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309 lines
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309 lines
8.2 KiB
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HREF="node85.html">Waveshaping using an exponential</A>
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HREF="node80.html">Examples</A>
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HREF="node83.html">Waveshaping and difference tones</A>
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<H2><A ID="SECTION00954000000000000000">
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Waveshaping using Chebychev polynomials</A>
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</H2>
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<A ID="sect5.chebychev"></A>
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<P>
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Example E05.chebychev.pd (Figure <A HREF="#fig05.12">5.12</A>) demonstrates how you can use waveshaping
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to generate pure harmonics. We'll limit ourselves to a specific example here
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in which we would like
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to generate the pure fifth harmonic,
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<DIV ALIGN="CENTER"><A ID="fig05.12"></A><A ID="5829"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.12:</STRONG>
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Using Chebychev polynomials as waveshaping transfer functions.</CAPTION>
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<TR><TD><IMG
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WIDTH="411" HEIGHT="313" BORDER="0"
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SRC="img494.png"
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ALT="\begin{figure}\psfig{file=figs/fig05.12.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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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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\cos(5 \omega n)
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\end{displaymath}
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-->
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<IMG
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WIDTH="60" HEIGHT="28" BORDER="0"
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SRC="img495.png"
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ALT="\begin{displaymath}
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\cos(5 \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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by waveshaping 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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x[n] = \cos(\omega n)
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\end{displaymath}
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-->
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<IMG
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WIDTH="100" HEIGHT="28" BORDER="0"
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SRC="img453.png"
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ALT="\begin{displaymath}
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x[n] = \cos (\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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We
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need to find a suitable transfer function <IMG
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WIDTH="34" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img460.png"
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ALT="$f(x)$">. First we recall
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the formula for the waveshaping function <IMG
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WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img496.png"
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ALT="$f(x) = x^5$">
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(Page <A HREF="node78.html#eq-waveshaping"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>), which gives first,
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third and fifth harmonics:
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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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16 {x^5} = \cos (5 \omega n) + 5 \cos(3 \omega n) + 10 \cos(\omega n)
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\end{displaymath}
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-->
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<IMG
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WIDTH="296" HEIGHT="28" BORDER="0"
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SRC="img497.png"
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ALT="\begin{displaymath}
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16 {x^5} = \cos (5 \omega n) + 5 \cos(3 \omega n) + 10 \cos(\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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Next we add a suitable multiple of <IMG
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WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img498.png"
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ALT="$x^3$"> to
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cancel the third harmonic:
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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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16 {x^5} - 20 {x^3} = \cos (5 \omega n) - 5 \cos(\omega n)
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\end{displaymath}
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-->
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<IMG
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WIDTH="248" HEIGHT="28" BORDER="0"
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SRC="img499.png"
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ALT="\begin{displaymath}
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16 {x^5} - 20 {x^3} = \cos (5 \omega n) - 5 \cos(\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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and then a multiple of <IMG
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WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img243.png"
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ALT="$x$"> to cancel the first harmonic:
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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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16 {x^5} - 20 {x^3} + 5 x = \cos (5 \omega n)
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\end{displaymath}
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-->
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<IMG
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WIDTH="201" HEIGHT="28" BORDER="0"
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SRC="img500.png"
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ALT="\begin{displaymath}
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16 {x^5} - 20 {x^3} + 5 x = \cos (5 \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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So for our waveshaping function we choose
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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(x) = 16 {x^5} - 20 {x^3} + 5 x
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\end{displaymath}
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-->
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<IMG
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WIDTH="170" HEIGHT="28" BORDER="0"
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SRC="img501.png"
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ALT="\begin{displaymath}
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f(x) = 16 {x^5} - 20 {x^3} + 5 x
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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 procedure allows us to isolate any desired harmonic; the resulting
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functions <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="$f$"> are known as
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<A ID="5840"></A><I>Chebychev polynomials</I> [<A
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HREF="node202.html#r-lebrun79">Leb79</A>].
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<P>
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To incorporate this in a waveshaping instrument, we simply build a patch
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that works as in Figure <A HREF="node78.html#fig05.05">5.5</A>, computing the expression
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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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x[n] = f( a[n] \cos(\omega n))
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\end{displaymath}
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-->
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<IMG
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WIDTH="151" HEIGHT="28" BORDER="0"
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SRC="img502.png"
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ALT="\begin{displaymath}
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x[n] = f( a[n] \cos(\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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where <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img503.png"
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ALT="$a[n]$"> is a suitable
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<A ID="5844"></A><I>index</I>
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which may vary as a function of the sample number <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">. When <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="$a$"> happens to
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be one in value, out comes the pure fifth harmonic. Other values of <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="$a$">
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give varying spectra which, in general, have first and third harmonics as
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well as the fifth.
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<P>
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By suitably combining Chebychev polynomials we can fix any desired
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superposition of components in the output waveform (again, as long as
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the waveshaping index is one). But the real
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promise of waveshaping--that by simply changing the index we can
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manufacture spectra that evolve in interesting but controllable ways--is
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not addressed, at least directly, in the Chebychev picture.
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<P>
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HREF="node85.html">Waveshaping using an exponential</A>
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HREF="node80.html">Examples</A>
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HREF="node83.html">Waveshaping and difference tones</A>
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HREF="node4.html">Contents</A></B>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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