The Riemann zeta function and tuning: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 250529746 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 250535182 - Original comment: ** |
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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 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. | ||
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). | |||
For any value of x, we can construct a p-limit [[patent val]]. We do so by rounding log2(q)*x to the nearest integer for each prime q up to p. The [[Tenney-Euclidean metrics|Tenney-Euclidean error]] for this val will be | |||
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Suppose that x can also be continuous, so that it can also represent fractional or &quot;nonoctave&quot; divisions as well. The Bohlen-Pierce scale, 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the &quot;octave&quot; (although the octave itself does not appear in this tuning), and would hence be represented by a value of x = 8.202.<br /> | Suppose that x can also be continuous, so that it can also represent fractional or &quot;nonoctave&quot; divisions as well. The Bohlen-Pierce scale, 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the &quot;octave&quot; (although the octave itself does not appear in this tuning), and would hence be represented by a value of x = 8.202.<br /> | ||
<br /> | <br /> | ||
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).<br /> | |||
<br /> | |||
For any value of x, we can construct a p-limit <a class="wiki_link" href="/patent%20val">patent val</a>. We do so by rounding log2(q)*x to the nearest integer for each prime q up to p. The <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean error</a> for this val will be<br /> | |||
<br /> | <br /> | ||
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