Beats vs cycles vs cents

[Sarah Fox]

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Hi Julia,

To get mathematical on ya', pitch (e.g. the note#) and frequency (cps or =
Hz) are exponentially related, where

F =3D 440 * 2^(n/12),

where n is the number of half steps above (positive value) or below =
(negative value) A440.  This can be represented on a log scale:

log(F) =3D log(440) + n/12 * log(2)

In this form of the equation, we see there is a linear relationship =
between log(F) and n.  Similarly, there's a linear relationship between =
log(F) and n/100 (which is a cent).  The frequency value of a cent is =
greater in the upper end of a note's nominal range and smaller in the =
lower end.  However, there really isn't *much* difference from lower to =
upper.  For instance, consider A440:

For this note, F=3D440 (obviously)
A cent at this frequency would be .254 Hz
One beat/sec would occur at a pitch difference of 3.9 cents.

For a half of a semitone higher (or 50 cents -- the division between A =
and A#):

F=3D452.9 Hz
A cent at this frequency would be .262 Hz
One beat/sec would occur at a pitch difference of 3.8 cents.

At the lower end of the nominal range for A, 50 cents lower:

F=3D 427.5 Hz
A cent at this frequency would be .247 Hz
One beat/sec would occur at a pitch difference of 4 cents.

So you see, there's really very little difference.  Furthermore, the =
frequency difference across this range is 25.4 Hz.  (Nothing implied =
here about conversion between imperial and metric -- just coincidence!)  =
A hundredth of that value is .254 Hz, which you should notice is the =
calculated value of a cent at 440 Hz.  It is also 97% of a cent at a =
half of a semitone higher, and 103% of a cent at a half of a semitone =
lower.

So TECHNICALLY a cent is NOT 1/100 of a the frequency span of a =
semitone, except at the center frequency.  But unless you are good =
enough to distinguish beat rate differences on the order of 0.0075 cps =
(and I challenge *anyone* on this list, even Andre, to be that =
accurate!), then yes, Julia, a cent is indeed 1/100 of the frequency =
difference between semitones (at least approximately enough for tuning). =
 ;-)

Happy tuning, and...

Peace,
Sarah


----- Original Message -----=20
  From: Alpha88x@aol.com=20
  To: pianotech@ptg.org=20
  Sent: Thursday, December 02, 2004 1:22 PM
  Subject: Re: Beats vs cycles vs cents


  Greetings,

              I know this is a delayed response on this thread,  but I =
am confused.=20

             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?=20

  Julia
  PA

  In a message dated 3/15/2004 7:08:48 PM Eastern Standard Time, =
eromlignod@kc.rr.com writes:


    There are 100 cents in a half-step (semitone), but a cent is *not* =
1/100th
    of a half-step.


  (100 cents is always a half-step no matter how high or low the
  frequency is.


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