2.189	+7.543
13/7	+2.219	+7.645
19/3	-2.821	-9.719
19/15	-2.896	-9.977
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
6/5	+3.632	+12.515
6/1	+3.708	+12.774
30/1	+3.783	+13.032
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
30/23	+4.404	+15.172
29/1	+4.424	+15.243
11/6	-4.467	-15.391
30/11	+4.542	+15.649
31/13	+4.786	+16.489
31/17	+4.816	+16.591
31/25	+4.915	+16.935
31/5	+4.990	+17.194
29/23	+5.046	+17.383
31/1	+5.066	+17.452
29/11	+5.184	+17.860
12/11	-5.513	-18.993
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
12/1	-6.272	-21.610
12/5	-6.347	-21.869
29/7	+6.364	+21.925
25/12	+6.422	+22.127
29/19	-6.442	-22.195
17/12	+6.522	+22.471
19/18	-6.528	-22.492
13/12	+6.552	+22.573
31/21	-6.682	-23.023
31/7	+7.005	+24.134
30/19	-7.084	-24.406
19/6	+7.159	+24.665
29/21	-7.324	-25.232
26/7	-7.761	-26.739
10/7	-7.965	-27.443
7/2	+8.040	+27.702
31/3	-8.622	-29.705
31/15	-8.697	-29.964
22/7	-8.800	-30.319
26/11	-8.941	-30.803
26/23	-9.079	-31.281
11/10	+9.145	+31.508
11/2	+9.220	+31.766
29/3	-9.263	-31.914
23/10	+9.284	+31.985
29/15	-9.338	-32.173
23/2	+9.359	+32.243
26/1	-9.700	-33.421
26/5	-9.775	-33.679
26/25	-9.850	-33.938
10/1	-9.905	-34.125
26/17	-9.950	-34.282
2/1	-9.980	-34.384
5/2	+10.055	+34.642
23/22	+10.118	+34.861
25/2	+10.130	+34.901
17/10	+10.155	+34.986
13/10	+10.184	+35.088
17/2	+10.230	+35.244
13/2	+10.259	+35.346
19/13	+10.587	+36.475
19/17	+10.617	+36.577
29/12	+10.697	+36.853
25/19	-10.716	-36.921
22/1	-10.739	-37.001
19/5	+10.791	+37.180
22/5	-10.814	-37.259
19/1	+10.866	+37.438
25/22	+10.890	+37.518
22/17	-10.989	-37.862
22/13	-11.019	-37.964
14/11	-11.160	-38.448
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
21/5	+11.673	+40.216
21/1	+11.748	+40.475
14/1	-11.919	-41.066
14/5	-11.994	-41.324
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
21/11	+12.507	+43.092
19/7	+12.806	+44.120
29/18	-12.970	-44.687
13/3	-13.408	-46.194
17/3	-13.437	-46.296
15/13	+13.483	+46.453
17/15	-13.513	-46.555
25/3	-13.537	-46.640
5/3	-13.612	-46.898
3/1	+13.687	+47.157
15/1	+13.762	+47.416
29/26	+14.125	+48.664
23/3	-14.308	-49.297
24/7	-14.313	-49.312
29/10	+14.329	+49.368
23/15	-14.384	-49.556
29/2	+14.404	+49.627
11/3	-14.447	-49.774
15/11	+14.522	+50.033
31/26	+14.766	+50.873
31/10	+14.970	+51.577
31/2	+15.045	+51.836
29/22	+15.164	+52.244
24/11	-15.492	-53.376
7/3	-15.627	-53.839
24/23	-15.631	-53.853
15/7	+15.702	+54.098
31/22	+15.805	+54.453
24/1	-16.252	-55.994
24/5	-16.327	-56.252
29/14	+16.344	+56.309
25/24	+16.402	+56.511
24/17	-16.502	-56.854
19/9	-16.508	-56.876
24/13	-16.532	-56.956
31/14	+16.985	+58.518
18/13	+17.115	+58.968
19/12	+17.139	+59.048
18/17	+17.145	+59.070
25/18	-17.245	-59.413
18/5	+17.320	+59.672
18/1	+17.395	+59.931
20/7	-17.945	-61.826
23/18	-18.016	-62.071
7/4	+18.020	+62.085
18/11	+18.154	+62.548
20/11	-19.125	-65.891
11/4	+19.200	+66.150
23/20	+19.263	+66.368
18/7	+19.334	+66.613
23/4	+19.338	+66.627
20/1	-19.884	-68.508
4/1	-19.960	-68.767
5/4	+20.035	+69.026
