Stretching

[Richard Moody]

Robert, 
	Some stretching does occur "naturally" in aural tuning particularly at
the octave level.  If you are tuning A880 (A5)to A440, (A4) what  you are
really tuning is the first partial of A5 to the second partial of A4.  Now
if the second partial of A4 is sharp, well you get the picture.  The first
partial is also the fundamental.  
	To give an example, suppose the second partial of A4 is not 880, but 881.
 Thus tuning A5 to A4, the fundamental of A5 would end up at at 881, or
one cps sharp when tuning this interval beatless.  Which brings up another
question about stretching, since the interval is beatless, all of the
other partials must be in tune or THEY would beat.  In our example tuning
A5 to A4 if sounding beatless, the other paritials involved, the 2nd of A5
to the 4th of A4 must also be in tune. Also the 3rd and 6th partials don't
beat.  If they are sharp, they are then all sharp by the same amount. 
This then would indicate that the third and the tenth tests should beat
the same. 
	A major test for aural tuners for "proving" octaves is the third - tenth
test.  The beat rate of a third is the same as the beat rate of the tenth
from the same root  note.  To determine what partials of a given interval
are heard when tuning, simply refer to that interval's arithmetic ratio. 
The ratio for a third is 4:5. So when hearing the beat rate of a third,
one is actually listening to the 5th partial of the root beating with the
4th partial of the upper note.  When listening to the beat rate of a
tenth, one is hearing the 5th partial of the root beating with the 2nd
partial of the note an octave and a third above.  The ratio of the tenth
then should be 2:5. 
	How does the third-tenth prove the octave? Take our A440 - A880 octave.  
the F below A440 beats beats 13.86 with A4.  The F3 to A5 will beat the
same if the A5 is really an octave above A4.  If it beats slower, the A4 -
A5 octave is narrow.  (A5 is flat).  If it beats faster the A5 is sharp.  

	If you tune an octave down from A440 beatless and measure it with a
device set to 220, it should read slightly flat, which is an indication of
how sharp the second partial of A3 is.  Thus your temperament octave is
already mathmatically a little wide. Which is one more reason to tune from
A440 if that is what orchestras are taking their pitch from. 

Richard Moody 

----------
> From: robert sadowski <rls@ncinter.net>
> To: pianotech@ptg.org
> Subject: Stretching
> Date: Wednesday, June 03, 1998 7:39 AM
> 
> Dear List,
>      I recall my instructor at the Chatauqua Institution stating that if
you
> tune beatless octaves above the temperament octave you will
automatically
> stretch these octaves.
> >From one who still uses the "human computer" I'm curious.  I'm ashamed
to
> admit I haven't spent much time studying these matters but do you
electronic
> folks impart a beat to these octaves to stretch them?
> 
> Bob Sadowski
> Erie, PA