217edo

← 216edo

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

217 equal divisions of the octave (abbreviated 217edo or 217ed2), also called 217-tone equal temperament (217tet) or 217 equal temperament (217et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 217 equal parts of about 5.53 ¢ each. Each step represents a frequency ratio of 2^1/217, or the 217th root of 2.

Theory

217edo is a strong 19-limit system, the smallest distinctly consistent in the 19-odd-limit and consistent to the 21-odd-limit as well as the no-23 31-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, excelling as a 2.3.5.13-subgroup temperament. It can be used as a decent approximation of the 31-limit, almost being consistent through the 31-odd-limit except for 23/14, 23/21, 29/23 and their octave complements, with errors below the melodic just-noticeable difference. 224edo, only a bit bigger, offers a much more accurate 13-limit, at the cost of worse higher limits. If one desires even higher consistency and precision, 311edo offers a much better palette.

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)

Subsets and supersets

Since 217 factors into primes as 7 × 31, a product of two Mersenne primes, 217edo contains 7edo and 31edo as subset edos.

Intervals

Here below is an algorithmically generated table of no-37 39-odd-limit intervals of 217edo using Godtone's code, with some manually added useful intervals outside that limit. Intervals in italics are inconsistently mapped.

Table of 217edo intervals

#	Cents	Marks	Approximate intervals
0	0	P1
1	5.53		352/351, 5120/5103, 32805/32768
2	11.06		144/143, 169/168, 225/224
3	16.59		4131/4096
4	22.12		81/80
5	27.65		64/63, 531441/524288
6	33.18		49/48
7	38.71		128/125
8	44.24		40/39, 39/38
9	49.77		36/35, 35/34, 34/33, 1053/1024
10	55.3		33/32, 32/31, 31/30
11	60.83		30/29, 29/28, 28/27
12	66.36		27/26, 26/25
13	71.89		25/24, 24/23
14	77.42		23/22
15	82.95		22/21, 21/20
16	88.48	m2	20/19, 256/243
17	94.01		19/18
18	99.54		18/17, 35/33
19	105.07		17/16
20	110.6		33/31, 16/15
21	116.13	A1	31/29, 2187/2048
22	121.66		15/14, 29/27
23	127.19		14/13
24	132.72		27/25
25	138.25		13/12
26	143.78		38/35, 25/23
27	149.31		12/11
28	154.84		35/32
29	160.37		23/21, 34/31
30	165.9		11/10
31	171.43		32/29, 21/19
32	176.96		31/28
33	182.49		10/9
34	188.02		39/35, 29/26
35	193.55		19/17, 28/25
37	204.61	M2	9/8
38	210.14		44/39, 35/31, 26/23
39	215.67		17/15
40	221.2		25/22
41	226.73		33/29
42	232.26		8/7
43	237.79		39/34, 31/27
44	243.32		23/20, 38/33
45	248.85		15/13
46	254.38		22/19, 29/25
47	259.91		36/31
48	265.44		7/6
50	276.5		34/29, 27/23
51	282.03		20/17
52	287.56		33/28, 46/39, 13/11
53	293.09	m3	32/27
54	298.62		19/16
55	304.15		25/21, 31/26
57	315.21		6/5
59	326.27		35/29, 29/24
60	331.8		23/19, 40/33
61	337.33		17/14, 28/23
62	342.86		39/32
63	348.39		11/9
64	353.92		38/31, 27/22
65	359.45		16/13
66	364.98		21/17
67	370.51		26/21, 31/25
