217edo

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217edo

218edo →

Prime factorization	7 × 31
Step size	5.52995 ¢
Fifth	127\217 (702.304 ¢)
Semitones (A1:m2)	21:16 (116.1 ¢ : 88.48 ¢)
Consistency limit	21
Distinct consistency limit	19

Theory

217edo is a strong 19-limit system, the smallest distinctly 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.

The equal temperament 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, 5632/5625, and 16384/16335 in the 11-limit; 364/363, 676/675, 1001/1000, 1575/1573, 2080/2079 and 4096/4095 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 allows minor minthmic chords, werckismic chords, and sinbadmic chords in the 13-odd-limit, in addition to island chords and nicolic chords in the 15-odd-limit. It provides the optimal patent val for the 11- and 13-limit arch and the 11- and 13-limit cotoneum.

Prime harmonics

Approximation of prime harmonics in 217edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.35	+0.78	-1.08	+1.68	+0.03	+0.11	+1.10	+2.14	-1.01	-0.34
	Relative (%)	+0.0	+6.3	+14.2	-19.6	+30.3	+0.5	+2.1	+20.0	+38.7	-18.2	-6.1
Steps (reduced)		217 (0)	344 (127)	504 (70)	609 (175)	751 (100)	803 (152)	887 (19)	922 (54)	982 (114)	1054 (186)	1075 (207)

Approximation to JI

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. The following tables show how 23-odd-limit intervals are represented in 217edo. Prime harmonics are in bold; inconsistent intervals are in italics.

23-odd-limit intervals in 217edo (direct approximation, even if inconsistent)

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
13/8, 16/13	0.025	0.5
19/15, 30/19	0.028	0.5
9/5, 10/9	0.085	1.5
17/13, 26/17	0.088	1.6
17/16, 32/17	0.114	2.1
17/12, 24/17	0.235	4.3
19/10, 20/19	0.321	5.8
13/12, 24/13	0.324	5.9
3/2, 4/3	0.349	6.3
19/18, 36/19	0.406	7.3
5/3, 6/5	0.434	7.8
23/22, 44/23	0.463	8.4
15/11, 22/15	0.545	9.9
19/11, 22/19	0.573	10.4
17/9, 18/17	0.585	10.6
17/10, 20/17	0.669	12.1
13/9, 18/13	0.673	12.2
9/8, 16/9	0.698	12.6
21/16, 32/21	0.735	13.3
19/12, 24/19	0.755	13.7
13/10, 20/13	0.758	13.7
21/13, 26/21	0.760	13.7
5/4, 8/5	0.783	14.2
21/17, 34/21	0.849	15.3
11/10, 20/11	0.894	16.2
11/9, 18/11	0.979	17.7
19/17, 34/19	0.991	17.9
23/15, 30/23	1.008	18.2
17/15, 30/17	1.018	18.4
23/19, 38/23	1.036	18.7
19/13, 26/19	1.079	19.5
7/4, 8/7	1.084	19.6
19/16, 32/19	1.104	20.0
15/13, 26/15	1.107	20.0
13/7, 14/13	1.109	20.1
15/8, 16/15	1.132	20.5
17/14, 28/17	1.198	21.7
11/6, 12/11	1.328	24.0
23/20, 40/23	1.357	24.5
7/6, 12/7	1.433	25.9
23/18, 36/23	1.442	26.1
21/20, 40/21	1.518	27.4
17/11, 22/17	1.564	28.3
13/11, 22/13	1.652	29.9
11/8, 16/11	1.677	30.3
9/7, 14/9	1.782	32.2
23/12, 24/23	1.791	32.4
21/19, 38/21	1.839	33.3
7/5, 10/7	1.867	33.8
23/17, 34/23	2.027	36.6
23/13, 26/23	2.115	38.2
23/16, 32/23	2.140	38.7
19/14, 28/19	2.188	39.6
15/14, 28/15	2.216	40.1
23/14, 28/23	2.306	41.7
21/11, 22/21	2.412	43.6
23/21, 42/23	2.655	48.0
11/7, 14/11	2.761	49.9

23-odd-limit intervals in 217edo (patent val mapping)

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
13/8, 16/13	0.025	0.5
19/15, 30/19	0.028	0.5
9/5, 10/9	0.085	1.5
17/13, 26/17	0.088	1.6
17/16, 32/17	0.114	2.1
17/12, 24/17	0.235	4.3
19/10, 20/19	0.321	5.8
13/12, 24/13	0.324	5.9
3/2, 4/3	0.349	6.3
19/18, 36/19	0.406	7.3
5/3, 6/5	0.434	7.8
23/22, 44/23	0.463	8.4
15/11, 22/15	0.545	9.9
19/11, 22/19	0.573	10.4
17/9, 18/17	0.585	10.6
17/10, 20/17	0.669	12.1
13/9, 18/13	0.673	12.2
9/8, 16/9	0.698	12.6
21/16, 32/21	0.735	13.3
19/12, 24/19	0.755	13.7
13/10, 20/13	0.758	13.7
21/13, 26/21	0.760	13.7
5/4, 8/5	0.783	14.2
21/17, 34/21	0.849	15.3
11/10, 20/11	0.894	16.2
11/9, 18/11	0.979	17.7
19/17, 34/19	0.991	17.9
23/15, 30/23	1.008	18.2
17/15, 30/17	1.018	18.4
23/19, 38/23	1.036	18.7
19/13, 26/19	1.079	19.5
7/4, 8/7	1.084	19.6
19/16, 32/19	1.104	20.0
15/13, 26/15	1.107	20.0
13/7, 14/13	1.109	20.1
15/8, 16/15	1.132	20.5
17/14, 28/17	1.198	21.7
11/6, 12/11	1.328	24.0
23/20, 40/23	1.357	24.5
7/6, 12/7	1.433	25.9
23/18, 36/23	1.442	26.1
21/20, 40/21	1.518	27.4
17/11, 22/17	1.564	28.3
13/11, 22/13	1.652	29.9
11/8, 16/11	1.677	30.3
9/7, 14/9	1.782	32.2
23/12, 24/23	1.791	32.4
21/19, 38/21	1.839	33.3
7/5, 10/7	1.867	33.8
23/17, 34/23	2.027	36.6
23/13, 26/23	2.115	38.2
23/16, 32/23	2.140	38.7
19/14, 28/19	2.188	39.6
15/14, 28/15	2.216	40.1
21/11, 22/21	2.412	43.6
11/7, 14/11	2.761	49.9
23/21, 42/23	2.875	52.0
23/14, 28/23	3.224	58.3

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 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	3\217	16.59	100/99	Quincy
1	5\217	27.65	64/63	Arch
1	9\217	49.77	36/35	Hemiquindromeda
1	10\217	55.30	16875/16384	Escapade
1	18\217	99.54	18/17	Quintagar / quintoneum / quinsandra
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	Cotoneum
1	101\217	558.53	112/81	Condor
7	94\217 (1\217)	519.82 (5.53)	27/20 (325/324)	Brahmagupta
31	90\217 (1\217)	497.70 (5.53)	4/3 (243/242)	Birds

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Scales

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