EDe: Difference between revisions

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'''Equal division of the [[natave]]''' ('''EDe''' or '''EDN''') is the equal division of ~~the constant~~ ''e'' (where ''e'' is treated as a musical interval in the same way as ''2'' is an octave or ''1.5'' is a perfect fifth).

'''Equal division of the [[natave]]''' ('''EDe''' or '''EDN''') is the equal division of [[Acoustic e|acoustic ''e'']] (where ''e'' is treated as a musical interval in the same way as ''2'' is an octave or ''1.5'' is a perfect fifth).

''e'' is of particular interest because of its relationship with logarithms, given the fact that pitch is perceived logarithmically, and the fact that equal divisions are logarithmic.

Sometimes it is convenient to treat [[equal-step tuning]]s as (possibly non-integer) EDes in mathematics and computer programs, since it makes the logarithm used in equations the natural logarithm.

== Correspondence of EDe to EDO ==

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=== 17-EDe ===

17-~~EDN~~ is very close to 12-EDO but with slightly sharp semitones (101.84 cents). This causes the octave to be far too sharp (1222.05 cents; essentially double a Pythagorean large tritone) and gives it a rather pleasant sharp fifth of 712.86 cents.

17-EDe is very close to 12-EDO but with slightly sharp semitones (101.84 cents). This causes the octave to be far too sharp (1222.05 cents; essentially double a Pythagorean large tritone) and gives it a rather pleasant sharp fifth of 712.86 cents.

{{Harmonics in equal|17|1457|536|title=Approximation of harmonics in 17-EDe}}

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24-EDe has third tones so far sharp of 17-EDO that it becomes a stretched 50-ED8 (50\24 is 3606.74 cents). However, 43\24 is essentially the 6th harmonic (1514.83+1586.965=3101.79 cents).

{{Harmonics in equal|24|1457|536|title=Approximation of harmonics in 24-EDe}}

== See also ==

* [[Edϕ]]

* [[Acoustic pi]]

* [[User:Eliora/Phi to the phi]]

[[Category:Transcendental]]

[[Category:Equal-step tuning]]

@@ Line 1: / Line 1: @@
 {{Mathematical interest}}
-'''Equal division of the [[natave]]''' ('''EDe''' or '''EDN''') is the equal division of the constant ''e'' (where ''e'' is treated as a musical interval in the same way as ''2'' is an octave or ''1.5'' is a perfect fifth).
+'''Equal division of the [[natave]]''' ('''EDe''' or '''EDN''') is the equal division of [[Acoustic e|acoustic ''e'']] (where ''e'' is treated as a musical interval in the same way as ''2'' is an octave or ''1.5'' is a perfect fifth).
 ''e'' is of particular interest because of its relationship with logarithms, given the fact that pitch is perceived logarithmically, and the fact that equal divisions are logarithmic.
+Sometimes it is convenient to treat [[equal-step tuning]]s as (possibly non-integer) EDes in mathematics and computer programs, since it makes the logarithm used in equations the natural logarithm.
 == Correspondence of EDe to EDO ==
@@ Line 191: / Line 193: @@
 === 17-EDe ===
--EDN is very close to 12-EDO but with slightly sharp semitones (101.84 cents). This causes the octave to be far too sharp (1222.05 cents; essentially double a Pythagorean large tritone) and gives it a rather pleasant sharp fifth of 712.86 cents.
+-EDe is very close to 12-EDO but with slightly sharp semitones (101.84 cents). This causes the octave to be far too sharp (1222.05 cents; essentially double a Pythagorean large tritone) and gives it a rather pleasant sharp fifth of 712.86 cents.
 {{Harmonics in equal|17|1457|536|title=Approximation of harmonics in 17-EDe}}
@@ Line 201: / Line 203: @@
 -EDe has third tones so far sharp of 17-EDO that it becomes a stretched 50-ED8 (50\24 is 3606.74 cents). However, 43\24 is essentially the 6th harmonic (1514.83+1586.965=3101.79 cents).
 {{Harmonics in equal|24|1457|536|title=Approximation of harmonics in 24-EDe}}
+== See also ==
+* [[Edϕ]]
+* [[Acoustic pi]]
+* [[User:Eliora/Phi to the phi]]
 [[Category:Transcendental]]
 [[Category:Equal-step tuning]]