68	376.04		36/29
70	387.1		5/4
72	398.16		44/35, 39/31, 34/27, 29/23
73	403.69		24/19
74	409.22	M3	19/15, 81/64
75	414.75		33/26, 14/11
77	425.81		23/18, 32/25
78	431.34		50/39
79	436.87		9/7
80	442.4		40/31, 31/24
81	447.93		22/17, 35/27
82	453.46		13/10
83	458.99		30/23
84	464.52		17/13
85	470.05		38/29, 21/16
86	475.58		46/35, 25/19, 29/22
87	481.11		33/25
90	497.7	P4	4/3
93	514.29		39/29, 35/26, 31/23
94	519.82		27/20
95	525.35		23/17, 42/31
96	530.88		19/14, 34/25
97	536.41		15/11
98	541.94		26/19
99	547.47		48/35
100	553.0		11/8
101	558.53		40/29, 29/21
102	564.06		18/13
103	569.59		25/18, 32/23
104	575.12		39/28, 46/33
105	580.65		7/5
106	586.18	d5	1024/729
107	591.71		38/27, 31/22
108	597.24		24/17
109	602.76		17/12
110	608.29		44/31, 27/19
111	613.82	A4	729/512
112	619.35		10/7
113	624.88		33/23, 56/39
114	630.41		23/16, 36/25
115	635.94		13/9
116	641.47		42/29, 29/20
117	647.0		16/11
118	652.53		35/24
119	658.06		19/13
120	663.59		22/15
121	669.12		25/17, 28/19
122	674.65		31/21, 34/23
123	680.18		40/27
124	685.71		46/31, 52/35, 58/39
127	702.3	P5	3/2
130	718.89		50/33
131	724.42		44/29, 38/25, 35/23
132	729.95		32/21, 29/19
133	735.48		26/17
134	741.01		23/15
135	746.54		20/13
136	752.07		54/35, 17/11
137	757.6		48/31, 31/20
138	763.13		14/9
139	768.66		39/25
140	774.19		25/16, 36/23
142	785.25		11/7, 52/33
143	790.78	m6	30/19, 128/81
144	796.31		19/12
145	801.84		46/29, 27/17, 62/39, 35/22
147	812.9		8/5
149	823.96		29/18
150	829.49		50/31, 21/13
151	835.02		34/21
152	840.55		13/8
153	846.08		44/27, 31/19
154	851.61		18/11
155	857.14		64/39
156	862.67		23/14, 28/17
157	868.2		33/20, 38/23
158	873.73		48/29, 58/35
160	884.79		5/3
162	895.85		52/31, 42/25
163	901.38		32/19
164	906.91	M6	27/16
165	912.44		22/13, 39/23, 56/33
166	917.97		17/10
167	923.5		46/27, 29/17
169	934.56		12/7
170	940.09		31/18
171	945.62		50/29, 19/11
172	951.15		26/15
173	956.68		33/19, 40/23
174	962.21		54/31, 68/39
175	967.74		7/4
176	973.27		58/33
177	978.8		44/25
178	984.33		30/17
179	989.86		23/13, 62/35, 39/22
180	995.39	m7	16/9
182	1006.45		25/14, 34/19
183	1011.98		52/29, 70/39
184	1017.51		9/5
185	1023.04		56/31
186	1028.57		38/21, 29/16
187	1034.1		20/11
188	1039.63		31/17, 42/23
189	1045.16		64/35
190	1050.69		11/6
191	1056.22		46/25, 35/19
192	1061.75		24/13
193	1067.28		50/27
194	1072.81		13/7
195	1078.34		54/29, 28/15
196	1083.87		58/31
197	1089.4		15/8, 62/33
198	1094.93		32/17
199	1100.46		66/35, 17/9
200	1105.99		36/19
201	1111.52	M7	19/10, 243/128
202	1117.05		40/21, 21/11
203	1122.58		44/23
204	1128.11		23/12, 48/25
205	1133.64		25/13, 52/27
206	1139.17		27/14, 56/29, 29/15
207	1144.7		60/31, 31/16, 64/33
208	1150.23		33/17, 68/35, 35/18
209	1155.76		76/39, 39/20
217	1200.	P8	2/1

Approximation to JI

Selected just intervals

The following tables show how 31-odd-limit intervals are represented in 217edo. Prime harmonics are in bold; inconsistent intervals are in italics.

