# Ed5

The equal division of the 5th harmonic (ed5) is a tuning obtained by dividing the 5th harmonic in a certain number of equal steps.

## Theory

The 5th harmonic is particularly wide as far as equivalences go. There are (at absolute most) ~4.8 pentaves within the human hearing range; imagine if that were the case with octaves. If one does indeed deal with pentave equivalence, this range restriction is a crucial consideration. Pentave equivalence itself may have a basis in Western music seeing as minor chords have an octave of 5 in their root (i.e. 10:12:15).

One way to treat 5/1 as an equivalence is by eliminating the primes 2 and 3. The most fundamental chord in this paradigm is 5:7:11. This chord can be approximated in a 5.7.11 (or "no-twos-or-threes 11-limit") subgroup regular temperament by eliminating the comma 859375/823543, equating a stack of 7 7/5 generators with 11/5. Other equivalences that could be used for such "no-two-or-threes" music include equal divisions of 11/5 and equal divisions of 11/7.

The quintessential example of a pentave based tuning is hyperpyth (see 17ed5). However, perhaps the more common reason to use these scales is in approximation with lower harmonic factors than 5. This approach is highlighted by Hieronymus (20ed5) which itself is a zeta peak tuning (not "no-fives", full on zeta). Other reasons for taking the n-th root of 5 include finding temperaments like orwell, meantone, and thuja. This approach can of course be used indiscriminately.

Some equal divisions of the pentave are known by alternate names or have special interest:

## Individual pages for ed5's

 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
 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199
 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299

## Ed5-edo correspondence

Following ed5's (up to 339) contain good correspondences to edo tunings[1].

