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<H1><A ID="SECTION00910000000000000000"></A>
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<A ID="sect5-spectra"></A>
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<BR>
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Taxonomy of spectra
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</H1>
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<P>
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Figure <A HREF="#fig05.01">5.1</A> introduces a way of visualizing the
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<A ID="5595"></A><I>spectrum</I> of an audio signal. The spectrum describes, roughly speaking,
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how the signal's power is distributed into frequencies. (Much more precise
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definitions can be given than those that we'll develop here, but they would
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require more mathematical background.)
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<P>
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<DIV ALIGN="CENTER"><A ID="fig05.01"></A><A ID="5599"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.1:</STRONG>
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A taxonomy of timbres. The spectral envelope describes the shape of
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the spectrum. The sound may be discretely or continuously distributed in
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frequency; if discretely, it may be harmonic or inharmonic.</CAPTION>
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<TR><TD><IMG
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WIDTH="411" HEIGHT="399" BORDER="0"
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SRC="img402.png"
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ALT="\begin{figure}\psfig{file=figs/fig05.01.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Part (a) of the figure shows the spectrum of a
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<A ID="5602"></A><I>harmonic signal</I>,
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which is a periodic signal whose fundamental frequency is in the range of
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perceptible pitches, roughly between 50 and 4000 Hertz.
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The Fourier series (Page <A HREF="node14.html#eq-fourierseries"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>) gives a description of
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a periodic signal as a sum of sinusoids. The frequencies of the sinusoids
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are in the ratio <IMG
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WIDTH="85" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img403.png"
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ALT="$0:1:2:\cdots$">. (The constant term in the Fourier series
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may be thought of as a sinusoid,
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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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{a_0} = {a_0}\cos(0 \cdot \omega n),
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\end{displaymath}
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-->
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<IMG
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WIDTH="131" HEIGHT="28" BORDER="0"
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SRC="img404.png"
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ALT="\begin{displaymath}
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{a_0} = {a_0}\cos(0 \cdot \omega n),
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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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whose frequency is zero.)
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<P>
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In a harmonic signal, the power shown in the spectrum is
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concentrated on a discrete subset of the frequency axis (a discrete
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set consists of isolated points, only finitely many in any bounded interval).
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We call
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this a
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<A ID="5607"></A><I>discrete</I> spectrum.
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Furthermore, the frequencies where the signal's power lies are in the
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<IMG
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WIDTH="75" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img405.png"
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ALT="$0:1:2\cdots$"> ratio that arises from a periodic signal. (It's not
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necessary for <I>all</I> of the harmonic frequencies to be present; some
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harmonics may have zero amplitude.)
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For a harmonic signal, the graph of the spectrum shows the amplitudes of the
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<A ID="5610"></A>partials of the signals.
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Knowing the amplitudes and phases of all the partials fully determines
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the original signal.
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<P>
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Part (b) of the figure shows a spectrum which is also discrete, so that the
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signal can again be considered as a sum of a series of
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partials. In this case, however, there is no fundamental frequency, i.e., no
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audible common submultiple of all the partials. This is called an
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<A ID="5611"></A>
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<I>inharmonic</I> signal. (The terms <IMG
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WIDTH="72" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img406.png"
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ALT="$harmonic$"> and
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<IMG
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WIDTH="87" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img407.png"
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ALT="$inharmonic$"> may be used to describe both the signals and their spectra.)
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<P>
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When dealing with discrete spectra, we report a partial's amplitude in a
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slightly non-intuitive way. Each component sinusoid,
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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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a \cos (\omega n + \phi)
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\end{displaymath}
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-->
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<IMG
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WIDTH="91" HEIGHT="28" BORDER="0"
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SRC="img408.png"
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ALT="\begin{displaymath}
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a \cos (\omega n + \phi)
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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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only counts as having amplitude <IMG
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WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img409.png"
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ALT="$a/2$"> as long as the angular frequency <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">
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is nonzero. But for a
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component of zero frequency, for which
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<!-- MATH
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$\omega = \phi = 0$
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-->
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<IMG
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WIDTH="73" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img410.png"
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ALT="$\omega = \phi = 0$">, the amplitude is given as <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$">--without dividing by two.
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(Components of zero frequency are often called
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<I>DC</I><A ID="5614"></A>
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components; "DC" is historically an acronym for "direct current").
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These conventions for amplitudes in spectra will simplify the mathematics later
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in this chapter; a deeper reason for them will become apparent in
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Chapter <A HREF="node104.html#chapter-delay">7</A>.
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<P>
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Part (c) of the figure shows a third possibility:
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the spectrum might not be concentrated into a discrete set of frequencies, but
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instead might be spread out among all possible frequencies. This can be called a
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<A ID="5616"></A><I>continuous</I>, or
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<A ID="5618"></A><I>noisy</I> spectrum.
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Spectra don't necessarily fall into either the discrete or continuous categories;
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real sounds, in particular, are usually somewhere in between.
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<P>
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Each of the three parts of the figure shows a continuous curve called the
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<A ID="5620"></A><I>spectral envelope</I>. In general, sounds don't have a single,
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well-defined
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spectral envelope; there may be many ways to draw a smooth-looking curve
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through a spectrum. On the other hand, a spectral envelope may be specified
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intentionally; in that case, it is usually
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clear how to make a spectrum conform to it. For
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a discrete spectrum, for example, we could
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simply read off, from the spectral envelope, the desired amplitude of each
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partial and make it so.
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<P>
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A sound's pitch can sometimes be inferred from its spectrum. For discrete
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spectra, the pitch is primarily encoded in the frequencies of the partials.
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Harmonic signals have a pitch determined by their fundamental frequency; for
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inharmonic ones, the pitch may be clear, ambiguous, or absent altogether,
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according to complex and incompletely understood rules. A noisy spectrum may
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also have a perceptible pitch if the spectral envelope contains one or more
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narrow peaks. In general, a sound's loudness and timbre depend more on its
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spectral envelope than on the frequencies in the spectrum, although the
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distinction between continuous and discrete spectra may also be heard as a
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difference in timbre.
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<P>
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Timbre, as well as pitch, may evolve over the life of a sound. We have been
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speaking of spectra here as static entities, not considering whether they
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change in time or not. If a signal's pitch and timbre are changing over time,
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we can think of the spectrum as a time-varying description of
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the signal's momentary behavior.
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<P>
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This way of viewing sounds is greatly oversimplified. The true behavior of
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audible pitch and timbre has many aspects which can't be explained in terms of
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this model. For instance, the timbral quality called "roughness" is sometimes
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thought of as being reflected in rapid changes in the spectral envelope over
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time. The simplified description developed here is useful nonetheless in
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discussions about how to construct discrete or continuous spectra for a wide
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variety of musical purposes, as we will begin to show in the rest of this
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chapter.
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<P>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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