209edt
← 208edt | 209edt | 210edt → |
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 31/209, 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 | 18.2 | |
3 | 27.3 | 63/62, 66/65 |
4 | 36.4 | 47/46 |
5 | 45.5 | 38/37, 39/38 |
6 | 54.6 | 65/63 |
7 | 63.7 | 28/27 |
8 | 72.8 | 49/47 |
9 | 81.9 | 65/62 |
10 | 91 | 39/37 |
11 | 100.1 | |
12 | 109.2 | 33/31, 49/46 |
13 | 118.3 | |
14 | 127.4 | 14/13 |
15 | 136.5 | |
16 | 145.6 | 37/34, 62/57 |
17 | 154.7 | |
18 | 163.8 | |
19 | 172.9 | 21/19 |
20 | 182 | 10/9 |
21 | 191.1 | 48/43 |
22 | 200.2 | 46/41, 55/49 |
23 | 209.3 | 35/31, 44/39 |
24 | 218.4 | 42/37 |
25 | 227.5 | 57/50, 65/57 |
26 | 236.6 | 39/34, 47/41 |
27 | 245.7 | |
28 | 254.8 | 22/19, 51/44 |
29 | 263.9 | |
30 | 273 | 41/35, 55/47 |
31 | 282.1 | 20/17 |
32 | 291.2 | |
33 | 300.3 | 44/37 |
34 | 309.4 | 49/41, 55/46 |
35 | 318.5 | |
36 | 327.6 | 29/24 |
37 | 336.7 | 17/14 |
38 | 345.8 | |
39 | 354.9 | 27/22, 70/57 |
40 | 364 | 37/30 |
41 | 373.1 | 31/25 |
42 | 382.2 | |
43 | 391.3 | |
44 | 400.4 | 63/50 |
45 | 409.5 | 19/15 |
46 | 418.6 | |
47 | 427.7 | |
48 | 436.8 | |
49 | 445.9 | 22/17 |
50 | 455 | 13/10 |
51 | 464.1 | 17/13 |
52 | 473.2 | 46/35 |
53 | 482.3 | 37/28 |
54 | 491.4 | |
55 | 500.5 | |
56 | 509.6 | 47/35, 51/38, 55/41 |
57 | 518.7 | 27/20, 58/43 |
58 | 527.8 | 19/14 |
59 | 536.9 | 15/11 |
60 | 546 | 37/27 |
61 | 555.1 | 51/37, 62/45 |
62 | 564.2 | 18/13 |
63 | 573.3 | 39/28 |
64 | 582.4 | 7/5 |
65 | 591.5 | 38/27 |
66 | 600.6 | |
67 | 609.7 | |
68 | 618.8 | |
69 | 627.9 | |
70 | 637 | 13/9 |
71 | 646.1 | 45/31 |
72 | 655.2 | 54/37 |
73 | 664.3 | 69/47 |
74 | 673.4 | 31/21 |
75 | 682.5 | 43/29, 46/31 |
76 | 691.6 | |
77 | 700.7 | |
78 | 709.8 | |
79 | 718.9 | 50/33 |
80 | 728 | |
81 | 737.1 | |
82 | 746.2 | 20/13 |
83 | 755.3 | 65/42 |
84 | 764.4 | 14/9 |
85 | 773.5 | |
86 | 782.6 | 11/7 |
87 | 791.7 | 30/19, 49/31 |
88 | 800.8 | 27/17 |
89 | 809.9 | |
90 | 819 | |
91 | 828.1 | 50/31 |
92 | 837.2 | 60/37 |
93 | 846.3 | 44/27 |
94 | 855.4 | 41/25 |
95 | 864.5 | 28/17 |
96 | 873.6 | |
97 | 882.7 | |
98 | 891.8 | 72/43 |
99 | 900.9 | 37/22, 69/41 |
100 | 910 | 22/13 |
101 | 919.1 | 17/10 |
102 | 928.2 | |
103 | 937.3 | |
104 | 946.4 | 19/11 |
105 | 955.5 | 33/19 |
