316edt

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316edt

317edt →

Prime factorization	22 × 79
Step size	6.01884¢
Octave	199\316edt (1197.75¢)
Consistency limit	2
Distinct consistency limit	2

316 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 316edt or 316ed3), is a nonoctave tuning system that divides the interval of 3/1 into 316 equal parts of about 6.02⁠ ⁠¢ each. Each step represents a frequency ratio of 3^1/316, or the 316th root of 3.

Intervals

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	6.02	4.11
2	12.04	8.23
3	18.06	12.34	99/98
4	24.08	16.46
5	30.09	20.57	58/57
6	36.11	24.68
7	42.13	28.8	42/41
8	48.15	32.91
9	54.17	37.03	65/63, 98/95
10	60.19	41.14
11	66.21	45.25
12	72.23	49.37
13	78.24	53.48	45/43, 91/87
14	84.26	57.59
15	90.28	61.71	98/93
16	96.3	65.82	37/35
17	102.32	69.94	35/33
18	108.34	74.05	33/31
19	114.36	78.16	47/44
20	120.38	82.28
21	126.4	86.39
22	132.41	90.51
23	138.43	94.62
24	144.45	98.73	25/23
25	150.47	102.85
26	156.49	106.96	81/74
27	162.51	111.08
28	168.53	115.19	43/39
29	174.55	119.3	52/47
30	180.57	123.42
31	186.58	127.53
32	192.6	131.65	19/17
33	198.62	135.76	37/33, 46/41
34	204.64	139.87
35	210.66	143.99	35/31
36	216.68	148.1	17/15
37	222.7	152.22	58/51
38	228.72	156.33
39	234.73	160.44	63/55
40	240.75	164.56	54/47
41	246.77	168.67	98/85
42	252.79	172.78	81/70
43	258.81	176.9
44	264.83	181.01
45	270.85	185.13
46	276.87	189.24
47	282.89	193.35
48	288.9	197.47	13/11
49	294.92	201.58	51/43
50	300.94	205.7	69/58
51	306.96	209.81
52	312.98	213.92
53	319	218.04
54	325.02	222.15	35/29
55	331.04	226.27	23/19
56	337.06	230.38
57	343.07	234.49	50/41
58	349.09	238.61
59	355.11	242.72	27/22, 70/57
60	361.13	246.84	85/69
61	367.15	250.95
62	373.17	255.06
63	379.19	259.18
64	385.21	263.29
65	391.22	267.41
66	397.24	271.52	39/31
67	403.26	275.63
68	409.28	279.75	19/15
69	415.3	283.86
70	421.32	287.97	37/29
71	427.34	292.09
72	433.36	296.2
73	439.38	300.32	58/45
74	445.39	304.43	75/58
75	451.41	308.54	74/57
76	457.43	312.66
77	463.45	316.77	98/75
78	469.47	320.89
79	475.49	325	25/19
80	481.51	329.11
81	487.53	333.23	57/43
82	493.55	337.34
83	499.56	341.46
84	505.58	345.57
85	511.6	349.68
86	517.62	353.8	58/43
87	523.64	357.91	23/17
88	529.66	362.03
89	535.68	366.14
90	541.7	370.25
91	547.71	374.37	70/51
92	553.73	378.48	95/69
93	559.75	382.59
94	565.77	386.71
95	571.79	390.82
96	577.81	394.94	81/58
97	583.83	399.05
98	589.85	403.16
99	595.87	407.28
100	601.88	411.39
101	607.9	415.51	27/19, 98/69
102	613.92	419.62
103	619.94	423.73	93/65
104	625.96	427.85
105	631.98	431.96
106	638	436.08
107	644.02	440.19	74/51
108	650.04	444.3
109	656.05	448.42
110	662.07	452.53	85/58
111	668.09	456.65	25/17
112	674.11	460.76	31/21
113	680.13	464.87
114	686.15	468.99	55/37
115	692.17	473.1	85/57
116	698.19	477.22
117	704.2	481.33
118	710.22	485.44	98/65
119	716.24	489.56	62/41
120	722.26	493.67
121	728.28	497.78	99/65
122	734.3	501.9
123	740.32	506.01	23/15
124	746.34	510.13
125	752.36	514.24
126	758.37	518.35
127	764.39	522.47	14/9
128	770.41	526.58	39/25
129	776.43	530.7
130	782.45	534.81	11/7
131	788.47	538.92	41/26
132	794.49	543.04	87/55
133	800.51	547.15	27/17
134	806.53	551.27
135	812.54	555.38
136	818.56	559.49	69/43
137	824.58	563.61	66/41
138	830.6	567.72	21/13
139	836.62	571.84
140	842.64	575.95
141	848.66	580.06
142	854.68	584.18	95/58
143	860.69	588.29	74/45
144	866.71	592.41
145	872.73	596.52
146	878.75	600.63
147	884.77	604.75	5/3
148	890.79	608.86
149	896.81	612.97	47/28
150	902.83	617.09
151	908.85	621.2	93/55
