Re: medieval fiddles -- all were carved from single block?

In article <3ff9cdfa$0$86327$d368eab@[EMAIL PROTECTED]
>,
 Margo Schulter <mschulter@[EMAIL PROTECTED]
> wrote:

> Todd Michel McComb <mccomb@[EMAIL PROTECTED]
> wrote:
> 
> > Not only that, but when Marchettus gets into these areas, I can't
> > help but wonder whether his discussion is reflecting practice or
> > advocating some new ideas he personally has.  I tend to get the
> > impression of the latter.  That's no reason not to try them, of
> > course, but there are some areas where I sometimes get the impression
> > that the theorists are speaking only for themselves.
> 
> Hi, there, Todd, and this is an excellent caution: I wouldn't want to 
> ****tray Marchettus as expounding "standard" 14th-century theory, and we 
> know that about a century later, Prosdocimus strongly condemns what he 
> takes to be the advocacy of an equal fivefold division of the tone.

Hi, Margo and Todd.

It's worth mentioning that what Prosdocimus is complaining about is not 
so much how an equal five-fold division of the tone might sound, as the 
fact that in standard Pythagorean music theory you simply *can't* divide 
a tone (or any other superparticular interval) into any number of equal 
parts.  

Pythagorean music theory only allows intervals to be ratios of whole 
numbers (e.g., a tone is 9:8).  A basic theorem of the system, in its 
simplest form, first proven by Archytas, is that you can't divide a tone 
exactly in half (or any other number).  In modern terminology, the 
reason is that any root of 9/8 is irrational.  The ancients used 
different terminology, but got to the same result.  Boethius has an 
incomplete proof of this fact, and zillions of other theorists, up 
through (I think at least) Zarlino quote the result and occasionally 
part of the proof.

-- 
Neal Plotkin
e-mail:  neal dot plotkin at nyu dot edu