EDF: Difference between revisions

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

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 ~~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~~.

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.

~~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.

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.

~~Interestingly~~, ~~there are not many harmonically useful JI~~ [[~~subgroup~~]]~~s including~~ [[~~3/2~~]]~~, aside from complicated fractional subgroups like the [[3L 2s~~ (3/2~~-equivalent~~)~~|uranian]]~~-~~related~~ 3/2.7/4.5/2, ~~and subgroups including~~ 2 ~~or 3 (which all contain octaves)~~. ~~This is because~~ the 3/2-~~reduced 5th and 11th harmonics are very near~~ 3/2 ~~(680 and 641 cents respectively), in contrast to~~ [[~~EDO~~]]~~s and~~ [[~~EDT~~]]~~s where most harmonics are closer~~ to ~~the middle. The smallest subgroup without octaves that could be compared~~ to ~~say, the~~ [[5~~-limit~~]] ~~in usability, would be 3/2~~.~~7.13~~, or ~~3/2.4.7 if non~~-~~primes~~ are ~~allowed~~.

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 ==

*[[2edf]]

{| class="wikitable center-all"

*[[3edf]]

|+ style=white-space:nowrap | 0…99

*[[4edf]]

| [[0edf|0]]

*[[5edf]]

| [[1edf|1]]

*[[6edf]]

| [[2edf|2]]

*[[7edf]]

| [[3edf|3]]

*[[8edf]]

| [[4edf|4]]

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

| [[5edf|5]]

*[[10edf]]

| [[6edf|6]]

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

| [[7edf|7]]

*[[12edf]]

| [[8edf|8]]

*[[13edf]]

| [[9edf|9]]/[[Carlos Alpha|α]]

*[[14edf]]

|-

*[[15edf]]

| [[10edf|10]]

*[[16edf]]

| [[11edf|11]]/[[Carlos Beta|β]]

*[[17edf]]

| [[12edf|12]]

*[[18edf]]

| [[13edf|13]]

*[[19edf]]

| [[14edf|14]]

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

| [[15edf|15]]

*[[21edf]]

| [[16edf|16]]

*[[22edf]]

| [[17edf|17]]

*[[23edf]]

| [[18edf|18]]

*[[24edf]]

| [[19edf|19]]

*[[25edf]]

|-

*[[26edf]]

| [[20edf|20]]/[[Carlos Gamma|γ]]

*[[27edf]]

| [[21edf|21]]

*[[28edf]]

| [[22edf|22]]

*[[29edf]]

| [[23edf|23]]

*[[30edf]]

| [[24edf|24]]

*[[31edf]]

| [[25edf|25]]

*[[32edf]]

| [[26edf|26]]

*[[33edf]]

| [[27edf|27]]

*[[34edf]]

| [[28edf|28]]

*[[35edf]]

| [[29edf|29]]

*[[36edf]]

|-

*[[37edf]]

| [[30edf|30]]

*[[38edf]]

| [[31edf|31]]

*[[39edf]]

| [[32edf|32]]

*[[40edf]]

| [[33edf|33]]

*[[41edf]]

| [[34edf|34]]

*[[42edf]]

| [[35edf|35]]

*[[43edf]]

| [[36edf|36]]

*[[44edf]]

| [[37edf|37]]

*[[45edf]]

| [[38edf|38]]

*[[46edf]]

| [[39edf|39]]

*[[47edf]]

|-

*[[48edf]]

| [[40edf|40]]

*[[49edf]]

| [[41edf|41]]

*[[50edf]]

| [[42edf|42]]

*[[51edf]]

| [[43edf|43]]

*[[52edf]]

| [[44edf|44]]

*[[53edf]]

| [[45edf|45]]

*[[54edf]]

| [[46edf|46]]

*[[55edf]]

| [[47edf|47]]

*[[56edf]]

| [[48edf|48]]

*[[57edf]]

| [[49edf|49]]

*[[58edf]]

|-

*[[59edf]]

| [[50edf|50]]

*[[60edf]]

| [[51edf|51]]

*[[61edf]]

| [[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]]

|}

== EDF-EDO correspondence ==

Line 317:

Line 371:

== See also ==

* [[Relative errors of small EDFs]]

* [[Ed9/4]]

~~[[Category:Equal-step tuning]]~~

[[Category:Edf| ]]

[[Category:Edf| ]]

[[Category:Lists of scales]]

[[Category:Acronyms]]

