219edt

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219edt

220edt →

Prime factorization	3 × 73
Step size	8.68473 ¢
Octave	138\219edt (1198.49 ¢) (→ 46\73edt)
Consistency limit	4
Distinct consistency limit	4

219 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 219edt or 219ed3), is a nonoctave tuning system that divides the interval of 3/1 into 219 equal parts of about 8.68 ¢ each. Each step represents a frequency ratio of 3^1/219, or the 219th root of 3.

Intervals

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	8.68	5.94
2	17.37	11.87
3	26.05	17.81	66/65, 69/68
4	34.74	23.74	50/49, 51/50
5	43.42	29.68	40/39, 41/40
6	52.11	35.62	34/33
7	60.79	41.55	29/28
8	69.48	47.49	51/49
9	78.16	53.42	45/43, 68/65
10	86.85	59.36	41/39
11	95.53	65.3	37/35
12	104.22	71.23	69/65
13	112.9	77.17
14	121.59	83.11	44/41, 74/69
15	130.27	89.04	55/51
16	138.96	94.98	13/12
17	147.64	100.91	49/45
18	156.33	106.85
19	165.01	112.79	11/10
20	173.69	118.72	21/19
21	182.38	124.66	10/9
22	191.06	130.59
23	199.75	136.53	46/41, 55/49
24	208.43	142.47	44/39
25	217.12	148.4	17/15
26	225.8	154.34	41/36, 49/43
27	234.49	160.27	63/55
28	243.17	166.21
29	251.86	172.15
30	260.54	178.08	43/37, 50/43
31	269.23	184.02
32	277.91	189.95	27/23, 74/63
33	286.6	195.89	46/39
34	295.28	201.83	51/43
35	303.97	207.76	56/47
36	312.65	213.7
37	321.33	219.63	65/54
38	330.02	225.57	23/19
39	338.7	231.51	45/37, 62/51
40	347.39	237.44	11/9
41	356.07	243.38	43/35, 70/57
42	364.76	249.32	58/47
43	373.44	255.25	36/29
44	382.13	261.19
45	390.81	267.12
46	399.5	273.06	34/27, 63/50
47	408.18	279	62/49
48	416.87	284.93	14/11
49	425.55	290.87	55/43
50	434.24	296.8	9/7
51	442.92	302.74
52	451.61	308.68	74/57
53	460.29	314.61	30/23
54	468.98	320.55
55	477.66	326.48	29/22, 54/41
56	486.34	332.42	49/37
57	495.03	338.36
58	503.71	344.29
59	512.4	350.23	39/29
60	521.08	356.16	50/37
61	529.77	362.1
62	538.45	368.04
63	547.14	373.97
64	555.82	379.91	40/29, 51/37
65	564.51	385.84
66	573.19	391.78	39/28
67	581.88	397.72	7/5
68	590.56	403.65
69	599.25	409.59	41/29, 65/46
70	607.93	415.53	27/19
71	616.62	421.46	10/7
72	625.3	427.4	33/23
73	633.99	433.33	62/43
74	642.67	439.27	29/20
75	651.35	445.21	51/35
76	660.04	451.14	41/28, 60/41
77	668.72	457.08
78	677.41	463.01	34/23
79	686.09	468.95	55/37
80	694.78	474.89
81	703.46	480.82
82	712.15	486.76
83	720.83	492.69	44/29
84	729.52	498.63
85	738.2	504.57	72/47
86	746.89	510.5
87	755.57	516.44	65/42
88	764.26	522.37	14/9
89	772.94	528.31
90	781.63	534.25	11/7
91	790.31	540.18	30/19
92	798.99	546.12	46/29
93	807.68	552.05
94	816.36	557.99
95	825.05	563.93	29/18, 66/41
96	833.73	569.86	34/21
97	842.42	575.8
98	851.1	581.74
99	859.79	587.67	23/14
100	868.47	593.61	38/23
101	877.16	599.54
102	885.84	605.48
103	894.53	611.42	57/34, 62/37
104	903.21	617.35
105	911.9	623.29
106	920.58	629.22	63/37
107	929.27	635.16	65/38
108	937.95	641.1	43/25
109	946.64	647.03	19/11
110	955.32	652.97	33/19
111	964	658.9
112	972.69	664.84
113	981.37	670.78	37/21
114	990.06	676.71	62/35
115	998.74	682.65