106 | 964.6 | |
107 | 973.7 | |
108 | 982.8 | 30/17 |
109 | 991.9 | 39/22, 55/31 |
110 | 1001 | 41/23, 66/37 |
111 | 1010.1 | 43/24, 52/29 |
112 | 1019.2 | |
113 | 1028.3 | |
114 | 1037.4 | 51/28 |
115 | 1046.5 | |
116 | 1055.6 | 46/25 |
117 | 1064.7 | 37/20 |
118 | 1073.8 | |
119 | 1082.9 | |
120 | 1092 | 47/25, 62/33 |
121 | 1101.1 | 17/9 |
122 | 1110.2 | 19/10 |
123 | 1119.3 | 21/11 |
124 | 1128.4 | |
125 | 1137.5 | 27/14 |
126 | 1146.6 | |
127 | 1155.7 | 39/20 |
128 | 1164.8 | 49/25 |
129 | 1173.9 | 65/33 |
130 | 1183 | |
131 | 1192.1 | |
132 | 1201.2 | |
133 | 1210.3 | |
134 | 1219.4 | |
135 | 1228.5 | 63/31 |
136 | 1237.6 | 47/23 |
137 | 1246.7 | 37/18 |
138 | 1255.8 | 31/15 |
139 | 1264.9 | 27/13 |
140 | 1274 | |
141 | 1283.1 | |
142 | 1292.2 | |
143 | 1301.3 | 70/33 |
144 | 1310.4 | |
145 | 1319.5 | 15/7 |
146 | 1328.6 | 28/13 |
147 | 1337.7 | 13/6 |
148 | 1346.8 | 37/17 |
149 | 1355.9 | |
150 | 1365 | 11/5 |
151 | 1374.1 | 42/19 |
152 | 1383.2 | 20/9 |
153 | 1392.3 | 38/17 |
154 | 1401.4 | |
155 | 1410.5 | 70/31 |
156 | 1419.6 | |
157 | 1428.7 | |
158 | 1437.8 | 39/17 |
159 | 1446.9 | 30/13 |
160 | 1456 | 51/22 |
161 | 1465.1 | |
162 | 1474.2 | |
163 | 1483.3 | |
164 | 1492.4 | 45/19 |
165 | 1501.5 | 50/21 |
166 | 1510.6 | |
167 | 1519.7 | |
168 | 1528.8 | |
169 | 1537.9 | |
170 | 1547 | 22/9 |
171 | 1556.1 | |
172 | 1565.2 | 42/17 |
173 | 1574.3 | 72/29 |
174 | 1583.4 | |
175 | 1592.5 | |
176 | 1601.6 | |
177 | 1610.7 | |
178 | 1619.8 | 51/20 |
179 | 1628.9 | |
180 | 1638 | |
181 | 1647.1 | 44/17, 57/22 |
182 | 1656.2 | |
183 | 1665.3 | 34/13 |
184 | 1674.4 | 50/19 |
185 | 1683.5 | 37/14 |
186 | 1692.6 | |
187 | 1701.7 | |
188 | 1710.8 | 43/16 |
189 | 1719.9 | 27/10 |
190 | 1729.1 | 19/7 |
191 | 1738.2 | |
192 | 1747.3 | |
193 | 1756.4 | |
194 | 1765.5 | |
195 | 1774.6 | 39/14 |
196 | 1783.7 | |
197 | 1792.8 | 31/11 |
198 | 1801.9 | |
199 | 1811 | 37/13 |
200 | 1820.1 | |
201 | 1829.2 | |
202 | 1838.3 | |
203 | 1847.4 | |
204 | 1856.5 | 38/13 |
205 | 1865.6 | |
206 | 1874.7 | 62/21, 65/22 |
207 | 1883.8 | |
208 | 1892.9 | |
209 | 1902 | 3/1 |
Harmonics
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) |
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) |