EDF: Difference between revisions

The '''equal division of the fifth''' ('''EDF''' or '''ED3/2''') is a [[tuning]] obtained by dividing the [[3/2|perfect fifth (3/2)]] in a certain number of [[equal]] steps.

Division of 3/2 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, EDF scales have a perceptually important [[Pseudo-octave|false octave]], with various degrees of accuracy.  

Arguments for the utility of 3/2 as a base - whether an equivalence or just a [[period]] - are its being one of the strongest [[consonance]]s after the [[octave]], as well as its use to form structurally important [[pentachord]]s in many musical traditions past and present.

Perhaps the first to divide the perfect fifth was [[Wendy Carlos]] ([http://www.wendycarlos.com/resources/pitch.html ''Three Asymmetric divisions of the octave'']). [[Carlo Serafini]] has also made much use of the alpha, beta and gamma scales.

One way to approach some EDF tunings 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.

Alternatively, [[User:CompactStar|CompactStar]] has also suggeted the usage of [[Half-prime subgroup|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 [[EDT]]s.

== Individual pages for EDFs ==

{| class="wikitable center-all"

|+ style=white-space:nowrap | 0…99

| [[2edf|2]]

| [[3edf|3]]

| [[4edf|4]]

| [[5edf|5]]

| [[6edf|6]]

| [[7edf|7]]

| [[8edf|8]]

| [[9edf|9]]/[[Carlos Alpha|α]]

| [[10edf|10]]

| [[11edf|11]]/[[Carlos Beta|β]]

| [[12edf|12]]

| [[13edf|13]]

| [[14edf|14]]

| [[15edf|15]]

| [[16edf|16]]

| [[17edf|17]]

| [[18edf|18]]

| [[19edf|19]]

| [[20edf|20]]/[[Carlos Gamma|γ]]

| [[21edf|21]]

| [[22edf|22]]

| [[23edf|23]]

| [[24edf|24]]

| [[25edf|25]]

| [[26edf|26]]

| [[27edf|27]]

| [[28edf|28]]

| [[29edf|29]]

| [[30edf|30]]

| [[31edf|31]]

| [[32edf|32]]

| [[33edf|33]]

| [[34edf|34]]

| [[35edf|35]]

| [[36edf|36]]

| [[37edf|37]]

| [[38edf|38]]

| [[39edf|39]]

| [[40edf|40]]

| [[41edf|41]]

| [[42edf|42]]

| [[43edf|43]]

| [[44edf|44]]

| [[45edf|45]]

| [[46edf|46]]

| [[47edf|47]]

| [[48edf|48]]

| [[49edf|49]]

| [[50edf|50]]

| [[51edf|51]]

| [[52edf|52]]

| [[53edf|53]]

| [[54edf|54]]

| [[55edf|55]]

| [[56edf|56]]

| [[57edf|57]]

| [[58edf|58]]

| [[59edf|59]]

| [[60edf|60]]

| [[61edf|61]]

| [[62edf|62]]

| [[63edf|63]]

| [[64edf|64]]

| [[65edf|65]]

| [[66edf|66]]

| [[67edf|67]]

| [[68edf|68]]

| [[69edf|69]]

| [[70edf|70]]

| [[71edf|71]]

| [[72edf|72]]

| [[73edf|73]]

| [[74edf|74]]

| [[75edf|75]]

| [[76edf|76]]

| [[77edf|77]]

| [[78edf|78]]

| [[79edf|79]]

| [[80edf|80]]

| [[81edf|81]]

| [[82edf|82]]

| [[83edf|83]]

| [[84edf|84]]

| [[85edf|85]]

| [[86edf|86]]

| [[87edf|87]]

| [[88edf|88]]

| [[89edf|89]]

| [[90edf|90]]

| [[91edf|91]]

| [[92edf|92]]

| [[93edf|93]]

| [[94edf|94]]

| [[95edf|95]]

| [[96edf|96]]

| [[97edf|97]]

| [[98edf|98]]

| [[99edf|99]]

| | 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.

| | 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.

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

| | 19edf falls 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.

 

[[Category:Edf| ]]

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[[Category:Lists of scales]]

[[Category:Acronyms]]