Many changes (all cosmetical) I had done before this repo existed.

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<!--Converted with LaTeX2HTML 2002-2-1 (1.71) <!--Converted with LaTeX2HTML 2002-2-1 (1.71)
original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <!DOCTYPE html>
<!--Converted with LaTeX2HTML 2002-2-1 (1.71) <html><head>
original version by: Nikos Drakos, CBLU, University of Leeds <title>Book</title>
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
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<TITLE>book</TITLE>
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<B> Next:</B> <A NAME="tex2html210"
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1]
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<LI><A NAME="tex2html211"
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HREF="node2.html">#1</A>
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<BR>
<LI><A NAME="tex2html213"
HREF="node3.html">#1</A>
<LI><A NAME="tex2html214"
HREF="node4.html">Contents</A>
<LI><A NAME="tex2html215"
HREF="node5.html">Foreword</A>
<LI><A NAME="tex2html216"
HREF="node6.html">Preface</A>
<LI><A NAME="tex2html217"
HREF="node7.html">Sinusoids, amplitude and frequency</A>
<UL>
<LI><A NAME="tex2html218"
HREF="node8.html">Measures of Amplitude</A>
<LI><A NAME="tex2html219"
HREF="node9.html">Units of Amplitude</A>
<LI><A NAME="tex2html220"
HREF="node10.html">Controlling Amplitude</A>
<LI><A NAME="tex2html221"
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<LI><A NAME="tex2html222"
HREF="node12.html">Synthesizing a sinusoid</A>
<LI><A NAME="tex2html223"
HREF="node13.html">Superposing Signals</A>
<LI><A NAME="tex2html224"
HREF="node14.html">Periodic Signals</A>
<LI><A NAME="tex2html225"
HREF="node15.html">About the Software Examples</A>
<UL>
<LI><A NAME="tex2html226"
HREF="node16.html">Quick Introduction to Pd</A>
<LI><A NAME="tex2html227"
HREF="node17.html">How to find and run the examples</A>
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HREF="node19.html">Constant amplitude scaler</A>
<LI><A NAME="tex2html230"
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<LI><A NAME="tex2html231"
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<LI><A NAME="tex2html232"
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<LI><A NAME="tex2html233"
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<LI><A NAME="tex2html234"
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<LI><A NAME="tex2html235"
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<BR>
<LI><A NAME="tex2html236"
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<UL>
<LI><A NAME="tex2html237"
HREF="node27.html">The Wavetable Oscillator</A>
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<LI><A NAME="tex2html247"
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<LI><A NAME="tex2html248"
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<LI><A NAME="tex2html249"
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</UL>
<BR>
<LI><A NAME="tex2html250"
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<UL>
<LI><A NAME="tex2html251"
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<LI><A NAME="tex2html252"
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<BR>
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<LI><A NAME="tex2html306"
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<UL>
<LI><A NAME="tex2html366"
HREF="node156.html">Prefabricated low-, high-, and band-pass filters</A>
<LI><A NAME="tex2html367"
HREF="node157.html">Prefabricated time-varying band-pass filter</A>
<LI><A NAME="tex2html368"
HREF="node158.html">Envelope followers</A>
<LI><A NAME="tex2html369"
HREF="node159.html">Single sideband modulation</A>
<LI><A NAME="tex2html370"
HREF="node160.html">Using elementary filters directly: shelving and peaking</A>
<LI><A NAME="tex2html371"
HREF="node161.html">Making and using all-pass filters</A>
</UL>
<LI><A NAME="tex2html372"
HREF="node162.html">Exercises</A>
</UL>
<BR>
<LI><A NAME="tex2html373"
HREF="node163.html">Fourier analysis and resynthesis</A>
<UL>
<LI><A NAME="tex2html374"
HREF="node164.html">Fourier analysis of periodic signals</A>
<UL>
<LI><A NAME="tex2html375"
HREF="node165.html">Periodicity of the Fourier transform</A>