116	1007.43	688.58	34/19
117	1016.11	694.52
118	1024.8	700.46	47/26
119	1033.48	706.39	69/38
120	1042.17	712.33	42/23
121	1050.85	718.26
122	1059.54	724.2
123	1068.22	730.14	63/34
124	1076.91	736.07	41/22, 54/29
125	1085.59	742.01
126	1094.28	747.95
127	1102.96	753.88	70/37
128	1111.64	759.82	19/10
129	1120.33	765.75	21/11
130	1129.01	771.69
131	1137.7	777.63	27/14
132	1146.38	783.56
133	1155.07	789.5
134	1163.75	795.43	47/24
135	1172.44	801.37
136	1181.12	807.31
137	1189.81	813.24
138	1198.49	819.18
139	1207.18	825.11
140	1215.86	831.05
141	1224.55	836.99	69/34
142	1233.23	842.92
143	1241.92	848.86	41/20
144	1250.6	854.79	35/17
145	1259.29	860.73	60/29
146	1267.97	866.67
147	1276.65	872.6	23/11
148	1285.34	878.54	21/10
149	1294.02	884.47	19/9
150	1302.71	890.41	70/33
151	1311.39	896.35
152	1320.08	902.28	15/7
153	1328.76	908.22	28/13
154	1337.45	914.16
155	1346.13	920.09	37/17
156	1354.82	926.03
157	1363.5	931.96
158	1372.19	937.9
159	1380.87	943.84
160	1389.56	949.77	29/13
161	1398.24	955.71	65/29, 74/33
162	1406.93	961.64
163	1415.61	967.58
164	1424.3	973.52	41/18, 66/29
165	1432.98	979.45
166	1441.66	985.39	23/10
167	1450.35	991.32
168	1459.03	997.26
169	1467.72	1003.2	7/3
170	1476.4	1009.13
171	1485.09	1015.07	33/14
172	1493.77	1021
173	1502.46	1026.94	50/21
174	1511.14	1032.88
175	1519.83	1038.81
176	1528.51	1044.75	29/12
177	1537.2	1050.68
178	1545.88	1056.62
179	1554.57	1062.56	27/11
180	1563.25	1068.49	37/15
181	1571.94	1074.43	57/23, 62/25
182	1580.62	1080.37
183	1589.3	1086.3
184	1597.99	1092.24
185	1606.67	1098.17	43/17
186	1615.36	1104.11
187	1624.04	1110.05	23/9
188	1632.73	1115.98
189	1641.41	1121.92
190	1650.1	1127.85	70/27
191	1658.78	1133.79
192	1667.47	1139.73	55/21
193	1676.15	1145.66
194	1684.84	1151.6	45/17
195	1693.52	1157.53
196	1702.21	1163.47
197	1710.89	1169.41
198	1719.58	1175.34	27/10
199	1728.26	1181.28	19/7
200	1736.95	1187.21	30/11
201	1745.63	1193.15	74/27
202	1754.31	1199.09
203	1763	1205.02	36/13
204	1771.68	1210.96
205	1780.37	1216.89
206	1789.05	1222.83
207	1797.74	1228.77	65/23
208	1806.42	1234.7
209	1815.11	1240.64
210	1823.79	1246.58	43/15
211	1832.48	1252.51	49/17
212	1841.16	1258.45
213	1849.85	1264.38
214	1858.53	1270.32
215	1867.22	1276.26	50/17
216	1875.9	1282.19	65/22, 68/23
217	1884.59	1288.13
218	1893.27	1294.06
219	1901.96	1300	3/1

Harmonics

Approximation of harmonics in 219edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-1.51	+0.00	-3.02	+1.48	-1.51	+0.85	+4.16	+0.00	-0.02	-0.02	-3.02
	Relative (%)	-17.4	+0.0	-34.7	+17.1	-17.4	+9.8	+47.9	+0.0	-0.3	-0.2	-34.7
Steps (reduced)		138 (138)	219 (0)	276 (57)	321 (102)	357 (138)	388 (169)	415 (196)	438 (0)	459 (21)	478 (40)	495 (57)

Approximation of harmonics in 219edt

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-2.63	-0.66	+1.48	+2.65	+1.91	-1.51	+0.42	-1.53	+0.85	-1.53	-0.32
	Relative (%)	-30.3	-7.6	+17.1	+30.6	+22.0	-17.4	+4.8	-17.6	+9.8	-17.6	-3.7
Steps (reduced)		511 (73)	526 (88)	540 (102)	553 (115)	565 (127)	576 (138)	587 (149)	597 (159)	607 (169)	616 (178)	625 (187)