Chord complexity: Difference between revisions

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

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>

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