217edo

The 217 equal divisions of the octave (217edo), or the 217(-tone) equal temperament (217tet, 217et) when viewed from a regular temperament perspective, is the equal division of the octave into 217 parts of 5.529954 cents each.

Theory

217edo is a strong 19-limit system, the smallest uniquely consistent in the 19-odd-limit and consistent to the 21-odd-limit. It shares the same 5th and 7th harmonics with 31edo (217 = 7 × 31), as well as the 11/9 interval (supporting the birds temperament). However, compared to 31EDO, its patent val differ on the mappings for 3, 11, 13, 17 and 19 – in fact, this EDO has a very accurate 13th harmonic, as well as the 19/15 interval.

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 11- and 13-limit arch and the 11- and 13-limit cotoneum.

JI approximation

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

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[344 -217⟩	[⟨217 344]]	-0.110	0.1101	1.99
2.3.5	[8 14 -13⟩, [32 -7 -9⟩	[⟨217 344 504]]	-0.186	0.1398	2.53
2.3.5.7	3136/3125, 4375/4374, 823543/819200	[⟨217 344 504 609]]	-0.043	0.2757	4.99
2.3.5.7.11	441/440, 3136/3125, 4000/3993, 4375/4374	[⟨217 344 504 609 751]]	-0.131	0.3034	5.49
2.3.5.7.11.13	364/363, 441/440, 676/675, 3136/3125, 4375/4374	[⟨217 344 504 609 751 803]]	-0.111	0.2808	5.08
2.3.5.7.11.13.17	364/363, 441/440, 595/594, 676/675, 1156/1155, 3136/3125	[⟨217 344 504 609 751 803 887]]	-0.099	0.2616	4.73
2.3.5.7.11.13.17.19	343/342, 364/363, 441/440, 476/475, 595/594, 676/675, 1216/1215	[⟨217 344 504 609 751 803 887 922]]	-0.119	0.2504	4.53

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	3\217	16.59	100/99	Quincy
1	5\217	27.65	64/63	Arch
1	10\217	55.30	16875/16384	Escapade
1	18\217	99.54	18/17	Quintagar / quintoneum / quintasandra
1	30\217	165.90	11/10	Satin
1	33\217	182.49	10/9	Mitonic / mineral
1	57\217	315.21	6/5	Parakleismic / paralytic
1	86\217	475.58	320/243	Vulture
1	90\217	497.70	4/3	Gary / cotoneum
1	101\217	558.53	112/81	Condor
7	94\217 (1\217)	519.82 (5.53)	27/20	Brahmagupta
31	90\217 (1\217)	497.70 (5.53)	4/3 (243/242)	Birds

Scales

• Cotoneum5
• Cotoneum7
• Cotoneum12
• Cotoneum17
• Cotoneum29
• Cotoneum41
• Cotoneum53