152	914.86	625.32
153	920.88	629.43	63/37
154	926.9	633.54
155	932.92	637.66
156	938.94	641.77	43/25
157	944.96	645.89
158	950.98	650
159	957	654.11
160	963.02	658.23	75/43, 82/47
161	969.03	662.34
162	975.05	666.46	65/37
163	981.07	670.57	37/21
164	987.09	674.68
165	993.11	678.8	55/31
166	999.13	682.91
167	1005.15	687.03	84/47
168	1011.17	691.14
169	1017.18	695.25	9/5
170	1023.2	699.37
171	1029.22	703.48
172	1035.24	707.59
173	1041.26	711.71
174	1047.28	715.82
175	1053.3	719.94
176	1059.32	724.05
177	1065.34	728.16
178	1071.35	732.28	13/7
179	1077.37	736.39	41/22, 95/51
180	1083.39	740.51	43/23
181	1089.41	744.62
182	1095.43	748.73
183	1101.45	752.85	17/9
184	1107.47	756.96	55/29
185	1113.49	761.08	78/41
186	1119.51	765.19	21/11
187	1125.52	769.3
188	1131.54	773.42	25/13
189	1137.56	777.53	27/14
190	1143.58	781.65
191	1149.6	785.76
192	1155.62	789.87
193	1161.64	793.99	45/23
194	1167.66	798.1
195	1173.67	802.22	65/33
196	1179.69	806.33	85/43
197	1185.71	810.44
198	1191.73	814.56
199	1197.75	818.67
200	1203.77	822.78
201	1209.79	826.9
202	1215.81	831.01
203	1221.83	835.13
204	1227.84	839.24	63/31
205	1233.86	843.35	51/25
206	1239.88	847.47
207	1245.9	851.58
208	1251.92	855.7
209	1257.94	859.81
210	1263.96	863.92
211	1269.98	868.04
212	1276	872.15
213	1282.01	876.27	65/31, 86/41
214	1288.03	880.38
215	1294.05	884.49	19/9
216	1300.07	888.61
217	1306.09	892.72
218	1312.11	896.84
219	1318.13	900.95
220	1324.15	905.06	58/27
221	1330.16	909.18
222	1336.18	913.29
223	1342.2	917.41
224	1348.22	921.52	85/39
225	1354.24	925.63
226	1360.26	929.75
227	1366.28	933.86
228	1372.3	937.97	95/43
229	1378.32	942.09	51/23
230	1384.33	946.2
231	1390.35	950.32
232	1396.37	954.43
233	1402.39	958.54
234	1408.41	962.66
235	1414.43	966.77	43/19
236	1420.45	970.89
237	1426.47	975	57/25, 98/43
238	1432.49	979.11
239	1438.5	983.23
240	1444.52	987.34
241	1450.54	991.46
242	1456.56	995.57	58/25
243	1462.58	999.68
244	1468.6	1003.8
245	1474.62	1007.91
246	1480.64	1012.03	87/37
247	1486.65	1016.14
248	1492.67	1020.25	45/19
249	1498.69	1024.37
250	1504.71	1028.48	31/13
251	1510.73	1032.59
252	1516.75	1036.71
253	1522.77	1040.82
254	1528.79	1044.94
255	1534.81	1049.05
256	1540.82	1053.16	95/39
257	1546.84	1057.28	22/9
258	1552.86	1061.39
259	1558.88	1065.51
260	1564.9	1069.62
261	1570.92	1073.73	57/23
262	1576.94	1077.85	87/35
263	1582.96	1081.96
264	1588.98	1086.08
265	1594.99	1090.19	98/39
266	1601.01	1094.3	58/23
267	1607.03	1098.42	43/17
268	1613.05	1102.53	33/13
269	1619.07	1106.65
270	1625.09	1110.76
271	1631.11	1114.87
272	1637.13	1118.99
273	1643.14	1123.1
274	1649.16	1127.22	70/27
275	1655.18	1131.33
276	1661.2	1135.44	47/18
277	1667.22	1139.56	55/21
278	1673.24	1143.67
279	1679.26	1147.78
280	1685.28	1151.9	45/17
281	1691.3	1156.01	93/35
282	1697.31	1160.13
283	1703.33	1164.24	99/37
284	1709.35	1168.35	51/19
285	1715.37	1172.47
286	1721.39	1176.58
287	1727.41	1180.7
288	1733.43	1184.81
289	1739.45	1188.92
290	1745.47	1193.04	74/27
291	1751.48	1197.15
292	1757.5	1201.27	69/25
293	1763.52	1205.38
294	1769.54	1209.49
295	1775.56	1213.61
296	1781.58	1217.72
297	1787.6	1221.84
298	1793.62	1225.95	31/11
299	1799.63	1230.06	99/35
300	1805.65	1234.18
301	1811.67	1238.29
302	1817.69	1242.41
303	1823.71	1246.52	43/15
304	1829.73	1250.63
305	1835.75	1254.75
306	1841.77	1258.86
307	1847.79	1262.97
308	1853.8	1267.09
309	1859.82	1271.2	41/14
310	1865.84	1275.32
311	1871.86	1279.43
312	1877.88	1283.54
313	1883.9	1287.66	98/33
314	1889.92	1291.77
315	1895.94	1295.89
316	1901.96	1300	3/1

Harmonics

Approximation of prime harmonics in 316edt

Harmonic		2	3	5	7	11	13	17	19	23
Error	Absolute (¢)	-2.25	+0.00	+0.41	+1.73	+1.69	+1.38	+0.40	+0.45	+0.72
	Relative (%)	-37.4	+0.0	+6.8	+28.7	+28.0	+22.9	+6.7	+7.5	+12.0
Steps (reduced)		199 (199)	316 (0)	463 (147)	560 (244)	690 (58)	738 (106)	815 (183)	847 (215)	902 (270)

Approximation of odd harmonics in 316edt

Harmonic		25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55
Error	Absolute (¢)	+0.82	+0.00	+2.68	+1.58	+1.69	+2.14	+2.24	+1.38	-0.94	+0.87	+0.41	-2.65	-2.56	+0.40	+0.02	+2.10
	Relative (%)	+13.7	+0.0	+44.6	+26.3	+28.0	+35.5	+37.1	+22.9	-15.6	+14.5	+6.8	-43.9	-42.6	+6.7	+0.3	+34.8
Steps (reduced)		926 (294)	948 (0)	969 (21)	988 (40)	1006 (58)	1023 (75)	1039 (91)	1054 (106)	1068 (120)	1082 (134)	1095 (147)	1107 (159)	1119 (171)	1131 (183)	1142 (194)	1153 (205)