Altering FAC settings

[FSSturm@aol.com]

Dear list,

Jim Coleman's series on this subject was very well thought out, and clearly
written. Many thanks, Jim! I have some thoughts on the same subject that I'd
like to share with the list. For the most part I agree with what Jim had to
say, but I have a few supplemental ideas, and a different slant which may
prove useful to some.

I'll start with the "A number," the number we measure as the difference
between the 2nd and 4th partials of A4, and input as part of the FAC process.
This number has two functions. The first is to place the pitch of A4 at A440.
The SAT tunes the 4th partial of A4, but we want the 1st partial to be at
A440 (A4 at 0.0 cents). The FAC program uses the "A number" to project the
difference between the 1st and 4th partials. Theoretically, the difference
between the 1st and 2nd partials should be 1/4 the difference between the 2nd
and 4th partials. Thus, if we multiply the difference between the 2nd and 4th
partials by 1 1/4 (1.25), we should produce the difference between the 1st
and 4th partials.

This is what the FAC program does (though perhaps not so directly). The value
it produces for A4 is the "A number" multiplied by 1.25. If we input 8.0, the
value for A4 will be 10.0, if 6.0, it will be 7.5, if 10.0, it will be 12.5,
etc. Most of the time this works quite well, and the 1st partial of A4 will
be within 0.5 cents of A440. However, there are anomolies, where the measured
difference between the 1st and 2nd partials is not in a 1:4 ratio to the
measured difference between the 2nd and 4th partials. I got into the habit of
measuring the fundamental of A4, and found that a significant number of
pianos, maybe as much as 10-20%, would have fundamentals off by as much as
1.0 to 2.5 cents. When this occurred, I would re-read the A5 to A6
difference, and often find I had made an error in my initial reading, due to
a false string or the like. Also sometimes there would be significant
variance between the three strings of the unison. Most often, though, my
re-measurement would be exactly the same as before.

To get around this problem, I now tune A4 to A4 at 0.0 cents (just turn on
the machine, push the tune button, and tune to stop the lights). I then read
A4 at A6 (reading the difference between the 1st and 4th partials). That
number multiplied by 0.8 is the number I use for my "A number" input.
(Actually I divide the number I read at A6 by 5, and subtract that result
from the the A6 number - easier to do in the head). In this way I can be
certain ahead of time that the fundamental of A4 will be right on A440. I
theorize that taking the difference between the 1st and 4th partials gives me
a larger sample, hence a more accurate inharmonicity constant, less subject
to anomolies. However, I keep a small notebook, in which I record FAC reading
for each model of piano I tune (in alphabetical order, with ample room for
each prominent maker. Rare makes are entered under miscellaneous, divided
between old uprights, other uprights, and grands). Thus I can check my
readings against earlier recorded ones to help guard against anomolous
readings. When I get a suspiciously strange reading, I doublecheck other
strings of the unison, and if the reading is still anomolous, I fudge in the
direction of my average reading.

The second function of the "A number" is to produce the tuning figures for C3
to B6, with minor help from the "F number". (The "F number" affects the
figures for C3 to F3 fairly significantly, and has a very minor affect on F#3
to B6, no more than 0.1 to 0.2 cents for any given note. The "C number" only
effects the values for C7 to C8). One thing the "A number" does is set the
width of the A3/A4 octave, as a 4:2 octave wide by 0.4 to 0.8 cents.

Since we know where the 4th partial of A4 is (that's what we tune), and we
know the difference between the 2nd and 4th partials of A4 (that's what we
read to get our "A number"), we also know where the 2nd partial of A4 is
(subtract the difference between the 2nd and 4th partials, the "A number",
from the number the FAC tuning produced for A4). To tune a pure 4:2 octave,
we would simply tune the 4th partial of A3 to the value for the 2nd partial
of A4. For a concrete example, let's take a piano where the "A number" is
8.0. A4's 4th partial is tuned to 10.0. Its 2nd partial must be 2.0 (10.0
minus 8.0). If we tune A3's 4th partial at 2.0, we will have a precisely pure
4:2 octave.

The FAC program makes A3/A4 a wide 4:2 octave. To check, make note of the
numbers for A3 and A4. The difference between those numbers will be equal to
the "A number" plus about 0.4 to 0.8 cents. When the "A number" is 6.0, it is
0.4 cents wide. This increases to 0.8 cents wide as the "A number" increases
to 12.0. The way the FAC program does this is much more complex, but this is
what it produces. Given the figures for the 4th partials of A3 and A4, we can
create a logarithmic curve to produce the values for the intervening notes.
This curve can then be projected down to C3 and up to B4 (with a little input
from the "F number"). Once again, this is not how the FAC program does it,
but describes the results.

Having created these values, the FAC program can then calculate where to
place C5 to B5 and C6 to B6. We know where the 4th partials of C3 to B3 are
(those are the values theFAC program just calculated). To make pure double
octaves, we simply need to place the first partials of C5 to B5 where the 4th
partials of C3 to B3 are. Since we tune the 2nd partial from C5 to B5, we
need to add a value equal to the difference between the 1st and 2nd partials
for each note. Since the FAC produces slightly wide double octaves, an
additional 0.5 or so cents is added as well. C6 to B6 is even simpler. We
know the 4th partials of C4 to B4 (those are likewise the values the FAC
program calculated). C6 to B6 are tuned to 1st partials, so they simply match
the numbers for C4 to B4, plus 0.5 cents or so (check the figure for C4
against the figure for C6 in any FAC tuning. You will see that C6 is about
0.4 to 0.8 cents larger).

Now increasing the "A number" has a fairly large effect on the width of
octaves from C3 to B4, but this effect diminishes in the C5 to B6 area. The
FAC program dampens the effect by insisting on 4:1 double octaves of a
certain width. Thus, while I like to increase my "A number" by about 0.3 to
0.5 cents (any wider produces octaves I find objectionable in the midrange),
I find it necessary to fudge C5 to B6 individually note by note to produce a
consistent stretch. I'll describe how I do that in my next post, where I also
will talk about fudging the "C number".

Best regards,
Fred Sturm
Albuquerque, NM