re: formula for 3-limit, 5-limit, and musical terminology

> From: "Carl Lumma" <ekin@lumma.org>
> Subject: Re: Dodekaphony
>
> > > What's a "3-limit comma"?
> >
> > In the sense I use the term, a rational number only a bit larger
> > than one, of the form 2^a 3^b where a and b are integers,
> > considered as an interval in music. The paradigm case is the
> > Pythagorean comma.
>
> Well put, Gene. For the curios, a 5-limit comma would then be:
>
> A rational number only a bit larger than 1, of the form
> 2^a 3^b 5^c where a, b and c are integers, considered as
> an interval in music. The paradigm case is the Syntonic
> comma (81/80).
>
> -Carl

There is something missing here. In the 3-limit case, there (maybe)
should have been a division, AND instead of a "bit larger", something
like

abs( 2^a/2^b - 1) < err

or some such.

Given that, I know that 2^a 3^b 5^c are all supposed to approximate
some number (which in musical terms, is convergence at some higher
octave, since 2 is a factor). Of course, I could very well be wrong.
In a program I wrote to look at this stuff, I considered the pairs,
i.e., 2^a 3^b approximating integer q
2^c 5^d approximating integer r
3^e 5^f approximating integer s

in musical terms, the fifths meet at some octave, the major thirds
meet at some octaves, and the the major thirds and fifths meet
somewhere else.

Bob

>> > > What's a "3-limit comma"?
>> >
>> > In the sense I use the term, a rational number only a bit larger
>> > than one, of the form 2^a 3^b where a and b are integers,
>> > considered as an interval in music. The paradigm case is the
>> > Pythagorean comma.
>>
>> Well put, Gene. For the curios, a 5-limit comma would then be:
>>
>> A rational number only a bit larger than 1, of the form
>> 2^a 3^b 5^c where a, b and c are integers, considered as
>> an interval in music. The paradigm case is the Syntonic
>> comma (81/80).
>>
>> -Carl
>
>There is something missing here. In the 3-limit case, there (maybe)
>should have been a division,

a b and c do not have to be positive.

>AND instead of a "bit larger", something
>like
>
> abs( 2^a/2^b - 1) < err
>
>or some such.

How does that help?

>Given that, I know that 2^a 3^b 5^c are all supposed to approximate
>some number (which in musical terms, is convergence at some higher
>octave, since 2 is a factor). Of course, I could very well be wrong.
>In a program I wrote to look at this stuff, I considered the pairs,
> i.e., 2^a 3^b approximating integer q
> 2^c 5^d approximating integer r
> 3^e 5^f approximating integer s
>
>in musical terms, the fifths meet at some octave, the major thirds
>meet at some octaves, and the the major thirds and fifths meet
>somewhere else.

Sorry, you lost me. It's just prime factorization. 9/8 is
a 5-limit comma 2^-3 * 3^2 * 5^0, but an extremely large one.

-C.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Sorry, you lost me. It's just prime factorization. 9/8 is
> a 5-limit comma 2^-3 * 3^2 * 5^0, but an extremely large one.

I'd call something a comma only if I thought someone was willing to
temper it out.