The Riemann zeta function and tuning: Difference between revisions

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__FORCETOC__

=Preliminaries=

Suppose x is a variable representing some equal division of the octave. For example, if x = 80, x reflects 80edo with a step size of 15 cents and with pure octaves. Suppose that x can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well. The Bohlen-Pierce scale, 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of x = 8.202.

Suppose x is a variable representing some equal division of the octave. For example, if x = 80, x reflects 80edo with a step size of 15 cents and with pure octaves. Suppose that x can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well. The [[Bohlen-Pierce|Bohlen-Pierce scale]], 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of x = 8.202.

Now suppose that ||x|| denotes the difference between x and the integer nearest to x. For example, ||8.202|| would be .202, since it's the difference between 8.202 and the nearest integer, which is 8. ||7.95|| would be .05, which is the difference between 7.95 and the nearest integer, which is 8. Mathematically speaking, ||x|| denotes the function |x - floor(x+1/2)|.

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 __FORCETOC__
 =Preliminaries=
-Suppose x is a variable representing some equal division of the octave. For example, if x = 80, x reflects 80edo with a step size of 15 cents and with pure octaves. Suppose that x can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well. The Bohlen-Pierce scale, 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of x = 8.202.
+Suppose x is a variable representing some equal division of the octave. For example, if x = 80, x reflects 80edo with a step size of 15 cents and with pure octaves. Suppose that x can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well. The [[Bohlen-Pierce|Bohlen-Pierce scale]], 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of x = 8.202.
 Now suppose that ||x|| denotes the difference between x and the integer nearest to x. For example, ||8.202|| would be .202, since it's the difference between 8.202 and the nearest integer, which is 8. ||7.95|| would be .05, which is the difference between 7.95 and the nearest integer, which is 8. Mathematically speaking, ||x|| denotes the function |x - floor(x+1/2)|.