45zpi

45 zeta peak index (abbreviated 45zpi), is the equal-step tuning system obtained from the 45th 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
45zpi	14.5944346577250	82.2231232756126	2.097730	0.344839	10.594800	15edo	1233.34684913419	2	2

Theory

45zpi is characterized by a very broad octave error, yet it maintains a quite decent zeta strength. This combination makes it an ideal candidate for no-octave tuning applications.

No other zeta peak indexes exhibit both a larger octave error and greater zeta height than 45zpi.

45zpi supports a complex chord structure with ratios of 1:3:4:5:7:9:13:15:18:19:20:21:22:23:24:25, which further exemplifies its capabilities.

The closest zeta peak indexes to 45zpi that exceed its strength are 42zpi and 47zpi, though 43zpi is nearly as strong as 45zpi.

Harmonic series

Approximation of harmonics in 45zpi

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	Absolute (¢)	+33.3	-10.8	-15.5	+9.3	+22.5	+2.3	+17.8	-21.6	-39.6	-40.2	-26.4	-0.5	+35.7	-1.6	-31.1
	Relative (%)	+40.6	-13.2	-18.9	+11.3	+27.4	+2.8	+21.7	-26.3	-48.2	-48.8	-32.1	-0.6	+43.4	-1.9	-37.8
Step		15	23	29	34	38	41	44	46	48	50	52	54	56	57	58

Approximation of harmonics in 45zpi

Harmonic		17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
Error	Absolute (¢)	+28.4	+11.7	+0.3	-6.3	-8.5	-6.8	-1.5	+7.0	+18.5	+32.9	-32.5	-13.2	+8.3	+31.8	-25.0	+2.3
	Relative (%)	+34.6	+14.2	+0.4	-7.6	-10.3	-8.3	-1.9	+8.5	+22.6	+40.0	-39.5	-16.1	+10.1	+38.7	-30.4	+2.8
Step		60	61	62	63	64	65	66	67	68	69	69	70	71	72	72	73

