Here next to the last of this series.
SAT Part 5 Altering the Stretch
Since the normal Stretch provided by the FAC true readings gives
a tuning curve which attempts to keep the double octaves almost
pure, the single octave tends to be just slightly wider than the
6-3 type octave in the upper Bass, so that the 3rd partial of the
upper note will be the standard to which the 6th partial of the
lower note of the octave is tuned. If this amount of stretch is
not sufficient for one's taste, the F3 stretch number can be
altered or increased to make the octaves more like an 8-4 type
match, that is where the 8th partial of the bottom note of the
octave is matched with the 4th partial of the upper note. There
may be certain pianos (usually 7' or 9') in concert situations
where one would prefer a 10-5 type match from the mid Bass on
down, and even a few situations where 12-6 matching of the
bottom few notes would be advantageous.
The effect of increasing the F3 stretch number is exponential, but
in the digressing direction. By that it is meant that there are
larger changes in the bottom of the scale than in the upper Bass.
When tuning downward in the Bass using the unaltered stretch
numbers, it may be noticed that when the upper note of an octave
is played, the dot rotation is usually slow and toward the sharp
side of where the 6th partial of the lower note is to be tuned.
If one alters or increases the F3 Stretch number, more difference
or rotation will be seen in this comparison. As one continues to
increase the F3 stretch number, there will come a time when the
lowest octaves will remind one of the Great Pipe Organs which can
rattle the stained glass windows of a large cathedral with that
peculiarly slow roll of the octave. When one is young and
learning to tune, there is a tendency to want to eliminate all
beating in all intervals. Eventually it is discovered that there
is something nice about hearing controlled beating , especially
in the low Bass. This is an effect which happily accompanies the
near matching of very high partials such as the 16th or the 32nd.
It often is the matching of higher partials like these rather than
the matching of lower coincident partials which gives one the
sense that the low Bass SOUNDS in tune.
For an example, let us take our hypothetical case where Page 1
had 8, 8, & 8 for stretch numbers F3, A4 and C6 respectively.
If on Page 7 we alter the F3 # to 10, you can see below how it
affects the widening of the lower octaves more than the upper
Bass octaves.
FAC #'s A0 A1 A2 A3 A4
Page 1 8.0 8.0 8.0 -14.8 -5.4 0.4 1.4 10.0
Page 7 10.0 8.0 8.0 -16.0 -5.6 0.4 1.4 10.0
However, it must be remembered that whenever we alter a stretch
number from the true measured number, we must adjust the pitch
at the border where the machine changes the partial to which it
listens. So, in the above case, since the partial change occurs
between B2 and C3 on the SAT, we have the following analysis:
B2 Partial 6 C3 partial 4 Diff.
Page 1 1.4 -2.4 3.8
Page 7 1.4 -2.6 4.0
=======
-0.2
Since the C3 is .2 cents lower, the B2 needs to be made .2 cents
lower in order to preserve the smooth progression across the break
which the true numbers projected on page 1. Another way of saying
this is that the relationship of the 6th partial of B2 needs to
remain the same toward the 4th partial of C3 in the wider stretch.
This requires resetting the pitch at the location of B2 by the
amount -0.2 cents as you tune downward from C3 to B2. This will
cause the values in the comparison of Page 1 and Page 7 to be as
follows:
FAC #'s A0 A1 A2 A3 A4
Page 1 8.0 8.0 8.0 -14.8 -5.4 0.4 1.4 10.0
Page 7 10.0 8.0 8.0 -16.2 -5.8 0.2 1.4 10.0
>From this example, one can see that there is greater change in
the lower octaves than in the upper octaves of the Bass when the
F3 stretch number is increased. I tend to think of this as being
a reverse exponential curve. There is probably a better name for
it.
Additional stretch of the Bass such as F3=12 and F3=14 would
provide (including .2 cent and .6 cent break corrections) values
respectively as follows:
FAC #'s A0 A1 A2 A3 A4
Page 1 8.0 8.0 8.0 -14.8 -5.4 0.4 1.4 10.0
Page 7 10.0 8.0 8.0 -16.2 -5.8 0.2 1.4 10.0
Page 8 12.0 8.0 8.0 -17.2 -6.0 0.2 1.4 10.0
Page 9 14.0 8.0 8.0 -18.8 -6.8 -0.4 1.4 10.0
There may be occasions where one would wish to have less stretch
in the Bass than is normally provided by the regular FAC numbers.
This can be easily accomplished by merely reducing the F3 number
and even possibly reducing the A4 number. First let's consider
just decreasing the F3 number and compare it with the normal
results in Page 1. We will put this new arrangement on Page 10.
FAC #'s A0 A1 A2 A3 A4
Page 1 8.0 8.0 8.0 -14.8 -5.4 0.4 1.4 10.0
Page 10 6.0 8.0 8.0 -13.6 -5.0 0.2 1.4 10.0
Next let's apply the correction at the border change from partial
4 to partial 6 between B2 and C3 on pages 1 and 10.
B2 Partial 6 C3 partial 4 Diff.
Page 1 1.4 -2.4 3.8
Page 10 1.2 -2.2 3.4
=======
0.4
Since the 4th partial of C3 is higher than the true values would
provide, the 6th partial of B2 is lower than the true values, B2
needs to be raised by .4 cents. You can see that when this
correction is applied, the Bass curve flattens out just a little
bit.
FAC #'s A0 A1 A2 A3 A4
Page 1 8.0 8.0 8.0 -14.8 -5.4 0.4 1.4 10.0
Page 10 6.0 8.0 8.0 -13.2 -4.6 0.6 1.4 10.0
Page 11 4.0 8.0 8.0 -11.4 -3.6 1.2 1.4 10.0
On Page 11 you can see the effect of even greater altering of the
F3 stretch number. It now begins to affect the upper Bass a
little more.
The next and final article (I think) will be on combinations of
stretching one and shrinking another of the normal stretch numbers
and why one might wish to do so.
Jim Coleman, Sr.
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