Chord complexity: Difference between revisions

Line 213:

By using the hyperbolic trigonometric identity <math>\cosh x = (e^x + e^{-x})/2</math>, that denominator can be rewritten in terms of the <math>\cosh</math> function as follows:

<math>\displaystyle (a/b)^{s/2} + (b/a)^{s/2} = 2 \cosh((s/2) \log(b/a))</math>

<math>\displaystyle (a/b)^{s/2} + (b/a)^{s/2} = 2 \cosh((s/2) \cdot \log(b/a))</math>

Now, we note that <math>\log(b/a)</math> can basically be thought of as a function of the span of the dyad. The span in cents would be <math>\text{cents}(b/a) = 1200\log_2(b/a) = 1200\log(b/a)/\log 2</math>, so we have <math>\log(b/a) = \text{cents}(b/a) \log(2)/1200</math>.<ref>In fact, this can also be thought of as a representation 'of' the span in terms of a different unit: rather than cents, we are using "nepers", where one "neper" is equal to <math>1200\log_2(e) = 1731.234</math> cents, rather than the typical units of cents or octaves - perfectly legitimate, if not a bit strange, and used rather frequently in the writings of the late [[Martin Gough]].</ref> Thus, the above expression is a monotonic function purely in terms of the span. Putting it all together, we have

<math>\displaystyle D_s(a, b) = \frac{(ab)^{s/2}}{2 \cosh((s/2) \log(b/a))}</math>

<math>\displaystyle D_s(a, b) = \frac{(ab)^{s/2}}{2 \cosh((s/2) \cdot \log(b/a))}</math>

The numerator is the Benedetti height raised to the power of s, but the denominator is an exponentially increasing monotonic function of the span! This is the basic issue: that intervals are literally being rewarded as the span increases.

@@ Line 213: / Line 213: @@
 By using the hyperbolic trigonometric identity <math>\cosh x = (e^x + e^{-x})/2</math>, that denominator can be rewritten in terms of the <math>\cosh</math> function as follows:
-<math>\displaystyle (a/b)^{s/2} + (b/a)^{s/2} = 2 \cosh((s/2) \log(b/a))</math>
+<math>\displaystyle (a/b)^{s/2} + (b/a)^{s/2} = 2 \cosh((s/2) \cdot \log(b/a))</math>
 Now, we note that <math>\log(b/a)</math> can basically be thought of as a function of the span of the dyad. The span in cents would be <math>\text{cents}(b/a) = 1200\log_2(b/a) = 1200\log(b/a)/\log 2</math>, so we have <math>\log(b/a) = \text{cents}(b/a) \log(2)/1200</math>.<ref>In fact, this can also be thought of as a representation 'of' the span in terms of a different unit: rather than cents, we are using "nepers", where one "neper" is equal to <math>1200\log_2(e) = 1731.234</math> cents, rather than the typical units of cents or octaves - perfectly legitimate, if not a bit strange, and used rather frequently in the writings of the late [[Martin Gough]].</ref> Thus, the above expression is a monotonic function purely in terms of the span. Putting it all together, we have
-<math>\displaystyle D_s(a, b) = \frac{(ab)^{s/2}}{2 \cosh((s/2) \log(b/a))}</math>
+<math>\displaystyle D_s(a, b) = \frac{(ab)^{s/2}}{2 \cosh((s/2) \cdot \log(b/a))}</math>
 The numerator is the Benedetti height raised to the power of s, but the denominator is an exponentially increasing monotonic function of the span! This is the basic issue: that intervals are literally being rewarded as the span increases.