Intervals

Intervals in 45zpi

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 %			⟨73 116] at every 5 steps Whole tone = 13 steps Limma = 4 steps Apotome = 9 steps
Degree	Cents	Ratios	Ups and Downs Notation	Step
0	0.000		P1	0
1	82.223	32/31, 31/30, 30/29, 29/28, 28/27, 27/26, 26/25, 25/24, 24/23, 23/22, 22/21, 21/20, 20/19, 19/18, 18/17, 17/16, 16/15, 31/29, 15/14	^m2	5
2	164.446	29/27, 14/13, 27/25, 13/12, 25/23, 12/11, 23/21, 11/10, 32/29, 21/19, 31/28, 10/9, 29/26, 19/17, 28/25, 9/8	vvvM2	10
3	246.669	26/23, 17/15, 25/22, 8/7, 31/27, 23/20, 15/13, 22/19, 29/25, 7/6, 27/23, 20/17	^^M2, vvm3	15
4	328.892	13/11, 32/27, 19/16, 25/21, 31/26, 6/5, 29/24, 23/19, 17/14, 28/23, 11/9, 27/22, 16/13, 21/17, 26/21	^^^m3	20
5	411.116	31/25, 5/4, 29/23, 24/19, 19/15, 14/11, 23/18, 32/25, 9/7, 31/24, 22/17	vM3	25
6	493.339	13/10, 30/23, 17/13, 21/16, 25/19, 29/22, 4/3, 31/23, 27/20, 23/17, 19/14	P4	30
7	575.562	15/11, 26/19, 11/8, 29/21, 18/13, 25/18, 32/23, 7/5, 31/22, 24/17, 17/12, 27/19	v4A4	35
8	657.785	10/7, 23/16, 13/9, 29/20, 16/11, 19/13, 22/15, 25/17, 28/19, 31/21	vvv5	40
9	740.008	3/2, 32/21, 29/19, 26/17, 23/15, 20/13, 17/11, 31/20, 14/9, 25/16	^^5, vvm6	45
10	822.231	11/7, 30/19, 19/12, 27/17, 8/5, 29/18, 21/13, 13/8, 31/19, 18/11, 23/14	^^^m6	50
11	904.454	28/17, 5/3, 32/19, 27/16, 22/13, 17/10, 29/17, 12/7, 31/18	vM6	55
12	986.677	19/11, 26/15, 7/4, 30/17, 23/13, 16/9, 25/14, 9/5	m7	60
13	1068.901	29/16, 20/11, 31/17, 11/6, 24/13, 13/7, 28/15, 15/8, 32/17, 17/9	v4M7	65
14	1151.124	19/10, 21/11, 23/12, 25/13, 27/14, 29/15, 31/16	^M7	70
15	1233.347	2/1, 31/15, 29/14, 27/13, 25/12	^^1 +1 oct, vvm2 +1 oct	75
16	1315.570	23/11, 21/10, 19/9, 17/8, 32/15, 15/7, 28/13, 13/6, 24/11	^^^m2 +1 oct	80
17	1397.793	11/5, 31/14, 20/9, 29/13, 9/4, 25/11, 16/7	vM2 +1 oct	85
18	1480.016	23/10, 30/13, 7/3, 26/11, 19/8, 31/13, 12/5	m3 +1 oct	90
19	1562.239	29/12, 17/7, 22/9, 27/11, 32/13, 5/2	v4M3 +1 oct	95
20	1644.462	28/11, 23/9, 18/7, 31/12, 13/5, 21/8, 29/11	^M3 +1 oct	100
21	1726.686	8/3, 27/10, 19/7, 30/11, 11/4	^^4 +1 oct	105
