Ed6
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ED6 means division of the sixth harmonic (6/1) into n equal parts.
Introduction
The sixth harmonic is particularly wide as far as equivalences go. There are (at absolute most) ~4.3 hexataves within the human hearing range; imagine if that were the case with octaves. If one does indeed deal with hexatave equivalence, this fact shapes 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 ED6s 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:
- 4ED6 squares generator (with octaves)
- 5ED6 tritonic generator (with octaves)
- 6ED6 harry generator (with octaves)
- 7ED6 sensi generator (with octaves)
- 8ED6 würschmidt generator (with octaves)
- 10ED6 myna generator (with octaves)
- 14ED6 hemisensi generator (with octaves)
- 16ED6 hemiwürschimdt generator (with octaves)
- 17ED6 Minortonic generator (with octaves)
- 19ED6 Porcupine generator (with octaves)
- 21ED6 progression generator (with octaves)
- 24ED6 twothirdtonic generator (with octaves)
- 26ED6 septidiasemi generator (with octaves)
- 35ED6 octacot generator (with octaves)
- 40ED6 valentine generator (with octaves)
Individual pages for ED6s
ED6-EDO correspondence
Following ED6s (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. |
- ↑ EDO with relative error of 6th harmonic below 1/3