155ed6

← 154ed6

155ed6

156ed6 →

Prime factorization	5 × 31
Step size	20.0126 ¢
Octave	60\155ed6 (1200.76 ¢) (→ 12\31ed6)
Twelfth	95\155ed6 (1901.2 ¢) (→ 19\31ed6)
Consistency limit	10
Distinct consistency limit	10

155 equal divisions of the 6th harmonic (abbreviated 155ed6) is a nonoctave tuning system that divides the interval of 6/1 into 155 equal parts of about 20 ¢ each. Each step represents a frequency ratio of 6^1/155, or the 155th root of 6.

Theory

155ed6 is related to 60edo (tenth-tone tuning), but with the 6th harmonic rather than the octave being just. This stretches the octave by about 0.757 ¢. Like 60edo, 155ed6 is consistent to the 10-integer-limit. While 155ed6 tunes prime 2 and 13 sharp, the 5, 7, and 17 remain flat but less so, which may be seen as an improvement in intonation over 60edo.

Harmonics

Approximation of harmonics in 155ed6

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.76	-0.76	+1.51	-4.56	+0.00	-6.71	+2.27	-1.51	-3.80	-8.71	+0.76
	Relative (%)	+3.8	-3.8	+7.6	-22.8	+0.0	-33.5	+11.3	-7.6	-19.0	-43.5	+3.8
Steps (reduced)		60 (60)	95 (95)	120 (120)	139 (139)	155 (0)	168 (13)	180 (25)	190 (35)	199 (44)	207 (52)	215 (60)

Approximation of harmonics in 155ed6 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+2.27	-5.95	-5.32	+3.03	-1.87	-0.76	+5.70	-3.05	-7.46	-7.95	-4.86	+1.51
	Relative (%)	+11.4	-29.7	-26.6	+15.1	-9.3	-3.8	+28.5	-15.2	-37.3	-39.7	-24.3	+7.6
Steps (reduced)		222 (67)	228 (73)	234 (79)	240 (85)	245 (90)	250 (95)	255 (100)	259 (104)	263 (108)	267 (112)	271 (116)	275 (120)

Subsets and supersets

Since 155 factors into primes as 5 × 31, 155ed6 has subset ed6's 5ed6 and 31ed6.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	20
2	40	43/42, 44/43, 45/44
3	60	29/28, 30/29
4	80.1	22/21, 45/43
5	100.1	18/17
6	120.1	15/14
7	140.1	51/47
8	160.1	34/31, 45/41, 57/52
9	180.1	51/46
10	200.1	46/41, 55/49
11	220.1	25/22, 42/37
12	240.2	31/27, 54/47
13	260.2	36/31, 43/37, 50/43
14	280.2	20/17, 47/40
15	300.2	44/37
16	320.2
17	340.2	28/23, 45/37
18	360.2	16/13
19	380.2
20	400.3	29/23, 34/27
21	420.3	51/40
22	440.3	40/31, 58/45
23	460.3	30/23, 47/36
24	480.3	33/25
25	500.3
26	520.3	27/20, 50/37
27	540.3	41/30, 56/41
28	560.4	47/34
29	580.4
30	600.4	41/29, 58/41
31	620.4
32	640.4	42/29
33	660.4	41/28
34	680.4	40/27
35	700.4
36	720.5	44/29, 47/31, 50/33
37	740.5	23/15
38	760.5	45/29
39	780.5
40	800.5	27/17
41	820.5	45/28
42	840.5	13/8
43	860.5	23/14, 51/31
44	880.6
45	900.6	37/22
46	920.6
47	940.6	31/18
48	960.6	47/27, 54/31
49	980.6	37/21
50	1000.6	41/23, 57/32
51	1020.6
52	1040.7	31/17
53	1060.7	24/13
54	1080.7	28/15
55	1100.7	17/9
56	1120.7	21/11
57	1140.7	29/15, 56/29
58	1160.7	43/22, 45/23
59	1180.7
60	1200.8	2/1
61	1220.8
62	1240.8	43/21
63	1260.8	29/14
64	1280.8	44/21
65	1300.8
66	1320.8	15/7
67	1340.8
68	1360.9
69	1380.9
70	1400.9
71	1420.9	25/11
72	1440.9	23/10
73	1460.9
74	1480.9	40/17
75	1500.9	50/21
76	1521
77	1541	56/23
78	1561
79	1581
80	1601	58/23
81	1621	51/20
82	1641
83	1661	47/18
84	1681.1	37/14
85	1701.1
86	1721.1
87	1741.1	41/15
88	1761.1	47/17
89	1781.1	14/5
90	1801.1
91	1821.1
92	1841.2
93	1861.2	41/14
94	1881.2
95	1901.2	3/1
96	1921.2
97	1941.2	43/14, 46/15
98	1961.2
99	1981.2	22/7
100	2001.3	54/17
101	2021.3	45/14
102	2041.3	13/4
103	2061.3
104	2081.3
105	2101.3	37/11
106	2121.3
107	2141.3	31/9
108	2161.4
109	2181.4
110	2201.4
111	2221.4
112	2241.4
113	2261.4	48/13
114	2281.4	56/15
115	2301.5	34/9
116	2321.5
117	2341.5	58/15
118	2361.5	43/11
119	2381.5
120	2401.5
121	2421.5
122	2441.5	41/10
123	2461.6	29/7
124	2481.6
125	2501.6
126	2521.6
127	2541.6
128	2561.6
129	2581.6	40/9
130	2601.6
131	2621.7	50/11
132	2641.7	23/5
133	2661.7
134	2681.7
135	2701.7
136	2721.7
137	2741.7	39/8
138	2761.7
139	2781.8
140	2801.8
141	2821.8	51/10
142	2841.8	31/6
143	2861.8	47/9
144	2881.8	37/7
145	2901.8
146	2921.8
147	2941.9
148	2961.9
149	2981.9	28/5
150	3001.9	17/3
151	3021.9
152	3041.9	29/5
153	3061.9
154	3081.9
155	3102	6/1

See also

• 35edf – relative edf
• 60edo – relative edo
• 95edt – relative edt
• 139ed5 – relative ed5