[CAUT] SAT numbers

[Jeff Tanner]

On Aug 20, 2007, at 7:33 PM, Fred Sturm wrote:

> I'm not talking about SAT III. THe original SAT I. Way back in the  
> mists of time. I have no idea what was done in the transition from  
> SAT I to SAT III. No doubt it was for the good.

The bass is much more stretched than the tunings in the library,  
which I think Paul told me were calculated by SAT II.  It is a wider  
bass, but more than just the bass is affected.  My recollection is  
that Paul told me the SAT I and SAT II used the same curve.  There  
were intentions of making changes but that didn't happen until SAT  
III.  Been a while since that conversation, and specifics are foggy,  
but that is the recollection of our conversation.


>
> The F number doesn't affect the temperament octave. At all. Try it.  
> Do two calculated tunings, one with F at 18, the other with F at 10  
> (A and C the same for both). Read the numbers and compare. I did  
> that (and all sorts of other permutations as well), and found out  
> there was no more than 0.1 and occasionally 0.2 cents difference  
> for any note above F3, based on changes of any amplitude in the F  
> number.
>

Depends on what you call the temperament section.  I'm referring to  
C3 to E4, but in particular the octave C3 to C4.  (My aural  
temperament is a "customized" combination of two temperament  
systems).  From F3 to F4, everything is usually pretty good either  
way.  C3-F3 is where things get squirrelly with the altered F number,  
though I've had some odd results (usually slow thirds - depending on  
where plain wire starts) from F3 to C4 with unaltered FAC -  
especially when F3 is a wound string.  The section below F3 is where  
the higher F number helps smooth out the "temperament".

I hope this works.  It is typed in Helvetica 12, so if it doesn't  
convert legibly to whatever font you may be using, try highlighting  
and changing the font.  But here are the results of your suggestion.

Temperament 41 is the library tuning for Baldwin Hamilton - F/A/C =  
17.9/6.2/6.0 (old curve)
Temperament 1 is the same FAC recalculated to SAT III curve
Temperament 2 is same as Temperament 1, with F number changed to 10.0

Note	Temperament
		1		2		41 (old curve)
A0		-20.6	-13.4	-11.9
C1		-16.5	-10.6	-10.5
C2		-4.3		-2.6		-4.7
B2		+3.2	+2.2	+3.0
C3		-3.6		-2.3		-5.0
F3		-1.0		-0.4		-2.0
A3		+0.8	+1.0	+0.3
C4		+2.2	+2.4	+2.1
F4		+5.0	+5.0	+5.1
A4		+7.7	+7.7	+7.8
C5		+2.6	+2.5	+2.5
C6		+4.4	+3.8	+3.7
C7		+11.4	+11.2	+10.7
C8		+35.0	+34.2	+33.4

Just changing the F number does affect every number in the scale,  
even if only so slightly that it doesn't round to a different tenth.   
But almost every number changes.  From C3 to C5 only 5 numbers are  
rounded to an identical 0.1.  Even C6 is different by 0.6 cent, so it  
is inaccurate to say that the F number only affects the scale below  
F3.  But it really changes the interval relationships of the C3-C4  
octave.  Negligible difference at C4, but 1.3 cents at C3 - the 3rds  
might not be so noticeably different, but the 6ths and 10ths will.   
And while F4 are identical, there is a 0.6 cent narrower difference  
at F3 -- not within the 0.1 or 0.2 deviation you claim.

I found the difference at B2 to be really interesting.  You're going  
to get a big difference in the speed of the B2-D#3 third.  With the  
altered F, the D# is 1.1 cents higher, and the B2 is a full cent  
flatter.

Jeff


Jeff Tanner, RPT
University of South Carolina



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