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 8ve	Step size (cents)	Height	Integral	Gap	Edo	Octave (cents)		Consistent	Distinct
							Size	Stretch
45zpi	14.594435	82.223123	2.09773	0.344839	10.5948	15edo	1233.346849	33.346849	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 25-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	25/24, 24/23, 23/22, 22/21, 21/20, 20/19, 19/18, 18/17, 17/16, 16/15, 15/14	^m2	5
2	164.446	14/13, 13/12, 25/23, 12/11, 23/21, 11/10, 21/19, 10/9, 19/17, 9/8	vvvM2	10
3	246.669	17/15, 25/22, 8/7, 23/20, 15/13, 22/19, 7/6, 20/17	^^M2, vvm3	15
4	328.892	13/11, 19/16, 25/21, 6/5, 23/19, 17/14, 11/9, 16/13, 21/17	^^^m3	20
5	411.116	5/4, 24/19, 19/15, 14/11, 23/18, 9/7, 22/17	vM3	25
6	493.339	13/10, 17/13, 21/16, 25/19, 4/3, 23/17, 19/14	P4	30
7	575.562	15/11, 11/8, 18/13, 25/18, 7/5, 24/17, 17/12	v4A4	35
8	657.785	10/7, 23/16, 13/9, 16/11, 19/13, 22/15, 25/17	vvv5	40
9	740.008	3/2, 23/15, 20/13, 17/11, 14/9, 25/16	^^5, vvm6	45
10	822.231	11/7, 19/12, 8/5, 21/13, 13/8, 18/11, 23/14	^^^m6	50
11	904.454	5/3, 22/13, 17/10, 12/7	vM6	55
12	986.677	19/11, 7/4, 23/13, 16/9, 25/14, 9/5	m7	60
13	1068.901	20/11, 11/6, 24/13, 13/7, 15/8, 17/9	v4M7	65
14	1151.124	19/10, 21/11, 23/12, 25/13	^M7	70
15	1233.347	2/1, 25/12	^^1 +1 oct, vvm2 +1 oct	75
16	1315.570	23/11, 21/10, 19/9, 17/8, 15/7, 13/6, 24/11	^^^m2 +1 oct	80
17	1397.793	11/5, 20/9, 9/4, 25/11, 16/7	vM2 +1 oct	85
18	1480.016	23/10, 7/3, 19/8, 12/5	m3 +1 oct	90
19	1562.239	17/7, 22/9, 5/2	v4M3 +1 oct	95
20	1644.462	23/9, 18/7, 13/5, 21/8	^M3 +1 oct	100
21	1726.686	8/3, 19/7, 11/4	^^4 +1 oct	105
22	1808.909	25/9, 14/5, 17/6, 20/7, 23/8	^^^d5 +1 oct	110
23	1891.132	3/1	v5 +1 oct	115
24	1973.355	25/8, 22/7, 19/6, 16/5	m6 +1 oct	120
25	2055.578	13/4, 23/7, 10/3	v4M6 +1 oct	125
26	2137.801	17/5, 24/7, 7/2	^M6 +1 oct	130
27	2220.024	25/7, 18/5, 11/3	^^m7 +1 oct	135
28	2302.247	15/4, 19/5, 23/6	vvM7 +1 oct	140
29	2384.471	4/1	v1 +2 oct	145
30	2466.694	25/6, 21/5, 17/4	m2 +2 oct	150
31	2548.917	13/3, 22/5	v4M2 +2 oct	155
32	2631.140	9/2, 23/5, 14/3	^M2 +2 oct	160
33	2713.363	19/4, 24/5	^^m3 +2 oct	165
34	2795.586	5/1	vvM3 +2 oct	170
35	2877.809	21/4, 16/3	v4 +2 oct	175
36	2960.032	11/2	^44 +2 oct	180
37	3042.256	17/3, 23/4	v45 +2 oct	185
38	3124.479	6/1	^5 +2 oct	190
39	3206.702	25/4, 19/3, 13/2	^^m6 +2 oct	195
40	3288.925	20/3	vvM6 +2 oct	200
41	3371.148	7/1	vm7 +2 oct	205
42	3453.371	22/3, 15/2	^4m7 +2 oct	210
43	3535.594	23/3	M7 +2 oct	215
44	3617.817	8/1	^1 +3 oct	220
45	3700.041	25/3, 17/2	^^m2 +3 oct	225
46	3782.264	9/1	vvM2 +3 oct	230
47	3864.487	19/2	vm3 +3 oct	235
48	3946.710	10/1	^4m3 +3 oct	240
49	4028.933		M3 +3 oct	245
50	4111.156	21/2, 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		M6 +3 oct	275
56	4604.495	14/1	^m7 +3 oct	280
57	4686.718	15/1	vvvM7 +3 oct	285
58	4768.941	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

