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.

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== 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.

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 {{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.
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 == 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.