<LI><A NAME="tex2html376"
HREF="node166.html">Fourier transform as additive synthesis</A>
</UL>
<LI><A NAME="tex2html377"
HREF="node167.html">Properties of Fourier transforms</A>
<UL>
<LI><A NAME="tex2html378"
HREF="node168.html">Fourier transform of DC</A>
<LI><A NAME="tex2html379"
HREF="node169.html">Shifts and phase changes</A>
<LI><A NAME="tex2html380"
HREF="node170.html">Fourier transform of a sinusoid</A>
</UL>
<LI><A NAME="tex2html381"
HREF="node171.html">Fourier analysis of non-periodic signals</A>
<LI><A NAME="tex2html382"
HREF="node172.html">Fourier analysis and reconstruction of audio signals</A>
<UL>
<LI><A NAME="tex2html383"
HREF="node173.html">Narrow-band companding</A>
<LI><A NAME="tex2html384"
HREF="node174.html">Timbre stamping (classical vocoder)</A>
</UL>
<LI><A NAME="tex2html385"
HREF="node175.html">Phase</A>
<UL>
<LI><A NAME="tex2html386"
HREF="node176.html">Phase relationships between channels</A>
</UL>
<LI><A NAME="tex2html387"
HREF="node177.html">Phase bashing</A>
<LI><A NAME="tex2html388"
HREF="node178.html">Examples</A>
<UL>
<LI><A NAME="tex2html389"
HREF="node179.html">Fourier analysis and resynthesis in Pd</A>
<LI><A NAME="tex2html390"
HREF="node180.html">Narrow-band companding: noise suppression</A>
<LI><A NAME="tex2html391"
HREF="node181.html">Timbre stamp (``vocoder")</A>
<LI><A NAME="tex2html392"
HREF="node182.html">Phase vocoder time bender</A>
</UL>
<LI><A NAME="tex2html393"
HREF="node183.html">Exercises</A>
</UL>
<BR>
<LI><A NAME="tex2html394"
HREF="node184.html">Classical waveforms</A>
<UL>
<LI><A NAME="tex2html395"
HREF="node185.html">Symmetries and Fourier series</A>
<UL>
<LI><A NAME="tex2html396"
HREF="node186.html">Sawtooth waves and symmetry</A>
</UL>
<LI><A NAME="tex2html397"
HREF="node187.html">Dissecting classical waveforms</A>
<LI><A NAME="tex2html398"
HREF="node188.html">Fourier series of the elementary waveforms</A>
<UL>
<LI><A NAME="tex2html399"
HREF="node189.html">Sawtooth wave</A>
<LI><A NAME="tex2html400"
HREF="node190.html">Parabolic wave</A>
<LI><A NAME="tex2html401"
HREF="node191.html">Square and symmetric triangle waves</A>
<LI><A NAME="tex2html402"
HREF="node192.html">General (non-symmetric) triangle wave</A>
</UL>
<LI><A NAME="tex2html403"
HREF="node193.html">Predicting and controlling foldover</A>
<UL>
<LI><A NAME="tex2html404"
HREF="node194.html">Over-sampling</A>
<LI><A NAME="tex2html405"
HREF="node195.html">Sneaky triangle waves</A>
<LI><A NAME="tex2html406"
HREF="node196.html">Transition splicing</A>
</UL>
<LI><A NAME="tex2html407"
HREF="node197.html">Examples</A>
<UL>
<LI><A NAME="tex2html408"
HREF="node198.html">Combining sawtooth waves</A>
<LI><A NAME="tex2html409"
HREF="node199.html">Strategies for band-limiting sawtooth waves</A>
</UL>
<LI><A NAME="tex2html410"
HREF="node200.html">Exercises</A>
</UL>
<BR>
<LI><A NAME="tex2html411"
HREF="node201.html">Index</A>
<LI><A NAME="tex2html412"
HREF="node202.html">Bibliography</A>
<LI><A NAME="tex2html413"
HREF="node203.html">About this document ...</A>
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<ADDRESS>
Miller Puckette
2006-12-30
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <!DOCTYPE html>
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -29,23 +33,23 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -70,7 +74,7 @@ Perhaps the most frequently used operation on electronic sounds is to change
their amplitudes. For example, a simple strategy for synthesizing sounds is by their amplitudes. For example, a simple strategy for synthesizing sounds is by
combining sinusoids, which can be generated by evaluating the formula on Page combining sinusoids, which can be generated by evaluating the formula on Page
<A HREF="node7.html#eq-realsinusoid"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" <A HREF="node7.html#eq-realsinusoid"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>, sample by sample. But the sinusoid has a constant SRC="crossref.png"></A>, sample by sample. But the sinusoid has a constant
nominal amplitude <IMG nominal amplitude <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png" SRC="img4.png"
@@ -165,23 +169,23 @@ considered later in Chapter <A HREF="node75.html#chapter-modulation">5</A>.
