User:Overthink/The 7-limit in 171edo

An excellent system for approximating the 7-limit is 171edo. The 9-odd-limit tonality diamond is approximated with just over 0.4 cents of error, and all 7-limit intervals with odd limit up to 10,000,000 are consistent (first odd to cause inconsistencies is 13671875). 171edo is thus an excellent system for classifying intervals and commas simply and accurately.

Approximation of prime harmonics in 171edo

Harmonic		2	3	5	7
Error	Absolute (¢)	+0.00000	-0.20061	-0.34880	-0.40485
	Relative (%)	+0.0	-2.9	-5.0	-5.8
Steps (reduced)		171 (0)	271 (100)	397 (55)	480 (138)

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	7.02		^D, ^4E♭♭
2	14.04		^^D, ^5E♭♭
3	21.05		^3D, ^6E♭♭
4	28.07		^4D, ^7E♭♭
5	35.09	49/48, 50/49, 51/50	^5D, v8E♭
6	42.11	41/40, 42/41	^6D, v7E♭
7	49.12	35/34, 36/35	^7D, v6E♭
8	56.14	31/30, 32/31	^8D, v5E♭
9	63.16	28/27	v7D♯, v4E♭
10	70.18	25/24, 51/49	v6D♯, v3E♭
11	77.19	23/22	v5D♯, vvE♭
12	84.21	21/20	v4D♯, vE♭
13	91.23	39/37, 58/55	v3D♯, E♭
14	98.25	18/17, 55/52	vvD♯, ^E♭
15	105.26	17/16	vD♯, ^^E♭
16	112.28	16/15	D♯, ^3E♭
17	119.3	15/14	^D♯, ^4E♭
18	126.32	43/40	^^D♯, ^5E♭
19	133.33	27/25	^3D♯, ^6E♭
20	140.35	51/47	^4D♯, ^7E♭
21	147.37	37/34, 49/45	^5D♯, v8E
22	154.39	35/32, 47/43	^6D♯, v7E
23	161.4	45/41, 56/51	^7D♯, v6E
24	168.42	43/39, 54/49	^8D♯, v5E
25	175.44	31/28, 52/47	v7D𝄪, v4E
26	182.46	10/9	v6D𝄪, v3E
27	189.47	29/26, 48/43	v5D𝄪, vvE
28	196.49	28/25	v4D𝄪, vE
29	203.51	9/8	E
30	210.53	35/31	^E, ^4F♭
31	217.54	17/15	^^E, ^5F♭
32	224.56	33/29, 41/36	^3E, ^6F♭
33	231.58	8/7	^4E, ^7F♭
34	238.6	31/27, 39/34	^5E, v8F
35	245.61		^6E, v7F
36	252.63	37/32	^7E, v6F
37	259.65	36/31, 43/37	^8E, v5F
38	266.67	7/6	v7E♯, v4F
39	273.68	41/35, 48/41	v6E♯, v3F
40	280.7	20/17	v5E♯, vvF
41	287.72		v4E♯, vF
42	294.74	32/27, 51/43	F
43	301.75	25/21	^F, ^4G♭♭
44	308.77	43/36, 49/41, 55/46	^^F, ^5G♭♭
45	315.79	6/5	^3F, ^6G♭♭
46	322.81	41/34, 47/39	^4F, ^7G♭♭
47	329.82	52/43	^5F, v8G♭
48	336.84	17/14, 62/51	^6F, v7G♭
49	343.86	50/41	^7F, v6G♭
50	350.88	49/40, 60/49	^8F, v5G♭
51	357.89		v7F♯, v4G♭
52	364.91	21/17, 58/47	v6F♯, v3G♭
53	371.93	31/25	v5F♯, vvG♭
54	378.95	51/41, 56/45	v4F♯, vG♭
55	385.96	5/4	v3F♯, G♭
56	392.98		vvF♯, ^G♭
57	400	29/23, 34/27	vF♯, ^^G♭
58	407.02	43/34, 62/49	F♯, ^3G♭
59	414.04	33/26, 47/37	^F♯, ^4G♭
60	421.05	37/29, 51/40	^^F♯, ^5G♭
61	428.07	32/25, 41/32	^3F♯, ^6G♭
62	435.09	9/7	^4F♯, ^7G♭
63	442.11	31/24, 40/31	^5F♯, v8G
64	449.12	35/27	^6F♯, v7G
65	456.14	56/43	^7F♯, v6G
66	463.16	17/13	^8F♯, v5G
67	470.18	21/16	v7F𝄪, v4G
68	477.19	29/22, 54/41	v6F𝄪, v3G
69	484.21	41/31, 45/34	v5F𝄪, vvG
70	491.23		v4F𝄪, vG
71	498.25	4/3	G
72	505.26		^G, ^4A♭♭