22	1808.909	25/9, 14/5, 31/11, 17/6, 20/7, 23/8, 26/9, 29/10, 32/11	^^^d5 +1 oct	110
23	1891.132	3/1	v5 +1 oct	115
24	1973.355	31/10, 28/9, 25/8, 22/7, 19/6, 16/5	m6 +1 oct	120
25	2055.578	29/9, 13/4, 23/7, 10/3	v4M6 +1 oct	125
26	2137.801	27/8, 17/5, 24/7, 31/9, 7/2	^M6 +1 oct	130
27	2220.024	32/9, 25/7, 18/5, 29/8, 11/3	^^m7 +1 oct	135
28	2302.247	26/7, 15/4, 19/5, 23/6, 27/7	vvM7 +1 oct	140
29	2384.471	31/8, 4/1	v1 +2 oct	145
30	2466.694	29/7, 25/6, 21/5, 17/4	m2 +2 oct	150
31	2548.917	30/7, 13/3, 22/5, 31/7	v4M2 +2 oct	155
32	2631.140	9/2, 32/7, 23/5, 14/3	^M2 +2 oct	160
33	2713.363	19/4, 24/5, 29/6	^^m3 +2 oct	165
34	2795.586	5/1	vvM3 +2 oct	170
35	2877.809	31/6, 26/5, 21/4, 16/3	v4 +2 oct	175
36	2960.032	27/5, 11/2, 28/5	^44 +2 oct	180
37	3042.256	17/3, 23/4, 29/5	v45 +2 oct	185
38	3124.479	6/1, 31/5	^5 +2 oct	190
39	3206.702	25/4, 19/3, 32/5, 13/2	^^m6 +2 oct	195
40	3288.925	20/3, 27/4	vvM6 +2 oct	200
41	3371.148	7/1	vm7 +2 oct	205
42	3453.371	29/4, 22/3, 15/2	^4m7 +2 oct	210
43	3535.594	23/3, 31/4	M7 +2 oct	215
44	3617.817	8/1	^1 +3 oct	220
45	3700.041	25/3, 17/2, 26/3	^^m2 +3 oct	225
46	3782.264	9/1	vvM2 +3 oct	230
47	3864.487	28/3, 19/2	vm3 +3 oct	235
48	3946.710	29/3, 10/1	^4m3 +3 oct	240
49	4028.933	31/3	M3 +3 oct	245
50	4111.156	21/2, 32/3, 11/1	^4 +3 oct	250
51	4193.379	23/2	vvvA4 +3 oct	255
52	4275.602	12/1	vv5 +3 oct	260
53	4357.826	25/2	vm6 +3 oct	265
54	4440.049	13/1	^4m6 +3 oct	270
55	4522.272	27/2	M6 +3 oct	275
56	4604.495	14/1, 29/2	^m7 +3 oct	280
57	4686.718	15/1	vvvM7 +3 oct	285
58	4768.941	31/2, 16/1	^^M7 +3 oct, vv1 +4 oct	290
59	4851.164		vm2 +4 oct	295
60	4933.387	17/1	^4m2 +4 oct	300
61	5015.611	18/1	M2 +4 oct	305
62	5097.834	19/1	^m3 +4 oct	310
63	5180.057	20/1	vvvM3 +4 oct	315
64	5262.280	21/1	^^M3 +4 oct, vv4 +4 oct	320
65	5344.503	22/1	^^^4 +4 oct	325
66	5426.726	23/1	^4d5 +4 oct	330
67	5508.949	24/1	P5 +4 oct	335
68	5591.172	25/1	^m6 +4 oct	340
69	5673.396	26/1, 27/1	vvvM6 +4 oct	345
70	5755.619	28/1	^^M6 +4 oct, vvm7 +4 oct	350
71	5837.842	29/1	^^^m7 +4 oct	355
72	5920.065	30/1, 31/1	vM7 +4 oct	360
73	6002.288	32/1	P1 +5 oct	365

