EDF
The equal division of the fifth (EDF or ED3/2) is a tuning obtained by dividing the perfect fifth in a certain number of equal steps.
Division of the 3:2 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of equivalence is still in its infancy. The utility of 3:2 as a base though, is apparent by being one of the strongest consonances after the octave. Many, if not all, of these scales have a perceptually important pseudo (false) octave, with various degrees of accuracy.
Perhaps the first to divide the perfect fifth was Wendy Carlos (Three Asymmetric divisions of the octave). Carlo Serafini has also made much use of the alpha, beta and gamma scales.
Incidentally, one way to treat 3/2 as an equivalence is the use of the 8:9:10:(12) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone, and conversely one way to treat secundal chords (relative to scales where the large step is no larger than 253¢) as the one true type of triad is the use of 3/2 as the (formal) equivalence. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes six 5/4 to get to 9/8 (tempering out the comma 15625/15552. So, doing this yields 9, 11, and 20 note MOS which the Carlos scales temper equally. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it if it hasn't been named yet, but in any case here is an example of it.
Alternatively, CompactStar has also suggeted the usage of half-prime (such as 3/2.5/2.7/2.11/2...) subgroups for a JI/RTT-based interpretation of EDFs. But such a system, even for the simplest case of 3/2.5/2.7/2, would require very high odd-limit intervals if we want everything to fit within 3/2. The simplest chord in the 7/2-limit which fits inside 3/2 is already quite complex as 1-28/27-10/9 (27:28:30) and that is a very dense tone cluster–to have a non-tone cluster it is required to go up to 1-10/9-7/5 (45:50:63). However this approach has the advantage, or disadvantage depending on your compositional approach, of completely avoiding octaves similar to no-twos subgroups that are used for EDTs.
Individual pages for EDFs
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9/α |
| 10 | 11/β | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
| 20/γ | 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 |
EDF-EDO correspondence
| EDF | EDO | Comments |
|---|---|---|
| 4edf | 7edo | 4edf is 7edo with 28.5 cent stretched octaves. Equivalently, 7edo is 4edf with 3/2s compressed by ~16 cents. Patent vals match through the 5 limit. Only a rough correspondence. |
| 5edf | 9edo | Very rough correspondence - patent vals disagree in the 5 limit. |
| 6edf | 10edo | Also very rough. |
| 7edf | 12edo | 7edf is 12edo with 3.4 cent stretched octaves. Equivalently, 12edo is 7edf with 2.0 cent compressed 3/2s. With the exception of 11 (which falls almost exactly halfway between steps in both cases), the patent vals match through the 31 limit, so the agreement is excellent. |
| 8edf | Since 88cET/octacot is well known to approximate some intervals quite accurately, it would be wrong to lump this in with 14edo. | |
| 9edf | The Carlos alpha scale is neither 15edo nor 16edo. | |
| 10edf | 17edo | 10edf is 17edo with 6.6 cent compressed octaves. Patent vals match through the 13 limit, with the exception of 5 (as expected). |
| 11edf | 19edo | 11edf is 19edo with 12.5 cent stretched octaves. Patent vals match through the 7 limit. If you don't think Carlos beta is accurately represented by 19edo then ignore this correspondence. |
| 12edf | The 4nedf~7nedo correspondence is already breaking down. 12edf falls halfway between 20 and 21 EDOs. It entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\41 of an octave. | |
| 13edf | Perhaps surprisingly, this is not very similar to 22edo. Patent vals differ in the 5 limit. | |
| 14edf | 24edo | Same 3.4 cent octave stretch as 7edf~12edo. Patent vals agree through the 19 limit. |
| 15edf | 26edo | Fairly rough correspondence. 15edf is 26edo with ~17 cent stretched octaves. Patent vals agree through the 5 limit, but not through the 7 limit. |
| 16edf | 16edf falls halfway between 27 and 28 EDOs. It entirely misses 2/1, and just barely does not miss the "double octave" 4/1. | |
| 17edf | 29edo | 17edf is 29edo with 2.5 cent compressed octaves. Patent vals disagree in the 7 limit. |
| 18edf | Perhaps surprisingly, this is not very similar to 31edo. Patent vals differ in the 5 limit. | |
| 19edf | 19edf falls halfway between 32 and 33 EDOs. | |
