Chord complexity: Difference between revisions

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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.

It is relatively easy to see that this denominator will be maximized when b/a = 1/1, meaning the span is zero, so that the cosh term ~~cancels out~~ and the denominator is 2. This is the basic behavior for relatively small intervals, which is what we want. For relatively large intervals, on the other hand, the entire thing tends to <math>\min(a,b)^s</math> - and since we have the identity <math>\log \min(a,b) = 1/2 \log(a \cdot b) - 1/2 |\log(b/a)|</math>, meaning we are subtracting the span from the Tenney height, this is definitely not what we want.

It is relatively easy to see that this denominator will be maximized when b/a = 1/1, meaning the span is zero, so that the cosh term equals 1 and thus the denominator is 2. This is the basic behavior for relatively small intervals, which is what we want. For relatively large intervals, on the other hand, the entire thing tends to <math>\min(a,b)^s</math> - and since we have the identity <math>\log \min(a,b) = 1/2 \log(a \cdot b) - 1/2 |\log(b/a)|</math>, meaning we are subtracting the span from the Tenney height, this 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

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 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.
-It is relatively easy to see that this denominator will be maximized when b/a = 1/1, meaning the span is zero, so that the cosh term cancels out and the denominator is 2. This is the basic behavior for relatively small intervals, which is what we want. For relatively large intervals, on the other hand, the entire thing tends to <math>\min(a,b)^s</math> - and since we have the identity <math>\log \min(a,b) = 1/2 \log(a \cdot b) - 1/2 |\log(b/a)|</math>, meaning we are subtracting the span from the Tenney height, this is definitely not what we want.
+It is relatively easy to see that this denominator will be maximized when b/a = 1/1, meaning the span is zero, so that the cosh term equals 1 and thus the denominator is 2. This is the basic behavior for relatively small intervals, which is what we want. For relatively large intervals, on the other hand, the entire thing tends to <math>\min(a,b)^s</math> - and since we have the identity <math>\log \min(a,b) = 1/2 \log(a \cdot b) - 1/2 |\log(b/a)|</math>, meaning we are subtracting the span from the Tenney height, this 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