34zpi

34 zeta peak index (abbreviated 34zpi), is the equal-step tuning system obtained from the 34th 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
34zpi	12.0231830072926	99.8071807833375	5.193290	1.269599	15.899282	12edo	1197.68616940005	10	6

Intervals

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 %			12edo Whole tone = 2 steps Limma = 1 step Apotome = 1 step
Degree	Cents	Ratios	Ups and Downs Notation
0	0.000		P1
1	99.807	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, 29/27, 14/13, 27/25, 13/12, 25/23	m2
2	199.614	12/11, 23/21, 11/10, 32/29, 21/19, 31/28, 10/9, 29/26, 19/17, 28/25, 9/8, 26/23, 17/15, 25/22, 8/7, 31/27, 23/20, 15/13	M2
3	299.422	22/19, 29/25, 7/6, 27/23, 20/17, 13/11, 32/27, 19/16, 25/21, 31/26, 6/5, 29/24, 23/19, 17/14, 28/23, 11/9	m3
4	399.229	27/22, 16/13, 21/17, 26/21, 31/25, 5/4, 29/23, 24/19, 19/15, 14/11, 23/18, 32/25, 9/7, 31/24, 22/17	M3
5	499.036	13/10, 30/23, 17/13, 21/16, 25/19, 29/22, 4/3, 31/23, 27/20, 23/17, 19/14, 15/11, 26/19	P4
6	598.843	11/8, 29/21, 18/13, 25/18, 32/23, 7/5, 31/22, 24/17, 17/12, 27/19, 10/7, 23/16, 13/9, 29/20, 16/11	A4, d5
7	698.650	19/13, 22/15, 25/17, 28/19, 31/21, 3/2, 32/21, 29/19, 26/17, 23/15, 20/13	P5
8	798.457	17/11, 31/20, 14/9, 25/16, 11/7, 30/19, 19/12, 27/17, 8/5, 29/18, 21/13, 13/8, 31/19	m6
9	898.265	18/11, 23/14, 28/17, 5/3, 32/19, 27/16, 22/13, 17/10, 29/17, 12/7, 31/18, 19/11	M6
10	998.072	26/15, 7/4, 30/17, 23/13, 16/9, 25/14, 9/5, 29/16, 20/11, 31/17	m7
11	1097.879	11/6, 24/13, 13/7, 28/15, 15/8, 32/17, 17/9, 19/10, 21/11, 23/12, 25/13, 27/14, 29/15, 31/16	M7
12	1197.686	2/1	P1 +1 oct
13	1297.493	31/15, 29/14, 27/13, 25/12, 23/11, 21/10, 19/9, 17/8, 32/15, 15/7, 28/13, 13/6	m2 +1 oct
14	1397.301	24/11, 11/5, 31/14, 20/9, 29/13, 9/4, 25/11, 16/7, 23/10	M2 +1 oct
15	1497.108	30/13, 7/3, 26/11, 19/8, 31/13, 12/5, 29/12, 17/7	m3 +1 oct
16	1596.915	22/9, 27/11, 32/13, 5/2, 28/11, 23/9, 18/7, 31/12	M3 +1 oct
17	1696.722	13/5, 21/8, 29/11, 8/3, 27/10, 19/7, 30/11	P4 +1 oct
18	1796.529	11/4, 25/9, 14/5, 31/11, 17/6, 20/7, 23/8, 26/9, 29/10	A4 +1 oct, d5 +1 oct
19	1896.336	32/11, 3/1	P5 +1 oct
20	1996.144	31/10, 28/9, 25/8, 22/7, 19/6, 16/5, 29/9, 13/4	m6 +1 oct
21	2095.951	23/7, 10/3, 27/8, 17/5, 24/7, 31/9	M6 +1 oct
22	2195.758	7/2, 32/9, 25/7, 18/5, 29/8	m7 +1 oct
23	2295.565	11/3, 26/7, 15/4, 19/5, 23/6, 27/7, 31/8	M7 +1 oct
24	2395.372	4/1	P1 +2 oct
25	2495.180	29/7, 25/6, 21/5, 17/4, 30/7, 13/3	m2 +2 oct
26	2594.987	22/5, 31/7, 9/2, 32/7, 23/5	M2 +2 oct
27	2694.794	14/3, 19/4, 24/5, 29/6	m3 +2 oct
28	2794.601	5/1, 31/6	M3 +2 oct
29	2894.408	26/5, 21/4, 16/3, 27/5	P4 +2 oct
30	2994.215	11/2, 28/5, 17/3, 23/4, 29/5	A4 +2 oct, d5 +2 oct
31	3094.023	6/1	P5 +2 oct
32	3193.830	31/5, 25/4, 19/3, 32/5, 13/2	m6 +2 oct
33	3293.637	20/3, 27/4	M6 +2 oct
34	3393.444	7/1, 29/4	m7 +2 oct
35	3493.251	22/3, 15/2, 23/3	M7 +2 oct
36	3593.059	31/4, 8/1	P1 +3 oct
37	3692.866	25/3, 17/2, 26/3	m2 +3 oct
38	3792.673	9/1	M2 +3 oct
39	3892.480	28/3, 19/2, 29/3	m3 +3 oct
40	3992.287	10/1	M3 +3 oct
41	4092.094	31/3, 21/2, 32/3	P4 +3 oct
42	4191.902	11/1, 23/2	A4 +3 oct, d5 +3 oct
43	4291.709	12/1	P5 +3 oct
44	4391.516	25/2, 13/1	m6 +3 oct
45	4491.323	27/2	M6 +3 oct
46	4591.130	14/1, 29/2	m7 +3 oct
47	4690.937	15/1	M7 +3 oct
48	4790.745	31/2, 16/1	P1 +4 oct
49	4890.552	17/1	m2 +4 oct
50	4990.359	18/1	M2 +4 oct
51	5090.166	19/1	m3 +4 oct
52	5189.973	20/1	M3 +4 oct
53	5289.781	21/1	P4 +4 oct
54	5389.588	22/1, 23/1	A4 +4 oct, d5 +4 oct
55	5489.395	24/1	P5 +4 oct
56	5589.202	25/1	m6 +4 oct
57	5689.009	26/1, 27/1	M6 +4 oct
58	5788.816	28/1, 29/1	m7 +4 oct
59	5888.624	30/1	M7 +4 oct
60	5988.431	31/1, 32/1	P1 +5 oct

