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<H1><A NAME="SECTION00540000000000000000">
Frequency</A>
</H1>
<P>
Frequencies, like amplitudes, are often measured on a logarithmic scale, in
order to emphasize proportions between them, which usually provide a better
description of the relationship between frequencies than do differences
between them. The frequency ratio between two musical tones determines
the musical interval between them.
<P>
The Western musical scale divides the
<A NAME="1095"></A><I>octave</I> (the musical interval associated with a ratio of 2:1) into
twelve equal sub-intervals, each of which therefore corresponds to a ratio
of <!-- MATH
${2 ^ {1/{12}}}$
-->
<IMG
WIDTH="37" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img110.png"
ALT="${2 ^ {1/{12}}}$">. For historical reasons this sub-interval is called a
<A NAME="1098"></A><I>half-step</I>.
A convenient logarithmic scale for pitch is simply to
count the number of half-steps from a reference pitch--allowing fractions to
permit us to specify pitches which don't fall on a note of the Western scale.
The most commonly used logarithmic pitch scale is
<A NAME="1100"></A>"MIDI pitch", in which the pitch 69 is assigned to a frequency of 440 cycles
per second--the A above middle C. To convert between a MIDI pitch <IMG
WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img111.png"
ALT="$m$">
and a frequency
in cycles per second <IMG
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img112.png"
ALT="$f$">, apply the
<A NAME="1101"></A>Pitch/Frequency Conversion formulas:
<P>
<A NAME="eq-pitchmidi"></A>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
m = 69 + 12 \cdot {\log _ 2} (f/440)
\end{displaymath}
-->
<IMG
WIDTH="178" HEIGHT="28" BORDER="0"
SRC="img113.png"
ALT="\begin{displaymath}
m = 69 + 12 \cdot {\log _ 2} (f/440)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
f = 440 \cdot {{2} ^ {(m - 69) / 12}}
\end{displaymath}
-->
<IMG
WIDTH="136" HEIGHT="27" BORDER="0"
SRC="img114.png"
ALT="\begin{displaymath}
f = 440 \cdot {{2} ^ {(m - 69) / 12}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Middle C, corresponding to MIDI pitch <IMG
WIDTH="54" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img115.png"
ALT="$m=60$">, comes to <IMG
WIDTH="86" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img116.png"
ALT="$f=261.626$"> cycles per
second.
<P>
MIDI itself is an old hardware protocol which has unfortunately insinuated
itself into a great deal of software design. In hardware, MIDI allows only
integer pitches between 0 and 127. However, the underlying scale is well
defined for any "MIDI" number, even negative ones; for example a "MIDI pitch"
of -4 is a decent rate of vibrato. The pitch scale cannot, however, describe
frequencies less than or equal to zero cycles per second. (For a clear
description of MIDI, its capabilities and limitations, see
[<A
HREF="node202.html#r-ballora03">Bal03</A>, ch.6-8]).
<P>
A half-step comes to a ratio of about 1.059 to 1, or about a six percent
increase in frequency. Half-steps are further divided into <A NAME="1107"></A><I>cents</I>, each cent being one hundredth of a half-step. As a rule of
thumb, it might take about three cents to make a discernible change in the
pitch of a musical tone. At middle C this comes to a difference of about 1/2
cycle per second. A graph of frequency as a function of MIDI pitch, over a
two-octave range, is shown in Figure <A HREF="node10.html#fig01.04">1.4</A>.
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2006-12-30
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