EDe: Difference between revisions

(5 intermediate revisions by 3 users not shown)

Line 1:

'''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 103:

Line 105:

== Zeta function and tuning ==

In [[Gene Ward Smith~~|Gene~~]]’s [[~~the~~ Riemann zeta function ~~and tuning#The Black Magic Formulas|black magic formulas~~]], it is mathematically more "natural" to consider the number of divisions to the natave rather than the octave, thus scaling the graph of |''Z''(''x'')| horizontally by a factor of 1 instead of 1/ln(2).

In [[Gene Ward Smith]]’s derivation of the [[Riemann zeta function]], it is mathematically more "natural" to consider the number of divisions to the natave rather than the octave, thus scaling the graph of |''Z''(''x'')| horizontally by a factor of 1 instead of 1/ln(2).

The sequence of non-[[stretched and compressed tuning|stretched]] zeta peak EDe's are 1, 2, 3, 10, 20, 36, 39, 72, 111, 163, 202, 264, 466, 538, 740, 1349, 1887... corresponding to {{EDOs|1, 1, 2, 7, 14, 25, 27, 50, 77, 113, 140, 183, 323, 373, 513, 935, 1308}}... edos.

Line 191:

Line 193:

=== 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}}

Line 201:

Line 203:

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 103: / Line 105: @@
 == Zeta function and tuning ==
-In [[Gene Ward Smith|Gene]]’s [[the Riemann zeta function and tuning#The Black Magic Formulas|black magic formulas]], it is mathematically more "natural" to consider the number of divisions to the natave rather than the octave, thus scaling the graph of |''Z''(''x'')| horizontally by a factor of 1 instead of 1/ln(2).
+In [[Gene Ward Smith]]’s derivation of the [[Riemann zeta function]], it is mathematically more "natural" to consider the number of divisions to the natave rather than the octave, thus scaling the graph of |''Z''(''x'')| horizontally by a factor of 1 instead of 1/ln(2).
 The sequence of non-[[stretched and compressed tuning|stretched]] zeta peak EDe's are 1, 2, 3, 10, 20, 36, 39, 72, 111, 163, 202, 264, 466, 538, 740, 1349, 1887... corresponding to {{EDOs|1, 1, 2, 7, 14, 25, 27, 50, 77, 113, 140, 183, 323, 373, 513, 935, 1308}}... edos.
@@ 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]]