97ed9

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97ed9

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Prime factorization	97 (prime)
Step size	39.2156 ¢
Octave	31\97ed9 (1215.68 ¢)
Twelfth	49\97ed9 (1921.56 ¢)
Consistency limit	2
Distinct consistency limit	2

97 equal divisions of the 9th harmonic (abbreviated 97ed9) is a nonoctave tuning system that divides the interval of 9/1 into 97 equal parts of about 39.2 ¢ each. Each step represents a frequency ratio of 9^1/97, or the 97th root of 9.

Theory

97ed9 corresponds to 30.6001…edo. Each step of 97ed9 corresponds closely to five steps of 153edo.

97ed9 features a well-balanced harmonic series segment from 4 to 9 and another from 39 to 50 (see table below). It performs well across all prime harmonics from 5 to 19, with the exception of 13, which is slightly flat.

97ed9 sets a height record on the Riemann zeta function with primes 2 and 3 removed, approximating 30.59745…edo. This record remains unbeaten until approximately 41.3478…edo.

Additionally, 97ed9 is close to 125zpi (see Zeta peak index).

Harmonics

Approximation of harmonics in 97ed9

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+15.7	+19.6	-7.9	-2.0	-3.9	+3.7	+7.8	+0.0	+13.7	+5.5	+11.8
	Relative (%)	+40.0	+50.0	-20.0	-5.1	-10.0	+9.5	+20.0	+0.0	+34.9	+14.1	+30.0
Steps (reduced)		31 (31)	49 (49)	61 (61)	71 (71)	79 (79)	86 (86)	92 (92)	97 (0)	102 (5)	106 (9)	110 (13)

Approximation of harmonics in 97ed9 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	-9.2	+19.4	+17.6	-15.7	-3.0	+15.7	+0.5	-9.9	-15.9	-18.0	-16.5	-11.8
	Relative (%)	-23.4	+49.5	+44.9	-40.0	-7.7	+40.0	+1.3	-25.1	-40.5	-45.9	-42.1	-30.0
Steps (reduced)		113 (16)	117 (20)	120 (23)	122 (25)	125 (28)	128 (31)	130 (33)	132 (35)	134 (37)	136 (39)	138 (41)	140 (43)

Approximation of harmonics in 97ed9 (39–50)

Harmonic		39	40	41	42	43	44	45	46	47	48	49	50
Error	Absolute (¢)	+10.4	+5.8	+2.3	-0.2	-1.7	-2.3	-2.0	-0.8	+1.1	+3.9	+7.4	+11.7
	Relative (%)	+26.6	+14.9	+5.8	-0.5	-4.4	-5.9	-5.1	-2.2	+2.9	+10.0	+18.9	+29.7
Steps (reduced)		162 (65)	163 (66)	164 (67)	165 (68)	166 (69)	167 (70)	168 (71)	169 (72)	170 (73)	171 (74)	172 (75)	173 (76)

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	39.2
2	78.4	22/21
3	117.6	15/14, 31/29
4	156.9
5	196.1
6	235.3	39/34
7	274.5	34/29, 41/35
8	313.7
9	352.9	38/31
10	392.2
11	431.4
12	470.6
13	509.8
14	549
15	588.2
16	627.4
17	666.7	25/17
18	705.9
19	745.1
20	784.3	11/7
21	823.5	37/23
22	862.7
23	902
24	941.2	43/25
25	980.4
26	1019.6
27	1058.8	35/19
28	1098
29	1137.3
30	1176.5
31	1215.7
32	1254.9	31/15
33	1294.1
34	1333.3	41/19
35	1372.5
36	1411.8	43/19
37	1451
38	1490.2	26/11
39	1529.4
40	1568.6
41	1607.8	38/15, 43/17
42	1647.1
43	1686.3
44	1725.5
45	1764.7
46	1803.9
47	1843.1	29/10
48	1882.3
49	1921.6
50	1960.8	31/10
51	2000
52	2039.2
53	2078.4
54	2117.6	17/5
55	2156.9
56	2196.1
57	2235.3
58	2274.5
59	2313.7	19/5
60	2352.9
61	2392.1
62	2431.4
63	2470.6
64	2509.8
65	2549
66	2588.2
67	2627.4
68	2666.7	14/3
69	2705.9
70	2745.1	44/9
71	2784.3	5/1
72	2823.5
73	2862.7
74	2902
75	2941.2
76	2980.4
77	3019.6
78	3058.8	41/7
79	3098
80	3137.2
81	3176.5
82	3215.7
83	3254.9
84	3294.1
85	3333.3
86	3372.5
87	3411.8
88	3451	22/3
89	3490.2	15/2
90	3529.4
91	3568.6
92	3607.8
93	3647
94	3686.3
95	3725.5	43/5
96	3764.7
97	3803.9