Chord complexity: Difference between revisions

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<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~~.

This shows us how this metric relates to the Benedetti height. The numerator is the geometric mean raised to the power of s, but the denominator is an exponentially increasing monotonic function of the span! This is the basic issue: if we view Benedetti height as a decent barometer for how things should scale, then relative to that, intervals are being rewarded for having larger spans.

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.

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 b/a = 1/1, meaning the span is zero, so that the cosh term equals 1 and thus the denominator is 2, leaving only the numerator of <math>(ab)^{s/2}</math>. 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 <math>\min(a,b)^s</math>. Since we have the identity <math>\log \min(a,b) = 1/2 \log(a \cdot b) - 1/2 |\log(b/a)|</math>, this means 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

@@ Line 219: / Line 219: @@
 <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.
+This shows us how this metric relates to the Benedetti height. The numerator is the geometric mean raised to the power of s, but the denominator is an exponentially increasing monotonic function of the span! This is the basic issue: if we view Benedetti height as a decent barometer for how things should scale, then relative to that, intervals are being rewarded for having larger spans.
-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.
+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 b/a = 1/1, meaning the span is zero, so that the cosh term equals 1 and thus the denominator is 2, leaving only the numerator of <math>(ab)^{s/2}</math>. 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 <math>\min(a,b)^s</math>. Since we have the identity <math>\log \min(a,b) = 1/2 \log(a \cdot b) - 1/2 |\log(b/a)|</math>, this means 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