SAT Part 2 Altering the stretch

[Jim]

Dear List:

Here is the second of the latest series from Jim Coleman, Sr.


SAT part 2 Varying the stretch

Although in general when one uses the FAC tuning program in the
Sanderson AccuTuner, the octaves around A3 - A4 will have .3 bps
stretch at the second coincident partials (the 4-2 relationship).
This means that the F3-A4 10th will be 1/3 beat faster than the
f3-A3 M3rd.  Sometimes this is not true when the inharmonicity
curve is not consistent due to the piano scaling.  In fact
occasionally the 10th will be slower than the M3rd.  There is a
way to change this, and that is by altering the A4 stretch number.
Any addition to the A4 stretch number will spread out the entire
tuning both ways; that is, flatter toward the Bass and sharper
toward the Treble.  You can then see how that by increasing the
stretch number of A4, the A4 would be tuned actually sharper and
the A3 would be tuned actually lower in some cases by a small
amount.

For an example, let us set up page 1 of memory by using FAC
numbers of 8.0, 8.0, and 8.0.  Also let us set up page 2 of memory
using #'s  8.0, 8.5, and 8.0.  The Accutuner presumes to  tune
each note between A4 and F3 on a smooth exponential curve.  You
can visualize the curve as below:

                                                  X
                                                  A4

                                           X
                                           F4

                                     X
                                     C#4

                              X
                              A3

                    X
                    F3


Now, suppose that instead, the A3 had a lower amount of inharm-
onicity than the SAT assumed.  Its 4th partial would still be
tuned by the numbers projected on the above curve.  The actual
curve might look something like this:

                                                  X
                                                  A4

                                           X
                                           F4


                                     X
                                     C#4


                              X
                    X         A3
                    F3

Can you see that the F3 is out of alignment with the above four
notes and that since the SAT smooth curves between F3 and A4,
the A3 is actually tuned slightly higher to fit on the upper
smooth curve and therefore the F3-A3 M3rd will beat faster than
it should in respect to the F3-A4 M10th?  Now, if you wish to
keep the A3-A4 10th faster, the A4 would need to be raised.
This is easily done by using the above page 2 memory tuning
instead of the page 1 tuning.     Here are the numbers
for         A3 (4th partial) and A4 (4th partial)

page 1:     1.4                  10.0

page 2:     1.4                  10.6

The diff:    .0                    .6 cents

Now, at the 4th partial of A4 that amounts to just over 1/2 bps,
and at the coincidence of the A3 and A4 octave, that amounts to
..3 bps.

It should be obvious that if in the page 1 tuning the 3rd-10th
tests were equal, then by adding .5 cents to the stretch number
of A4, you can have an approximate additional beat rate difference
of .3 bps. If you prefer wider octaves than that, such as 1/2 bps
difference in the M3rd-10th test, you could add perhaps .9 cents
to the A4 stretch number, assuming that the true measurements
produced equal beating M3rd-10ths.

A note of caution needs to be made here:  As the octave is spread
wider, the 5ths get slower, the 4ths get faster.  FAC is able to
keep a smooth graduation of all of these beats, barring other
quirks in the scaling of the piano.

Another caution is that the A4 fundamental is no longer exactly at
A440.  See Part 1 of this series for the correction to be made.

Another caution is that when the A4 number is changed from the true
reading, a compensation has to be made at the change of partials
which occurs between B4 and C5.  This will be dealt with in Part 3.

One other thing to be noted is the additional stretch which occurs
at A0 and C8.  For these 2 notes:

                 on page 1 you have -14.8 and 43.8
                 on page 2 you have -15.7 and 44.4

In future articles we will deal with corrections due to partial
changes, Additional stretch of the high Treble, additional stretch
of the Bass, as well as decreases of the Bass stretch.

Jim Coleman, Sr.