source file: m1455.txt
Date: Tue, 23 Jun 1998 15:10:49 +0000

Subject: Re: Commas and Consistency

From: kollos@cavehill.dnet.co.uk (Jonathan Walker)

Paul H. Erlich wrote:

> ... The consistency issue which
> Graham Breed is referring to involves the traditional definition of 
> the syntonic comma as the difference between an octave and a major 
> sixth up, and three perfect fifths up. Many equal temperaments have a 
> better approximation to the major sixth than the best approximation of 
> a major third plus the best approximation of a perfect fourth. 
> Therefore two distinct syntonic commas can be defined for such equal 
> temperaments. 5-limit consistency means that the best approximations 
> of the perfect fifth/fourth, the major third, and the major sixth are 
> all compatible. Given 5-limit consistency, then, there can be only one 
> syntonic comma. ETs that are not 5-limit consistent include 11, 17, 
> 20, and 21.

Thanks, Paul, for your definition of the "consistency" which was at
issue. A 5-limit consistent ET, unless I misunderstand you, would then
be definable as any ET of which the proposition

abs ( 4f(3/2) modn - f(5/4) ) = abs ( f(4/3) + f(5/4) - f(5/3) )

is true, where for any real x, f(x) = [1/2 + log_2 x], and presumably
where the value of the expressions on each side is greater than 0. Is
this what you effectively mean?

Jonathan Walker

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End of TUNING Digest 1455
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