| 20edf | 34edo | Same 6.6 cent octave compression as 10edf~17edo. Patent vals match through the 5 limit, but not the 7 limit. If you don't think Carlos gamma is accurately represented by 34edo then ignore this correspondence. |
| 21edf | 36edo | Same 3.4 cent octave stretch as 7edf~12edo. Patent vals differ in the 5 limit. |
| 22edf | 38edo | Only rough correspondence. Patent vals differ in the 5 limit. |
| 23edf | 39edo | Only rough correspondence. 23edf is 39edo with ~9.7 cent compressed octaves. Patent vals differ in the 7-limit. |
| 24edf | 41edo | 24edf is 41edo with 0.83 cent compressed octaves. Patent vals match through the 19 limit. |
| 25edf | 43edo | 25edf is 43edo with 7.4 cent stretched octaves, but a rough correspondence. Patent vals differ in the 5 limit. |
| 26edf | Perhaps surprisingly, this is not very similar to 44edo or 45edo. Patent vals differ in the 5 limit. | |
| 27edf | 46edo | 27edf is 46edo with 4.1 cent compressed octaves. Patent vals match through the 5 limit, but not the 7 limit. |
| 28edf | 48edo | Same 3.4 cent octave stretch as 7edf~12edo. Patent vals match through the 5 limit, but not the 7 limit. |
| 29edf | 50edo | 29edf is 50edo with 10.27 cent stretched octaves. |
| 30edf | 51edo | Same 6.6 cent octave compression as 10edf~17edo. |
| 31edf | 53edo | 31edf is 53edo with 0.12 cent stretched octaves. Patent vals match through the 61 limit. |
| 32edf | 55edo | 32edf is 55edo with 6.485 cent stretched octaves. |
| 33edf | 56edo | 33edf is 56edo with 8.8 cent compressed octaves. |
| 34edf | 58edo | Same 2.5 cent octave compression as 17edf~29edo. Patent vals match through the 13 limit. |
| 35edf | 60edo | Same 3.4 cent octave stretch as 7edf~12edo. Patent vals match through the 7 limit. |
| 36edf | Perhaps surprisingly, this is halfway between 61edo and 62edo. | |
| 37edf | 63edo | 37edf is 63edo with 4.78 cent compressed octaves. |
| 38edf | 65edo | 38edf is 65edo with 0.71 cent stretched octaves. Patent vals match through the 11 limit. |
| 39edf | 67edo | Surprisingly, 39edf is actually 67edo with 5.92 cent stretched octaves |
| 40edf | 68edo | Same 6.6 cent octave compression as 10edf~17edo. |
| 41edf | 70edo | 41edf is 70edo with 1.5 cent compressed octaves. Patent vals match through the 17 limit. |
| 42edf | 72edo | This is a rough correspondence, as the (7n)edf ~ (12n)edo sequence begins to break down. Patent vals match through the 7 limit. |
| 43edf | 74edo | 43edf is 74edo with 2.57 cent stretched octaves. In other words, it is an extended meantone with a just 3/2 |
| 44edf | 75edo | 44edf is 75edo with 3.49 cent compressed octaves. |
| 45edf | 77edo | 45edf is 77edo with 1.1 cent stretched octaves. Patent vals match through the 13 limit. |
| 46edf | 79edo | 46edf is 79edo with ~9.7 cent compressed octaves. Patent vals differ in the 7-limit. |
| 47edf | 80edo | 47edf is 80edo with 5.18 cent compressed octaves. |
| 48edf | 82edo | Same 0.83 cent octave compression as 24edf~41edo. Patent vals match through the 11 limit, with the exception of 5. |
| 49edf | 84edo | This is a rough correspondence, as the (7n)edf ~ (12n)edo sequence continues to break down. Patent vals match through the 3 limit. |
| 50edf | The (10n)edf ~ (17n)edo sequence has broken down completely, 50edf falls halfway between 85 and 86 edos. Technically, it may not entirely miss 2/1 (it falls within 7.4 cents on either side), but it nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\171 of an octave. | |
| 51edf | 87edo | Same 2.5 cent octave compression as 17edf~29edo. Patent vals match through the 5 limit, but not the 7 limit. |
| 52edf | 89edo | 52edf is 89edo with 1.4 cent stretched octaves. Patent vals match through the 13 limit, with the exception of 5. |
| 53edf | 91edo | 53edf is 91edo with 5.24 cent stretched octaves. |
| 54edf | 92edo | Same 4.1 cent octave compression as 27edf~46edo. Patent vals also match through the same limit. |
| 55edf | 94edo | 55edf is 94edo with 0.3 cent compressed octaves. Patent vals match through the 47 limit. |
| 56edf | 96edo | This is a rough correspondence, as the (7n)edf ~ (12n)edo sequence momentarily ceases to break down further. Patent vals match through the 3 limit. |
| 57edf | 97edo | 57edo is 97edo with 4.455 cent copmressed octave. |
| 58edf | 99edo | 58edf is 99edo with 1.8 cent compressed octaves. Patent vals match through the 7 limit. |
| 59edf | 101edo | 59edf is 101edo with 1.495 cent stretched octaves. |
| 60edf | 103edo | 60edf is 103edo with 5.02 cent stretched octaves. |
| 61edf | 104edo | 61edf is 104edo with 3.22 cent compressed octaves. |