[CAUT] Shulze tuning research

[Fred Sturm]

On Oct 23, 2008, at 3:44 PM, rwest1 at unl.edu wrote:

> The question is this:  Does tuning pure 12ths over-stretch the  
> thirds especially as they are expressed in 10ths and 17ths.


	A tuning based on the 12th (nominally, setting aside inharmonicity)  
will produce an octave wider by 1.2 cents than a tuning based on the  
octave. (Inharmonicity will have an effect, but this is a good  
starting point - it gives a good idea of the relative size we are  
looking at). This means that each half step is expanded by  
approximately 0.1 cents. The M3 is thus expanded by 0.4 cents. At A4,  
it takes 4 cents to equal a hertz. Thus, the difference in beat rate  
for M3s in the temperament area will be an increase on the order of  
0.1 beats per second. Hardly clangorous or even noticeable. To put  
that into perspective, I would ask how many among us are always able  
to tune every note of the mid range within + or - 0.2 cents (at total  
range of 0.4 cents) and then also tune all those unisons within the  
same parameters. To my way of thinking, "unison smear" will trump any  
change in M3 speed in this area.
	Now it is true that the difference in beat rate will multiply as you  
move upward in the piano. Enough to make a significant difference in  
sound attributable to M10 and M17 beat rate? Well, perhaps you will  
argue it is. Personally I don't think that this would be a significant  
factor in the overall sound of the piano. I think that the greater  
degree of coincidence and cohesion among the 1, 2, 3, 4, 6, and 8  
partials is of far greater consequence.
Regards,
Fred Sturm
University of New Mexico
fssturm at unm.edu