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<TITLE>Strategies for band-limiting sawtooth waves</TITLE>
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<BR>
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<B> Next:</B> <A NAME="tex2html3540"
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HREF="node200.html">Exercises</A>
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HREF="node197.html">Examples</A>
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<B> Previous:</B> <A NAME="tex2html3530"
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HREF="node198.html">Combining sawtooth waves</A>
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<H2><A NAME="SECTION001452000000000000000">
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Strategies for band-limiting sawtooth waves</A>
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</H2>
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig10.14"></A><A NAME="14512"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.14:</STRONG>
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Alternative techniques for making waveforms with corners: (a) a
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triangle wave as the minimum of two line segments; (b) clipping a triangle wave to
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make an "envelope".</CAPTION>
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<TR><TD><IMG
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WIDTH="617" HEIGHT="359" BORDER="0"
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ALT="\begin{figure}\psfig{file=figs/fig10.14.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 J05.triangle.pd (Figure <A HREF="#fig10.14">10.14</A>, part a) shows a simple way to make a
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triangle wave, in which only the slope of the rising and falling segment are
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specified. A phasor supplies the rising shape (its amplitude being the slope),
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and the same phasor, subtracted from one, gives the decaying shape. The
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minimum of the two linear functions follows the rising phasor up to the
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intersection of the two, and then follows the falling phasor back down to zero
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at the end of the cycle.
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<P>
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A triangle wave can be clipped above and below to make a trapezoidal wave,
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which can be used either as an audio-frequency pulse or, at a lower fundamental
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frequency, as a repeating ASR (attack/sustain/release) envelope. Patch
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J06.enveloping.pd (Figure <A HREF="#fig10.14">10.14</A> part b) demonstrates this. The same rising
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shape is used as in the previous example, and the falling shape differs only
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in that its phase is set so that it falls to zero at a controllable point (not
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necessarily at the end of the cycle as before). The <TT>clip~</TT> object prevents
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it from rising above 1 (so that, if the intersection of the two segments is
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higher than one, we get a horizontal "sustain" segment), and also from falling
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below zero, so that once the falling shape reaches zero, the output is zero for
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the rest of the cycle.
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<P>
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Example J07.oversampling.pd shows how to use up-sampling to reduce foldover when using a
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<TT>phasor~</TT> object as an audio sawtooth wave. A subpatch, running at 16
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times the base sample rate, contains the <TT>phasor~</TT> object and a
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three-pole, three-zero Butterworth filter to reduce the amplitudes of partials
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above the Nyquist frequency of the parent patch (running at the original sample
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rate) so that the output won't fold over when it is down-sampled at the
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<TT>outlet~</TT> object. Example J08.classicsynth.pd demonstrates using up-sampled
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phasors as signal generators to make an imitation of a classic synthesizer
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doing subtractive synthesis.
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<P>
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Example J09.bandlimited.pd shows how to use transition splicing as an alternative way to
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generate a sawtooth wave with controllable foldover. This has the advantage of
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being more direct (and usually less compute-intensive) than the up-sampling
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method. On the other hand, this technique depends on using the reciprocal of
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the fundamental frequency as an audio signal in its own right (to control the
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amplitude of the sweeping signal that reads the transition table) and, in the
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same way as for the PAF technique of Chapter <A HREF="node89.html#chapter-paf">6</A>, care must be
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taken to avoid clicks if the fundamental frequency changes discontinuously.
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