Approximation to JI

Interval mappings

The following tables show how 32-integer-limit intervals are represented in 45zpi. Prime harmonics are in bold; inconsistent intervals are in italics.

32-integer-limit intervals in 45zpi (by direct approximation)

Ratio	Error (abs, ¢)	Error (rel, %)
23/15	+0.002	+0.003
32/7	-0.034	-0.042
19/1	+0.321	+0.390
13/1	-0.479	-0.583
11/10	-0.558	-0.679
25/8	+0.728	+0.885
19/13	+0.800	+0.973
29/5	-1.008	-1.226
23/13	-1.069	-1.300
15/13	-1.072	-1.303
29/24	+1.270	+1.545
31/12	+1.382	+1.681
27/16	-1.411	-1.716
23/1	-1.548	-1.883
15/1	-1.551	-1.886
22/21	+1.686	+2.051
23/19	-1.869	-2.273
19/15	+1.871	+2.276
32/19	+1.967	+2.393
19/7	-2.002	-2.434
21/20	-2.244	-2.729
24/5	-2.278	-2.771
32/1	+2.288	+2.783
7/1	+2.322	+2.824
28/3	-2.384	-2.900
18/5	+2.428	+2.953
32/13	+2.767	+3.365
13/7	-2.801	-3.407
31/9	-3.324	-4.043
30/17	+3.364	+4.092
29/18	-3.436	-4.179
32/23	+3.836	+4.666
32/15	+3.839	+4.669
23/7	-3.870	-4.707
15/7	-3.873	-4.710
25/6	-3.979	-4.839
22/3	+4.008	+4.875
26/17	+4.436	+5.395
20/3	+4.566	+5.553
24/7	+4.672	+5.682
4/3	-4.706	-5.724
23/20	+4.709	+5.727
17/2	-4.915	-5.977
22/15	-5.264	-6.402
23/22	+5.267	+6.405
20/13	-5.778	-7.027
17/6	+5.908	+7.186
29/7	+5.942	+7.227
32/29	-5.977	-7.269
31/16	+6.088	+7.404
9/4	-6.117	-7.439
20/1	-6.257	-7.610
22/13	-6.336	-7.706
14/11	-6.392	-7.774
20/19	-6.578	-8.000
24/19	+6.674	+8.116
22/1	-6.815	-8.288
25/18	+6.844	+8.324
7/5	-6.950	-8.453
23/21	+6.953	+8.456
32/5	-6.984	-8.495
24/1	+6.994	+8.506
21/4	+7.028	+8.548
27/10	+7.134	+8.677
22/19	-7.136	-8.678
17/14	-7.237	-8.802
24/13	+7.473	+9.089
31/27	+7.499	+9.120
27/11	+7.692	+9.355
29/19	+7.944	+9.661
21/13	-8.022	-9.756
29/1	+8.265	+10.051
28/9	+8.439	+10.264
21/1	-8.501	-10.339
24/23	+8.542	+10.389
8/5	+8.545	+10.392
20/7	-8.579	-10.434
11/2	+8.714	+10.599
29/13	+8.744	+10.634
21/19	-8.822	-10.729
19/5	-8.952	-10.887
16/11	+9.103	+11.071
22/7	-9.137	-11.113
26/11	-9.194	-11.181
5/1	+9.272	+11.277
23/3	+9.275	+11.280
18/7	+9.378	+11.406
16/9	-9.413	-11.448
31/4	-9.441	-11.483
29/8	-9.553	-11.618
13/5	-9.751	-11.860
29/23	+9.813	+11.934
29/15	+9.815	+11.937
25/17	-9.887	-12.025
30/11	-10.265	-12.485
29/25	-10.280	-12.503
13/3	+10.344	+12.581
17/8	+10.615	+12.909
32/21	+10.789	+13.122
23/5	-10.821	-13.160
3/1	-10.823	-13.163
19/3	+11.144	+13.553
19/18	-11.380	-13.840
25/24	+11.551	+14.048