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML> <HTML>
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -230,23 +234,23 @@ the fundamental dropping, not rising, in amplitude as the string decays.
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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Recall the formula for a (real-valued) sinusoid from Page Recall the formula for a (real-valued) sinusoid from Page
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <!DOCTYPE html>
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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@@ -595,23 +599,23 @@ over the entire range of possible delay times.
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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@@ -281,23 +285,23 @@ and lengthy tuning by trial, error, and critical listening.
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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@@ -262,7 +266,7 @@ appropriately in case <IMG
SRC="img2.png" SRC="img2.png"
ALT="$y[n]$"> is not an integer. This is exactly the formula ALT="$y[n]$"> is not an integer. This is exactly the formula
for wavetable lookup (Page <A HREF="node26.html#chapter-wavetable"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" for wavetable lookup (Page <A HREF="node26.html#chapter-wavetable"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>). We can use all the SRC="crossref.png"></A>). We can use all the
properties of wavetable lookup of recorded sounds to predict the behavior properties of wavetable lookup of recorded sounds to predict the behavior
of variable delay lines. of variable delay lines.
@@ -390,23 +394,23 @@ diagonal region.
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -78,7 +82,7 @@ Section <A HREF="node31.html#sect2.interpolation">2.5</A> apply. To use them, w
delay line input can be broken down into sinusoids and consider separately what delay line input can be broken down into sinusoids and consider separately what
happens to each individual sinusoid. We can use Table 2.1 happens to each individual sinusoid. We can use Table 2.1
(Page <A HREF="node31.html#tab02.1"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" (Page <A HREF="node31.html#tab02.1"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>) to predict SRC="crossref.png"></A>) to predict
the RMS level of the combined distortion products for an interpolated the RMS level of the combined distortion products for an interpolated
variable delay line. variable delay line.
@@ -220,23 +224,23 @@ the way up to 20000 Hertz.
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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@@ -129,7 +133,7 @@ the average delay, and <IMG
SRC="img27.png" SRC="img27.png"
ALT="$\omega $"> is an angular frequency. The Momentary ALT="$\omega $"> is an angular frequency. The Momentary
Transposition (Page <A HREF="node113.html#eq.momentarydel"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" Transposition (Page <A HREF="node113.html#eq.momentarydel"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>), is approximately SRC="crossref.png"></A>), is approximately
<BR><P></P> <BR><P></P>
<DIV ALIGN="CENTER"> <DIV ALIGN="CENTER">
<!-- MATH <!-- MATH
@@ -233,7 +237,7 @@ looping sampler (Section <A HREF="node28.html#sect2.sampling">2.2</A>).
The sawtooth output is also used to envelope the output in exactly the same The sawtooth output is also used to envelope the output in exactly the same
way as in the enveloped wavetable sampler of Figure <A HREF="node29.html#fig02.07">2.7</A> way as in the enveloped wavetable sampler of Figure <A HREF="node29.html#fig02.07">2.7</A>
(Page <A HREF="node29.html#fig02.07"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" (Page <A HREF="node29.html#fig02.07"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>). The SRC="crossref.png"></A>). The
envelope is zero at the points where the sawtooth wraps around, and in envelope is zero at the points where the sawtooth wraps around, and in
between, rises smoothly to a maximum value of 1 (for unit gain). between, rises smoothly to a maximum value of 1 (for unit gain).
@@ -302,7 +306,7 @@ Adding the constant <IMG
SRC="img801.png" SRC="img801.png"
ALT="$d_0$"> has no effect on this slope. The Momentary ALT="$d_0$"> has no effect on this slope. The Momentary
Transposition (Page <A HREF="node113.html#eq.momentarydel"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" Transposition (Page <A HREF="node113.html#eq.momentarydel"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>) is then: SRC="crossref.png"></A>) is then:
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<!-- MATH <!-- MATH
@@ -421,7 +425,7 @@ this causes hard-to-control Doppler shifts.
A good choice of envelope is one half cycle of a sinusoid. If we assume A good choice of envelope is one half cycle of a sinusoid. If we assume
on average that the two delay outputs are uncorrelated (Page on average that the two delay outputs are uncorrelated (Page
<A HREF="node13.html#eq-meanpowersum"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" <A HREF="node13.html#eq-meanpowersum"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>), the signal power from the two delay lines, after SRC="crossref.png"></A>), the signal power from the two delay lines, after
enveloping, will add to a constant (since the sum of squares of the two enveloping, will add to a constant (since the sum of squares of the two
envelopes is one). envelopes is one).