73	512.28	39/29, 43/32	^^G, ^5A♭♭
74	519.3	27/20, 58/43	^3G, ^6A♭♭
75	526.32	42/31	^4G, ^7A♭♭
76	533.33	34/25, 49/36	^5G, v8A♭
77	540.35	41/30, 56/41	^6G, v7A♭
78	547.37	48/35	^7G, v6A♭
79	554.39	51/37, 62/45	^8G, v5A♭
80	561.4	47/34	v7G♯, v4A♭
81	568.42	25/18	v6G♯, v3A♭
82	575.44	46/33, 60/43	v5G♯, vvA♭
83	582.46	7/5	v4G♯, vA♭
84	589.47	45/32, 52/37	v3G♯, A♭
85	596.49	24/17	vvG♯, ^A♭
86	603.51	17/12	vG♯, ^^A♭
87	610.53	37/26	G♯, ^3A♭
88	617.54	10/7	^G♯, ^4A♭
89	624.56	33/23, 43/30	^^G♯, ^5A♭
90	631.58	36/25, 49/34	^3G♯, ^6A♭
91	638.6		^4G♯, ^7A♭
92	645.61	45/31	^5G♯, v8A
93	652.63	35/24, 51/35	^6G♯, v7A
94	659.65	41/28, 60/41	^7G♯, v6A
95	666.67	25/17, 47/32	^8G♯, v5A
96	673.68	31/21	v7G𝄪, v4A
97	680.7	40/27, 43/29	v6G𝄪, v3A
98	687.72	58/39	v5G𝄪, vvA
99	694.74		v4G𝄪, vA
100	701.75	3/2	A
101	708.77		^A, ^4B♭♭
102	715.79	62/41	^^A, ^5B♭♭
103	722.81	41/27, 44/29	^3A, ^6B♭♭
104	729.82	32/21	^4A, ^7B♭♭
105	736.84	26/17, 49/32	^5A, v8B♭
106	743.86	43/28	^6A, v7B♭
107	750.88	54/35	^7A, v6B♭
108	757.89	31/20, 48/31	^8A, v5B♭
109	764.91	14/9	v7A♯, v4B♭
110	771.93	25/16	v6A♯, v3B♭
111	778.95	58/37	v5A♯, vvB♭
112	785.96	52/33	v4A♯, vB♭
113	792.98	49/31	v3A♯, B♭
114	800	27/17, 46/29	vvA♯, ^B♭
115	807.02	51/32	vA♯, ^^B♭
116	814.04	8/5	A♯, ^3B♭
117	821.05	45/28	^A♯, ^4B♭
118	828.07	50/31	^^A♯, ^5B♭
119	835.09	34/21, 47/29	^3A♯, ^6B♭
120	842.11		^4A♯, ^7B♭
121	849.12	49/30	^5A♯, v8B
122	856.14	41/25	^6A♯, v7B
123	863.16	28/17, 51/31	^7A♯, v6B
124	870.18	43/26	^8A♯, v5B
125	877.19		v7A𝄪, v4B
126	884.21	5/3	v6A𝄪, v3B
127	891.23		v5A𝄪, vvB
128	898.25	42/25	v4A𝄪, vB
129	905.26	27/16	B
130	912.28		^B, ^4C♭
131	919.3	17/10	^^B, ^5C♭
132	926.32	41/24	^3B, ^6C♭
133	933.33	12/7	^4B, ^7C♭
134	940.35	31/18	^5B, v8C
135	947.37		^6B, v7C
136	954.39		^7B, v6C
137	961.4	54/31	^8B, v5C
138	968.42	7/4	v7B♯, v4C
139	975.44	58/33	v6B♯, v3C
140	982.46	30/17	v5B♯, vvC
141	989.47	62/35	v4B♯, vC
142	996.49	16/9	C
143	1003.51	25/14	^C, ^4D♭♭
144	1010.53	43/24, 52/29	^^C, ^5D♭♭
145	1017.54	9/5	^3C, ^6D♭♭
146	1024.56	47/26, 56/31	^4C, ^7D♭♭
147	1031.58	49/27	^5C, v8D♭
148	1038.6	51/28	^6C, v7D♭
149	1045.61		^7C, v6D♭
150	1052.63		^8C, v5D♭
151	1059.65		v7C♯, v4D♭
152	1066.67	50/27	v6C♯, v3D♭
153	1073.68		v5C♯, vvD♭
154	1080.7	28/15	v4C♯, vD♭
155	1087.72	15/8	v3C♯, D♭
156	1094.74	32/17	vvC♯, ^D♭
157	1101.75	17/9	vC♯, ^^D♭
158	1108.77	55/29	C♯, ^3D♭
159	1115.79	40/21	^C♯, ^4D♭
160	1122.81	44/23	^^C♯, ^5D♭
161	1129.82	48/25	^3C♯, ^6D♭
162	1136.84	27/14	^4C♯, ^7D♭
163	1143.86	31/16, 60/31	^5C♯, v8D
164	1150.88	35/18	^6C♯, v7D
165	1157.89	41/21	^7C♯, v6D
166	1164.91	49/25	^8C♯, v5D
167	1171.93		v7C𝄪, v4D
168	1178.95		v6C𝄪, v3D
169	1185.96		v5C𝄪, vvD
170	1192.98		v4C𝄪, vD
171	1200	2/1	D