31-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
27/19, 38/27	0.057	1.0
29/28, 56/29	0.078	1.4
9/5, 10/9	0.085	1.5
17/13, 26/17	0.088	1.6
25/22, 44/25	0.111	2.0
17/16, 32/17	0.114	2.1
17/12, 24/17	0.235	4.3
27/20, 40/27	0.264	4.8
29/21, 42/29	0.271	4.9
19/10, 20/19	0.321	5.8
13/12, 24/13	0.324	5.9
31/16, 32/31	0.335	6.1
3/2, 4/3	0.349	6.3
31/26, 52/31	0.360	6.5
31/21, 42/31	0.400	7.2
19/18, 36/19	0.406	7.3
5/3, 6/5	0.434	7.8
31/17, 34/31	0.449	8.1
25/19, 38/25	0.462	8.3
23/22, 44/23	0.463	8.4
27/25, 50/27	0.519	9.4
15/11, 22/15	0.545	9.9
19/11, 22/19	0.573	10.4
25/23, 46/25	0.574	10.4
17/9, 18/17	0.585	10.6
27/22, 44/27	0.630	11.4
17/10, 20/17	0.669	12.1
31/29, 58/31	0.671	12.1
13/9, 18/13	0.673	12.2
31/24, 48/31	0.684	12.4
9/8, 16/9	0.698	12.6
21/16, 32/21	0.735	13.3
31/28, 56/31	0.749	13.5
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
25/18, 36/25	0.868	15.7
11/10, 20/11	0.894	16.2
27/17, 34/27	0.934	16.9
11/9, 18/11	0.979	17.7
19/17, 34/19	0.991	17.9
29/16, 32/29	1.006	18.2
23/15, 30/23	1.008	18.2
17/15, 30/17	1.018	18.4
27/26, 52/27	1.022	18.5
29/26, 52/29	1.031	18.6
31/18, 36/31	1.033	18.7
23/19, 38/23	1.036	18.7
27/16, 32/27	1.047	18.9
19/13, 26/19	1.079	19.5
7/4, 8/7	1.084	19.6
27/23, 46/27	1.093	19.8
19/16, 32/19	1.104	20.0
15/13, 26/15	1.107	20.0
13/7, 14/13	1.109	20.1
31/20, 40/31	1.118	20.2
29/17, 34/29	1.119	20.2
15/8, 16/15	1.132	20.5
17/14, 28/17	1.198	21.7
25/24, 48/25	1.217	22.0
11/6, 12/11	1.328	24.0
29/24, 48/29	1.355	24.5
23/20, 40/23	1.357	24.5
31/27, 54/31	1.383	25.0
7/6, 12/7	1.433	25.9
31/19, 38/31	1.440	26.0
23/18, 36/23	1.442	26.1
25/17, 34/25	1.452	26.3
31/30, 60/31	1.467	26.5
21/20, 40/21	1.518	27.4
25/13, 26/25	1.541	27.9
17/11, 22/17	1.564	28.3
25/16, 32/25	1.566	28.3
13/11, 22/13	1.652	29.9
11/8, 16/11	1.677	30.3
29/18, 36/29	1.704	30.8
9/7, 14/9	1.782	32.2
29/20, 40/29	1.789	32.3
23/12, 24/23	1.791	32.4
21/19, 38/21	1.839	33.3
7/5, 10/7	1.867	33.8
31/25, 50/31	1.901	34.4
31/22, 44/31	2.013	36.4
23/17, 34/23	2.027	36.6
29/27, 54/29	2.053	37.1
29/19, 38/29	2.110	38.2
23/13, 26/23	2.115	38.2
27/14, 28/27	2.131	38.5
29/15, 30/29	2.138	38.7
23/16, 32/23	2.140	38.7
19/14, 28/19	2.188	39.6
15/14, 28/15	2.216	40.1
25/21, 42/25	2.301	41.6
23/14, 28/23	2.306	41.7
29/23, 46/29	2.384	43.1
21/11, 22/21	2.412	43.6
31/23, 46/31	2.476	44.8
29/25, 50/29	2.572	46.5
25/14, 28/25	2.650	47.9
23/21, 42/23	2.655	48.0
29/22, 44/29	2.683	48.5
11/7, 14/11	2.761	49.9