@@ -444,23 +448,23 @@ artifacts such as difference tones.
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -79,7 +83,7 @@ which essentially corresponds to the synthesis algorithm to be used, and then
worries about how to control the various unit generators in time. In this worries about how to control the various unit generators in time. In this
section, we'll use abstract block diagrams to describe patches, but in the section, we'll use abstract block diagrams to describe patches, but in the
``examples" section (Page <A HREF="node18.html#sect1.examples"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" ``examples" section (Page <A HREF="node18.html#sect1.examples"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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specific implementation environment and show some of the software-dependent specific implementation environment and show some of the software-dependent
details. details.
@@ -265,23 +269,23 @@ Using an envelope generator to control amplitude.</CAPTION>
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -140,23 +144,23 @@ sonic qualities described as ``presence", ``warmth", ``clarity", and so on.
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -114,7 +118,7 @@ head rotation speed" is the same as the formula for <IMG
SRC="img112.png" SRC="img112.png"
ALT="$f$"> given on Page ALT="$f$"> given on Page
<A HREF="node115.html#sect7.psformula"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" <A HREF="node115.html#sect7.psformula"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>. Here the positive interval (seven half-steps) SRC="crossref.png"></A>. Here the positive interval (seven half-steps)
gives rise to a transposition factor greater than one, and therefore to a gives rise to a transposition factor greater than one, and therefore to a
negative value for <IMG negative value for <IMG
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
@@ -129,7 +133,7 @@ Once <IMG
the corresponding envelopes parallels exactly that of the corresponding envelopes parallels exactly that of
the overlapping sample looper (example B10.sampler.overlap.pd, Page the overlapping sample looper (example B10.sampler.overlap.pd, Page
<A HREF="node37.html#sect2.example.overlap"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" <A HREF="node37.html#sect2.example.overlap"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>). The minimum delay SRC="crossref.png"></A>). The minimum delay
is added to each of the two sawtooth signals to make delay inputs for the is added to each of the two sawtooth signals to make delay inputs for the
<TT>vd~</TT> objects, whose outputs are multiplied by the corresponding <TT>vd~</TT> objects, whose outputs are multiplied by the corresponding
envelopes and summed. envelopes and summed.
@@ -140,23 +144,23 @@ envelopes and summed.
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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@@ -259,23 +263,23 @@ getting a quick response and a smooth result.
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML> <HTML>
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -72,7 +76,7 @@ Among the applications of filters discussed in Chapter <A HREF="node127.html#cha
saw how to use heterodyning, combined with a low-pass filter, to find the saw how to use heterodyning, combined with a low-pass filter, to find the
amplitude and phase of a sinusoidal component of a signal (Page amplitude and phase of a sinusoidal component of a signal (Page
<A HREF="node158.html#sect8.heterodyning"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" <A HREF="node158.html#sect8.heterodyning"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>). In this chapter we will refine this technique SRC="crossref.png"></A>). In this chapter we will refine this technique
into what is called into what is called
<A NAME="12298"></A><I>Fourier analysis</I>. <A NAME="12298"></A><I>Fourier analysis</I>.
In its simplest form, Fourier analysis takes as input any periodic In its simplest form, Fourier analysis takes as input any periodic

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@@ -88,7 +92,7 @@ sinusoid were better localized around the corresponding value of <IMG
ALT="$k$">. We ALT="$k$">. We
can accomplish this using the enveloping technique first introduced in Figure can accomplish this using the enveloping technique first introduced in Figure
<A HREF="node29.html#fig02.07">2.7</A> (Page <A HREF="node29.html#fig02.07"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" <A HREF="node29.html#fig02.07">2.7</A> (Page <A HREF="node29.html#fig02.07"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>). Applying this technique to Fourier SRC="crossref.png"></A>). Applying this technique to Fourier
analysis will not only improve our analyses, but will also shed new light on analysis will not only improve our analyses, but will also shed new light on
the enveloping looping sampler of Chapter 2. the enveloping looping sampler of Chapter 2.
@@ -506,23 +510,23 @@ four-period rule may be relaxed to three or even slightly less.
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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original version by: Nikos Drakos, CBLU, University of Leeds original version by: Nikos Drakos, CBLU, University of Leeds
@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others --> Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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@@ -7,6 +7,10 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -131,23 +135,23 @@ unchanged, and values in between give a smooth interpolation between the two.
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