208 lines
7.0 KiB
HTML
208 lines
7.0 KiB
HTML
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<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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* with significant contributions from:
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<TITLE>Frequency</TITLE>
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<B> Next:</B> <A NAME="tex2html708"
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HREF="node12.html">Synthesizing a sinusoid</A>
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HREF="node7.html">Sinusoids, amplitude and frequency</A>
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<B> Previous:</B> <A NAME="tex2html696"
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HREF="node10.html">Controlling Amplitude</A>
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<B> <A NAME="tex2html704"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html706"
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HREF="node201.html">Index</A></B>
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<!--End of Navigation Panel-->
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<H1><A NAME="SECTION00540000000000000000">
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Frequency</A>
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</H1>
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<P>
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Frequencies, like amplitudes, are often measured on a logarithmic scale, in
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order to emphasize proportions between them, which usually provide a better
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description of the relationship between frequencies than do differences
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between them. The frequency ratio between two musical tones determines
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the musical interval between them.
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<P>
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The Western musical scale divides the
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<A NAME="1095"></A><I>octave</I> (the musical interval associated with a ratio of 2:1) into
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twelve equal sub-intervals, each of which therefore corresponds to a ratio
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of <!-- MATH
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${2 ^ {1/{12}}}$
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-->
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<IMG
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WIDTH="37" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img110.png"
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ALT="${2 ^ {1/{12}}}$">. For historical reasons this sub-interval is called a
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<A NAME="1098"></A><I>half-step</I>.
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A convenient logarithmic scale for pitch is simply to
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count the number of half-steps from a reference pitch--allowing fractions to
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permit us to specify pitches which don't fall on a note of the Western scale.
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The most commonly used logarithmic pitch scale is
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<A NAME="1100"></A>``MIDI pitch", in which the pitch 69 is assigned to a frequency of 440 cycles
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per second--the A above middle C. To convert between a MIDI pitch <IMG
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WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img111.png"
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ALT="$m$">
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and a frequency
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in cycles per second <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$">, apply the
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<A NAME="1101"></A>Pitch/Frequency Conversion formulas:
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<P>
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<A NAME="eq-pitchmidi"></A>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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m = 69 + 12 \cdot {\log _ 2} (f/440)
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\end{displaymath}
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-->
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<IMG
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WIDTH="178" HEIGHT="28" BORDER="0"
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SRC="img113.png"
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ALT="\begin{displaymath}
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m = 69 + 12 \cdot {\log _ 2} (f/440)
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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f = 440 \cdot {{2} ^ {(m - 69) / 12}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="136" HEIGHT="27" BORDER="0"
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SRC="img114.png"
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ALT="\begin{displaymath}
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f = 440 \cdot {{2} ^ {(m - 69) / 12}}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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Middle C, corresponding to MIDI pitch <IMG
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WIDTH="54" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img115.png"
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ALT="$m=60$">, comes to <IMG
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WIDTH="86" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img116.png"
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ALT="$f=261.626$"> cycles per
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second.
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<P>
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MIDI itself is an old hardware protocol which has unfortunately insinuated
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itself into a great deal of software design. In hardware, MIDI allows only
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integer pitches between 0 and 127. However, the underlying scale is well
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defined for any ``MIDI" number, even negative ones; for example a ``MIDI pitch"
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of -4 is a decent rate of vibrato. The pitch scale cannot, however, describe
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frequencies less than or equal to zero cycles per second. (For a clear
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description of MIDI, its capabilities and limitations, see
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[<A
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HREF="node202.html#r-ballora03">Bal03</A>, ch.6-8]).
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<P>
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A half-step comes to a ratio of about 1.059 to 1, or about a six percent
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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
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thumb, it might take about three cents to make a discernible change in the
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pitch of a musical tone. At middle C this comes to a difference of about 1/2
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cycle per second. A graph of frequency as a function of MIDI pitch, over a
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two-octave range, is shown in Figure <A HREF="node10.html#fig01.04">1.4</A>.
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<P>
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HREF="node201.html">
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<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
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SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
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<BR>
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<B> Next:</B> <A NAME="tex2html708"
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HREF="node12.html">Synthesizing a sinusoid</A>
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<B> Up:</B> <A NAME="tex2html702"
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HREF="node7.html">Sinusoids, amplitude and frequency</A>
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<B> Previous:</B> <A NAME="tex2html696"
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HREF="node10.html">Controlling Amplitude</A>
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<B> <A NAME="tex2html704"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html706"
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HREF="node201.html">Index</A></B>
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<!--End of Navigation Panel-->
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<ADDRESS>
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Miller Puckette
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2006-12-30
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</ADDRESS>
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