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<H2><A ID="SECTION001054000000000000000">
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The PAF generator</A>
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</H2>
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<P>
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<DIV ALIGN="CENTER"><A ID="fig06.17"></A><A ID="7005"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.18:</STRONG>
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The phase-aligned formant (PAF) synthesis algorithm.</CAPTION>
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<TR><TD><IMG
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WIDTH="482" HEIGHT="525" BORDER="0"
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SRC="img621.png"
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ALT="\begin{figure}\psfig{file=figs/fig06.17.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Example F12.paf.pd (Figure <A HREF="#fig06.17">6.18</A>) is a realization of the PAF generator,
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described in Section <A HREF="node96.html#sect6.paf">6.4</A>.
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The control inputs specify the fundamental frequency, the center frequency, and
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the bandwidth, all in "MIDI" units. The first steps taken in the realization
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are to divide center frequency by fundamental (to get the center frequency quotient)
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and bandwidth by fundamental to get the index of modulation for the
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waveshaper. The center frequency quotient is sampled-and-held so that it is
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only updated at periods of the fundamental.
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<P>
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The one oscillator (the <TT>phasor~</TT> object) runs at the fundamental
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frequency. This is used both to control a <TT>samphold~</TT> object which
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synchronizes updates to the center frequency quotient (labeled "C.F. relative
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to fundamental" in the figure), and to compute phases for both <TT>cos~</TT> objects which operate as shown earlier in Figure <A HREF="node100.html#fig06.16">6.17</A>.
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<P>
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The waveshaping portion of the patch uses a half period of a sinusoid
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as a lookup function (to compensate for the frequency doubling because
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of the symmetry of the lookup function). To get a half-cycle of the sine function
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we multiply the phase by 0.5 and subtract 0.25, so that the adjusted phase
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runs from -0.25 to +0.25, once each period. This scans the positive half
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of the cycle defined by the <TT>cos~</TT> object.
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<P>
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The amplitude of the half-sinusoid is then adjusted by an index of modulation
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(which is just the bandwidth quotient <!-- MATH
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${\omega_b}/\omega$
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-->
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<IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img588.png"
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ALT="${\omega_b}/\omega$">). The table
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("bell-curve") holds an unnormalized Gaussian curve sampled
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from -4 to 4 over 200 points (25 points per unit), so the center of the table,
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at point 100, corresponds to the central peak of the bell curve. Outside the
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interval from -4 to 4 the Gaussian curve is negligibly small.
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<P>
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Figure <A HREF="#fig06.18">6.19</A> shows how the Gaussian wavetable is prepared. One new
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control object is needed:
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<P>
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<DIV ALIGN="CENTER"><A ID="fig06.18"></A><A ID="7047"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.19:</STRONG>
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Filling in the wavetable for Figure <A HREF="#fig06.17">6.18</A>.</CAPTION>
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<TR><TD><IMG
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WIDTH="476" HEIGHT="240" BORDER="0"
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SRC="img622.png"
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ALT="\begin{figure}\psfig{file=figs/fig06.18.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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<BR><!-- MATH
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$\fbox{ $\mathrm{until}$\ }$
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-->
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<IMG
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WIDTH="57" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
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SRC="img623.png"
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ALT="\fbox{ $\mathrm{until}$\ }"> :
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<A ID="7054"></A>When the left, "start" inlet is banged, output sequential bangs (with no
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elapsed time between them) iteratively, until the right, "stop" inlet is
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banged. The stopping "bang" message must originate somehow from the
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<TT>until</TT> object's outlet; otherwise, the outlet will send "bang" messages
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forever, freezing out any other object which could break the loop.
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<P>
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As used here, a loop driven by an <TT>until</TT> object
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counts from 0 to 199, inclusive. The loop count is maintained by the
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"<TT>f</TT>" and "<TT>+ 1</TT>" objects, each of which feeds the other. But
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since the "<TT>+ 1</TT>" object's output goes to the right inlet of the
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"<TT>f</TT>", its result (one greater) will only emerge from the
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"<TT>f</TT>" the next time it is banged by "<TT>until</TT>". So each bang
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from "<TT>until</TT>" increments the value by one.
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<P>
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The order in which the loop is started matters: the upper "<TT>t b b</TT>"
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object (short for "trigger bang bang") must first send zero to the
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"<TT>f</TT>", thus initializing it, and then set the <TT>until</TT> object sending
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bangs, incrementing the value, until stopped. To stop it when the value
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reaches 199, a <TT>select</TT> object checks the value and, when it sees the
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match, bangs the "stop" inlet of the <TT>until</TT> object.
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<P>
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Meanwhile, for every number from 0 to 199 that comes out of the "<TT>f</TT>"
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object, we create an ordered pair of messages to the <TT>tabwrite</TT> object.
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First, at right, goes the index itself, from 0 to 199. Then for the left inlet,
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the first <TT>expr</TT> object adjusts the index to range from -4 to 4 (it
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previously ranged from 0 to 199) and the second one evaluates the
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Gaussian function.
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<P>
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In this patch we have not fully addressed the issue of updating the
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center frequency quotient at the appropriate times. Whenever the carrier
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frequency is changed the sample-and-hold step properly delays the update
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of the quotient. But if, instead or in addition, the fundamental itself
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changes abruptly, then for a fraction of a period the <TT>phasor~</TT> object's frequency and the quotient are out of sync. Pd does not allow
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the <TT>samphold~</TT> output to be connected back into the <TT>phasor~</TT> input without the inclusion of an explicit delay (see the next chapter) and
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there is no simple way to modify the patch to solve this problem.
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<P>
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Assuming that we <I>did</I> somehow clock the <TT>phasor~</TT> object's input
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synchronously with its own wraparound points, we would then have to do the
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same for the bandwidth/fundamental quotient on the right side of the patch
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as well. In the current scenario, however, there is no problem updating
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that value continuously.
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<P>
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A practical solution to this updating problem could be simply to rewrite the
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entire patch in C as a Pd class; this also turns out to use much less CPU time
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than the pictured patch, and is the more practical solution overall--as long
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as you don't want to experiment with making embellishments or other changes to
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the algorithm. Such embellishments might include: adding an inharmonic upward
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or downward shift in the partials; allowing to switch between smooth and
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sampled-and-held center frequency updates; adding separate gain controls for
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even and odd partials; introducing gravel by irregularly modulating the phase;
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allowing mixtures of two or more waveshaping functions; or making sharper
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percussive attacks by aligning the phase of the oscillator with the timing of
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an amplitude envelope generator.
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<P>
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One final detail about amplitude is in order: since the amplitude of the
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strongest partial decreases roughly as <IMG
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WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img572.png"
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ALT="$1/(1+b)$"> where <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$"> is the index of
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modulation, it is sometimes (but not always) desirable to correct the amplitude
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of the output by multiplying by <IMG
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WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img606.png"
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ALT="$1+b$">. This is only an option if <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$"> is
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smoothly updated (as in this example), not if it is sampled-and-held.
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One situation in which this is appropriate is in simulating plucked strings (by
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setting center frequency to the fundamental, starting with a high index of
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modulation and dropping it exponentially); it would be appropriate to hear
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the fundamental dropping, not rising, in amplitude as the string decays.
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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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