Beats vs cycles vs cents

[Don A. Gilmore]

There are 100 cents in a half-step (semitone), but a cent is *not* 1/100th
of a half-step.  In fact the size of a cent (in Hz) grows by double going
from the lower note to the semitone above.  This is because cents are
logarithmic, just like notes.  So the ratio between two pitches that are a
cent apart is

2^(1/1200) = 1.0005778...

As the pitch gets higher, the cents get larger.

As you mentioned, the beat frequency is the difference in frequency between
the two pitches.  It's due to the waves adding and subtracting from each
other.  That's why using beats to tune two notes together is less accurate
at the low end than at the high end of the piano.  When the bottom A (~27.5
Hz) is one cent sharp it is

27.5 x 1.0005778 = 27.516 Hz

The difference in frequency is 0.016 Hz, which would produce one beat every
62 seconds!  But at the top C, one cent has become

4186 x 1.005778 = 4188.4

for a difference of about 2.4 Hz, or 2.4 beats per second.

Cents are really a more accurate indicator of how far out-of-tune a note is
than beats are since they are based on the actual perceived musical pitch of
a note (100 cents is always a half-step no matter how high or low the
frequency is.

Don A. Gilmore
Mechanical Engineer
Kansas City

----------
Sarah wrote:

A beat happens every time two frequencies drift past each other by one cycle
(one complete vibration).  In other words, if you have two notes at 440 and
441 Hz (i.e. 440 and 441 cycles per second), respectively, there will be a
resultant beat frequency of one beat per second.

For those of you with heads stuck in physics or acoustics books, take heed
that there is no acoustic energy at the beat frequency, provided there is
linearity in the system, but that's been argued back and forth before!

And yes, there are 100 cents between one half step and the next, where each
note is roughly 2^(1/12) or 1.0595 times the frequency of the previous half
step and 2^(-1/12) or 0.944 times the frequency of the next half step.  A
cent, at any frequency, would be roughly 1-2^((1/12)/100) or 0.0005778 times
that frequency.  At 440 Hz, a cent would be roughly 0.25 Hz.

Peace,
Sarah