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

----- 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




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