[CAUT] SAT numbers

[Fred Sturm]

On Aug 20, 2007, at 8:40 AM, Keith Roberts wrote:

> My experience with the high strech numbers was that you had to  
> check the A stretch number. It has the most effect on the curve.
	Yes, quite right, the A number is the key. It sets up the size of  
octave in the center of the piano.
	The FAC formula for the SAT I/II works quite well, given accurate  
input. (Works well within its limitations and purposes: to produce a  
tuning with slightly wide 4:1 double octaves from about C3 up, and  
slightly wide 6:3 octaves down. And I guess SAT III gives some  
flexibility from that. I am not familiar enough with SAT III to have  
any specific opinion about it). The math is sound, and functional.  
Really, its essence lies in the fact that one is tuning 4th partials  
from C3 to B4, which gives excellent progression of M3s and whatnot.  
Any even curve of numbers will do that. There isn't any black magic  
or even very complicated mathematics involved.
	And the fact that only three notes are sampled really isn't a major  
problem, as long as the data from those three notes is accurate. And  
therein lie some problems: unfortunately, the output of partials from  
a string through a piano has anomalies. Make a string sound, and  
occasionally one of the partials will read "wrong," out of phase with  
the rest. Doesn't happen too often, but often enough to cause  
problems. From my own experience, I think that 2nd partials are  
particularly vulnerable to being "wrong."
	What causes this "wrongness?" Probably not the string itself. I  
think the cause usually lies elsewhere in the piano. Could be  
soundboard resonance. Could be coupling with not quite damped strings  
or aliquot/duplex segments. Could be standing waves. At any rate, it  
is a real phenomenon, and not a "mistake" made by the ETD. The ETD is  
simply reading a pitch output of the piano, and doesn't know whether  
it is a "good one" or not.
	To get around this problem, RCT does full partial ladders of several  
A's. So problem partials tend to be averaged out. With the SAT, this  
is where the reading of only three sample notes becomes a problem:  
when one of the six partials read is an anomaly, the tuning produced  
won't fit the piano, especially when the anomaly is in an A reading.  
I found a way around this problem that worked quite well, when I was  
using an SAT. I simply didn't read any 2nd partials, but calculated  
them instead. For both A4 and C6, I read the difference between the  
first and 4th partials, and extrapolated where the 2nd partial  
"should be."
	This isn't very hard. The ratio is a simple one. The difference  
between the 4th partial and the 2nd partial is four times the  
difference between the 2nd and 1st partials. So if the difference  
between the 1st and 4th partials has a value of 5, the first partial  
will be at 0, the 2nd partial will be at 1, and the 4th partial will  
be at 5.
	In practical terms, you start by turning on the SAT and hitting  
tune. You are at A4, 0.0 cents. Tune the string. Up octave to A6 and  
measure. Take the measured value and multiply it by 0.8. enter that  
result (standard method of entering, use cents down to get to the  
calculated value and enter it). For C6, tune or offset the C6 string  
to 0.0 at C6. Up octave to C8 and measure. It often helps to do a  
cents up to around 25 - 30 cents and then use the measure button, to  
make it easier for SAT to find that pitch. Take the measured value  
and divide by 5. Octave down to C7, cents down to the result of your  
division, and enter. Then on to the F number as usual.
	This is a wee bit of extra work, but is offset by the fact that
1) A4 will be dead on 0.0 cents when you have tuned it (guaranteed,  
assuming a high level of skill and care).
2) You save a bit of trouble doing octave and note buttoning,  
starting right in where the SAT puts you when you turn it on rather  
than navigating to F5.
3) You actually start by tuning a note (A4) where you want it, not a  
couple cents flat.
4) C8 will definitely be a slightly wide double octave from C6 once  
tuned (guaranteed, again assuming a high level of skill and care).
	All of this applies to SAT I/II. I think it probably also applies  
pretty well to SAT III. It's a short cut ahead of the technique Keith  
Roberts outlined to make sure the A stretch number was correct: get  
it right the first time, and Keith's check will be a formality.
Regards,
Fred Sturm
University of New Mexico
fssturm at unm.edu