EDF: Difference between revisions

[[de:Edf]]

'''EDF''' or '''Ed3/2''' means '''Division of the Perfect Fifth ([[3/2]]) into n equal parts'''.  

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]] ( http://www.wendycarlos.com/resources/pitch.html). [[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 [http://www.youtube.com/watch?v=x_HSMND6RnA example] of it.

 

*[[2edf]]

*[[3edf]]

*[[4edf]]

*[[5edf]]

*[[6edf]]

*[[7edf]]

*[[8edf]]

*[[Carlos_Alpha|9edf]] : Carlos Alpha

*[[10edf]]

*[[Carlos_Beta|11edf]] : Carlos Beta

*[[12edf]]

*[[13edf]]

*[[14edf]]

*[[15edf]]

*[[16edf]]

*[[17edf]]

*[[18edf]]

*[[19edf]]

*[[Carlos_Gamma|20edf]] : Carlos Gamma

*[[21edf]]

*[[22edf]]

*[[23edf]]

*[[24edf]]

*[[25edf]]

*[[26edf]]

*[[27edf]]

*[[28edf]]

*[[29edf]]

*[[30edf]]

*[[31edf]]

*[[32edf]]

*[[33edf]]

*[[34edf]]

*[[35edf]]

*[[36edf]]

*[[37edf]]

*[[38edf]]

*[[39edf]]

*[[40edf]]

*[[41edf]]

*[[42edf]]

*[[43edf]]

*[[44edf]]

*[[45edf]]

*[[46edf]]

*[[47edf]]

*[[48edf]]

*[[49edf]]

*[[50edf]]

*[[51edf]]

*[[52edf]]

*[[53edf]]

*[[54edf]]

*[[55edf]]

*[[56edf]]

*[[57edf]]

*[[58edf]]

| | 4edf is 7edo with 28.5 cent stretched octaves.<br>Equivalently, 7edo is 4edf with 3/2s compressed by ~16 cents.<br>Patent vals match through the 5 limit. Only a rough correspondence.

| | 11edf is 19edo with 12.5 cent stretched octaves. Patent vals match through the 7 limit.<br>If you don't think Carlos beta is accurately represented by 19edo then ignore this correspondence.

| | 12edf falls exactly 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.

| | 16edf falls exactly halfway between 27 and 28 edos. It entirely misses 2/1, and just barely does not miss the "double octave" 4/1.

| | 19edf falls exactly halfway between 32 and 33 edos.

| | Same 6.6 cent octave compression as 10edf~17edo. Patent vals match through the 5 limit, but not the 7 limit.<br>If you don't think Carlos gamma is accurately represented by 34edo then ignore this correspondence.

|51edo

|55edo

|32edf is 55edo with 6.485 cent stretched octaves

|The (10n)edf ~ (17n)edo sequence has broken down completely, 50edf falls exactly 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.

* [[Relative Errors of Small EDFs]]

[[Category:Edf| ]] <!-- main article -->