7ed5 3edo 7ed5 is 3edo with ~5.9 cent compressed octaves.
Equivalently, 3edo is 7ed5 with pentaves stretched by ~13.7 cents.
Patent vals match through the 67-limit.
9ed5 4edo Very rough correspondence (~38 cent octave stretch), but patent vals agree through the 7-limit.
14ed5 6edo Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 17-limit.
16ed5 7edo Only rough correspondence (~19 cent octave stretch). Patent vals differ in the 7-limit.
21ed5 9edo Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 59-limit.
23ed5 10edo 23ed5 is 10edo with ~11 cent stretched octaves. Patent vals match through the 23-limit, with the exception of 11 and 17.
28ed5 12edo Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 19-limit, with the exception of 13 (as expected).
30ed5 13edo 30ed5 is 13edo with ~7.4 cent stretched octaves. Patent vals match through the 19-limit, with the exception of 3 (which neither represents well).
35ed5 15edo Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 13-limit.
37ed5 16edo 37ed5 is 16edo with ~4.9 cent stretched octaves. Patent vals match through the 23-limit.
42ed5 18edo Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 17-limit, with the exception of 11.
44ed5 19edo 44ed5 is 19edo with ~3.2 cent stretched octaves. Patent vals match through the 13-limit.
49ed5 21edo Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 17-limit.
51ed5 22edo 51ed5 is 22edo with ~1.9 cent stretched octaves. Patent vals match through the 19-limit.
56ed5 24edo This is a rough correspondence, as the (7n)ed5 ~ (3n)edo sequence begins to break down. Patent vals match through the 13-limit, with the exception of 7.
58ed5 25edo 58ed5 is 25edo with ~1.0 cent stretched octaves. Patent vals match through the 73-limit, with the exception of 13.
63ed5 27edo This is a rough correspondence, but the (7n)ed5 ~ (3n)edo sequence is momentarily fixed. Patent vals match through the 13-limit.
65ed5 28edo 65ed5 is 28edo with ~0.26 cent stretched octaves. Patent vals match through the 43-limit.
72ed5 31edo 72ed5 is 31edo with ~0.34 cent compressed octaves. Patent vals match through the 31-limit.
74ed5 32edo Same 4.9 cent octave stretch as 37ed5~16edo. Patent vals match through the 13-limit with the exception of 7 and 11.
79ed5 34edo 79ed5 is 34edo with ~0.83 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 7 (which neither represents well).
81ed5 35edo 81ed5 is 35edo with ~4.0 cent stretched octaves. Patent vals match through the 11-limit.
84ed5 36edo Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 11-limit, with the exception of 7.
86ed5 37edo 86ed5 is 37edo with ~1.2 cent compressed octaves. Patent vals match through the 19-limit.
88ed5 38edo Same 3.2 cent octave stretch as 44ed5~19edo. Patent vals differ in the 7-limit.
93ed5 40edo 93ed5 is 40edo with ~1.6 cent compressed octaves. Patent vals match through the 7-limit.
95ed5 41edo 95ed5 is 41edo with ~2.5 cent stretched octaves. Patent vals match through the 11-limit.
100ed5 43edo 100ed5 is 43edo with ~1.9 cent compressed octaves. Patent vals match through the 37-limit.
102ed5 44edo Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 23-limit, with the exception of 7.
107ed5 46edo 107ed5 is 46edo with ~2.1 cent compressed octaves. Patent vals match through the 11-limit.
109ed5 47edo 109ed5 is 47edo with ~1.4 cent stretched octaves. Patent vals match through the 17-limit, with the exception of 11.
114ed5 49edo 114ed5 is 49edo with ~2.4 cent compressed octaves. Patent vals match through the 11-limit.
116ed5 50edo Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals agree through the 43-limit.
121ed5 52edo 121ed5 is 52edo with ~2.6 cent compressed octaves, but the correspondence is rough. Patent vals match through the 11-limit, with the exception of 3 (which neither represents well).
123ed5 53edo 123ed5 is 53edo with ~0.61 cent stretched octaves. Patent vals match through the 29-limit.
128ed5 55edo 128ed5 is 55edo with ~2.8 cent compressed octaves, but the correspondence is rough. Patent vals differ in the 7-limit.
130ed5 56edo Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals agree through the 103-limit.
135ed5 58edo 135ed5 is 58edo with ~2.9 cent compressed octaves. Patent vals match through the 13-limit.
137ed5 59edo 137ed5 is 59edo with ~0.05 cent compressed octaves. Patent vals match through the 179-limit.
139ed5 60edo 139ed5 is 60edo with ~2.7 cent stretched octaves. Pantent vals match through the 31-limit with the exception of 11.
144ed5 62edo Same 0.34 cent octave compression as 72ed5~31edo. Patent vals match through the 19-limit, with the exception of 11.
146ed5 63edo 146ed5 is 63edo with ~2.3 cent stretched octaves. Patent vals match through the 13-limit.
151ed5 65edo 151ed5 is 65edo with ~0.59 cent compressed octaves. Patent vals match through the 53-limit, with the exception of 7 (as expected).
153ed5 66edo Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 13-limit, with the exception of 3.
158ed5 68edo Same 0.83 cent octave compression as 79ed5~34edo. Patent vals agree through the 23-limit.
160ed5 69edo 160ed5 is 69edo with ~1.6 cent stretched octaves. Patent vals differ in the 7-limit.
165ed5 71edo 165ed5 is 71edo with ~1.0 cent compressed octaves. Patent vals match through the 37-limit.
167ed5 72edo 167ed5 is 72edo with ~1.3 cent stretched octaves. Patent vals agree through the 19-limit.
172ed5 74edo Same 1.2 cent octave compression as 86ed5~37edo. Patent vals agree through the 13-limit.
174ed5 75edo Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals differ in the 7-limit.