28/15	-11.657	-14.177
28/23	-11.659	-14.180
18/1	+11.701	+14.230
31/28	-11.763	-14.307
18/13	+12.180	+14.813
28/13	-12.728	-15.480
32/3	+13.111	+15.946
7/3	+13.145	+15.987
28/1	-13.207	-16.063
23/18	-13.249	-16.113
6/5	+13.251	+16.116
28/19	-13.528	-16.453
17/11	-13.629	-16.576
12/11	+13.809	+16.795
15/4	+13.979	+17.001
23/4	+13.981	+17.004
27/14	+14.085	+17.130
31/3	-14.148	-17.206
17/10	-14.187	-17.255
29/6	-14.259	-17.342
26/25	+14.323	+17.420
29/20	+14.522	+17.661
31/10	+14.633	+17.797
25/2	-14.802	-18.002
22/9	+14.831	+18.038
13/4	+15.050	+18.304
29/22	+15.079	+18.340
31/11	+15.191	+18.475
20/9	+15.389	+18.717
8/7	+15.495	+18.845
4/1	-15.529	-18.887
19/4	+15.850	+19.277
22/5	-16.087	-19.566
25/7	+16.223	+19.730
32/25	-16.257	-19.772
27/2	+16.407	+19.954
31/21	-16.470	-20.030
18/17	-16.731	-20.349
29/21	+16.766	+20.390
27/26	+16.886	+20.537
27/4	-16.940	-20.603
25/14	-17.124	-20.826
19/8	-17.497	-21.280
21/5	-17.773	-21.616
8/1	+17.817	+21.670
7/4	+17.852	+21.711
10/9	-17.957	-21.840
31/22	-18.156	-22.081
25/19	+18.224	+22.164
13/8	-18.296	-22.252
11/9	-18.515	-22.519
25/1	+18.545	+22.554
31/20	-18.714	-22.760
25/13	+19.024	+23.137
29/3	+19.088	+23.215
17/5	+19.160	+23.302
28/27	+19.262	+23.427
23/8	-19.366	-23.553
15/8	-19.368	-23.556
11/6	+19.538	+23.762
25/23	+20.093	+24.437
5/3	+20.096	+24.440
23/9	+20.098	+24.443
29/17	-20.167	-24.528
7/6	-20.202	-24.569
16/3	-20.236	-24.611
13/9	+21.167	+25.744
27/17	+21.322	+25.931
24/17	-21.438	-26.073
29/28	+21.472	+26.114
31/14	+21.583	+26.250
9/1	-21.646	-26.326
19/9	+21.967	+26.716
19/6	-22.203	-27.003
28/5	-22.480	-27.340
6/1	+22.524	+27.393
21/16	+22.558	+27.435
17/16	-22.732	-27.647
13/6	-23.003	-27.976
31/15	-23.420	-28.483
31/23	-23.422	-28.486
25/11	-23.516	-28.601
30/29	+23.532	+28.619
26/5	+23.595	+28.697
29/4	+23.794	+28.938
31/2	+23.906	+29.074
32/9	+23.934	+29.109
9/7	-23.968	-29.151
23/6	-24.072	-29.276
5/2	-24.074	-29.279
11/8	+24.244	+29.486
31/26	+24.385	+29.657
31/13	-24.492	-29.787
29/26	-24.603	-29.923
11/4	-24.632	-29.958
5/4	+24.802	+30.164
23/12	+24.804	+30.167
14/9	-24.908	-30.293
31/1	-24.971	-30.369
29/2	-25.082	-30.505
31/19	-25.291	-30.759
25/22	+25.360	+30.843
31/30	+25.456	+30.960
27/22	-25.655	-31.201
13/12	+25.874	+31.468
17/7	+26.110	+31.755
32/17	-26.144	-31.796
27/20	-26.213	-31.880
21/8	-26.318	-32.009
12/1	-26.353	-32.050