31-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
27/19, 38/27	0.057	1.0
29/28, 56/29	0.078	1.4
9/5, 10/9	0.085	1.5
17/13, 26/17	0.088	1.6
25/22, 44/25	0.111	2.0
17/16, 32/17	0.114	2.1
17/12, 24/17	0.235	4.3
27/20, 40/27	0.264	4.8
29/21, 42/29	0.271	4.9
19/10, 20/19	0.321	5.8
13/12, 24/13	0.324	5.9
31/16, 32/31	0.335	6.1
3/2, 4/3	0.349	6.3
31/26, 52/31	0.360	6.5
31/21, 42/31	0.400	7.2
19/18, 36/19	0.406	7.3
5/3, 6/5	0.434	7.8
31/17, 34/31	0.449	8.1
25/19, 38/25	0.462	8.3
23/22, 44/23	0.463	8.4
27/25, 50/27	0.519	9.4
15/11, 22/15	0.545	9.9
19/11, 22/19	0.573	10.4
25/23, 46/25	0.574	10.4
17/9, 18/17	0.585	10.6
27/22, 44/27	0.630	11.4
17/10, 20/17	0.669	12.1
31/29, 58/31	0.671	12.1
13/9, 18/13	0.673	12.2
31/24, 48/31	0.684	12.4
9/8, 16/9	0.698	12.6
21/16, 32/21	0.735	13.3
31/28, 56/31	0.749	13.5
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
25/18, 36/25	0.868	15.7
11/10, 20/11	0.894	16.2
27/17, 34/27	0.934	16.9
11/9, 18/11	0.979	17.7
19/17, 34/19	0.991	17.9
29/16, 32/29	1.006	18.2
23/15, 30/23	1.008	18.2
17/15, 30/17	1.018	18.4
27/26, 52/27	1.022	18.5
29/26, 52/29	1.031	18.6
31/18, 36/31	1.033	18.7
23/19, 38/23	1.036	18.7
27/16, 32/27	1.047	18.9
19/13, 26/19	1.079	19.5
7/4, 8/7	1.084	19.6
27/23, 46/27	1.093	19.8
19/16, 32/19	1.104	20.0
15/13, 26/15	1.107	20.0
13/7, 14/13	1.109	20.1
31/20, 40/31	1.118	20.2
29/17, 34/29	1.119	20.2
15/8, 16/15	1.132	20.5
17/14, 28/17	1.198	21.7
25/24, 48/25	1.217	22.0
11/6, 12/11	1.328	24.0
29/24, 48/29	1.355	24.5
23/20, 40/23	1.357	24.5
31/27, 54/31	1.383	25.0
7/6, 12/7	1.433	25.9
31/19, 38/31	1.440	26.0
23/18, 36/23	1.442	26.1
25/17, 34/25	1.452	26.3
31/30, 60/31	1.467	26.5
21/20, 40/21	1.518	27.4
25/13, 26/25	1.541	27.9
17/11, 22/17	1.564	28.3
25/16, 32/25	1.566	28.3
13/11, 22/13	1.652	29.9
11/8, 16/11	1.677	30.3
29/18, 36/29	1.704	30.8
9/7, 14/9	1.782	32.2
29/20, 40/29	1.789	32.3
23/12, 24/23	1.791	32.4
21/19, 38/21	1.839	33.3
7/5, 10/7	1.867	33.8
31/25, 50/31	1.901	34.4
31/22, 44/31	2.013	36.4
23/17, 34/23	2.027	36.6
29/27, 54/29	2.053	37.1
29/19, 38/29	2.110	38.2
23/13, 26/23	2.115	38.2
27/14, 28/27	2.131	38.5
29/15, 30/29	2.138	38.7
23/16, 32/23	2.140	38.7
19/14, 28/19	2.188	39.6
15/14, 28/15	2.216	40.1
25/21, 42/25	2.301	41.6
21/11, 22/21	2.412	43.6
31/23, 46/31	2.476	44.8
29/25, 50/29	2.572	46.5
25/14, 28/25	2.650	47.9
29/22, 44/29	2.683	48.5
11/7, 14/11	2.761	49.9
23/21, 42/23	2.875	52.0
29/23, 46/29	3.146	56.9
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
2.3.5.7.11.13.17.19.23	343/342, 364/363, 392/391, 441/440, 476/475, 507/506, 595/594, 676/675	[⟨217 344 504 609 751 803 887 922 982]]	−0.158	0.2610	4.72