25/4	+20.110	+69.284
20/17	-20.134	-69.369
20/13	-20.164	-69.471
17/4	+20.209	+69.628
13/4	+20.239	+69.730
26/19	-20.567	-70.859
29/24	+20.676	+71.237
19/10	+20.771	+71.563
19/2	+20.846	+71.822
28/11	-21.139	-72.832
28/23	-21.278	-73.309
31/24	+21.318	+73.446
26/21	-21.448	-73.896
22/19	-21.606	-74.439
21/10	+21.653	+74.600
21/2	+21.728	+74.859
28/1	-21.899	-75.449
28/5	-21.974	-75.708
28/25	-22.049	-75.966
28/17	-22.149	-76.310
28/13	-22.178	-76.412
31/9	-22.309	-76.862
22/21	-22.487	-77.476
19/14	+22.786	+78.504
29/9	-22.950	-79.071
26/3	-23.388	-80.578
26/15	-23.463	-80.836
10/3	-23.592	-81.282
3/2	+23.667	+81.541
15/2	+23.742	+81.799
29/20	+24.309	+83.752
29/4	+24.384	+84.010
22/3	-24.427	-84.158
22/15	-24.502	-84.417
31/20	+24.950	+85.961
31/4	+25.025	+86.219
14/3	-25.607	-88.223
15/14	+25.682	+88.481
29/28	+26.323	+90.692
31/28	+26.965	+92.901
13/9	-27.095	-93.351
24/19	-27.119	-93.432
17/9	-27.125	-93.453
25/9	-27.224	-93.797
9/5	+27.300	+94.056
9/1	+27.375	+94.314
23/9	-27.996	-96.454
8/7	-28.000	-96.469
11/9	-28.134	-96.931
11/8	+29.180	+100.533
9/7	+29.314	+100.996
23/8	+29.318	+101.011
8/1	-29.939	-103.151
8/5	-30.014	-103.409
25/8	+30.090	+103.668
17/8	+30.189	+104.012
27/19	+30.195	+104.033
13/8	+30.219	+104.114
20/19	-30.751	-105.947
19/4	+30.826	+106.205
21/20	+31.632	+108.984
21/4	+31.707	+109.242
28/19	-32.765	-112.887
20/3	-33.572	-115.666
4/3	-33.647	-115.924
15/4	+33.722	+116.183
29/8	+34.364	+118.394
31/8	+35.005	+120.603
28/3	-35.586	-122.606
28/15	-35.661	-122.865
31/27	-35.996	-124.019
29/27	-36.638	-126.228
26/9	-37.075	-127.735
10/9	-37.279	-128.439
9/2	+37.354	+128.698
16/7	-37.980	-130.852
22/9	-38.114	-131.315
16/11	-39.160	-134.917
14/9	-39.294	-135.380
23/16	+39.298	+135.394
16/1	-39.919	-137.534
16/5	-39.994	-137.793
25/16	+40.069	+138.052
17/16	+40.169	+138.395
16/13	-40.199	-138.497
27/13	+40.782	+140.508
19/8	+40.806	+140.589
27/17	+40.812	+140.610
27/25	+40.912	+140.954
27/5	+40.987	+141.213
27/1	+41.062	+141.471
27/23	+41.683	+143.611
21/8	+41.687	+143.626
27/11	+41.822	+144.089
27/7	+43.001	+148.153
8/3	-43.627	-150.308
15/8	+43.702	+150.566
29/16	+44.343	+152.777
31/16	+44.985	+154.986
20/9	-47.259	-162.823
9/4	+47.334	+163.081
32/7	-47.959	-165.236
32/11	-49.139	-169.301
28/9	-49.274	-169.763
32/23	-49.278	-169.778
32/1	-49.899	-171.918
32/5	-49.974	-172.176
32/25	-50.049	-172.435
32/17	-50.149	-172.779
32/13	-50.178	-172.881
27/26	+50.762	+174.892
19/16	+50.786	+174.973
27/10	+50.967	+175.596
27/2	+51.042	+175.855
21/16	+51.667	+178.009
27/22	+51.801	+178.472
27/14	+52.981	+182.537
16/3	-53.606	-184.691
16/15	-53.682	-184.950
32/29	-54.323	-187.161
32/31	-54.964	-189.370
9/8	+57.314	+197.465
32/19	-60.765	-209.356
27/20	+60.946	+209.980
27/4	+61.021	+210.238
32/21	-61.647	-212.393
28/27	-62.961	-216.920
32/3	-63.586	-219.075
32/15	-63.661	-219.334
16/9	-67.294	-231.848
27/8	+71.001	+244.622
32/9	-77.274	-266.232
27/16	+80.981	+279.006
32/27	-90.961	-313.389