[plt-scheme] music theory (was Natural numbers)

On Thu, Mar 12, 2009 at 05:26:13PM +0100, Jos Koot wrote:
> 
> ----- Original Message ----- 
> From: "Prabhakar Ragde" <plragde at uwaterloo.ca>
> To: <plt-scheme at list.cs.brown.edu>
> Sent: Thursday, March 12, 2009 3:47 PM
> Subject: Re: [plt-scheme] music theory (was Natural numbers)
> 
> 
> >Jos Koot wrote:
> >
> >>I am not sure that bach used the nowadays equally tempered scale for Das 
> >>Whol Temperierte Klavier (I have been told differently once by a music 
> >>teacher)
> >
> >Bach, a man of taste, probably realized that well-tempered frequencies and 
> >string divisions could be expressed with exact numbers in Scheme, whereas 
> >equal-tempered frequencies and string divisions would require inexact 
> >numbers. --PR
> 
> That depends on the representation. In quantum mechanics (or rather group 
> theory) there are lots of important coefficients (Clebsch Gordon 
> coefficients) that are exactly the square root of a rational number. Long 
> ago I wrote a program that did most of the computation exactly and 
> read/printed the positive square root of 2/3 like (2 -1) meaning the square 
> root of ( prime 2 squared times the reprocal of prime 3). This 
> representation allows easy multiplication/division. Addition/subtraction 
> give problems of course, but can be avoided in many cases by clever use of 
> orthogonalities and recursive relations.
> 
> For the equally tempered scale you could represent the twelfth root of 2 by 
> (1) and that of 1/2 by (-1). An octave interval would be written as (12) 
> and a quint as (7) For an equally tempered scale with quarts of a whole 
> secund, the twenty-fourth root could be used.
> 
> There even is an exact representation for pi, namely the word 'pi'.
> 
> Jos

If any one wants to look at some serious numerical representatino 
problems, have a look at the following conjecture:

Every positive integer can be represented as an expression using only 
the number 3, factorial, and integer square root (i.e., square root with 
the fraction part chopped off).

The challenge is simply to search the space of such expressions to find 
integers.  3 is pretty easy, of course,  But you rapidly get to the 
point where you have to use iterated factorials (things like 3!!!!) and 
your calculations get difficult.

-- hendrik

P.S.  This was one of the problems in the old MIT memo HAKMEM.