• 217et has lower relative errors than any previous equal temperaments in the 19- and 23-limit. It is the first to beat 72 in the 19-limit and 193 in the 23-limit. The next equal temperament that does better in either subgroup is 243e for absolute error and 270 for relative error.
• 23-limit is not the subgroup it does best, with the no-23 29- and 31-limit approximated even better.
• It is also notable in the 17-limit, with a lower absolute error than any previous equal temperaments, beating 183 and superseded by 224.

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
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	33/32	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	25/19	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 distinct

Notation

Sagittal

217edo can be written in Sagittal using almost the entire Athenian extension by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian. It is identical to 224edo's Sagittal notation, but it uses the 11/7C for the +6/-6 alteration instead of 55C.

It shares the same exact symbol system as the Athenian notation for just intonation or Medium-precision JI notation.

Sagittal notation

Steps		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
Symbol	Evo																					
	Revo																					

Because it uses the entire Athenian system (except for     since it tempers out 1240029/1239040), it allows no accidental enharmonic respellings.

Ups-and-downs notation

The 5-up (quup) alteration neatly maps to the pythagorean-septimal comma.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43
Sharp symbol
Flat symbol

31edo-based meantone notation

Since 217 = 31 × 7, one could base the notation on its inherited meantone fifth 126\217 (18\31) instead of its best fifth.

This could be useful when 31edo is used as a base tuning, where the whole palette of 217edo is only used to provide subtle inflections of the 31edo pitches, similar to how one might use 159edo to provide subtle corrections of 53edo pitches. It also can be written with SZ half-sharps and up/down. The main drawback of this notation is that 3/2 is no longer P5 but ^P5 , so that ~4:5:6 is from C written as C-E-^G, compared to C-^<E-G to the patent val chain-of-fifths ups-and-downs notation.

Alternative 31edo-based notation

Steps	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Symbol	^	^^	^^^	vvvt	vvt	vt	t	^t	^^t	^^^t	v#	vv#	vvv#	#
				v>	>	^>	^^>	^^^>	v>>	>>	^>>	^^>>	^^^>>	v>>>

7edo-based whitewood notation

Since 217 = 7 × 31, one could use the inherited whitewood fifth 124\217, ditch sharps and flats, and instead use ups and downs to represent pitch deviations from the 7edo nominals. Since 31=15*2+1, all pitches can be notated with ups, downs, quips and quids. ~4:5:6 becomes C-^^^>E-^^^G.