14/5	+26.397	+32.104
19/12	+26.673	+32.440
28/11	+26.955	+32.782
25/21	+27.046	+32.893
9/2	+27.230	+33.117
32/31	+27.259	+33.152
31/7	-27.293	-33.194
29/14	-27.404	-33.329
17/12	-27.439	-33.371
26/9	-27.709	-33.700
19/17	-28.111	-34.189
17/1	+28.432	+34.579
8/3	+28.641	+34.833
12/7	-28.675	-34.874
10/3	-28.781	-35.003
31/17	+28.820	+35.052
17/13	+28.911	+35.162
11/3	-29.339	-35.682
25/3	+29.368	+35.718
30/7	+29.474	+35.846
16/15	-29.508	-35.888
23/16	+29.511	+35.891
29/9	+29.911	+36.378
23/17	-29.980	-36.462
17/15	+29.983	+36.465
18/11	-30.361	-36.925
26/7	+30.546	+37.150
16/13	-30.580	-37.191
9/5	-30.919	-37.604
27/23	-30.921	-37.607
7/2	-31.025	-37.732
16/1	-31.059	-37.774
21/10	+31.103	+37.827
27/25	+31.209	+37.956
19/16	+31.379	+38.164
30/19	+31.476	+38.281
21/11	+31.661	+38.506
28/25	-31.752	-38.617
30/1	+31.796	+38.671
27/8	+31.936	+38.841
31/24	-31.965	-38.876
27/13	-31.990	-38.907
17/9	-32.145	-39.095
30/13	+32.275	+39.253
27/1	-32.469	-39.489
26/19	+32.547	+39.584
25/16	-32.619	-39.672
11/5	+32.789	+39.878
27/19	-32.790	-39.879
26/1	+32.868	+39.974
19/2	-33.026	-40.167
31/29	-33.235	-40.421
30/23	+33.344	+40.554
2/1	+33.347	+40.557
16/7	-33.381	-40.598
29/11	-33.797	-41.104
13/2	-33.826	-41.139
20/11	+33.905	+41.235
25/4	+34.074	+41.441
31/5	-34.243	-41.647
29/10	-34.355	-41.782
26/23	+34.416	+41.857
26/15	+34.419	+41.860
29/12	+34.617	+42.101
20/17	-34.689	-42.189
31/6	+34.729	+42.237
32/27	+34.757	+42.272
27/7	-34.792	-42.314
23/2	-34.895	-42.439
15/2	-34.898	-42.442
24/11	-35.067	-42.649
22/17	-35.247	-42.867
19/14	-35.348	-42.991
12/5	-35.625	-43.327
14/1	+35.669	+43.381
14/3	-35.731	-43.456
14/13	+36.148	+43.963
31/18	-36.671	-44.600
21/17	-36.933	-44.918
23/14	-37.217	-45.264
15/14	-37.220	-45.267
25/12	-37.326	-45.395
3/2	+38.053	+46.280
23/10	+38.056	+46.283
17/4	-38.262	-46.534
26/3	-38.532	-46.863
15/11	+38.611	+46.959
23/11	+38.614	+46.962
31/25	+38.707	+47.076
13/10	+39.125	+47.584
17/3	+39.255	+47.742
29/16	+39.323	+47.825
31/8	+39.435	+47.961
9/8	-39.464	-47.996
10/1	-39.604	-48.166
13/11	+39.683	+48.262
11/7	+39.739	+48.331
32/11	-39.773	-48.372
19/10	+39.924	+48.556
11/1	-40.162	-48.845
25/9	+40.191	+48.881
10/7	+40.297	+49.010
16/5	-40.331	-49.051
21/2	+40.375	+49.105
27/5	+40.481	+49.233
19/11	+40.482	+49.235
28/17	+40.584	+49.358
29/27	+40.734	+49.541
26/21	-40.854	-49.687

