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

Intervals in 34zpi

JI ratios are comprised of 16-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 %			⟨12 19] Whole tone = 2 steps Limma = 1 step Apotome = 1 step
Degree	Cents	Ratios	Ups and Downs Notation
0	0.000		P1
1	99.807	16/15, 15/14, 14/13, 13/12	m2
2	199.614	12/11, 11/10, 10/9, 9/8, 8/7, 15/13	M2
3	299.422	7/6, 13/11, 6/5, 11/9	m3
4	399.229	16/13, 5/4, 14/11, 9/7	M3
5	499.036	13/10, 4/3, 15/11	P4
6	598.843	11/8, 7/5, 10/7, 13/9, 16/11	A4, d5
7	698.650	3/2	P5
8	798.457	14/9, 11/7, 8/5, 13/8	m6
9	898.265	5/3, 12/7	M6
10	998.072	7/4, 16/9, 9/5	m7
11	1097.879	11/6, 13/7, 15/8	M7
12	1197.686	2/1	P1 +1 oct
13	1297.493	15/7, 13/6	m2 +1 oct
14	1397.301	11/5, 9/4, 16/7	M2 +1 oct
15	1497.108	7/3, 12/5	m3 +1 oct
16	1596.915	5/2	M3 +1 oct
17	1696.722	13/5, 8/3	P4 +1 oct
18	1796.529	11/4, 14/5	A4 +1 oct, d5 +1 oct
19	1896.336	3/1	P5 +1 oct
20	1996.144	16/5, 13/4	m6 +1 oct
21	2095.951	10/3	M6 +1 oct
22	2195.758	7/2	m7 +1 oct
23	2295.565	11/3, 15/4	M7 +1 oct
24	2395.372	4/1	P1 +2 oct
25	2495.180	13/3	m2 +2 oct
26	2594.987	9/2	M2 +2 oct
27	2694.794	14/3	m3 +2 oct
28	2794.601	5/1	M3 +2 oct
29	2894.408	16/3	P4 +2 oct
30	2994.215	11/2	A4 +2 oct, d5 +2 oct
31	3094.023	6/1	P5 +2 oct
32	3193.830	13/2	m6 +2 oct
33	3293.637		M6 +2 oct
34	3393.444	7/1	m7 +2 oct
35	3493.251	15/2	M7 +2 oct
36	3593.059	8/1	P1 +3 oct
37	3692.866		m2 +3 oct
38	3792.673	9/1	M2 +3 oct
39	3892.480		m3 +3 oct
40	3992.287	10/1	M3 +3 oct
41	4092.094		P4 +3 oct
42	4191.902	11/1	A4 +3 oct, d5 +3 oct
43	4291.709	12/1	P5 +3 oct
44	4391.516	13/1	m6 +3 oct
45	4491.323		M6 +3 oct
46	4591.130	14/1	m7 +3 oct
47	4690.937	15/1	M7 +3 oct
48	4790.745	16/1	P1 +4 oct

Approximation to JI

Interval mappings

The following tables show how 16-integer limit ratios are represented in 34zpi. Prime harmonics are in bold; inconsistent intervals are in italics.

16-integer limit ratios in 34zpi by direct approximation

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 %

16-integer limit ratios in 34zpi by patent val mapping

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 %
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 %
7/2	+26.932 ¢	+26.984 %
14/3	+27.923 ¢	+27.977 %
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 %
11/2	+42.897 ¢	+42.980 %
13/3	-43.393 ¢	-43.477 %
13/4	-44.384 ¢	-44.470 %
11/4	+45.211 ¢	+45.299 %
11/3	+46.202 ¢	+46.291 %
13/2	-46.698 ¢	-46.788 %
11/8	+47.525 ¢	+47.617 %
11/6	+48.516 ¢	+48.610 %
13/1	-49.012 ¢	-49.106 %
16/11	-49.839 ¢	-49.935 %
12/11	-50.830 ¢	-50.928 %
15/13	+51.680 ¢	+51.780 %
11/9	+51.821 ¢	+51.921 %
13/10	-54.985 ¢	-55.091 %
13/5	-57.299 ¢	-57.410 %
14/13	+71.316 ¢	+71.454 %
13/7	-73.630 ¢	-73.772 %
13/11	-89.595 ¢	-89.768 %

See also

• 12edo
• 19edt
• 28ed5

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