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>>Pecan tree patterns, in a nutshell
>
>this is a concept that comes up now and again, for instance k.
>pepper brought it up here . . . on my trip to europe i saw these
>large groves of trees planted in perfect grids for the first
>time, outside milano and then in holland . . . very interesting

Here's my reply to that message...

>Keenan Pepper wrote,
>
>>Okay, how about if some madman went around and tacked a sign
>>on each tree with two numbers, how many trees up from you and
>>how many over, with a slash between them, like a fraction. The
>>tree in front of you has "1/0", the one two your left has "0/1"
>>the one up and to the right of you has "1/1", the one to the
>>right of that one has "1/2", you get the idea.
>
>How do you plan on dealing with negatives here? S'thinks you
>only need the upper-right (+,+) quadrant. In which case, you've
>got a strict "lambdoma" (like Partch's tonality diamond, but
>starting with 0/0 and proceeding by integer-limit instead of
>odd-limit). And since this structure has a plane of mirror
>symmetry intersecting its line of 1/1's, you really only need
>the lower 45-degrees of the quadrant.
>
>>Also, the closest tree in between two others has the mediant
>>of the their two fractions.
>
>Closest tree? Isn't it the middle tree, only working when there
>are an odd number of trees involved?
>
>>Get this, the distance a tree is from you exactly corresponds
>>to its musical consonance.
>
>Inversely, I'm sure you meant.
>
>For a fraction, a/b, that's sqrt(a^2 + b^2) -- an integer-limit
>version of the prime-factor measure you've posted here before.
>This new version also requires we ignore ratios that aren't in
>lowest form.
>
>One interesting thing about this metric is that, like a numerator
>limit for proper fractions, it assigns very low consonance to
>some very wide intervals (like 7/1). I guess it's a matter of
>definition what happens to consonance as intervals get very wide.
>We're used to thinking of consonance and dissonance as mutually-
>exclusive inverses, but perhaps wide intervals are neither
>consonant or dissonant. . .
>
>>How well you can see through the trees is harmonic entropy.
>
>Hurm! This leaves a lot to the imagination -- perhaps you
>suggest using your distance metric of a given radius, instead of
>a Farey series of a given order, to obtain the initial template
>of ratios. Since how well you see through depends on how far the
>target tree is from you (as you pointed out), this is not
>unreasonable. In fact, it's been proposed on this list before.
>
>IIRC, the Farey series was liked because the entropy curves it
>produces are well-behaved as its order is changed, and because
>a Farey series of a given order is a list of all the intervals
>in a harmonic series of a certain limit, and it is tempting to
>think that the ear/mind uses a harmonic series as _its_ template.
>The a*b series also had some things going for it... none of which
>I can remember. And I can't recall if sqrt(a^2 + b^2) was ever
>tried. Paul?

-Carl

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