Chord complexity: Difference between revisions
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\displaystyle T_s\left(x_1, x_2, \ldots, x_N\right) = \log B_s\left(x_1, x_2, \ldots, x_N\right) = \frac{ | \displaystyle T_s\left(x_1, x_2, \ldots, x_N\right) = \log B_s\left(x_1, x_2, \ldots, x_N\right) = \frac{\log\left(x_1 \cdot x_2 \cdot \ldots \cdot x_N\right)}{N} - \frac{\log N}{s} | ||
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\displaystyle \log W_s\left(x_1, x_2, \ldots, x_N\right) = \log \max\left(x_1, x_2, \ldots, x_N\right) - \frac{ | \displaystyle \log W_s\left(x_1, x_2, \ldots, x_N\right) = \log \max\left(x_1, x_2, \ldots, x_N\right) - \frac{\log N}{s} | ||
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\log \max\left(n, d\right) = \frac{ | \log \max\left(n, d\right) = \frac{\log\left(n\cdot d\right) + \left|\log\left(\frac{n}{d}\right)\right|}{2} | ||
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\displaystyle \log W_s\left(x_1, x_2, \ldots, x_N\right) = \log B_s\left(x_1, x_2, \ldots, x_N\right) + \frac{ | \displaystyle \log W_s\left(x_1, x_2, \ldots, x_N\right) = \log B_s\left(x_1, x_2, \ldots, x_N\right) + \frac{\log\frac{x_N}{x_1} + \log\frac{x_N}{x_2} + \ldots + \log\frac{x_N}{x_{N-1}}}{N} | ||
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Now, it is interesting to look at how the above expression scales for intervals that are fairly small. It is relatively easy to see that this denominator will be maximized when {{nowrap|''b''/''a'' {{=}} 1}}, meaning the span is zero, so that the cosh term equals 1 and thus the denominator is 2, leaving only the numerator of {{nowrap|(''ab'')<sup>''s''/2</sup>}}. For relatively small intervals, we'll get something close to this, meaning span is irrelevant—perhaps what we want. | Now, it is interesting to look at how the above expression scales for intervals that are fairly small. It is relatively easy to see that this denominator will be maximized when {{nowrap|''b''/''a'' {{=}} 1}}, meaning the span is zero, so that the cosh term equals 1 and thus the denominator is 2, leaving only the numerator of {{nowrap|(''ab'')<sup>''s''/2</sup>}}. For relatively small intervals, we'll get something close to this, meaning span is irrelevant—perhaps what we want. | ||
For relatively large intervals, on the other hand, the entire thing simply tends to {{nowrap|min(''a'', ''b'')<sup>''s''</sup>}}. Since we have the identity | For relatively large intervals, on the other hand, the entire thing simply tends to {{nowrap|min(''a'', ''b'')<sup>''s''</sup>}}. Since we have the identity | ||
$$ | |||
\log \min(a, b) = \frac{\log(ab) - \left|\log\left(\frac{b}{a}\right)\right|}{2} | |||
$$ | |||
this means meaning we are subtracting the span from the Tenney height, which is definitely not what we want. | |||
So what we will do is simply modify our formula so that the behavior for relatively small intervals is preserved across the entire interval spectrum, thus "span-correcting" our original formula. Doing so, we simply keep the numerator (the "complexity" part) the same, while pretending that we have always plugged 1/1 into the denominator. Thus, we simply get | So what we will do is simply modify our formula so that the behavior for relatively small intervals is preserved across the entire interval spectrum, thus "span-correcting" our original formula. Doing so, we simply keep the numerator (the "complexity" part) the same, while pretending that we have always plugged 1/1 into the denominator. Thus, we simply get | ||