209edt

← 208edt

209edt

210edt →

Prime factorization	11 × 19
Step size	9.10026¢
Octave	132\209edt (1201.23¢) (→12\19edt)
Consistency limit	7
Distinct consistency limit	7

209 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 209edt or 209ed3), is a nonoctave tuning system that divides the interval of 3/1 into 209 equal parts of about 9.1 ¢ each. Each step represents a frequency ratio of 3^1/209, or the 209th root of 3.

Since 209 factors into 11 × 19, 209edt has 11edt and 19edt as subsets; it inherits its mapping of the 11th harmonic from the former and the 17th (along with the octave, corresponding to 19edt being a close octave stretch of 12edo) from the latter. It otherwise represents a very strong system in the 19-limit (even including prime 2), being the sum of 78edt which has a flat tendency in the 19-limit and 131edt which has a slight sharp tendency. Its most accurate simple intervals are 7/5, 11/5, and 17/9, all of which it approximates to within about 0.1 cents.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	9.1
2	18.201
3	27.301	63/62, 66/65
4	36.401	47/46
5	45.501	38/37, 39/38
6	54.602	65/63
7	63.702	28/27
8	72.802	49/47
9	81.902	65/62
10	91.003	39/37
11	100.103
12	109.203	33/31, 49/46
13	118.303
14	127.404	14/13
15	136.504
16	145.604	37/34, 62/57
17	154.704
18	163.805
19	172.905	21/19
20	182.005	10/9
21	191.106	48/43
22	200.206	46/41, 55/49
23	209.306	35/31, 44/39
24	218.406	42/37
25	227.507	57/50, 65/57
26	236.607	39/34, 47/41
27	245.707
28	254.807	22/19, 51/44
29	263.908
30	273.008	41/35, 55/47
31	282.108	20/17
32	291.208
33	300.309	44/37
34	309.409	49/41, 55/46
35	318.509
36	327.609	29/24
37	336.71	17/14
38	345.81
39	354.91	27/22, 70/57
40	364.011	37/30
41	373.111	31/25
42	382.211
43	391.311
44	400.412	63/50
45	409.512	19/15
46	418.612
47	427.712
48	436.813
49	445.913	22/17
50	455.013	13/10
51	464.113	17/13
52	473.214	46/35
53	482.314	37/28
54	491.414
55	500.514
56	509.615	47/35, 51/38, 55/41
57	518.715	27/20, 58/43
58	527.815	19/14
59	536.916	15/11
60	546.016	37/27
61	555.116	51/37, 62/45
62	564.216	18/13
63	573.317	39/28
64	582.417	7/5
65	591.517	38/27
66	600.617
67	609.718
68	618.818
69	627.918
70	637.018	13/9
71	646.119	45/31
72	655.219	54/37
73	664.319	69/47
74	673.419	31/21
75	682.52	43/29, 46/31
76	691.62
77	700.72
78	709.821
79	718.921	50/33
80	728.021
81	737.121
82	746.222	20/13
83	755.322	65/42
84	764.422	14/9
85	773.522
86	782.623	11/7
87	791.723	30/19, 49/31
88	800.823	27/17
89	809.923
90	819.024
91	828.124	50/31
92	837.224	60/37
93	846.324	44/27
94	855.425	41/25
95	864.525	28/17
96	873.625
97	882.726
98	891.826	72/43
99	900.926	37/22, 69/41
100	910.026	22/13
101	919.127	17/10
102	928.227
103	937.327
104	946.427	19/11
105	955.528	33/19
106	964.628
107	973.728
108	982.828	30/17
109	991.929	39/22, 55/31
110	1001.029	41/23, 66/37
111	1010.129	43/24, 52/29
112	1019.229
113	1028.33
114	1037.43	51/28
115	1046.53
116	1055.631	46/25
117	1064.731	37/20
118	1073.831
119	1082.931
120	1092.032	47/25, 62/33
121	1101.132	17/9
122	1110.232	19/10
123	1119.332	21/11
124	1128.433
125	1137.533	27/14
126	1146.633
127	1155.733	39/20
128	1164.834	49/25
129	1173.934	65/33
130	1183.034
131	1192.134
132	1201.235
133	1210.335
134	1219.435
135	1228.536	63/31
136	1237.636	47/23
137	1246.736	37/18
138	1255.836	31/15
139	1264.937	27/13
140	1274.037
141	1283.137
142	1292.237
143	1301.338	70/33
144	1310.438
145	1319.538	15/7
146	1328.638	28/13
147	1337.739	13/6
148	1346.839	37/17
149	1355.939
150	1365.039	11/5
151	1374.14	42/19
152	1383.24	20/9
153	1392.34	38/17
154	1401.441
155	1410.541	70/31
156	1419.641
157	1428.741
158	1437.842	39/17
159	1446.942	30/13
160	1456.042	51/22
161	1465.142
162	1474.243
163	1483.343
164	1492.443	45/19
165	1501.543	50/21
166	1510.644
167	1519.744
168	1528.844
169	1537.944
170	1547.045	22/9
171	1556.145
172	1565.245	42/17
173	1574.346	72/29
174	1583.446
175	1592.546
176	1601.646
177	1610.747
178	1619.847	51/20
179	1628.947
180	1638.047
181	1647.148	44/17, 57/22
182	1656.248
183	1665.348	34/13
184	1674.448	50/19
185	1683.549	37/14
186	1692.649
187	1701.749
188	1710.849	43/16
189	1719.95	27/10
190	1729.05	19/7
191	1738.15
192	1747.251
193	1756.351
194	1765.451
195	1774.551	39/14
196	1783.652
197	1792.752	31/11
198	1801.852
199	1810.952	37/13
200	1820.053
201	1829.153
202	1838.253
203	1847.353
204	1856.454	38/13
205	1865.554
206	1874.654	62/21, 65/22
207	1883.754
208	1892.855
209	1901.955	3/1

Harmonics

Approximation of prime harmonics in 209edt

Harmonic		2	3	5	7	11	13	17	19	23
Error	Absolute (¢)	+1.23	+0.00	-1.63	-1.73	-1.60	+0.40	+0.09	-1.37	-4.52
	Relative (%)	+13.6	+0.0	-17.9	-19.0	-17.6	+4.4	+0.9	-15.0	-49.6
Steps (reduced)		132 (132)	209 (0)	306 (97)	370 (161)	456 (38)	488 (70)	539 (121)	560 (142)	596 (178)

Approximation of odd harmonics in 209edt

Harmonic		25	27	29	31	33	35	37	39	41	43	45
Error	Absolute (¢)	-3.27	+0.00	+3.69	-2.56	-1.60	-3.36	+0.54	+0.40	-4.28	+4.27	-1.63
	Relative (%)	-35.9	+0.0	+40.6	-28.2	-17.6	-36.9	+5.9	+4.4	-47.0	+46.9	-17.9
Steps (reduced)		612 (194)	627 (0)	641 (14)	653 (26)	665 (38)	676 (49)	687 (60)	697 (70)	706 (79)	716 (89)	724 (97)