135ed4

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135 equal divisions of the 4th harmonic (abbreviated 135ed4) is a nonoctave tuning system that divides the interval of 4/1 into 135 equal parts of about 17.8 ¢ each. Each step represents a frequency ratio of 4^1/135, or the 135th root of 4.

← 133ed4

135ed4

137ed4 →

Prime factorization	33 × 5
Step size	17.7778 ¢
Octave	68\135ed4 (1208.89 ¢)
Twelfth	107\135ed4 (1902.22 ¢) (semiconvergent)
Consistency limit	1
Distinct consistency limit	1

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	17.8
2	35.6	47/46
3	53.3	34/33
4	71.1
5	88.9
6	106.7
7	124.4	29/27
8	142.2	51/47
9	160	45/41
10	177.8	41/37, 51/46
11	195.6	28/25, 47/42
12	213.3
13	231.1
14	248.9	15/13
15	266.7
16	284.4	33/28, 46/39
17	302.2
18	320
19	337.8	17/14, 45/37
20	355.6	43/35
21	373.3
22	391.1
23	408.9	19/15
24	426.7
25	444.4
26	462.2
27	480	33/25
28	497.8
29	515.6	31/23
30	533.3	34/25
31	551.1
32	568.9	25/18
33	586.7
34	604.4
35	622.2
36	640	42/29
37	657.8	19/13
38	675.6	31/21
39	693.3
40	711.1
41	728.9
42	746.7
43	764.4	14/9
44	782.2
45	800	27/17, 46/29
46	817.8
47	835.6	47/29
48	853.3	18/11
49	871.1
50	888.9
51	906.7
52	924.4	29/17
53	942.2
54	960	47/27
55	977.8	51/29
56	995.6
57	1013.3
58	1031.1
59	1048.9	11/6
60	1066.7
61	1084.4	43/23
62	1102.2	17/9
63	1120
64	1137.8	27/14
65	1155.6	37/19
66	1173.3
67	1191.1
68	1208.9
69	1226.7
70	1244.4	39/19
71	1262.2	29/14
72	1280
73	1297.8
74	1315.6
75	1333.3
76	1351.1
77	1368.9
78	1386.7
79	1404.4
80	1422.2	25/11
81	1440
82	1457.8
83	1475.6
84	1493.3	45/19
85	1511.1
86	1528.9
87	1546.7
88	1564.4	37/15, 42/17
89	1582.2
90	1600
91	1617.8	28/11
92	1635.6
93	1653.3	13/5
94	1671.1
95	1688.9
96	1706.7
97	1724.4	46/17
98	1742.2	41/15
99	1760	47/17
100	1777.8
101	1795.6
102	1813.3
103	1831.1
104	1848.9
105	1866.7
106	1884.4
107	1902.2	3/1
108	1920
109	1937.8
110	1955.6
111	1973.3
112	1991.1
113	2008.9
114	2026.7	29/9
115	2044.4
116	2062.2
117	2080
118	2097.8	47/14
119	2115.6
120	2133.3
121	2151.1	45/13
122	2168.9
123	2186.7	46/13
124	2204.4
125	2222.2
126	2240
127	2257.8
128	2275.6
129	2293.3
130	2311.1	19/5
131	2328.9
132	2346.7
133	2364.4
134	2382.2
135	2400

Harmonics

Approximation of harmonics in 135ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+8.89	+0.27	+0.00	+4.80	-8.62	-8.83	+8.89	+0.53	-4.09	+8.68	+0.27
	Relative (%)	+50.0	+1.5	+0.0	+27.0	-48.5	-49.6	+50.0	+3.0	-23.0	+48.8	+1.5
Steps (reduced)		68 (68)	107 (107)	135 (0)	157 (22)	174 (39)	189 (54)	203 (68)	214 (79)	224 (89)	234 (99)	242 (107)

Approximation of harmonics in 135ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+3.92	+0.06	+5.06	+0.00	+1.71	-8.35	+4.71	+4.80	-8.56	-0.21	-6.05
	Relative (%)	+22.0	+0.4	+28.5	+0.0	+9.6	-47.0	+26.5	+27.0	-48.1	-1.2	-34.0
Steps (reduced)		250 (115)	257 (122)	264 (129)	270 (0)	276 (6)	281 (11)	287 (17)	292 (22)	296 (26)	301 (31)	305 (35)