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<TITLE>Additive synthesis: Risset's bell</TITLE>
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<H2><A NAME="SECTION00883000000000000000">
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Additive synthesis: Risset's bell</A>
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</H2>
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<A NAME="4778"></A>
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<P>
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The abstraction mechanism of Pd, which we used above to make a reusable ADSR
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generator, is also useful for making voice banks. Here we will use
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abstractions to organize banks of oscillators for additive synthesis. There
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are many possible ways of organizing oscillator banks besides
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those shown here.
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<P>
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The simplest and most direct organization of the sinusoids is to form
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partials to add up to a note. The result is monophonic, in the sense that
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the patch will play only one note at a time, which, however, will consist of
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several sinusoids whose individual frequencies and amplitudes might depend both
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on those of the note we're playing, and also on their individual placement
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in a harmonic (or inharmonic) overtone series.
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<P>
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Example D07.additive.pd (Figure <A HREF="#fig04.16">4.16</A>) uses a bank of 11 copies
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of an abstraction named <TT>partial</TT> (Figure <A HREF="#fig04.17">4.17</A>) in an
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imitation of a well-known bell instrument by Jean-Claude Risset. As described
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in [<A
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HREF="node202.html#r-dodge85">DJ85</A>, p. 94], the bell sound has 11 partials, each with its own
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relative amplitude, frequency, and duration.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig04.16"></A><A NAME="4890"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4.16:</STRONG>
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A Pd realization of Jean-Claude Risset's bell instrument. The
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bell sound is made by summing 11 sinusoids, each made by a copy of the
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<TT>partial</TT> abstraction.</CAPTION>
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<TR><TD><IMG
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WIDTH="498" HEIGHT="389" BORDER="0"
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SRC="img384.png"
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ALT="\begin{figure}\psfig{file=figs/fig04.16.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig04.17"></A><A NAME="4891"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4.17:</STRONG>
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The <TT>partial</TT> abstraction used by Risset's bell instrument
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from Figure <A HREF="#fig04.16">4.16</A>.</CAPTION>
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<TR><TD><IMG
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WIDTH="487" HEIGHT="376" BORDER="0"
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SRC="img385.png"
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ALT="\begin{figure}\psfig{file=figs/fig04.17.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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For each note, the <TT>partial</TT> abstraction computes a simple (quartic)
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amplitude envelope consisting only of an attack and a decay segment; there is
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no sustain or release segment. This is multiplied by a sinusoid, and the
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product is added into a summing bus. Two new object classes are introduced
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to implement the summing bus:
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<P>
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<BR><!-- MATH
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$\fbox{ \texttt{catch\~}}$
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-->
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<IMG
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WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img386.png"
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ALT="\fbox{ \texttt{catch\~}}">:
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<A NAME="4908"></A>define and output a summing bus. The creation argument ("sum-bus" in this example)
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gives the summing bus a name so that <TT>throw~</TT> objects below can refer
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to it. You may have as many summing busses (and hence <TT>catch~</TT> objects)
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as you like but they must all have different names.
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<P>
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<BR><!-- MATH
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$\fbox{ \texttt{throw\~}}$
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-->
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<IMG
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WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img388.png"
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ALT="\fbox{ \texttt{throw\~}}">:
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<A NAME="4909"></A>add to a summing bus. The creation argument selects which summing bus to
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use.
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<P>
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The control interface is crude: number boxes control the "fundamental"
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frequency of the bell and its duration. Sending a "bang" message to
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the <TT>s trigger</TT> object starts a note. (The note then decays over
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the period of time controlled by the duration parameter; there is no
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separate trigger to stop the note). There is no amplitude
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control except via the <TT>output~</TT> object.
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<P>
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The four arguments to each invocation of the <TT>partial</TT> abstraction specify:
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<OL>
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<LI>amplitude. The amplitude of the partial at its peak, at the
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end of the attack and the beginning of the decay of the note.
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<P>
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</LI>
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<LI>relative duration. This is multiplied by the overall note duration
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(controlled in the main patch) to
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determine the duration of the decay portion of the sinusoid. Individual partials
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may thus have different decay times, so that some partials die out faster than
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others, under the main patch's overall control.
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<P>
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</LI>
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<LI>relative frequency. As with the relative duration, this controls each
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partial's frequency as a multiple of the overall frequency controlled in the
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main patch.
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<P>
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</LI>
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<LI>detune. A frequency in Hertz to be added to the product of the global
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frequency and the relative frequency.
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<P>
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</LI>
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</OL>
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Inside the <TT>partial</TT> abstraction, the amplitude is simply taken
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directly from the "$1" argument (multiplying by 0.1 to adjust for
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the high individual amplitudes); the duration is calculated from the
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<TT>r duration</TT> object, multiplying it by the "$2" argument.
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The frequency is computed as <IMG
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WIDTH="48" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img390.png"
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ALT="$fp+d$"> where <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$"> is the global frequency (from
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the <TT>r frequency</TT> object), <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$"> is the relative frequency of the
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partial, and <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$"> is the detune frequency.
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<P>
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Miller Puckette
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2006-12-30
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