Riemann zeta function: Difference between revisions

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Since the second is a simple monotonic transformation of the first, we can see that the same function basically measures both the relative error on just the prime powers, and also on all unreduced rationals, at least in the sense that EDOs will be ranked identically by both measures. The third function is really just the second function divided by a constant, since we only really care about letting <math>t</math> vary—we instead typically set <math>\sigma</math> to some value which represents the weighting "rolloff" on rationals. So, all three of these functions will rank EDOs identically.

We also note that, above, Gene tended to look at things in terms of the Z(''t'') function, which is defined so that we have {{nowrap|{{!}}Z(''t'') {{=}} {{!}}ζ(''t''){{!}}}}. So, the absolute value of the Z function is also monotonically equivalent to the above set of expressions, so that any one of these things will produce the same ranking on EDOs.

We also note that, above, Gene tended to look at things in terms of the Z(''t'') function, which is defined so that we have {{nowrap|{{!}}Z(''t''){{!}} {{=}} {{!}}ζ(''t''){{!}}}}. So, the absolute value of the Z function is also monotonically equivalent to the above set of expressions, so that any one of these things will produce the same ranking on EDOs.

It turns out that using the same principles of derivation above, we can also derive another expression, this time for the relative error on only the harmonics—i.e. those intervals of the form <math>1/1, 2/1, 3/1, ... n/1, ...</math>. This was studied in a paper by Peter Buch called [[:File:Zetamusic5.pdf|"Favored cardinalities of scales"]]. The expression is:

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 Since the second is a simple monotonic transformation of the first, we can see that the same function basically measures both the relative error on just the prime powers, and also on all unreduced rationals, at least in the sense that EDOs will be ranked identically by both measures. The third function is really just the second function divided by a constant, since we only really care about letting <math>t</math> vary&mdash;we instead typically set <math>\sigma</math> to some value which represents the weighting "rolloff" on rationals. So, all three of these functions will rank EDOs identically.
-We also note that, above, Gene tended to look at things in terms of the Z(''t'') function, which is defined so that we have {{nowrap|{{!}}Z(''t'') {{=}} {{!}}&zeta;(''t''){{!}}}}. So, the absolute value of the Z function is also monotonically equivalent to the above set of expressions, so that any one of these things will produce the same ranking on EDOs.
+We also note that, above, Gene tended to look at things in terms of the Z(''t'') function, which is defined so that we have {{nowrap|{{!}}Z(''t''){{!}} {{=}} {{!}}&zeta;(''t''){{!}}}}. So, the absolute value of the Z function is also monotonically equivalent to the above set of expressions, so that any one of these things will produce the same ranking on EDOs.
 It turns out that using the same principles of derivation above, we can also derive another expression, this time for the relative error on only the harmonics&mdash;i.e. those intervals of the form <math>1/1, 2/1, 3/1, ... n/1, ...</math>. This was studied in a paper by Peter Buch called [[:File:Zetamusic5.pdf|"Favored cardinalities of scales"]]. The expression is: