[CAUT] Measuring FAC (was Re: Re. Link to Young Paper)

[Fred Sturm]

On Jan 23, 2007, at 10:19 AM, Jim Busby wrote:

> Thanks Fred.
>
>
>
> Thant’s exactly what I was looking for. When I first heard this Al  
> Sanderson was in the class and agreed that it was a good formula. I  
> guess the reason the FAC measurements are taken the way the SAT  
> “manual” says is because Dr. Sanderson chose a particular piano  
> faculty to “test” their taste of stretch (Harvard??) and came up  
> with their recipe. Yours seems to give much more stretch, and when  
> I compare my aural tunings with what you suggest below it comes out  
> much closer than the “out of the box” SAT tuning
>
> Thanks for your work. I’ll buy lunch in Kansas!
>
> Jim Busby BYU
>
>
Hi Jim,
	It shouldn't increase the stretch at all, at least consistently, to  
measure the way I do. It just helps eliminate problem readings,   
coming up with more reliable inharmonicity constants. And, as a side  
effect, it means that A4 will be right at A4 every time (assuming no  
operator error <G>). And, in fact, that's what got me started  
measuring the A number that way. Sometimes you would end up, after  
tuning precisely to the calculated tuning, with A4 being  
significantly sharp or flat. The literature that came with the  
machine more or less said "Live with it [pianos aren't perfect], or,  
if you care, tune A4 according to the calculated tuning, measure it,  
and create a pitch offset to account for the difference." I preferred  
a third approach, where the A number input reliably put A4  
fundamental right at 440.
	The problem arises when, say, there is a "false" reading for the 2nd  
partial of A4. It could read sharp or flat (presumably due to some  
kind of coupling with a resonance in the whole soundboard/strung back  
system). The formula relies on the difference between A4 and A5 being  
in the predicted ratio of 1:4 to the difference between A5 and A6.  
Sometimes the ratio of the measured partials is different. The FAC  
formula predicts where A4 will be based on that 4:1 ratio. If the  
reading of the 2nd partial isn't at that precise point in comparison  
to the 1st and 4th partials, the prediction will be wrong.
	Sanderson's FAC stretch (pre-SAT III) is basically somewhat wide 4:1  
throughout, aiming at some amount of cents wide double octaves. At  
some point, early on, I think I heard he altered the formula for the  
bass to give it a bit more stretch. This mathematical fix, I think,  
may have been a source with the "F problem" I complain about. I'm not  
familiar with the first generation FAC, so I'm just speculating based  
on a few snippets of memory of things I read some time or other,  
maybe 15 years or more ago.
	The C number is basically used to place C7 to C8, starting with a  
target C8, at just wide of 4:1 to C6. Again, it relies on that 4:1  
ratio between the C7C8 cents difference and the C6C7 cents  
difference. Again, there are occasionally (and fairly often) pianos  
where the measurements won't be in that ratio. If you actually  
measure the C8 partial, and do the calculation I outlined, the FAC  
formula will reliably give you that slightly wide 4:1 stretch up  
through that top octave.
	Dr. Al says he personally prefers a wider stretch. If you want more  
stretch (I did), you need to add to those FAC numbers. I added a  
fairly constant 0.5 cents to the A number (and decided I didn't care  
if I was consistently tuning all pianos 0.5 cents sharp), and I think  
I added about 2.5 to 5.0 to the C number. And then I checked the top  
notes as I tuned, playing notes a 12th, double octave, 19th and  
triple octave below, and comparing (looking at the lights for each),  
coming up with a compromise usually pretty close to 19th and triple  
octave.
Regards,
Fred Sturm
University of New Mexico
fssturm at unm.edu



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