32-integer-limit intervals in 45zpi (by patent val mapping)

Ratio	Error (abs, ¢)	Error (rel, %)
23/15	+0.002	+0.003
19/1	+0.321	+0.390
13/1	-0.479	-0.583
19/13	+0.800	+0.973
29/5	-1.008	-1.226
23/13	-1.069	-1.300
15/13	-1.072	-1.303
23/1	-1.548	-1.883
15/1	-1.551	-1.886
22/21	+1.686	+2.051
23/19	-1.869	-2.273
19/15	+1.871	+2.276
19/7	-2.002	-2.434
7/1	+2.322	+2.824
18/5	+2.428	+2.953
13/7	-2.801	-3.407
31/9	-3.324	-4.043
30/17	+3.364	+4.092
29/18	-3.436	-4.179
23/7	-3.870	-4.707
15/7	-3.873	-4.710
25/6	-3.979	-4.839
22/3	+4.008	+4.875
26/17	+4.436	+5.395
17/2	-4.915	-5.977
22/15	-5.264	-6.402
23/22	+5.267	+6.405
17/6	+5.908	+7.186
29/7	+5.942	+7.227
22/13	-6.336	-7.706
22/1	-6.815	-8.288
25/18	+6.844	+8.324
7/5	-6.950	-8.453
23/21	+6.953	+8.456
22/19	-7.136	-8.678
17/14	-7.237	-8.802
31/27	+7.499	+9.120
27/11	+7.692	+9.355
29/19	+7.944	+9.661
21/13	-8.022	-9.756
29/1	+8.265	+10.051
21/1	-8.501	-10.339
29/13	+8.744	+10.634
21/19	-8.822	-10.729
19/5	-8.952	-10.887
22/7	-9.137	-11.113
5/1	+9.272	+11.277
23/3	+9.275	+11.280
18/7	+9.378	+11.406
13/5	-9.751	-11.860
29/23	+9.813	+11.934
29/15	+9.815	+11.937
25/17	-9.887	-12.025
29/25	-10.280	-12.503
13/3	+10.344	+12.581
23/5	-10.821	-13.160
3/1	-10.823	-13.163
19/3	+11.144	+13.553
19/18	-11.380	-13.840
18/1	+11.701	+14.230
18/13	+12.180	+14.813
7/3	+13.145	+15.987
23/18	-13.249	-16.113
6/5	+13.251	+16.116
31/3	-14.148	-17.206
17/10	-14.187	-17.255
29/6	-14.259	-17.342
26/25	+14.323	+17.420
25/2	-14.802	-18.002
22/9	+14.831	+18.038
29/22	+15.079	+18.340
31/11	+15.191	+18.475
22/5	-16.087	-19.566
25/7	+16.223	+19.730
31/21	-16.470	-20.030
18/17	-16.731	-20.349
29/21	+16.766	+20.390
25/14	-17.124	-20.826
21/5	-17.773	-21.616
31/22	-18.156	-22.081
25/19	+18.224	+22.164
11/9	-18.515	-22.519
25/1	+18.545	+22.554
25/13	+19.024	+23.137
29/3	+19.088	+23.215
17/5	+19.160	+23.302
25/23	+20.093	+24.437
5/3	+20.096	+24.440
23/9	+20.098	+24.443
29/17	-20.167	-24.528
7/6	-20.202	-24.569
13/9	+21.167	+25.744
9/1	-21.646	-26.326
19/9	+21.967	+26.716
19/6	-22.203	-27.003
6/1	+22.524	+27.393
13/6	-23.003	-27.976
31/15	-23.420	-28.483
31/23	-23.422	-28.486
30/29	+23.532	+28.619
26/5	+23.595	+28.697
9/7	-23.968	-29.151
23/6	-24.072	-29.276
5/2	-24.074	-29.279
31/13	-24.492	-29.787
29/26	-24.603	-29.923
31/1	-24.971	-30.369
29/2	-25.082	-30.505
31/19	-25.291	-30.759
25/22	+25.360	+30.843
27/22	-25.655	-31.201
17/7	+26.110	+31.755
14/5	+26.397	+32.104
25/21	+27.046	+32.893
31/7	-27.293	-33.194
29/14	-27.404	-33.329
17/12	-27.439	-33.371
19/17	-28.111	-34.189
17/1	+28.432	+34.579
17/13	+28.911	+35.162
11/3	-29.339	-35.682
25/3	+29.368	+35.718
30/7	+29.474	+35.846
29/9	+29.911	+36.378
23/17	-29.980	-36.462
17/15	+29.983	+36.465
26/7	+30.546	+37.150
9/5	-30.919	-37.604
27/23	-30.921	-37.607
7/2	-31.025	-37.732
30/19	+31.476	+38.281
21/11	+31.661	+38.506
30/1	+31.796	+38.671
27/13	-31.990	-38.907
30/13	+32.275	+39.253
27/1	-32.469	-39.489
26/19	+32.547	+39.584
27/19	-32.790	-39.879
26/1	+32.868	+39.974
19/2	-33.026	-40.167
31/29	-33.235	-40.421
30/23	+33.344	+40.554
2/1	+33.347	+40.557
13/2	-33.826	-41.139
31/5	-34.243	-41.647
29/10	-34.355	-41.782
26/23	+34.416	+41.857
26/15	+34.419	+41.860
27/7	-34.792	-42.314
23/2	-34.895	-42.439
15/2	-34.898	-42.442
22/17	-35.247	-42.867
19/14	-35.348	-42.991
14/1	+35.669	+43.381
14/13	+36.148	+43.963
31/18	-36.671	-44.600
21/17	-36.933	-44.918
23/14	-37.217	-45.264
15/14	-37.220	-45.267
25/12	-37.326	-45.395
17/4	-38.262	-46.534
15/11	+38.611	+46.959
23/11	+38.614	+46.962
17/3	+39.255	+47.742
13/11	+39.683	+48.262
11/1	-40.162	-48.845
