Ed6

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Equal divisions of the 6th harmonic (ED6) are tunings obtained by dividing the 6th harmonic in a certain number of equal steps.

Introduction

The 6th harmonic, or hexatave, is particularly wide as far as equivalences go, and there are only about 3.9 hexataves in the human hearing range. If one does indeed deal with hexatave equivalence, this will alter one's musical approach dramatically. Even so, the hexatave is one of the three particularly interesting composite harmonics whereof there are enough within the human hearing range to fill three periods of keyboard (the 10th, and to a lesser extent, the 12th share this property). Following this, the quintessential reason for using a hexatave based tuning is that it will split the difference between octave and tritave based tunings, which is a potentially very desirable thing for a tuning to do given the importance of these harmonics in the musics of much of the world (see 44ed6 and 49ed6). However, this is not to say of ed6's not supporting this important 13 & 18 temperament that they can be dismissed out of hand as entirely worthless, for to do that would shut off all non-patent musical approaches to this equivalence. In fact, taking the n-th root of 6 is itself an approach to finding temperaments like squares, tritonic, and sensi. This approach can of course be used indiscriminately.

Some equal divisions of the hexatave serve as generators for octave temperaments:

Individual pages for ed6's

0…99
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 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
100…199
100 101 102 103 104 105 106 107 108 109
110 111 112 113 114 115 116 117 118 119
120 121 122 123 124 125 126 127 128 129
130 131 132 133 134 135 136 137 138 139
140 141 142 143 144 145 146 147 148 149
150 151 152 153 154 155 156 157 158 159
160 161 162 163 164 165 166 167 168 169
170 171 172 173 174 175 176 177 178 179
180 181 182 183 184 185 186 187 188 189
190 191 192 193 194 195 196 197 198 199

Ed6-edo correspondence

Following ed6's (up to 200) contain good correspondences to edo tunings[1].

