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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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<TITLE>Control</TITLE>
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<BR>
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<B> Next:</B> <A NAME="tex2html1176"
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HREF="node43.html">Control streams</A>
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<B> Up:</B> <A NAME="tex2html1170"
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HREF="node40.html">Audio and control computations</A>
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<B> Previous:</B> <A NAME="tex2html1164"
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HREF="node41.html">The sampling theorem</A>
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<B> <A NAME="tex2html1172"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html1174"
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HREF="node201.html">Index</A></B>
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<H1><A NAME="SECTION00720000000000000000"></A>
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<A NAME="sect3.control"></A>
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<BR>
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Control
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</H1>
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<P>
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So far we have dealt with audio signals, which are just sequences <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$">
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defined for integers <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="$n$">, which correspond to regularly spaced points in time.
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This is often an adequate framework for describing synthesis techniques, but
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real electronic music applications usually also entail other computations
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which have to be made at irregular points in time. In this section we'll
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develop a framework for describing what we will call
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<A NAME="3576"></A><I>control</I>
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computations. We will always require that any computation correspond to
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a specific
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<A NAME="3578"></A><I>logical time</I>.
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The logical time controls which sample of audio output will be the first to
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reflect the result of the computation.
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<P>
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In a non-real-time system (such as Csound in its classical form),
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this means that logical time proceeds from zero to the length of the
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output soundfile. Each ``score card" has an associated logical time (the
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time in the score), and is acted upon once the audio computation has reached
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that time. So audio and control calculations (grinding out the samples and
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handling note cards) are each handled in turn, all in increasing order of
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logical time.
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<P>
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In a real-time system, logical time, which still corresponds to the time of the
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next affected sample of audio output, is always slightly in advance of
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<A NAME="3580"></A><I>real time</I>,
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which is measured by the sample that is actually leaving the computer. Control
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and audio computations still are carried out in alternation, sorted by logical
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time.
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<P>
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The reason for using logical time and not real time in computer music
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computations is to keep the calculations independent of the actual execution
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time of the computer, which can vary for a variety of reasons, even for two
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seemingly identical calculations. When we are calculating a new value of an
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audio signal or processing some control input, real time may pass but we
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require that the logical time stay the same through the whole calculation, as
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if it took place instantaneously. As a result of this, electronic music
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computations, if done correctly, are deterministic: two runs of the same
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real-time or non-real-time audio computation, each having the same inputs,
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should have identical results.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig03.02"></A><A NAME="3584"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3.2:</STRONG>
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Timeline for digital audio and control computation: (a) with a
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block size of one sample; (b) with a block size of four samples.</CAPTION>
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<TR><TD><IMG
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WIDTH="533" HEIGHT="279" BORDER="0"
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SRC="img310.png"
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ALT="\begin{figure}\psfig{file=figs/fig03.02.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Figure <A HREF="#fig03.02">3.2</A> (part a) shows schematically how logical time and
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sample computation are lined up. Audio samples are computed at regular
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periods
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(marked as wavy lines), but
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before the calculation of each sample we do all the control calculations
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that might affect it
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(marked as straight line segments). First we do the control computations
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associated with logical times starting at zero, up to but not including one;
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then we compute the first audio sample (of index zero), at logical time
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one. We then do all control calculations up to but not including logical
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time 2, then the sample of index one, and so on. (Here we are adopting
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certain conventions about labeling that could be chosen differently. For
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instance, there is no fundamental reason control should be pictured as
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coming ``before" audio computation but it is easier to think that way.)
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<P>
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Part (b) of the figure shows the situation if we wish to compute the audio
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output in blocks of more than one sample at a time. Using the variable
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<IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img311.png"
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ALT="$B$"> to denote the number of elements in a block (so <IMG
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WIDTH="45" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img312.png"
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ALT="$B=4$"> in the figure),
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the first audio computation will output samples <IMG
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WIDTH="87" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img313.png"
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ALT="$0, 1, ... B-1$"> all at once
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in a block computed at logical time <IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img311.png"
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ALT="$B$">. We have
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to do the relevant control computations for all <IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img311.png"
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ALT="$B$"> periods of time in
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advance. There is a delay of <IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img311.png"
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ALT="$B$"> samples between logical time and the
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appearance of audio output.
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<P>
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Most computer music software computes audio in blocks. This is done to
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increase the efficiency of individual audio operations (such as Csound's unit
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generators and Max/MSP and Pd's tilde objects). Each unit generator or tilde
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object incurs overhead each time it is called, equal to perhaps twenty times
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the cost of computing one sample on average. If the block size is one, this
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means an overhead of 2,000%; if it is sixty-four (as in Pd by default), the
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overhead is only some 30%.
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<B> Next:</B> <A NAME="tex2html1176"
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HREF="node43.html">Control streams</A>
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<B> Up:</B> <A NAME="tex2html1170"
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<B> <A NAME="tex2html1172"
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