25/9	+40.191	+48.881
10/7	+40.297	+49.010
19/11	+40.482	+49.235
28/17	+40.584	+49.358
29/27	+40.734	+49.541
26/21	+41.369	+50.313
27/5	-41.742	-50.767
21/2	-41.848	-50.895
19/10	-42.299	-51.444
11/7	-42.484	-51.669
10/1	+42.619	+51.834
13/10	-43.098	-52.416
31/25	-43.516	-52.924
26/3	+43.691	+53.137
23/10	-44.168	-53.717
3/2	-44.170	-53.720
14/3	+46.492	+56.544
12/5	+46.598	+56.673
31/6	-47.494	-57.763
20/17	+47.534	+57.811
29/12	-47.606	-57.899
25/4	-48.149	-58.559
29/11	+48.426	+58.896
11/5	-49.434	-60.122
17/9	+50.078	+60.905
28/25	+50.471	+61.383
27/25	-51.014	-62.044
21/10	-51.120	-62.173
18/11	+51.862	+63.075
31/17	-53.403	-64.948
10/3	+53.442	+64.997
12/7	+53.548	+65.126
26/9	+54.514	+66.300
9/2	-54.993	-66.883
19/12	-55.550	-67.560
12/1	+55.871	+67.950
13/12	-56.350	-68.532
31/30	-56.767	-69.040
14/9	+57.315	+69.707
23/12	-57.419	-69.833
5/4	-57.421	-69.836
31/26	-57.839	-70.343
31/2	-58.318	-70.926
29/4	-58.429	-71.062
25/11	+58.707	+71.399
28/5	+59.743	+72.660
31/14	-60.640	-73.750
29/28	-60.751	-73.886
24/17	+60.785	+73.927
27/17	-60.901	-74.069
11/6	-62.685	-76.238
10/9	+64.266	+78.160
7/4	-64.372	-78.289
27/26	-65.337	-79.463
27/2	-65.816	-80.046
19/4	-66.373	-80.723
4/1	+66.694	+81.113
13/4	-67.173	-81.696
31/10	-67.590	-82.203
29/20	-67.702	-82.339
27/14	-68.138	-82.870
23/4	-68.242	-82.996
15/4	-68.244	-82.999
17/11	+68.594	+83.424
28/19	+68.695	+83.547
28/1	+69.016	+83.937
28/13	+69.495	+84.520
28/23	+70.564	+85.820
28/15	+70.567	+85.823
25/24	-70.672	-85.952
17/8	-71.609	-87.091
30/11	+71.958	+87.515
26/11	+73.030	+88.819
11/2	-73.509	-89.401
20/7	+73.644	+89.566
27/10	-75.089	-91.323
21/4	-75.195	-91.452
20/19	+75.646	+92.000
14/11	+75.831	+92.226
20/1	+75.966	+92.390
20/13	+76.445	+92.973
23/20	-77.514	-94.273
4/3	+77.517	+94.276
28/3	+79.839	+97.100
24/5	+79.945	+97.229
31/12	-80.841	-98.319
29/24	-80.953	-98.455
25/8	-81.496	-99.115
11/10	-82.781	-100.679
21/20	-84.467	-102.729
20/3	+86.789	+105.553
24/7	+86.895	+105.682
9/4	-88.340	-107.439
24/19	+88.897	+108.116
24/1	+89.217	+108.506
24/13	+89.696	+109.089
28/9	+90.662	+110.264
24/23	+90.766	+110.389
8/5	+90.768	+110.392
31/4	-91.664	-111.483
29/8	-91.776	-111.618
31/28	-93.987	-114.307
12/11	+96.032	+116.795
20/9	+97.613	+118.717
8/7	+97.718	+118.845
27/4	-99.163	-120.603
19/8	-99.720	-121.280
8/1	+100.041	+121.670
13/8	-100.520	-122.252
31/20	-100.937	-122.760
28/27	+101.485	+123.427
23/8	-101.589	-123.553
15/8	-101.591	-123.556
17/16	-104.955	-127.647
11/4	-106.855	-129.958
27/20	-108.436	-131.880
21/8	-108.542	-132.009
28/11	+109.178	+132.782
8/3	+110.864	+134.833
31/24	-114.188	-138.876
25/16	-114.842	-139.672
20/11	+116.128	+141.235
9/8	-121.687	-147.996
16/5	+124.115	+150.949
31/8	-125.011	-152.039
29/16	-125.123	-152.175
24/11	+129.379	+157.351
16/7	+131.065	+159.402
27/8	-132.510	-161.159
19/16	-133.067	-161.836
16/1	+133.387	+162.226
16/13	+133.866	+162.809
23/16	-134.936	-164.109
16/15	+134.938	+164.112
32/17	+138.302	+168.204
11/8	-140.202	-170.514
21/16	-141.888	-172.565
16/3	+144.211	+175.389
32/25	+148.189	+180.228
16/9	+155.034	+188.552
32/5	+157.462	+191.505
31/16	-158.358	-192.596
32/29	+158.470	+192.731
32/7	+164.412	+199.958
27/16	-165.857	-201.716
32/19	+166.414	+202.393
32/1	+166.734	+202.783
32/13	+167.213	+203.365
32/23	+168.282	+204.666
32/15	+168.285	+204.669
16/11	+173.549	+211.071
32/21	+175.235	+213.122
32/3	+177.557	+215.946
32/9	+188.381	+229.109
32/31	+191.705	+233.152
32/27	+199.204	+242.272
32/11	+206.896	+251.628

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