179ed5 77edo 179ed5 is 77edo with ~1.4 cent compressed octaves. Patent vals match through the 19-limit, with the exception of 11.
181ed5 78edo 181ed5 is 78edo with ~0.73 cent stretched octaves. Patent vals match through the 59-limit, with the exception of 13.
186ed5 80edo Same 1.6 cent octave compression as 93ed5~40edo. Patent vals agree through the 29-limit.
188ed5 81edo 188ed5 is 81edo with ~0.49 cent stretched octaves. Patent vals match through the 41-limit.
193ed5 83edo 193ed5 is 83edo with ~1.7 cent compressed octaves. Patent vals match through the 7-limit.
195ed5 84edo Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals agree through the 31-limit.
200ed5 86edo Same 1.9 cent octeave compression as 100ed5~43edo. Patent vals match alternate primes through the 17-limit.
202ed5 87edo 202ed5 is 87edo with ~0.05 cent stretched octaves. Patent vals match through the 71-limit.
204ed5 88edo Same 1.9 cent octave stretch as 51ed5~22edo. Patent vals match through the 7-limit.
209ed5 90edo 209ed5 is 90edo with ~0.15 cent compressed octaves. Patent vals match through the 53-limit.
211ed5 91edo 211ed5 is 91edo with ~1.7 cent stretched octaves. Patent vals match through the 7-limit.
216ed5 93edo Same 0.34 cent octave compression as 72ed5~31edo. Patent vals agree through the 31-limit.
218ed5 94edo Same 1.4 cent octave stretch as 109ed5~47edo. Patent vals match through the 23-limit, with the exception of 13.
223ed5 96edo 223ed5 is 96edo with ~0.51 cent compressed octaves. Patent vals match through the 13-limit.
225ed5 97edo 225ed5 is 97edo with ~1.2 cent stretched octaves. Patent vals match through the 19-limit, with the exception of 11.
230ed5 99edo 230ed5 is 99edo with ~0.67 cent compressed octaves. Patent vals match through the 7-limit.
232ed5 100edo Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals match through the 13-limit.
237ed5 102edo Same 0.83 cent octave compression as 79ed5~34edo. Patent vals differ in the 7-limit.
239ed5 103edo 239ed5 is 103edo with ~0.80 cent stretched octaves. Patent vals match through the 17-limit.
244ed5 105edo 244ed5 is 105edo with ~0.97 cent compressed octaves. Patent vals disagree in the 5-limit and higher prime limit.
246ed5 106edo Same 0.61 cent octave stretch as 123ed5~53edo. Patent vals match through the 43-limit, with the exception of 7.
251ed5 108edo 251ed5 is 108edo with ~1.1 cent compressed octaves. Patent vals match through the 13-limit.
253ed5 109edo 253ed5 is 109edo with ~0.43 cent stretched octaves. Patent vals match through the 13-limit.
258ed5 111edo Same 1.2 cent octave compression as 86ed5~37edo. Patent vals agree through the 19-limit.
260ed5 112edo Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals match through the 47-limit, with the exception of 3.
265ed5 114edo 265ed5 is 114edo with ~2.2 cent compressed octaves. Patent vals match through the 43-limit, with the exception of 11, 19 and 37.
267ed5 115edo 267ed5 is 115edo with ~0.10 cent stretched octaves. Patent vals match through the 17-limit.
274ed5 118edo Same 0.05 cent octave compression as 137ed5~59edo. Patent vals agree through the 103-limit.
276ed5 119edo 276ed5 is 119edo with ~1.35 cent stretched octaves. Patent vals match through the 23-limit with the exception of 13 and 19.
281ed5 121edo 281ed5 is 121edo with ~0.20 cent compressed octaves. Patent vals match through the 29-limit.
283ed5 122edo 283ed5 is 122edo with ~1.2 cent stretched octaves. Patent vals match through the 13-limit.
288ed5 124edo Same 0.34 cent octave compression as 72ed5~31edo. Patent vals agree through the 23-limit.
290ed5 125edo Same 1.0 cent octave stretch as 58ed5~25edo. Patent vals match through the 37-limit, with the exception of 13.
295ed5 127edo 295ed5 is 127edo with ~0.47 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 11.
297ed5 128edo 297ed5 is 128edo with ~0.84 cent stretched octaves. Patent vals match through the 7-limit.
302ed5 130edo Same 0.59 cent octave compression as 151ed5~65edo. Patent vals match through the 41-limit, with the exception of 11, 19 and 23.
304ed5 131edo 304ed5 is 131edo with ~0.68 cent stretched octaves. Patent vals match through the 41-limit, with the exception of 13 and 23.
309ed5 133edo 309ed5 is 133edo with ~0.73 cent compressed octaves. Patent vals match through the 71-limit, with the exception of 7 and 59.
311ed5 134edo 311ed5 is 134edo with ~0.53 cent stretched octaves. Patent vals match through the 59-limit, with the exception of 11, 19 and 53.
316ed5 136edo Same 0.83 cent octave compression as 79ed5~34edo. Patent vals match through the 41-limit, with the exception of 13 and 23.
318ed5 137edo 318ed5 is 137edo with ~0.39 cent stretched octaves. Patent vals match through the 23-limit, with the exception of 7.
323ed5 139edo 323ed5 is 139edo with ~0.94 cent compressed compressed octaves. Patent vals match alternating primes through the 11-limit.
325ed5 140edo Same 0.26 cent octave stretch as 65ed5~28edo. Patent vals match through the 73-limit, with the exception of 31 and 67.
330ed5 142edo Same 1.0 cent octave compression as 165ed5~71edo. Patent vals match through the 13-limit, with the exception of 11.
332ed5 143edo 32ed5 is 143edo with ~0.13 cent stretched octaves. Patent vals match through the 29-limit, with the exception of 7, 17 and 19.
337ed5 145edo 337ed5 is 145edo with ~1.1 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 7.
339ed5 146edo 339ed5 is practically identical to 146edo, with a slight octave stretch (~0.0053 cents). Patent vals match through the 827-limit.
1. Edo with relative error of 5th harmonic below 1/3