[CAUT] The Origins of P12ths tuning.

[Fred Sturm]

On Oct 22, 2008, at 12:14 AM, Richard Brekne wrote:

> In the treble the stretch starts to develop a bit steeper then  
> traditional octave priority approaches... but the top octave ends up  
> quite moderately stretched.  I rarely see C8 over 36 cents...  
> depends on the pianos built in inharmonicity.

	I find that on smoothly scaled pianos (certainly including most  
quality grands), the 6:3:1 and the 8:4:1 relations coincide pretty  
closely through about C6. In octave 6, there begins to be a divergence  
between 3:1 and 6:1, and a similar divergence between 4:1 and 8:1 (3:1  
and 4:1 stay close to one another, as do 6:1 and 8:1). At C7, the  
divergence accelerates, and there is commonly an additional  
significant divergence between 8:1 and 6:1 in the top six or so notes.
	So if you are doing a stretch that compromises between 3:1 and 6:1,  
and favor 3:1 a bit, I agree that 36 cents may be the highest for C8  
for the most part. If you favor 6:1, though, that number becomes  
higher. If you ignore 3:1 and just follow 6:1, C8 will often be as  
high as 45 to 50 cents. If you follow 8:1, C8 can get to 60 or  
occasionally higher.
	While this sounds like a big difference, it appears almost entirely  
in octaves 6 and 7, with the bulk of the difference in octave 7, and  
the bulk of the octave 7 difference in the top few notes. It's a  
matter of a curve that becomes fairly steep at the end.
	I prefer to follow the 6:1 and 8:1 (if they diverge too much, I favor  
6:1)  because I get very good support for octaves 5 and 6 (from higher  
partials of all the notes tuned below - as there is very little  
difference from 3:1 and 4:1 at that point), but then get more of a  
correspondence to the "psycho-acoustic" phenomenon ("the desire for  
high notes to be sharp") in the top of the range, where single octave  
beats and the like become pretty much irrelevant (hard to pick out and  
hear). Of course, if the customer prefers less stretch in the top, it  
is easy to drop those top two octaves - the rest of the tuning is  
"reasonably conservative." But I have yet to come across a customer  
who asked for less stretch. Maybe I lead a sheltered life <G>.
Regards,
Fred Sturm
University of New Mexico
fssturm at unm.edu