User:Overthink/The 7-limit in 171edo: Difference between revisions

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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.
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.
{{Harmonics in equal|171|columns=4|prec=5}}
{{Harmonics in equal|171|columns=4|prec=5}}
{{#invoke: Interval_table | interval_table
{| class="wikitable center-1 right-2 mw-collapsible mw-collapsed"
| tuning=171edo
|+{{nowrap|7-limit intervals in 171edo}}
| additional=
|-
| max_error=35
! Steps
| debug={{lc:}}
! Cents
}}
! Approximate ratios
!Name
|-
| 0
| 0.00
| 1/1
|Unison
|-
| 1
| 7.02
|225/224, 5120/5103, 15625/15552
|Kleisma
|-
| 2
| 14.04
|126/125, 245/243, 78732/78125
|Semicomma
|-
| 3
| 21.05
|81/80, 875/864, 3125/3087
|
|-
| 4
| 28.07
|64/63, 3125/3072, 20000/19683
|
|-
| 5
| 35.09
|49/48, 50/49
|
|-
| 6
| 42.11
|128/125, 525/512
|
|-
| 7
| 49.12
|36/35, 250/243
|
|-
| 8
| 56.14
|405/392, 1323/1280
|
|-
| 9
| 63.16
|28/27, 648/625
|
|-
| 10
| 70.18
|25/24, 20480/19683
|
|-
| 11
| 77.19
|256/245
|
|-
| 12
| 84.21
|21/20
|
|-
| 13
| 91.23
|135/128, 256/243
|
|-
| 14
| 98.25
|
|
|-
| 15
| 105.26
|
|
|-
| 16
| 112.28
|16/15, 2187/2048
|
|-
| 17
| 119.30
|
|
|-
| 18
| 126.32
|
|
|-
| 19
| 133.33
|
|
|-
| 20
| 140.35
|
|
|-
| 21
| 147.37
|
|
|-
| 22
| 154.39
|
|
|-
| 23
| 161.40
|
|
|-
| 24
| 168.42
|
|
|-
| 25
| 175.44
|
|
|-
| 26
| 182.46
|10/9
|
|-
| 27
| 189.47
|
|
|-
| 28
| 196.49
|
|
|-
| 29
| 203.51
|9/8
|
|-
| 30
| 210.53
|
|
|-
| 31
| 217.54
|
|
|-
| 32
| 224.56
|
|
|-
| 33
| 231.58
|8/7
|
|-
| 34
| 238.60
|
|
|-
| 35
| 245.61
|
|
|-
| 36
| 252.63
|
|
|-
| 37
| 259.65
|
|
|-
| 38
| 266.67
|7/6
|
|-
| 39
| 273.68
|
|
|-
| 40
| 280.70
|
|
|-
| 41
| 287.72
|
|
|-
| 42
| 294.74
|
|
|-
| 43
| 301.75
|
|
|-
| 44
| 308.77
|
|
|-
| 45
| 315.79
|6/5
|
|-
| 46
| 322.81
|
|
|-
| 47
| 329.82
|
|
|-
| 48
| 336.84
|
|
|-
| 49
| 343.86
|
|
|-
| 50
| 350.88
|
|
|-
| 51
| 357.89
|
|
|-
| 52
| 364.91
|
|
|-
| 53
| 371.93
|
|
|-
| 54
| 378.95
|
|
|-
| 55
| 385.96
|5/4
|
|-
| 56
| 392.98
|
|
|-
| 57
| 400.00
|
|
|-
| 58
| 407.02
|
|
|-
| 59
| 414.04
|
|
|-
| 60
| 421.05
|
|
|-
| 61
| 428.07
|
|
|-
| 62
| 435.09
|9/7
|
|-
| 63
| 442.11
|
|
|-
| 64
| 449.12
|
|
|-
| 65
| 456.14
|
|
|-
| 66
| 463.16
|
|
|-
| 67
| 470.18
|
|
|-
| 68
| 477.19
|
|
|-
| 69
| 484.21
|
|
|-
| 70