Steps	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Sharp Symbol	h	^	^^	^^^ vv>	v>	>	^>	^^>	^^^> vv>>	v>>	>>	^>>	^^>>	vv>>> ^^^>>	v>>>	>>>
Flat symbol		v	vv	vvv ^^<	v<	<	v<	vv<	vvv< vv<<	^<<	<<	v<<	vv<<	vvv<< ^^<<<	^<<<	<<<

Scales

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

Detemperaments

Ringer 217

217edo is the basis for an exceptional Ringer scale that maps an unusually very large amount of the harmonic series (without having to omit any other harmonics) compared to other edos in this size range. Specifically, it maps the complete mode 167 of the harmonic series, corresponding to the entire 333-odd-limit. As 217 = 31 × 7, this can be used to derive 7 possible 31nejis.

167:168:337/2:169:339/2:170:341/2:171:687/4:172:173:347/2:174:349/2:175:351/2:176:353/2:177:178:357/2:179:359/2:180:361/2:181:182:365/2:183:367/2:184:369/2:185:186:373/2:187:375/2:188:189:379/2:190:191:383/2:192:385/2:193:194:389/2:195:196:393/2:197:395/2:198:199:399/2:200:401/2:201:202:203:813/4:204:409/2:205:206:413/2:207:208:417/2:209:210:421/2:211:212:425/2:213:214:429/2:215:216:217:435/2:218:219:439/2:220:221:443/2:222:223:224:449/2:225:226:227:455/2:228:229:459/2:230:231:232:465/2:233:234:469/2:235:236:237:238:239:479/2:240:241:483/2:242:243:244:245:491/2:246:247:248:497/2:249:250:251:252:505/2:253:254:255:256:257:515/2:258:259:260:261:262:263:527/2:264:265:266:267:535/2:268:269:270:271:272:273:274:549/2:275:276:277:278:279:280:281:563/2:282:283:284:285:286:287:288:289:290:291:292:293:294:589/2:295:296:297:298:299:300:301:302:303:304:305:306:307:308:309:310:311:312:313:314:315:316:317:318:319:320:321:322:323:324:325:326:327:328:329:330:331:332:333:334

Deriving 31nejis

This section shows how one can programmatically derive the 7 possible 31nejis aforementioned through use of User:Godtone's copyleft Python 3 code:

>>> r217text = '[paste the above Ringer 217 data here]'
>>> r217=toneji(r217text) # r217
>>> r31s = [ [r217[7*i+j] for i in range(31)]+[r217[j]*2] for j in range(7) ]
>>> r31s2 = [ toneji(':'.join([ str(h) for h in r31 ]),True) for r31 in r31s ]
>>> for i in range(7):
  print(str(i)+'th: ',':'.join([ str(h) for h in r31s2[i] ]))
0th:  274:280:286:293:299:306:313:320:327:334:342:350:358:366:374:383:392:400:409:418:428:438:448:458:468:479:490:500:512:524:535:548
1th:  351:359:367:375:384:393:401:410:420:429:439:449:459:469:480:491:502:514:526:536:549:562:574:588:600:614:628:642:656:672:687:702
2th:  301:308:315:322:329:337:344:352:360:368:376:385:394:402:412:421:430:440:450:460:470:482:492:504:515:527:538:550:563:576:589:602
3th:  258:264:270:276:282:289:295:302:309:316:323:330:338:346:353:361:369:378:386:395:404:413:422:432:442:452:462:472:483:494:505:516
4th:  227:232:237:242:248:253:259:265:271:277:283:290:296:303:310:317:324:331:339:347:354:362:370:379:388:396:406:414:424:434:443:454
5th:  416:425:435:444:455:465:476:486:497:508:520:532:544:556:568:582:594:608:622:636:650:664:680:696:712:728:744:760:778:796:813:832
6th:  213:218:223:228:233:239:244:249:255:261:267:273:279:285:292:298:305:312:319:326:333:341:349:357:365:373:382:390:399:408:417:426
>>> # using the below code can be used to show that only the 0th and 1th 31nejis are mapped correctly by 31edo's patent val
>>> for i in range(7): # (output omitted to avoid spam)
  print(str(i)+'th:\n')
  worstneji(r31s2[i],9)
  print('\n'*2)