# [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