Formulae

[Robert Scott]

 Niklas Eliasson wrote:

>A cent is also a logarithmic unit!!!! That means we logaritmically
>divide an octave in 1200 parts. That is: one cent is
>
>2^(1/1200)=1.00057778950655....
>
>The reason why most of us use % as cents, is that it is a pretty
>good approximation!!! And usually it is OK - but if we want to nail
>down the definition, we have to do it.

  Perhaps this can be clarified a little with an analogy.  Suppose a piano costs
$2000.  Then suppose the price is raised 5%.  Then it is raised 5% again.
Is this the same as raising it 10%?.  No.  The first price rise results in
$2100.
The second price rise results in $2205.  But a single 10% price rise would
be $2200.

  So Niklas is correct in saying that 1 cent is not exactly the same thing as
1/100 of the difference in frequency of a semitone.  If you think it is,
then ask
yourself this question:  If you raise E4 by 50 cents, do you get exactly the
same pitch as if you lower F4 by 50 cents?  If you treat cents as logarithmic
units you do.  But if you calculate the frequencies by offsetting them by

       (f)  x  2^(1/12)/100  x 50

where f is the nominal frequency for  E4 or F4, then raising E4 by 50 cents will
give a slightly different frequency than lowering F4 by 50 cents.

  When Niklas says that  1 cent  is 1.00057778950655..., it means that
if you want to offset E4 by 50 cents, you would have to multiply the
usual frequency for E4 by  (1.00057778950655...) ^ 50.  But what we often
do for convenience is multiply by  1 + (.00059463094) x 50  instead.  These two
methods agree exactly at 100 cents, and are darn close for offsets less than
100 cents, but they become dramatically different for offsets higher than
100 cents.
Fortunately, most pianotech work involves small offsets where the approximation
is good enough.

  -Bob Scott
   Ann Arbor, Michigan