[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.

Thus you are numering pitch differences with exact integers.  They are 
the logarithms of frequencies.

-- hendrik