The logarithmic arithmetic developed by Napier and Briggs in the 17th
century was soon applied to tuning the musical scale: the result was
the "Equal Temperment Tuning", which was adopted immediately by
composers who enjoyed lots of key-changes.  True, the new tuning
algorithm didn't sound quite as sweet with simple "diatonic" music, and
so the Traditionalists continued to compose and perform using older
methods of "Just Tuning".  Others simply observed: "With the new
tuning, now the Klavier sounds equally _bad_ in all keys!"   Well, it's
not a panacea, it's a compromise, with a sound mathematical
description.

(Since I enjoy playing with lots of key- and chord-changes myself, I
tune my piano with the logarithmic Equal Temperment, like practically
everyone else.  Perhaps someday I'll get to play a fancy synthesizer,
which dynamically adjusts the temperment for the "sweetest sound" for
whichever chord (or dischord) I play at the moment!)

Here's the mathematics in simple form, and you don't have to worry
about logarithms if you have an engineering calculator (or equivalent
in a computer "assistant").

The basis is simply that the frequency of each adjacent note on the
piano keyboard differs by a constant _ratio_.  Since the "octave" of
the keyboard has 12 different notes, and since the frequency doubles in
each octave, the ration is the twelfth root of two, which is 1.059463094.

So, if I define my standard pitch as 220 Hz for the A below middle C,
then when I multiply 220 by 1.056... _twelve_ times, I get the next
octave higher, which is A = 440 Hz.

   220.00  233.08  246.94  261.63  ....  369.9994  391.995  415.30 440.000

The piano tuner also geometrically sub-divides the ratio between
adjacent tones into 100 parts, called "Cents".  1 Cent is the
twelve-hundredth root of two, which is 1.000577790.  Not much of a
change, is it!   But when I multiply 220.00 Hz by this tiny ratio one
hundred times, I get the correct frequency of the next note, A# 233.08 Hz.

Suppose, for example, the piano tuner reports that my "new" old piano
is 20 Cents flat compared to concert pitch.  Is this a lot?  Well,
since 100 Cents flat would amount to one note (key), the piano is
1/5-note flat.  5 times 12 is 60, so I find the sixtyth root of two,
which is 1.0116, and then I divide 440 Hz by 1.0116 to discover that
"A" on my piano is really at 434.95 Hz.  That's 5 Hz (cycles per
second) difference from concert pitch, and I can easily hear the 5 Hz
beat frequency when the tuner strikes his A440 tuning fork and the
piano key together.

So, I instruct the tuner to "wind it up to concert pitch" and then I
leave him alone for an hour or so.  The result -- a properly tuned
piano -- always pleases me!

-- Robbie Rhodes