| 491.23
|
|
|-
| 71
| 498.25
|4/3
|
|-
| 72
| 505.26
|
|
|-
| 73
| 512.28
|
|
|-
| 74
| 519.30
|
|
|-
| 75
| 526.32
|
|
|-
| 76
| 533.33
|
|
|-
| 77
| 540.35
|
|
|-
| 78
| 547.37
|
|
|-
| 79
| 554.39
|
|
|-
| 80
| 561.40
|
|
|-
| 81
| 568.42
|
|
|-
| 82
| 575.44
|
|
|-
| 83
| 582.46
|7/5
|
|-
| 84
| 589.47
|
|
|-
| 85
| 596.49
|
|
|-
| 86
| 603.51
|
|
|-
| 87
| 610.53
|
|
|-
| 88
| 617.54
|10/7
|
|-
| 89
| 624.56
|
|
|-
| 90
| 631.58
|
|
|-
| 91
| 638.60
|
|
|-
| 92
| 645.61
|
|
|-
| 93
| 652.63
|
|
|-
| 94
| 659.65
|
|
|-
| 95
| 666.67
|
|
|-
| 96
| 673.68
|
|
|-
| 97
| 680.70
|
|
|-
| 98
| 687.72
|
|
|-
| 99
| 694.74
|
|
|-
| 100
| 701.75
|3/2
|
|-
| 101
| 708.77
|
|
|-
| 102
| 715.79
|
|
|-
| 103
| 722.81
|
|
|-
| 104
| 729.82
|
|
|-
| 105
| 736.84
|
|
|-
| 106
| 743.86
|
|
|-
| 107
| 750.88
|
|
|-
| 108
| 757.89
|
|
|-
| 109
| 764.91
|14/9
|
|-
| 110
| 771.93
|
|
|-
| 111
| 778.95
|
|
|-
| 112
| 785.96
|
|
|-
| 113
| 792.98
|
|
|-
| 114
| 800.00
|
|
|-
| 115
| 807.02
|
|
|-
| 116
| 814.04
|8/5
|
|-
| 117
| 821.05
|
|
|-
| 118
| 828.07
|
|
|-
| 119
| 835.09
|
|
|-
| 120
| 842.11
|
|
|-
| 121
| 849.12
|
|
|-
| 122
| 856.14
|
|
|-
| 123
| 863.16
|
|
|-
| 124
| 870.18
|
|
|-
| 125
| 877.19
|
|
|-
| 126
| 884.21
|5/3
|
|-
| 127
| 891.23
|
|
|-
| 128
| 898.25
|
|
|-
| 129
| 905.26
|
|
|-
| 130
| 912.28
|
|
|-
| 131
| 919.30
|
|
|-
| 132
| 926.32
|
|
|-
| 133
| 933.33
|12/7
|
|-
| 134
| 940.35
|
|
|-
| 135
| 947.37
|
|
|-
| 136
| 954.39
|
|
|-
| 137
| 961.40
|
|
|-
| 138
| 968.42
|7/4
|
|-
| 139
| 975.44
|
|
|-
| 140
| 982.46
|
|
|-
| 141
| 989.47
|
|
|-
| 142
| 996.49
|16/9
|
|-
| 143
| 1003.51
|
|
|-
| 144
| 1010.53
|
|
|-
| 145
| 1017.54
|9/5
|
|-
| 146
| 1024.56
|
|
|-
| 147
| 1031.58
|
|
|-
| 148
| 1038.60
|
|
|-
| 149
| 1045.61
|
|
|-
| 150
| 1052.63
|
|
|-
| 151
| 1059.65
|
|
|-
| 152
| 1066.67
|
|
|-
| 153
| 1073.68
|
|
|-
| 154
| 1080.70
|
|
|-
| 155
| 1087.72
|
|
|-
| 156
| 1094.74
|
|
|-
| 157
| 1101.75
|
|
|-
| 158
| 1108.77
|
|
|-
| 159
| 1115.79
|
|
|-
| 160
| 1122.81
|
|
|-
| 161
| 1129.82
|
|
|-
| 162
| 1136.84
|
|
|-
| 163
| 1143.86
|
|
|-
| 164
| 1150.88
|
|
|-
| 165
| 1157.89
|
|
|-
| 166
| 1164.91
|
|
|-
| 167
| 1171.93
|
|
|-
| 168
| 1178.95
|
|
|-
| 169
| 1185.96
|
|
|-
| 170
| 1192.98
|
|
|-
| 171
| 1200.00
| 2/1
|
|}

Latest revision as of 09:49, 25 December 2025

