7 ms·

The Golden Ratio is Evil (and why, in Clojure)

30 points by ihodes 16y ago

gjm11 16y ago
Meh. It's "evil", supposedly, because if you add up the first N decimal digits of its fractional part then, for carefully chosen N, you get 666, the Number of the Beast. I say we should be using binary rather than decimal -- ten is a terribly arbitrary number -- in which case, of course, all numbers are "evil" other than a few dyadic rationals. Here's a better reason why the golden ratio is evil. As you know, Bob, in mediaeval times the musical interval of a tritone -- three tones, or six semitones -- was called "diabolus in musica", the devil in music. It was called that because it's a highly discordant interval; in equal temperament (which of course they didn't have back then, but hush) it corresponds to a frequency ratio of sqrt(2). And why does that sound highly discordant? Well, basically because there's no simple rational number that's a really good approximation to sqrt(2); for much much more about what's going on here, put "William Sethares" into Google and consider buying his book. In fact, in a certain sense sqrt(2) is the second-worst-approximable-by-rationals irrational number, which is related to the fact that its continued fraction is [1,2,2,2,2,2,2,...]. And what's the worst-approximable irrational number, which would produce an even more dissonant musical interval -- a maximally dissonant one -- something even worse than the devil himself in music? Why, the golden ratio, of course. Continued fraction [1,1,1,1,...]. Evil incarnate. (Note: there are a few half-truths in the above, but only for convenience of exposition.)
- ihodes 16y ago
  Oh that's pretty neat! I didn't realize the connection it had to even temperament tuning.
japherwocky 16y ago
I think there are about 666 parentheses in those code examples.. I don't get the appeal of Clojure?
- rickmode 16y ago
  It won't make sense until you make a concerted effort to learn lisp. Dig into "The Little Schemer". It's enlightening. And fun.
- ihodes 16y ago
  You don't really see the parens after a while. Clojure's appeal goes beyond its syntax, as I'm sure does whichever language you prefer. Curiously, the equivalent code in most other languages has about the same number of parens.
  - Groxx 16y ago
    Just to be troublesome: Ruby has implicit parens where possible, and none for defining loops :) Even including {} where applicable (single-line loops / anonymous functions), we're still talking likely significantly below 1/2.
  - substack 16y ago
    For fun I was whipping up some haskell one-liners as I read along. You can just do: let fibs = 1 : 1 : [ fibs !! n + fibs !! (n - 1) | n <- [1..] ] for the Fibonacci sequence and take 10 $ iterate (succ . recip) 1 for the first 10 iterations of the continued fraction. To me these read much more clearly but then I haven't written anything big in a lispy language yet.
    - ihodes 16y ago
      There are all sorts of ways to do it; for instance, I really like the following: (def fibo (map second (iterate (fn [[x y]] [y (+ x y)]) [0 1]))) I just went with a different version for the article to show another way of doing it that might be clearer to someone new to Clojure. As for understanding/ease of reading, the Haskell version is a little more alien to me, but that's because I haven't spent much time in Haskell. Once again, for your CF ex: (take 10 (iterate (comp inc /) 1)) Which gives ratios in Clojure, not doubles. Both elegant solutions, definitely. I just wanted to use my GCF, which could probably be made to look a little nicer as well. This post was more about playing around with the Golden Ratio than writing sexy code (though admittedly, I probably could have done better in many places!) Haskell is a really cool language: I'm making my way through Real World Haskell right now!
onoj 16y ago
"golden ratio" nonsense. The words never link to the correct maths. It is evil, but not rational. http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm
cicada 16y ago
The first 144 digits of the fractional part of pi sum to 666.

30 points by ihodes 16y ago

gjm11 16y ago

Meh. It's "evil", supposedly, because if you add up the first N decimal digits of its fractional part then, for carefully chosen N, you get 666, the Number of the Beast. I say we should be using binary rather than decimal -- ten is a terribly arbitrary number -- in which case, of course, all numbers are "evil" other than a few dyadic rationals. Here's a better reason why the golden ratio is evil. As you know, Bob, in mediaeval times the musical interval of a tritone -- three tones, or six semitones -- was called "diabolus in musica", the devil in music. It was called that because it's a highly discordant interval; in equal temperament (which of course they didn't have back then, but hush) it corresponds to a frequency ratio of sqrt(2). And why does that sound highly discordant? Well, basically because there's no simple rational number that's a really good approximation to sqrt(2); for much much more about what's going on here, put "William Sethares" into Google and consider buying his book. In fact, in a certain sense sqrt(2) is the second-worst-approximable-by-rationals irrational number, which is related to the fact that its continued fraction is [1,2,2,2,2,2,2,...]. And what's the worst-approximable irrational number, which would produce an even more dissonant musical interval -- a maximally dissonant one -- something even worse than the devil himself in music? Why, the golden ratio, of course. Continued fraction [1,1,1,1,...]. Evil incarnate. (Note: there are a few half-truths in the above, but only for convenience of exposition.)

ihodes 16y ago
Oh that's pretty neat! I didn't realize the connection it had to even temperament tuning.

japherwocky 16y ago

I think there are about 666 parentheses in those code examples.. I don't get the appeal of Clojure?

rickmode 16y ago
It won't make sense until you make a concerted effort to learn lisp. Dig into "The Little Schemer". It's enlightening. And fun.
ihodes 16y ago
You don't really see the parens after a while. Clojure's appeal goes beyond its syntax, as I'm sure does whichever language you prefer. Curiously, the equivalent code in most other languages has about the same number of parens.
- Groxx 16y ago
  Just to be troublesome: Ruby has implicit parens where possible, and none for defining loops :) Even including {} where applicable (single-line loops / anonymous functions), we're still talking likely significantly below 1/2.
- substack 16y ago
  For fun I was whipping up some haskell one-liners as I read along. You can just do: let fibs = 1 : 1 : [ fibs !! n + fibs !! (n - 1) | n <- [1..] ] for the Fibonacci sequence and take 10 $ iterate (succ . recip) 1 for the first 10 iterations of the continued fraction. To me these read much more clearly but then I haven't written anything big in a lispy language yet.
  - ihodes 16y ago
    There are all sorts of ways to do it; for instance, I really like the following: (def fibo (map second (iterate (fn [[x y]] [y (+ x y)]) [0 1]))) I just went with a different version for the article to show another way of doing it that might be clearer to someone new to Clojure. As for understanding/ease of reading, the Haskell version is a little more alien to me, but that's because I haven't spent much time in Haskell. Once again, for your CF ex: (take 10 (iterate (comp inc /) 1)) Which gives ratios in Clojure, not doubles. Both elegant solutions, definitely. I just wanted to use my GCF, which could probably be made to look a little nicer as well. This post was more about playing around with the Golden Ratio than writing sexy code (though admittedly, I probably could have done better in many places!) Haskell is a really cool language: I'm making my way through Real World Haskell right now!

onoj 16y ago

"golden ratio" nonsense. The words never link to the correct maths. It is evil, but not rational. http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm

cicada 16y ago

The first 144 digits of the fractional part of pi sum to 666.