Beats vs cycles vs cents

[Robert Scott]

Julia from PA wrote:

> ...Granted that 1/100th of a half step is a different numeric value for 
> each note of the piano, but 1/100th of a half step is always 100 divided 
> into the munber of cycles (or distance) from one half step to the next 
> succesive half step up (or down) isnt it?

Very close, but not exactly.  The division by 100 takes place on a 
logarithmic scale, not an arithmetic scale.  To illustrate the (small) 
difference in your example:

  A4  =               440.000 Hz
  A4 + 100 cents =    466.164 Hz
  Change in cycles =   26.164 Hz
   ...divide by 100 =   0.26164 Hz

So you might conclude that 1 cent at 440 Hz is 0.26164.  But in fact:

  A4 + 1 cent = 440.25423 Hz.

Now, granted that .25423 and .26164 are very close, they are not exactly 
the same.  The difference between these two ways of calculating gets 
larger when you deal with larger jumps, like 300 cents or 400 cents.  As 
long as you are dealing with small enough offsets, the difference between 
the logarithmic scale and the arithmetic scale will be negligible.  But 
for the purist, that difference never goes away totally.

Robert Scott
Ypsilanti, Michigan