Approximation to JI

Interval mappings

The following tables show how 25-integer-limit intervals are represented in 45zpi. Prime harmonics are in bold; inconsistent intervals are in italics.

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

Ratio	Error (abs, ¢)	Error (rel, %)
23/15	+0.002	+0.003
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
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
21/20	-2.244	-2.729
24/5	-2.278	-2.771
7/1	+2.322	+2.824
18/5	+2.428	+2.953
13/7	-2.801	-3.407
23/7	-3.870	-4.707
15/7	-3.873	-4.710
25/6	-3.979	-4.839
22/3	+4.008	+4.875
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
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
24/1	+6.994	+8.506
21/4	+7.028	+8.548
22/19	-7.136	-8.678
17/14	-7.237	-8.802
24/13	+7.473	+9.089
21/13	-8.022	-9.756
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
21/19	-8.822	-10.729
19/5	-8.952	-10.887
16/11	+9.103	+11.071
22/7	-9.137	-11.113
5/1	+9.272	+11.277
23/3	+9.275	+11.280
18/7	+9.378	+11.406
16/9	-9.413	-11.448
13/5	-9.751	-11.860
25/17	-9.887	-12.025
13/3	+10.344	+12.581
17/8	+10.615	+12.909
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
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
17/11	-13.629	-16.576
12/11	+13.809	+16.795
15/4	+13.979	+17.001
23/4	+13.981	+17.004
17/10	-14.187	-17.255
25/2	-14.802	-18.002
22/9	+14.831	+18.038
13/4	+15.050	+18.304
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
18/17	-16.731	-20.349
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
25/19	+18.224	+22.164
13/8	-18.296	-22.252
11/9	-18.515	-22.519
25/1	+18.545	+22.554
25/13	+19.024	+23.137
17/5	+19.160	+23.302
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
7/6	-20.202	-24.569
16/3	-20.236	-24.611
13/9	+21.167	+25.744
24/17	-21.438	-26.073
9/1	-21.646	-26.326
19/9	+21.967	+26.716
19/6	-22.203	-27.003
6/1	+22.524	+27.393
21/16	+22.558	+27.435
17/16	-22.732	-27.647
13/6	-23.003	-27.976
25/11	-23.516	-28.601
9/7	-23.968	-29.151
23/6	-24.072	-29.276
5/2	-24.074	-29.279
11/8	+24.244	+29.486
11/4	-24.632	-29.958
5/4	+24.802	+30.164
23/12	+24.804	+30.167
14/9	-24.908	-30.293
25/22	+25.360	+30.843
13/12	+25.874	+31.468
17/7	+26.110	+31.755
21/8	-26.318	-32.009
12/1	-26.353	-32.050
14/5	+26.397	+32.104
19/12	+26.673	+32.440
25/21	+27.046	+32.893
9/2	+27.230	+33.117
17/12	-27.439	-33.371
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
17/13	+28.911	+35.162
11/3	-29.339	-35.682
25/3	+29.368	+35.718
16/15	-29.508	-35.888
23/16	+29.511	+35.891
23/17	-29.980	-36.462
17/15	+29.983	+36.465
18/11	-30.361	-36.925
16/13	-30.580	-37.191
9/5	-30.919	-37.604
7/2	-31.025	-37.732
16/1	-31.059	-37.774
21/10	+31.103	+37.827
19/16	+31.379	+38.164
21/11	+31.661	+38.506
17/9	-32.145	-39.095
25/16	-32.619	-39.672
11/5	+32.789	+39.878
19/2	-33.026	-40.167
2/1	+33.347	+40.557
16/7	-33.381	-40.598
13/2	-33.826	-41.139
20/11	+33.905	+41.235
25/4	+34.074	+41.441
20/17	-34.689	-42.189
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
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
15/11	+38.611	+46.959
23/11	+38.614	+46.962
13/10	+39.125	+47.584
17/3	+39.255	+47.742
9/8	-39.464	-47.996
10/1	-39.604	-48.166
13/11	+39.683	+48.262
11/7	+39.739	+48.331
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
19/11	+40.482	+49.235

