[CAUT] P-12ths was: Tuning a Steinway D and a Bosendorfer Imperial together

[Fred Sturm]

Hi Ric,
  Comments below.
On Oct 13, 2008, at 4:49 AM, Richard Brekne wrote:

>   Fred writes
>   I frankly don't think dividing a 12th evenly into 19 is
>   significantly different from dividing an octave evenly into 12
>
>
> Actually it is... tho the distinction in the temperament area is a  
> bit subtle. If you require that D3 and A4 have a pure 3:1  
> relationship... then require that A4 and A3 have a pure 6:3  
> relationship then your 3:2 D3 and A3 is not necessarily going to  
> work out quite like you want it... and you may be forced to fudge.

My statement assumes that the same stretch is selected for both the  
12th and the octave which are to be divided. How you arrive at that  
"sameness of stretch" may be a bit convoluted, but it can certainly be  
done. We are talking about beginning with a bottom pitch and a top  
pitch and generating the half steps between them, nothing more and  
nothing less. The differences may be significant to a mathematician,  
but they are in the range of hundredth of a hertz or smaller, hence  
utterly insignificant in any practical way. Your statement about "then  
require A4 and A3 have a pure 6:3 relationship" is directly opposed to  
what I was trying to convey. The 6:3 relationship is almost certainly  
wider than the stretch called for in 3:1.

>  Course that kind of thing develops around the temperament octave as  
> you go.
> But more then this, and evenly divided 12ths temperament extended  
> into the treble so that all notes above this area are tuned to pure  
> 3:1 to their respective 12ths below then the treble curve around the  
> F5 - F6 area is significantly different then in an Octave type based  
> curve. Jim Coleman commented on this when my own P 12ths for Tune  
> Lab was released back around the turn of the century. Indeed, it was  
> a characteristic he liked quite abit.


I think that a great deal of this difference you describe can be  
attributed to the partial level at which the tuning is being done.  
Tuning first partials to the 3rd partial of notes a 12th lower will  
yield a different curve than tuning using a smooth curve at the 4th  
partial, for instance, particularly if large inharmonic jumps occur.  
But this is a separate issue. It is certainly possible to emulate  
closely the same stretch as a 3:1 using other parameters. A 4:1 plus  
0.5 cents or thereabouts will do a reasonable job.
	Focusing on a single interval is a way of achieving a consistent  
sound. Octave, 12th, double octave, 19th, triple octave, all can be  
used. And all can be used in more than one way (eg, 3:1 plus 0.3 cents  
as opposed to "pure"). It is also possible to achieve a consistent  
sound using some sort of progression (eg, narrower in the middle  
becoming wider in the extremes). Lots of possibilities out there.
	

Regards,
Fred Sturm
University of New Mexico
fssturm at unm.edu