The Riemann zeta function and tuning: Difference between revisions

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So far we have shown the following:

Error on prime powers: <math>\log |\zeta(\sigma+it)|<math>

* Error on prime powers: <math>\log |\zeta(\sigma+it)|</math>

Error on unreduced rationals: <math>|\zeta(\sigma+it)|^2<math>

* Error on unreduced rationals: <math>|\zeta(\sigma+it)|^2</math>

Error on reduced rationals: <math>|\zeta(\sigma+it)|^2/\zeta(2\sigma)<math>

* Error on reduced rationals: <math>\frac{|\zeta(\sigma+it)|^2}{\zeta(2\sigma)}</math>

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. 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.

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 <math>Z(t)</math> function, which is defined so that we have <math>|Z(t)| = |\zeta(t)|</math>. So, the absolute value of the <math>Z</math> 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.

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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:

Error on harmonics only: <math>|\textbf{Re}[zeta(\sigma+it)]|<math>

Error on harmonics only: <math>|\textbf{Re}[\zeta(\sigma+it)]|</math>

Note that, although the last four expressions were all monotonic transformations of one another, this one is not - this is the 'real part' of the zeta function, whereas the others were all some simple monotonic function of the 'absolute value' of the zeta function. The results, however, are very similar - in particular, the peaks are approximately to one another, shifted by only a small amount (at least for reasonably-sized EDOs up to a few hundred).

@@ Line 412: / Line 412: @@
 So far we have shown the following:
-Error on prime powers: <math>\log |\zeta(\sigma+it)|<math>
+* Error on prime powers: <math>\log |\zeta(\sigma+it)|</math>
-Error on unreduced rationals: <math>|\zeta(\sigma+it)|^2<math>
+* Error on unreduced rationals: <math>|\zeta(\sigma+it)|^2</math>
-Error on reduced rationals: <math>|\zeta(\sigma+it)|^2/\zeta(2\sigma)<math>
+* Error on reduced rationals: <math>\frac{|\zeta(\sigma+it)|^2}{\zeta(2\sigma)}</math>
-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. 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.
+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 <math>Z(t)</math> function, which is defined so that we have <math>|Z(t)| = |\zeta(t)|</math>. So, the absolute value of the <math>Z</math> 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.
@@ Line 422: / Line 422: @@
 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:
-Error on harmonics only: <math>|\textbf{Re}[zeta(\sigma+it)]|<math>
+Error on harmonics only: <math>|\textbf{Re}[\zeta(\sigma+it)]|</math>
 Note that, although the last four expressions were all monotonic transformations of one another, this one is not - this is the 'real part' of the zeta function, whereas the others were all some simple monotonic function of the 'absolute value' of the zeta function. The results, however, are  very similar - in particular, the peaks are approximately to one another, shifted by only a small amount (at least for reasonably-sized EDOs up to a few hundred).