217edo

217EDO is the equal division of the octave into 217 parts of 5.529954 cents each. It is a strong 19-limit system, the smallest uniquely consistent in the 19-limit and consistent to the 21-limit. It tempers out the parakleisma, |8 14 -13>, and the escapade comma, |32 -7 -9> in the 5-limit; 3136/3125, 4375/4374, 10976/10935 and 823543/819200 in the 7-limit; 441/440, 4000/3993 and 5632/5625 in the 11-limit; 364/363, 676/675, 1001/1000, 1575/1573 and 2080/2079 in the 13-limit; 595/594, 833/832, 936/935, 1156/1155, 1225/1224, 1701/1700 in the 17-limit; 343/342, 476/475, 969/968, 1216/1215, 1445/1444, 1521/1520 and 1540/1539 in the 19-limit. It provides the optimal patent val for the arch temperament in the 11 and 13 limits.

Just approximation

Approximation of primary intervals in 217 EDO

Prime number		2	3	5	7	11	13	17	19	23	29	31
Error	absolute (¢)	0.0	+0.349	+0.783	-1.084	+1.677	+0.025	+0.114	+1.104	+2.140	-1.006	-0.335
	relative (%)	0.0	+6.31	+14.16	-19.60	+30.33	+0.46	+2.06	+19.97	+38.71	-18.19	-6.06
Degree (reduced)		217 (0)	344 (127)	504 (70)	609 (175)	751 (100)	803 (152)	887 (19)	922 (54)	982 (114)	1054 (186)	1075 (207)

Selected just intervals

The following table shows how 23-odd-limit intervals are represented in 217EDO. Prime harmonics are in bold; inconsistent intervals are in italic.

Direct mapping (even if inconsistent)

Interval, complement	Error (abs, ¢)
16/13, 13/8	0.025
19/15, 30/19	0.028
10/9, 9/5	0.085
17/13, 26/17	0.088
17/16, 32/17	0.114
24/17, 17/12	0.235
20/19, 19/10	0.321
13/12, 24/13	0.324
4/3, 3/2	0.349
19/18, 36/19	0.406
6/5, 5/3	0.434
23/22, 44/23	0.463
15/11, 22/15	0.545
22/19, 19/11	0.573
18/17, 17/9	0.585
20/17, 17/10	0.669
18/13, 13/9	0.673
9/8, 16/9	0.698
21/16, 32/21	0.735
24/19, 19/12	0.755
26/21, 21/13	0.760
13/10, 20/13	0.758
5/4, 8/5	0.783
21/17, 34/21	0.849
11/10, 20/11	0.894
11/9, 18/11	0.979
19/17, 34/19	0.991
30/23, 23/15	1.008
17/15, 30/17	1.018
23/19, 38/23	1.036
26/19, 19/13	1.079
8/7, 7/4	1.084
19/16, 32/19	1.104
15/13, 26/15	1.107
14/13, 13/7	1.109
16/15, 15/8	1.132
17/14, 28/17	1.198
12/11, 11/6	1.328
23/20, 40/23	1.357
7/6, 12/7	1.433
23/18, 36/23	1.442
21/20, 40/21	1.518
22/17, 17/11	1.564
13/11, 22/13	1.652
11/8, 16/11	1.677
9/7, 14/9	1.782
24/23, 23/12	1.791
21/19, 38/21	1.839
7/5, 10/7	1.867
23/17, 34/23	2.027
26/23, 23/13	2.115
32/23, 23/16	2.140
19/14, 28/19	2.188
15/14, 28/15	2.216
28/23, 23/14	2.306
22/21, 21/11	2.412
23/21, 42/23	2.655
14/11, 11/7	2.761