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	Hekts	Approximate ratios
0	0	0	1/1
1	9.1	6.22
2	18.2	12.44
3	27.3	18.66	63/62, 66/65
4	36.4	24.88	47/46
5	45.5	31.1	38/37, 39/38
6	54.6	37.32	65/63
7	63.7	43.54	28/27
8	72.8	49.76	49/47
9	81.9	55.98	65/62
10	91	62.2	39/37
11	100.1	68.42
12	109.2	74.64	33/31, 49/46
13	118.3	80.86
14	127.4	87.08	14/13
15	136.5	93.3
16	145.6	99.52	37/34, 62/57
17	154.7	105.74
18	163.8	111.96
19	172.91	118.18	21/19
20	182.01	124.4	10/9
21	191.11	130.62	48/43
22	200.21	136.84	46/41, 55/49
23	209.31	143.06	35/31, 44/39
24	218.41	149.28	42/37
25	227.51	155.5	57/50, 65/57
26	236.61	161.72	39/34, 47/41
27	245.71	167.94
28	254.81	174.16	22/19, 51/44
29	263.91	180.38
30	273.01	186.6	41/35, 55/47
31	282.11	192.82	20/17
32	291.21	199.04
33	300.31	205.26	44/37
34	309.41	211.48	49/41, 55/46
35	318.51	217.7
36	327.61	223.92	29/24
37	336.71	230.14	17/14
38	345.81	236.36
39	354.91	242.58	27/22, 70/57
40	364.01	248.8	37/30
41	373.11	255.02	31/25
42	382.21	261.24
43	391.31	267.46
44	400.41	273.68	63/50
45	409.51	279.9	19/15
46	418.61	286.12
47	427.71	292.34
48	436.81	298.56
49	445.91	304.78	22/17
50	455.01	311	13/10
51	464.11	317.22	17/13
52	473.21	323.44	46/35
53	482.31	329.67	37/28
54	491.41	335.89
55	500.51	342.11
56	509.61	348.33	47/35, 51/38, 55/41
57	518.72	354.55	27/20, 58/43
58	527.82	360.77	19/14
59	536.92	366.99	15/11
60	546.02	373.21	37/27
61	555.12	379.43	51/37, 62/45
62	564.22	385.65	18/13
63	573.32	391.87	39/28
64	582.42	398.09	7/5
65	591.52	404.31	38/27
66	600.62	410.53
67	609.72	416.75
68	618.82	422.97
69	627.92	429.19
70	637.02	435.41	13/9
71	646.12	441.63	45/31
72	655.22	447.85	54/37
73	664.32	454.07	69/47
74	673.42	460.29	31/21
75	682.52	466.51	43/29, 46/31
76	691.62	472.73
77	700.72	478.95
78	709.82	485.17
79	718.92	491.39	50/33
80	728.02	497.61
81	737.12	503.83
82	746.22	510.05	20/13
83	755.32	516.27	65/42
84	764.42	522.49	14/9
85	773.52	528.71
86	782.62	534.93	11/7
87	791.72	541.15	30/19, 49/31
88	800.82	547.37	27/17
89	809.92	553.59
90	819.02	559.81
91	828.12	566.03	50/31
92	837.22	572.25	60/37
93	846.32	578.47	44/27
94	855.42	584.69	41/25
95	864.53	590.91	28/17
96	873.63	597.13
97	882.73	603.35
98	891.83	609.57	72/43
99	900.93	615.79	37/22, 69/41
100	910.03	622.01	22/13
101	919.13	628.23	17/10
102	928.23	634.45
103	937.33	640.67
104	946.43	646.89	19/11
105	955.53	653.11	33/19
106	964.63	659.33
107	973.73	665.55
108	982.83	671.77	30/17
109	991.93	677.99	39/22, 55/31
110	1001.03	684.21	41/23, 66/37
111	1010.13	690.43	43/24, 52/29
112	1019.23	696.65
113	1028.33	702.87
114	1037.43	709.09	51/28
115	1046.53	715.31
116	1055.63	721.53	46/25
117	1064.73	727.75	37/20
118	1073.83	733.97
119	1082.93	740.19
120	1092.03	746.41	47/25, 62/33
121	1101.13	752.63	17/9
122	1110.23	758.85	19/10
123	1119.33	765.07	21/11
124	1128.43	771.29
125	1137.53	777.51	27/14
126	1146.63	783.73
127	1155.73	789.95	39/20
128	1164.83	796.17	49/25
129	1173.93	802.39	65/33
130	1183.03	808.61
131	1192.13	814.83
132	1201.23	821.05
133	1210.34	827.27
134	1219.44	833.49
135	1228.54	839.71	63/31
136	1237.64	845.93	47/23
137	1246.74	852.15	37/18
138	1255.84	858.37	31/15
139	1264.94	864.59	27/13
140	1274.04	870.81
141	1283.14	877.03
142	1292.24	883.25
143	1301.34	889.47	70/33
144	1310.44	895.69
145	1319.54	901.91	15/7
146	1328.64	908.13	28/13
147	1337.74	914.35	13/6
148	1346.84	920.57	37/17
149	1355.94	926.79
150	1365.04	933.01	11/5
151	1374.14	939.23	42/19
152	1383.24	945.45	20/9
153	1392.34	951.67	38/17
154	1401.44	957.89
155	1410.54	964.11	70/31
156	1419.64	970.33
157	1428.74	976.56
158	1437.84	982.78	39/17
159	1446.94	989	30/13
160	1456.04	995.22	51/22
161	1465.14	1001.44
162	1474.24	1007.66
163	1483.34	1013.88
164	1492.44	1020.1	45/19
165	1501.54	1026.32	50/21
166	1510.64	1032.54
167	1519.74	1038.76
168	1528.84	1044.98
169	1537.94	1051.2
170	1547.04	1057.42	22/9
171	1556.15	1063.64
172	1565.25	1069.86	42/17
173	1574.35	1076.08	72/29
174	1583.45	1082.3
175	1592.55	1088.52
176	1601.65	1094.74
177	1610.75	1100.96
178	1619.85	1107.18	51/20
179	1628.95	1113.4
180	1638.05	1119.62
181	1647.15	1125.84	44/17, 57/22
182	1656.25	1132.06
183	1665.35	1138.28	34/13
184	1674.45	1144.5	50/19
185	1683.55	1150.72	37/14
186	1692.65	1156.94
187	1701.75	1163.16
188	1710.85	1169.38	43/16
189	1719.95	1175.6	27/10
190	1729.05	1181.82	19/7
191	1738.15	1188.04
192	1747.25	1194.26
193	1756.35	1200.48
194	1765.45	1206.7
195	1774.55	1212.92	39/14
196	1783.65	1219.14
197	1792.75	1225.36	31/11
198	1801.85	1231.58
199	1810.95	1237.8	37/13
200	1820.05	1244.02
201	1829.15	1250.24
202	1838.25	1256.46
203	1847.35	1262.68
204	1856.45	1268.9	38/13
205	1865.55	1275.12
206	1874.65	1281.34	62/21, 65/22
207	1883.75	1287.56
208	1892.85	1293.78
209	1901.96	1300	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
Error	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
Error	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)