Hertz and Cents

[Richard Moody]

    Cents is a measure of the distance between the frequencies of two notes,
or pitches.  Cents can only measure an interval.  When you say 4 cents
equals 1hz at A-440 you mean the distance between 440 and 441 is aprox 4
cents.   Now at 880 a 1 hertz difference means an interval of 880 - 881.
Which is the larger interval...440 - 441, or 880-881?
The larger the interval the more cents. The size of 440/441 is larger than
880-881 mainly because when reduced, 1/4 is greater than 1/8.    Or you can
compare the decimal equilivants.
441/440 = 1.00227,    881/880 = 1.001135.   Hmmm it appears one interval is
twice as large as the other (the decimal part anyhow) , hence its cents
difference should be twice as much.   So if 440/441 is 4 cents, then 880/881
being the smaller (and twice as small perhaps being an octave apart) should
be 2 cents.
    This  gives food for thought about etd's that only indicate cents
differences.  As we go up the octaves, (and up into the coincident partials)
the same number of cents indicate a larger and larger interval.  At 1760 one
cent difference is about 1 hz.
Hertz stands for cycles per second, don't we all know.   Of course a 1 hz
difference at any frequency means a beat of one per second. So machines
showing cents should less accurate in the treble than in the bass.  At 110
hz a one cent difference would give aprox 118, or  8 beats per second.  This
is a supposition, I do not know how the machines actually work.
    To actually calculate cents involves logarithms but using modern hand
held calculators  or  the Windows 95 calculator,  logs are a breeze.   The
formula is

                Log(F/f) / Log(2) * 1200 = cents.

This is telling you to take the log of the ratio of the interval, divide it
by the log of 2 and multiply that by 1200.  The results is cents between
between the frequencies of the notes in the interval of Fand f.

Lets take 440 and 441.    By convention the larger number is on top so we
have Log(441/440) / Log(2) * 1200 =  3.930158439433.   Or 3.93, or 4 cents.
And 881/880 in cents equals  1.966194474905  or 2 cents.    The 8 cents
difference we  expect would come between 220 and 221 .  You can check it out
on the W95 calculator.  You enter 221 / 220  then hit the log key then  =
next press /  then  2 then the log key  press = then * 1200.   and you
should get 7.85 something.

    For further explanations of cents and how they relate to logs, and how
logs (log 2) relates to music scales and a little history of how and by whom
cents came about,  I have an article I would gladly email out for "peer
review".  I am hoping the math part is written for "left brainers" such as
musicians and many of us  technicians, ---ric


>My understanding is that at A440 4 cents is >aproximately one
> hertz.
>
>If 4 cents
> equalled 1 hertz at A-440 then it would take 8 >cents to equal 1 hertz at
> A-880. (Unless I have it exactly backwards.)
>

> Larry Messerly, RPT
> Phoenix/Prescott
>