153edt

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153edt

154edt →

Prime factorization	32 × 17
Step size	12.4311 ¢
Octave	97\153edt (1205.81 ¢)
Consistency limit	3
Distinct consistency limit	3

153 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 153edt or 153ed3), is a nonoctave tuning system that divides the interval of 3/1 into 153 equal parts of about 12.4 ¢ each. Each step represents a frequency ratio of 3^1/153, or the 153rd root of 3.

153edt is notable for being the denominator of a convergent to log₃(7/3), after 9edt, 13edt and 35edt, and the last before 3401edt, and therefore has an extremely accurate approximation to 7/3, a mere 0.0036 cents flat. In fact, 153edt demonstrates 11-strong 7-3 telicity, due to the next term in the continued fraction expansion being large (note how much larger 3401 is than 153), although 3401edt in fact surpasses it, demonstrating 16-strong 7-3 telicity.

In the no-twos 7-limit, 153edt supports canopus temperament, which gives it a rather accurate approximation of the 5th harmonic; and it additionally is accurate in the 11-limit, tempering out the comma 387420489/386683451 in the 3.7.11 subgroup. Harmonics 19 and 29 are also notably good.

However, 153edt's approximation of 2/1 is close to maximally bad, meaning that it is as far from an octave-equivalent tuning that an EDT of this size can be (though by this point, it is only 6 or so cents off).

Intervals

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	12.4	8.5
2	24.9	17
3	37.3	25.5
4	49.7	34
5	62.2	42.5	57/55
6	74.6	51	47/45
7	87	59.5	41/39
8	99.4	68	18/17
9	111.9	76.5
10	124.3	85	29/27
11	136.7	93.5
12	149.2	102
13	161.6	110.5	45/41
14	174	119	21/19
15	186.5	127.5	39/35
16	198.9	135.9	37/33, 55/49
17	211.3	144.4	26/23, 35/31
18	223.8	152.9	33/29, 58/51
19	236.2	161.4	47/41
20	248.6	169.9	15/13
21	261.1	178.4	43/37, 50/43, 57/49
22	273.5	186.9	41/35, 55/47
23	285.9	195.4
24	298.3	203.9
25	310.8	212.4
26	323.2	220.9	47/39
27	335.6	229.4	17/14
28	348.1	237.9	11/9
29	360.5	246.4
30	372.9	254.9	31/25
31	385.4	263.4
32	397.8	271.9	39/31
33	410.2	280.4	19/15
34	422.7	288.9	37/29
35	435.1	297.4	9/7
36	447.5	305.9	22/17
37	459.9	314.4	30/23
38	472.4	322.9
39	484.8	331.4	41/31, 49/37
40	497.2	339.9
41	509.7	348.4	47/35, 51/38, 55/41
42	522.1	356.9	23/17, 50/37
43	534.5	365.4
44	547	373.9	37/27
45	559.4	382.4	29/21
46	571.8	390.8	57/41
47	584.3	399.3
48	596.7	407.8
49	609.1	416.3	27/19
50	621.6	424.8
51	634	433.3
52	646.4	441.8	45/31
53	658.8	450.3
54	671.3	458.8
55	683.7	467.3	49/33
56	696.1	475.8
57	708.6	484.3
58	721	492.8	47/31
59	733.4	501.3	29/19
60	745.9	509.8
61	758.3	518.3
62	770.7	526.8	39/25
63	783.2	535.3	11/7
64	795.6	543.8
65	808	552.3
66	820.5	560.8
67	832.9	569.3
68	845.3	577.8	57/35
69	857.7	586.3	41/25
70	870.2	594.8	38/23, 43/26
71	882.6	603.3
72	895	611.8
73	907.5	620.3	49/29
74	919.9	628.8	17/10
75	932.3	637.3
76	944.8	645.8	19/11
77	957.2	654.2	33/19
78	969.6	662.7
79	982.1	671.2	30/17, 37/21
80	994.5	679.7
81	1006.9	688.2
82	1019.3	696.7
83	1031.8	705.2	49/27
84	1044.2	713.7
85	1056.6	722.2	35/19
86	1069.1	730.7
87	1081.5	739.2
88	1093.9	747.7	47/25
89	1106.4	756.2
90	1118.8	764.7	21/11
91	1131.2	773.2	25/13
92	1143.7	781.7
93	1156.1	790.2
94	1168.5	798.7	57/29
95	1181	807.2
96	1193.4	815.7
97	1205.8	824.2
98	1218.2	832.7
99	1230.7	841.2	55/27
100	1243.1	849.7
101	1255.5	858.2	31/15
102	1268	866.7
103	1280.4	875.2
104	1292.8	883.7	19/9
105	1305.3	892.2
106	1317.7	900.7
107	1330.1	909.2	41/19
108	1342.6	917.6
109	1355	926.1
110	1367.4	934.6
111	1379.8	943.1	51/23
112	1392.3	951.6	38/17
113	1404.7	960.1
114	1417.1	968.6
115	1429.6	977.1
116	1442	985.6	23/10
117	1454.4	994.1	51/22
118	1466.9	1002.6	7/3
119	1479.3	1011.1
120	1491.7	1019.6	45/19
121	1504.2	1028.1	31/13
122	1516.6	1036.6
123	1529	1045.1
124	1541.5	1053.6
125	1553.9	1062.1	27/11
126	1566.3	1070.6	42/17
127	1578.7	1079.1
128	1591.2	1087.6
129	1603.6	1096.1
130	1616	1104.6
131	1628.5	1113.1
132	1640.9	1121.6	49/19
133	1653.3	1130.1	13/5
134	1665.8	1138.6	55/21
135	1678.2	1147.1	29/11
136	1690.6	1155.6
137	1703.1	1164.1
138	1715.5	1172.5	35/13
139	1727.9	1181	19/7
140	1740.4	1189.5	41/15
141	1752.8	1198
142	1765.2	1206.5
143	1777.6	1215
144	1790.1	1223.5
145	1802.5	1232	17/6
146	1814.9	1240.5
147	1827.4	1249
148	1839.8	1257.5	55/19
149	1852.2	1266
150	1864.7	1274.5
151	1877.1	1283
152	1889.5	1291.5
153	1902	1300	3/1

Harmonics

Approximation of harmonics in 153edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+5.81	+0.00	-0.80	-1.75	+5.81	-0.00	+5.01	+0.00	+4.06	+0.66	-0.80
	Relative (%)	+46.8	+0.0	-6.5	-14.1	+46.8	-0.0	+40.3	+0.0	+32.7	+5.3	-6.5
Steps (reduced)		97 (97)	153 (0)	193 (40)	224 (71)	250 (97)	271 (118)	290 (137)	306 (0)	321 (15)	334 (28)	346 (40)

Approximation of harmonics in 153edt

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-2.63	+5.81	-1.75	-1.60	+5.32	+5.81	-0.77	-2.55	-0.00	-5.95	+4.11
	Relative (%)	-21.2	+46.7	-14.1	-12.9	+42.8	+46.8	-6.2	-20.5	-0.0	-47.9	+33.0
Steps (reduced)		357 (51)	368 (62)	377 (71)	386 (80)	395 (89)	403 (97)	410 (104)	417 (111)	424 (118)	430 (124)	437 (131)