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234 lines
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234 lines
8.1 KiB
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<H1><A ID="SECTION001000000000000000000"></A>
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<A ID="chapter-paf"></A>
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<BR>
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Designer spectra
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</H1>
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<P>
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As suggested at the beginning of the previous chapter, a powerful way to
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synthesize musical sounds is to specify--and then realize--specific
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trajectories of pitch (or more generally, frequencies of partials), along
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with trajectories of
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spectral envelope [<A
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HREF="node202.html#r-puckette01a">Puc01</A>].
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The
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spectral envelope is used to determine the amplitude of the individual
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partials, as a function of their frequencies, and is thought of as controlling
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the sound's (possibly time-varying) timbre.
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<P>
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A simple example of this would be to imitate a plucked
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string by constructing a sound with harmonically spaced partials in which the spectral
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envelope starts out rich but then dies away exponentially with higher
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frequencies decaying faster than lower ones, so that the timbre mellows over
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time. Spectral-evolution
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models for various acoustic instruments have been proposed
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[<A
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HREF="node202.html#r-grey77">GM77</A>] [<A
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HREF="node202.html#r-risset69">RM69</A>]
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. A more complicated example is the
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spoken or sung voice, in which vowels appear as spectral envelopes, dipthongs
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and many consonants appear as time variations in the spectral envelopes, and
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other consonants appear as spectrally shaped noise.
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<P>
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Spectral envelopes may be obtained from analysis of recorded sounds (developed
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in Chapter <A HREF="node163.html#chapter-fourier">9</A>) or from purely synthetic criteria.
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To specify a spectral envelope from scratch for every possible frequency
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would be tedious, and in most cases you would want to describe them in
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terms of their salient features. The most popular way of doing this is
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to specify the size and shape of the spectral envelope's peaks, which are
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called
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<A ID="6795"></A><I>formants</I>. Figure <A HREF="#fig06.01">6.1</A> shows a spectral envelope with two
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formants. Although the shapes of the two peaks in the spectral envelope
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are different, they can both be roughly described by giving the coordinates
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of each apex (which give the formant's
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<A ID="6798"></A><I>center frequency</I>
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and amplitude) and each formant's
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<A ID="6800"></A><I>bandwidth</I>. A typical measure of bandwidth would be the width of the
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peak at a level 3 decibels below its apex.
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Note that if the peak is at (or near) the <IMG
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WIDTH="42" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img11.png"
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ALT="$f=0$"> axis, we pretend it falls
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off to the left at the same rate as (in reality) it falls off to the right.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig06.01"></A><A ID="6804"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.1:</STRONG>
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A spectral envelope showing the frequencies, amplitudes, and
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bandwidths of two formants.</CAPTION>
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<TR><TD><IMG
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WIDTH="467" HEIGHT="261" BORDER="0"
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SRC="img551.png"
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ALT="\begin{figure}\psfig{file=figs/fig06.01.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Suppose we wish to generate a harmonic sound with a specified collection of
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formants. Independently of the fundamental frequency desired, we wish the
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spectrum to have peaks with prescribed center frequencies, amplitudes, and
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bandwidths. Returning to the phase modulation spectra shown in Figure
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<A HREF="node87.html#fig05.16">5.16</A>, we see that, at small indices of modulation at least, the
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result has a single, well-defined spectral peak. We can imagine adding several
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of these, all sharing a fundamental (modulating) frequency but with carriers
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tuned to different harmonics to select the various desired center frequencies,
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and with indices of modulation chosen to give the desired bandwidths. This was
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first explored by Chowning [<A
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HREF="node202.html#r-chowning89">Cho89</A>] who arranged formants generated
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by phase modulation to synthesize singing voices.
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In this chapter we'll establish a general framework for building
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harmonic spectra with desired, possibly time-varying, formants.
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<P>
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<BR><HR>
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<!--Table of Child-Links-->
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<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A ID="tex2html1870"
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HREF="node90.html">Carrier/modulator model</A>
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<LI><A ID="tex2html1871"
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HREF="node91.html">Pulse trains</A>
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<UL>
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<LI><A ID="tex2html1872"
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HREF="node92.html">Pulse trains via waveshaping</A>
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<LI><A ID="tex2html1873"
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HREF="node93.html">Pulse trains via wavetable stretching</A>
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<LI><A ID="tex2html1874"
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HREF="node94.html">Resulting spectra</A>
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</UL>
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<BR>
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<LI><A ID="tex2html1875"
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HREF="node95.html">Movable ring modulation</A>
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<LI><A ID="tex2html1876"
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HREF="node96.html">Phase-aligned formant (PAF) generator</A>
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<LI><A ID="tex2html1877"
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HREF="node97.html">Examples</A>
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<UL>
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<LI><A ID="tex2html1878"
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HREF="node98.html">Wavetable pulse train</A>
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<LI><A ID="tex2html1879"
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HREF="node99.html">Simple formant generator</A>
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<LI><A ID="tex2html1880"
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HREF="node100.html">Two-cosine carrier signal</A>
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<LI><A ID="tex2html1881"
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HREF="node101.html">The PAF generator</A>
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<LI><A ID="tex2html1882"
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HREF="node102.html">Stretched wavetables</A>
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</UL>
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<BR>
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<LI><A ID="tex2html1883"
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HREF="node103.html">Exercises</A>
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</UL>
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HREF="node90.html">Carrier/modulator model</A>
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HREF="book.html">book</A>
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<B> Previous:</B> <A ID="tex2html1857"
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HREF="node88.html">Exercises</A>
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<B> <A ID="tex2html1865"
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<ADDRESS>
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Miller Puckette
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2006-12-30
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