Ed6 Edo Comments
13ed6 5edo 13ed6 is 5edo with ~6.9 cent compressed octaves.
Equivalently, 5edo is 13ed6 with hexataves stretched by ~18 cents.
Patent vals match through the 13-limit.
18ed6 7edo 18ed6 is 7edo with ~6.3 cent stretched octaves. Patent vals match through the 13-limit.
21ed6 8edo 21ed6 is 8edo with ~18 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 7.
23ed6 9edo 23ed6 is 9edo with ~14 cent stretched octaves. Patent vals match through the 13-limit.
26ed6 10edo Same ~6.9 cent octave compression as 13ed6~5edo. Patent vals match through the 17-limit.
31ed6 12edo 31ed6 is 12edo with ~0.76 cent stretched octaves. Patent vals match through the 31-limit, with the exception of 11.
36ed6 14edo Same ~6.3 cent octave stretch as 18ed6~7edo. Patent vals match through the 43-limit, with the exception of 5.
39ed6 15edo Same ~6.9 cent octave compression as 13ed6~5edo. Patent vals match through the 13-limit.
44ed6 17edo 44ed6 is 17edo with ~1.5 cent compressed octaves. Patent vals match through the 13-limit, with the exception of 5.
49ed6 19edo 49ed6 is 19edo with ~2.8 cent stretched octaves. Patent vals match through the 13-limit.
52ed6 20edo Same ~6.9 cent octave compression as 13ed6~5edo. Patent vals differ in the 5-limit.
54ed6 21edo Same ~6.3 cent octave stretch as 18ed6~7edo. Patent vals match through the 7-limit.
57ed6 22edo 57ed6 is 22edo with ~2.8 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 13.
62ed6 24edo Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 19-limit.
67ed6 26edo 67ed6 is 26edo with ~3.7 cent stretched octaves. Patent vals match through the 19-limit.
70ed6 27edo 70ed6 is 27edo with ~3.5 cent compressed octaves. Patent vals match through the 13-limit, with the exception of 11.
75ed6 29edo 75ed6 is 29edo with ~0.58 cent compressed octaves. Patent vals match through the 137-limit.
80ed6 31edo 80ed6 is 31edo with ~2.0 cent stretched octaves. Patent vals match through the 13-limit.
83ed6 32edo 83ed6 is 32edo with ~4.1 cent compressed octaves. Patent vals match through the 11-limit, with the exception of 5.
85ed6 33edo 85ed6 is 33edo with ~4.3 cent stretched octaves. Patent vals differ in the 5 and 7-limit.
88ed6 34edo Same ~1.5 cent octave compression as 44ed6~17edo. Patent vals match through the 17-limit, with the exception of 7.
93ed6 36edo Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 31-limit, with the exception of 11.
98ed6 38edo Same ~2.8 cent octave stretch as 49ed6~19edo. Patent vals match through the 11-limit, with the exception of 7.
101ed6 39edo 101ed6 is 39edo with ~2.2 cent compressed octaves. Patent vals match through the 11-limit, with the exception of 7.
106ed6 41edo 106ed6 is 41edo with ~0.19 cent compressed octaves. Patent vals match through the 41-limit.
111ed6 43edo 111ed6 is 43edo with ~1.7 cent stretched octaves. Patent vals match through the 17-limit.
114ed6 44edo Same ~2.8 cent octave compression as 57ed6~22edo. Patent vals match through the 31-limit, with the exception of 11.
116ed6 45edo 116ed6 is 45edo with ~3.3 cent stretched octaves. Patent vals match through the 7-limit.
119ed6 46edo 119ed6 is 46edo with ~0.92 cent compressed octaves. Patent vals match through the 17-limit.
124ed6 48edo Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 43-limit.
129ed6 50edo 129ed6 is 50edo with ~2.3 cent stretched octaves. Patent vals match through the 23-limit.
132ed6 51edo Same ~1.5 cent octave compression as 44ed6~17edo. Patent vals match through the 13-limit, with the exceptions of 5 and 11.
137ed6 53edo 137ed6 is 53edo with ~0.026 cent stretched octaves. Patent vals match through the 61-limit.
142ed6 55edo 142ed6 is 55edo with ~1.46 cent stretched octaves. Patent vals match through the 17-limit, with the exception of 13.
145ed6 56edo 145ed6 is 56edo with ~2.0 cent compressed octaves. Patent vals match through the 19-limit, with the exception of 13.
150ed6 58edo Same ~0.58 cent octave compression as 75ed6~29edo. Patent vals match through the 113-limit.
155ed6 60edo Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 31-limit, with the exception of 11.
158ed6 61edo 158ed6 is 61edo with ~2.4 cent compressed octaves. Patent vals match through the 13-limit, with the exception of 7.
160ed6 62edo Same ~2.0 cent octave stretch as 80ed6~31edo. Patent vals match through the 31-limit.
163ed6 63edo 163ed6 is 63edo with ~1.1 cent compressed octaves. Patent vals match through the 61-limit.
168ed6 65edo 168ed6 is 65edo with ~0.16 cent stretched octaves. Patent vals match through the 157-limit, with the exception of 13.
173ed6 67edo 173ed6 is 67edo with ~1.3 cent stretched octaves. Patent vals match through the 17-limit, with the exception of 5.
176ed6 68edo Same ~1.5 cent octave compression as 44ed6~17edo. Patent vals match through the 23-limit, with the exception of 11.
181ed6 70edo 181ed6 is 70edo with ~0.35 cent compressed octaves. Patent vals match through the 67-limit.
186ed6 72edo Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 19-limit.
189ed6 73edo 189ed6 is 73edo with ~1.9 cent compressed octaves. Patent vals match through the 11-limit.
191ed6 74edo 191ed6 is 74edo with ~1.8 cent stretched octaves. Patent vals match through the 11-limit, with the exception of 7.
194ed6 75edo 194ed6 is 75edo with ~0.79 cent compressed octaves. Patent vals match through the 23-limit, with the exception of 11.
199ed6 77edo 199ed6 is 77edo with ~0.25 cent stretched octaves. Patent vals match through the 37-limit.
  1. Edo with relative error of 6th harmonic below 1/3

See also