Approximation to JI

Intervals by direct approximation (even if inconsistent)

Ratio	Error (abs, ¢)	Error (rel, %)
4/3	-0.991	-0.993
8/3	+1.323	+1.325
16/9	-1.982	-1.986
2/1	+2.314	+2.318
15/1	-2.669	-2.674
3/2	+3.305	+3.311
16/3	+3.637	+3.644
9/8	+4.296	+4.304
4/1	+4.628	+4.637
15/2	-4.983	-4.992
3/1	+5.619	+5.629
10/1	-5.974	-5.985
9/4	+6.609	+6.622
8/1	+6.941	+6.955
15/4	-7.296	-7.311
6/1	+7.932	+7.948
5/1	-8.287	-8.303
9/2	+8.923	+8.941
16/1	+9.255	+9.273
15/8	-9.610	-9.629
13/11	-10.212	-10.232
12/1	+10.246	+10.266
5/2	-10.601	-10.622
9/1	+11.237	+11.259
10/3	-11.592	-11.614
16/15	+11.924	+11.947
5/4	-12.915	-12.940
5/3	-13.906	-13.933
14/5	-14.017	-14.044
8/5	+15.229	+15.258
11/7	-15.965	-15.996
6/5	+16.220	+16.251
7/5	-16.331	-16.362
10/9	-17.211	-17.244
16/5	+17.543	+17.577
14/11	+18.279	+18.315
12/5	+18.534	+18.569
10/7	+18.645	+18.681
9/5	+19.524	+19.562
15/14	+19.636	+19.674
15/7	+21.949	+21.992
14/1	-22.304	-22.347
7/1	-24.618	-24.666
13/7	-26.177	-26.228
7/2	-26.932	-26.984
14/3	-27.923	-27.977
14/13	+28.491	+28.546
7/4	-29.246	-29.302
7/3	-30.237	-30.295
8/7	+31.560	+31.621
11/5	-32.296	-32.359
7/6	-32.551	-32.614
14/9	-33.542	-33.606
16/7	+33.874	+33.939
11/10	-34.610	-34.677
12/7	+34.864	+34.932
9/7	+35.855	+35.925
13/9	+37.775	+37.848
15/11	+37.915	+37.988
13/12	+38.765	+38.840
16/13	-39.756	-39.833
11/1	-40.584	-40.662
13/6	+41.079	+41.159
13/8	+42.070	+42.151
13/5	-42.508	-42.590
11/2	-42.897	-42.980
13/3	+43.393	+43.477
13/4	+44.384	+44.470
13/10	-44.822	-44.909
11/4	-45.211	-45.299
11/3	-46.202	-46.291
13/2	+46.698	+46.788
11/8	-47.525	-47.617
11/9	+47.986	+48.079
15/13	+48.127	+48.220
11/6	-48.516	-48.610
12/11	-48.977	-49.072
13/1	+49.012	+49.106
16/11	+49.839	+49.935

See also

• 12edo
• 19edt
• 28ed5

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