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)
7-limit intervals in 171edo
Steps Cents Approximate ratios Name
0 0.00 1/1 Unison
1 7.02 225/224, 5120/5103, 15625/15552 Kleisma
2 14.04 126/125, 245/243, 78732/78125 Semicomma
3 21.05 81/80, 875/864, 3125/3087
4 28.07 64/63, 3125/3072, 20000/19683
5 35.09 49/48, 50/49
6 42.11 128/125, 525/512
7 49.12 36/35, 250/243
8 56.14 405/392, 1323/1280
9 63.16 28/27, 648/625
10 70.18 25/24, 20480/19683
11 77.19 256/245
12 84.21 21/20
13 91.23 135/128, 256/243
14 98.25
15 105.26
16 112.28 16/15, 2187/2048
17 119.30
18 126.32
19 133.33
20 140.35
21 147.37
22 154.39
23 161.40
24 168.42
25 175.44
26 182.46 10/9
27 189.47
28 196.49
29 203.51 9/8
30 210.53
31 217.54
32 224.56
33 231.58 8/7
34 238.60
35 245.61
36 252.63
37 259.65
38 266.67 7/6
39 273.68
40 280.70
41 287.72
42 294.74
43 301.75
44 308.77
45 315.79 6/5
46 322.81
47 329.82
48 336.84
49 343.86
50 350.88
51 357.89
52 364.91
53 371.93
54 378.95
55 385.96 5/4
56 392.98
57 400.00
58 407.02
59 414.04
60 421.05
61 428.07
62 435.09 9/7
63 442.11
64 449.12
65 456.14
66 463.16
67 470.18
68 477.19
69 484.21
70 491.23
71 498.25 4/3
72 505.26
73 512.28
74 519.30
75 526.32
76 533.33
77 540.35
78 547.37
79 554.39
80 561.40
81 568.42
82 575.44
83 582.46 7/5
84 589.47
85 596.49
86 603.51
87 610.53
88 617.54 10/7
89 624.56
90 631.58
91 638.60
92 645.61
93 652.63
94 659.65
95 666.67
96 673.68
97 680.70
98 687.72
99 694.74
100 701.75 3/2
101 708.77
102 715.79
103 722.81
104 729.82
105 736.84
106 743.86
107 750.88
108 757.89
109 764.91 14/9
110 771.93
111 778.95
112 785.96
113 792.98
114 800.00
115 807.02
116 814.04 8/5
117 821.05
118 828.07
119 835.09
120 842.11
121 849.12
122 856.14
123 863.16
124 870.18
125 877.19
126 884.21 5/3
127 891.23
128 898.25
129 905.26
130 912.28
131 919.30
132 926.32
133 933.33 12/7
134 940.35
135 947.37
136 954.39
137 961.40
138 968.42 7/4
139 975.44
140 982.46
141 989.47
142 996.49 16/9
143 1003.51
144 1010.53
145 1017.54 9/5
146 1024.56
147 1031.58
148 1038.60
149 1045.61
150 1052.63
151 1059.65
152 1066.67
153 1073.68
154 1080.70
155 1087.72
156 1094.74
157 1101.75
158 1108.77
159 1115.79
160 1122.81
161 1129.82
162 1136.84
163 1143.86
164 1150.88
165 1157.89
166 1164.91
167 1171.93
168 1178.95
169 1185.96
170 1192.98
171 1200.00 2/1
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