25-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
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
23/7	-3.870	-4.707
15/7	-3.873	-4.710
25/6	-3.979	-4.839
22/3	+4.008	+4.875
17/2	-4.915	-5.977
22/15	-5.264	-6.402
23/22	+5.267	+6.405
17/6	+5.908	+7.186
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
21/13	-8.022	-9.756
21/1	-8.501	-10.339
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
25/17	-9.887	-12.025
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
17/10	-14.187	-17.255
25/2	-14.802	-18.002
22/9	+14.831	+18.038
22/5	-16.087	-19.566
25/7	+16.223	+19.730
18/17	-16.731	-20.349
25/14	-17.124	-20.826
21/5	-17.773	-21.616
25/19	+18.224	+22.164
11/9	-18.515	-22.519
25/1	+18.545	+22.554
25/13	+19.024	+23.137
17/5	+19.160	+23.302
25/23	+20.093	+24.437
5/3	+20.096	+24.440
23/9	+20.098	+24.443
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
9/7	-23.968	-29.151
23/6	-24.072	-29.276
5/2	-24.074	-29.279
25/22	+25.360	+30.843
17/7	+26.110	+31.755
14/5	+26.397	+32.104
25/21	+27.046	+32.893
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
23/17	-29.980	-36.462
17/15	+29.983	+36.465
9/5	-30.919	-37.604
7/2	-31.025	-37.732
21/11	+31.661	+38.506
19/2	-33.026	-40.167
2/1	+33.347	+40.557
13/2	-33.826	-41.139
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
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
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
23/10	-44.168	-53.717
3/2	-44.170	-53.720
14/3	+46.492	+56.544
12/5	+46.598	+56.673
20/17	+47.534	+57.811
25/4	-48.149	-58.559
11/5	-49.434	-60.122
17/9	+50.078	+60.905
21/10	-51.120	-62.173
18/11	+51.862	+63.075
10/3	+53.442	+64.997
12/7	+53.548	+65.126
9/2	-54.993	-66.883
19/12	-55.550	-67.560
12/1	+55.871	+67.950
13/12	-56.350	-68.532
14/9	+57.315	+69.707
23/12	-57.419	-69.833
5/4	-57.421	-69.836
25/11	+58.707	+71.399
24/17	+60.785	+73.927
11/6	-62.685	-76.238
10/9	+64.266	+78.160
7/4	-64.372	-78.289
19/4	-66.373	-80.723
4/1	+66.694	+81.113
13/4	-67.173	-81.696
23/4	-68.242	-82.996
15/4	-68.244	-82.999
17/11	+68.594	+83.424
25/24	-70.672	-85.952
17/8	-71.609	-87.091
11/2	-73.509	-89.401
20/7	+73.644	+89.566
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
24/5	+79.945	+97.229
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
24/23	+90.766	+110.389
8/5	+90.768	+110.392
12/11	+96.032	+116.795
20/9	+97.613	+118.717
8/7	+97.718	+118.845
19/8	-99.720	-121.280
8/1	+100.041	+121.670
13/8	-100.520	-122.252
23/8	-101.589	-123.553
15/8	-101.591	-123.556
17/16	-104.955	-127.647
11/4	-106.855	-129.958
21/8	-108.542	-132.009
8/3	+110.864	+134.833
25/16	-114.842	-139.672
20/11	+116.128	+141.235
9/8	-121.687	-147.996
16/5	+124.115	+150.949
24/11	+129.379	+157.351
16/7	+131.065	+159.402
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
11/8	-140.202	-170.514
21/16	-141.888	-172.565
16/3	+144.211	+175.389
16/9	+155.034	+188.552
16/11	+173.549	+211.071

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