@@ Line 3: / Line 3: @@
 |de = Edt
 }}
-The '''equal division of the fifth''' ('''EDF''' or '''ED3/2''') is a [[tuning]] obtained by dividing the [[3/2|perfect fifth]] in a certain number of [[equal]] steps.
+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 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.
+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.
-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.
+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.
-Interestingly, there are not many harmonically useful JI [[subgroup]]s including [[3/2]], aside from complicated fractional subgroups like the [[3L 2s (3/2-equivalent)|uranian]]-related 3/2.7/4.5/2, and subgroups including 2 or 3 (which all contain octaves). This is because the 3/2-reduced 5th and 11th harmonics are very near 3/2 (680 and 641 cents respectively), in contrast to [[EDO]]s and [[EDT]]s where most harmonics are closer to the middle. The smallest subgroup without octaves that could be compared to say, the [[5-limit]] in usability, would be 3/2.7.13, or 3/2.4.7 if non-primes are allowed.
+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 ==
-*[[2edf]]
+{| class="wikitable center-all"
-*[[3edf]]
+|+ style=white-space:nowrap | 0…99
-*[[4edf]]
+| [[0edf|0]]
-*[[5edf]]
+| [[1edf|1]]
-*[[6edf]]
+| [[2edf|2]]
-*[[7edf]]
+| [[3edf|3]]
-*[[8edf]]
+| [[4edf|4]]
-*[[9edf]] : [[Carlos Alpha]]
+| [[5edf|5]]
-*[[10edf]]
+| [[6edf|6]]
-*[[11edf]] : [[Carlos Beta]]
+| [[7edf|7]]
-*[[12edf]]
+| [[8edf|8]]
-*[[13edf]]
+| [[9edf|9]]/[[Carlos Alpha|α]]
-*[[14edf]]
+|-
-*[[15edf]]
+| [[10edf|10]]
-*[[16edf]]
+| [[11edf|11]]/[[Carlos Beta|β]]
-*[[17edf]]
+| [[12edf|12]]
-*[[18edf]]
+| [[13edf|13]]
-*[[19edf]]
+| [[14edf|14]]
-*[[20edf]] : [[Carlos Gamma]]
+| [[15edf|15]]
-*[[21edf]]
+| [[16edf|16]]
-*[[22edf]]
+| [[17edf|17]]
-*[[23edf]]
+| [[18edf|18]]
-*[[24edf]]
+| [[19edf|19]]
-*[[25edf]]
+|-
-*[[26edf]]
+| [[20edf|20]]/[[Carlos Gamma|γ]]
-*[[27edf]]
+| [[21edf|21]]
-*[[28edf]]
+| [[22edf|22]]
-*[[29edf]]
+| [[23edf|23]]
-*[[30edf]]
+| [[24edf|24]]
-*[[31edf]]
+| [[25edf|25]]
-*[[32edf]]
+| [[26edf|26]]
-*[[33edf]]
+| [[27edf|27]]
-*[[34edf]]
+| [[28edf|28]]
-*[[35edf]]
+| [[29edf|29]]
-*[[36edf]]
+|-
-*[[37edf]]
+| [[30edf|30]]
-*[[38edf]]
+| [[31edf|31]]
-*[[39edf]]
+| [[32edf|32]]
-*[[40edf]]
+| [[33edf|33]]
-*[[41edf]]
+| [[34edf|34]]
-*[[42edf]]
+| [[35edf|35]]
-*[[43edf]]
+| [[36edf|36]]
-*[[44edf]]
+| [[37edf|37]]
-*[[45edf]]
+| [[38edf|38]]
-*[[46edf]]
+| [[39edf|39]]
-*[[47edf]]
+|-
-*[[48edf]]
+| [[40edf|40]]
-*[[49edf]]
+| [[41edf|41]]
-*[[50edf]]
+| [[42edf|42]]
-*[[51edf]]
+| [[43edf|43]]
-*[[52edf]]
+| [[44edf|44]]
-*[[53edf]]
+| [[45edf|45]]
-*[[54edf]]
+| [[46edf|46]]
-*[[55edf]]
+| [[47edf|47]]
-*[[56edf]]
+| [[48edf|48]]
-*[[57edf]]
+| [[49edf|49]]
-*[[58edf]]
+|-
-*[[59edf]]
+| [[50edf|50]]
-*[[60edf]]
+| [[51edf|51]]
-*[[61edf]]
+| [[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]]
+|}
 == EDF-EDO correspondence ==
@@ Line 317: / Line 371: @@
 == See also ==
 * [[Relative errors of small EDFs]]
+* [[Ed9/4]]
-[[Category:Equal-step tuning]]
+[[Category:Edf| ]]
-[[Category:Edf| ]] <!-- main article -->
+<!-- main article -->
+[[Category:Lists of scales]]
 [[Category:Acronyms]]
+{{todo|inline=1|cleanup|improve layout}}