[CAUT] The Origins of P12ths tuning.

[Fred Sturm]

On Oct 21, 2008, at 7:48 AM, Richard Brekne wrote:

> Ok... as promised I hunted down the article that I have said all  
> along contains the information that prompted me to create the  
> P-12ths tuning on Tunelab in 2000 along with all subsequent  
> discussion on both lists prior to Bernhard Stoppers appearance on  
> these lists some years later. As it turns out, this article PTG  
> Journal May 1982 by Gary Shulze RTT, six years before Stoppers  
> publication,  contains the entire basis for the P12th tuning along  
> with the mathematical definition for the a Perfect 12ths stretch.

Thanks for posting that, Ric.
	I think it is pretty obvious that, in practical terms, Shulze and  
Stopper describe precisely the same thing. The differences lie in  
emphasis and approach. Shulze more or less derives 3:1 (and 6:1) from  
inharmonically stretched octaves (and double octaves, etc), noting  
that 3:1 and 6:1 represent points along a continuum of stretch, with  
certain attributes and advantages. Stopper is more focused on 3:1 as a  
discrete, special system (I guess, from his recent post, modified in  
practical terms by 6:2, 9:3, etc, but without direct reference to  
other intervals/partial alignments like 2:1, 4:2, 4:1, 6:1, etc).
	Perhaps when we see the research Stopper has said will be published  
in 2009, which he says will show some special symmetries that result  
from his 3:1 system, we will be better able to judge the validity of  
his claims.
	Pending that, my own tendency is to hold with Shulze's view, that  
this is only one part of a complex picture.
Regards,
Fred Sturm
University of New Mexico
fssturm at unm.edu