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267 lines
8.8 KiB
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<H1><A ID="SECTION00610000000000000000"></A>
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<A ID="sect2.oscillator"></A>
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<BR>
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The Wavetable Oscillator
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</H1>
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<P>
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Figure <A HREF="node26.html#fig02.02">2.2</A> suggests an easy way to synthesize any desired fixed
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waveform at any desired frequency, using the block diagram shown in Figure
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<A HREF="#fig02.03">2.3</A>. The upper block is an oscillator--not the sinusoidal
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oscillator we saw earlier, but one that produces sawtooth waves instead. Its
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output values, as indicated at the left of the block, should range from <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img179.png"
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ALT="$0$"> to
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the wavetable size <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">. This is used as an index into the wavetable lookup
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block (introduced in Figure <A HREF="node26.html#fig02.01">2.1</A>), resulting in a periodic waveform.
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Figure <A HREF="#fig02.03">2.3</A> (part b) adds an envelope generator and a multiplier to control
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the output amplitude in the same way as for the sinusoidal oscillator shown in
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Figure <A HREF="node12.html#fig01.07">1.7</A> (Page <A HREF="node12.html#fig01.07"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>). Often, one uses a wavetable with (RMS or peak) amplitude 1, so that
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the amplitude of the output is just the magnitude of the envelope generator's
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output.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig02.03"></A><A ID="2190"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2.3:</STRONG>
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Block diagrams: (a) for a wavetable lookup oscillator; (b) with
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amplitude control by an envelope generator.</CAPTION>
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<TR><TD><IMG
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WIDTH="427" HEIGHT="346" BORDER="0"
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SRC="img183.png"
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ALT="\begin{figure}\psfig{file=figs/fig02.03.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Wavetable oscillators are often used to synthesize sounds with specified,
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static spectra. To do this, you can pre-compute <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> samples of any waveform
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of period <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> (angular frequency <IMG
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WIDTH="42" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img184.png"
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ALT="$2\pi/N$">) by adding up the elements of the
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Fourier Series (Page <A HREF="node14.html#sect1.fourier"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>). The computation involved in
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setting up the wavetable at first might be significant, but this may be done
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in advance of the synthesis process, which might take place in real time.
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<P>
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While direct additive synthesis of complex waveforms, as shown in Chapter 1,
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is in principle infinitely flexible as a technique for producing time-varying
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timbres, wavetable synthesis is much less expensive in terms of computation
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but requires switching wavetables to change the timbre. An intermediate
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technique, more flexible and expensive than simple wavetable synthesis but
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less flexible and less expensive than additive synthesis, is to create
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time-varying mixtures between a small number of fixed wavetables. If the
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number of wavetables is only two, this is in effect a cross-fade between
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the two waveforms, as diagrammed in Figure <A HREF="#fig02.04">2.4</A>. Suppose we wish to
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use some signal <!-- MATH
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$0 \le x[n] \le 1$
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-->
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<IMG
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WIDTH="89" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img185.png"
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ALT="$0 \le x[n] \le 1$"> to control the relative strengths of the two
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waveforms, so that, if <IMG
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WIDTH="60" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img186.png"
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ALT="$x[n] = 0$">, we get the first one and if <IMG
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WIDTH="60" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img187.png"
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ALT="$x[n] = 1$"> we
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get the second. Denoting the two signals to be cross-faded by <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> and
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<IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img167.png"
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ALT="$z[n]$">, we compute the signal
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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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(1 - x[n])y[n] + x[n]z[n]
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\end{displaymath}
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-->
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<IMG
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WIDTH="164" HEIGHT="28" BORDER="0"
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SRC="img188.png"
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ALT="\begin{displaymath}
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(1 - x[n])y[n] + x[n]z[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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or, equivalently and usually more efficient to calculate,
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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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y[n] + x[n](z[n]-y[n])
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\end{displaymath}
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-->
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<IMG
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WIDTH="158" HEIGHT="28" BORDER="0"
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SRC="img189.png"
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ALT="\begin{displaymath}
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y[n] + x[n](z[n]-y[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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This computation is diagrammed in Figure <A HREF="#fig02.04">2.4</A>.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig02.04"></A><A ID="2198"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2.4:</STRONG>
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Block diagram for cross-fading between two wavetables.</CAPTION>
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<TR><TD><IMG
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WIDTH="316" HEIGHT="353" BORDER="0"
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SRC="img190.png"
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ALT="\begin{figure}\psfig{file=figs/fig02.04.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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When using this technique to cross-fade between wavetable oscillators, it might
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be desirable to keep the phases of corresponding partials the same across the
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wavetables, so that their amplitudes combine additively when they are mixed. On
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the other hand, if arbitrary wavetables are used (borrowed, for instance, from
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a recorded sound) there will be a phasing effect as the different waveforms are
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mixed.
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<P>
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This scheme can be extended in a daisy chain to move along a continuous path
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between a succession of timbres. Alternatively, or in combination with
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daisy-chaining, cross-fading may be used to interpolate between two different
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timbres, for example as a function of musical dynamic. To do this you
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would prepare two or even several waveforms of a single synthetic voice
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played at
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different dynamics, and interpolate between successive ones as a function
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of the output dynamic you want.
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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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