Harmonic entropy: Difference between revisions

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If we discretize this to an integer array of cents, give S a standard deviation of 12 cents, and represent all delta functions in K as a vertical line of height 1, we can visualize S and K like so:

If we discretize this to an integer array of cents, give S a standard deviation of 17 cents, and represent all delta functions in K as a vertical line of height 1, we can visualize S and K like so:

[[File:S function.png|alt=S function with a standard deviation of 17 cents|frameless]][[File:K function.png|alt=Visualization of K(c) on 1201 samples for intervals of up to numerator/denominator of 200|frameless]]

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$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in J} \frac{e^{i t \log (j_n/j_d)}}{(j_n \cdot j_d)^{w}}$$

Now, suppose we want to analytically continue this so that the set ''J'' is the set of all reduced rational numbers. We can first do so by starting again with unreduced rationals, but expressing each rational not as {{sfrac|''n''|''d''}}, but rather as {{nowrap|{{sfrac|''n''{{'}}|''d''{{-'}}}} · {{sfrac|''c''|''c''}}}}, where ''n''{{'}} and ''d''{{-'}} are coprime, and ''c'' is the ~~gcd~~ of both. For example, we would express {{sfrac|6|4}} as {{nowrap|{{sfrac|3|2}} · {{sfrac|2|2}}}}. Doing so, and assuming that we denote the set of unreduced rationals by ''U'', we get the following equivalent expression of the same convolution kernel above:

Now, suppose we want to analytically continue this so that the set ''J'' is the set of all reduced rational numbers. We can first do so by starting again with unreduced rationals, but expressing each rational not as {{sfrac|''n''|''d''}}, but rather as {{nowrap|{{sfrac|''n''{{``}}|''d''{{-`}}}} · {{sfrac|''c''|''c''}}}}, where ''n''{{``}} and ''d''{{-`}} are coprime, and ''c'' is the GCD of both. For example, we would express {{sfrac|6|4}} as {{nowrap|{{sfrac|3|2}} · {{sfrac|2|2}}}}. Doing so, and assuming that we denote the set of unreduced rationals by ''U'', we get the following equivalent expression of the same convolution kernel above:

$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in \mathbb{U}} \frac{e^{i t \log (\frac{j_c j_{n'}}{j_c j_{d'}})}}{(j_c j_{n'} \cdot j_c j_{d'})^{w}} = |\zeta(w+i t)|^2$$

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-If we discretize this to an integer array of cents, give S a standard deviation of 12 cents, and represent all delta functions in K as a vertical line of height 1, we can visualize S and K like so:
+If we discretize this to an integer array of cents, give S a standard deviation of 17 cents, and represent all delta functions in K as a vertical line of height 1, we can visualize S and K like so:
 [[File:S function.png|alt=S function with a standard deviation of 17 cents|frameless]][[File:K function.png|alt=Visualization of K(c) on 1201 samples for intervals of up to numerator/denominator of 200|frameless]]
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 $$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in J} \frac{e^{i  t \log (j_n/j_d)}}{(j_n \cdot j_d)^{w}}$$
-Now, suppose we want to analytically continue this so that the set ''J'' is the set of all reduced rational numbers. We can first do so by starting again with unreduced rationals, but expressing each rational not as {{sfrac|''n''|''d''}}, but rather as {{nowrap|{{sfrac|''n''{{'}}|''d''{{-'}}}} · {{sfrac|''c''|''c''}}}}, where ''n''{{'}} and ''d''{{-'}} are coprime, and ''c'' is the gcd of both. For example, we would express {{sfrac|6|4}} as {{nowrap|{{sfrac|3|2}} · {{sfrac|2|2}}}}. Doing so, and assuming that we denote the set of unreduced rationals by ''U'', we get the following equivalent expression of the same convolution kernel above:
+Now, suppose we want to analytically continue this so that the set ''J'' is the set of all reduced rational numbers. We can first do so by starting again with unreduced rationals, but expressing each rational not as {{sfrac|''n''|''d''}}, but rather as {{nowrap|{{sfrac|''n''{{``}}|''d''{{-`}}}} · {{sfrac|''c''|''c''}}}}, where ''n''{{``}} and ''d''{{-`}} are coprime, and ''c'' is the GCD of both. For example, we would express {{sfrac|6|4}} as {{nowrap|{{sfrac|3|2}} · {{sfrac|2|2}}}}. Doing so, and assuming that we denote the set of unreduced rationals by ''U'', we get the following equivalent expression of the same convolution kernel above:
 $$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in \mathbb{U}} \frac{e^{i  t \log (\frac{j_c j_{n'}}{j_c j_{d'}})}}{(j_c j_{n'} \cdot j_c j_{d'})^{w}} = |\zeta(w+i t)|^2$$