Adam wrote:
>I have heard, though may be incorrectly, that this is an approximation
>(much like using 3.141592653), and that the actual formulae used to
>develop the scale are somewhat more complex.


No, that is the definition of equal temperament.  The intervals are divided
up between the octaves in an exponential manner so that the 12 intervals
are in equal proportion to the preceding note and the octave represents
a ratio of 2;1.  Like I said in the previous post, in actual practice
a piano tuner will probably deviate from exact equal temperament in
order to reduce upper order harmonics (which produce the sensation of
dissonance) on some of the higher keys.

Harold Hallikainen had an interesting note about Helmholtz in response
to my post.  Helmholtz was one of those interesting multi-disciplinary
scientists in the 19th century whose interests spanned many areas.  It
is interesting to me to see discussions of music such as this, since I
find many software engineers are also musicians.  I know that since
microcontrollers were introduced, music applications have been one of
main areas of interest --it's always seemed a natural to me.  Both
software and music deal with abstractions, symbols, patterns and
structures of many dimensions.

Darrel Johansen