You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
545 lines
17 KiB
545 lines
17 KiB
<!DOCTYPE html>
|
|
|
|
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
|
original version by: Nikos Drakos, CBLU, University of Leeds
|
|
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
|
|
* with significant contributions from:
|
|
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
|
<HTML>
|
|
<HEAD>
|
|
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0">
|
|
|
|
|
|
<TITLE>Timbre stretching</TITLE>
|
|
<META NAME="description" CONTENT="Timbre stretching">
|
|
<META NAME="keywords" CONTENT="book">
|
|
<META NAME="resource-type" CONTENT="document">
|
|
<META NAME="distribution" CONTENT="global">
|
|
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2002-2-1">
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
<LINK REL="STYLESHEET" HREF="book.css">
|
|
|
|
<LINK REL="next" HREF="node31.html">
|
|
<LINK REL="previous" HREF="node29.html">
|
|
<LINK REL="up" HREF="node26.html">
|
|
<LINK REL="next" HREF="node31.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html988"
|
|
HREF="node31.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A ID="tex2html982"
|
|
HREF="node26.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A ID="tex2html976"
|
|
HREF="node29.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A ID="tex2html984"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A ID="tex2html986"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A ID="tex2html989"
|
|
HREF="node31.html">Interpolation</A>
|
|
<B> Up:</B> <A ID="tex2html983"
|
|
HREF="node26.html">Wavetables and samplers</A>
|
|
<B> Previous:</B> <A ID="tex2html977"
|
|
HREF="node29.html">Enveloping samplers</A>
|
|
<B> <A ID="tex2html985"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A ID="tex2html987"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H1><A ID="SECTION00640000000000000000"></A>
|
|
<A ID="sect2.stretching"></A>
|
|
<BR>
|
|
Timbre stretching
|
|
</H1>
|
|
|
|
<P>
|
|
The wavetable oscillator of Section <A HREF="node27.html#sect2.oscillator">2.1</A>, which we extended in
|
|
Section <A HREF="node28.html#sect2.sampling">2.2</A> to encompass grabbing waveforms from arbitrary
|
|
wavetables such as recorded sounds, may additionally be extended in
|
|
a complementary way, that we'll refer to as
|
|
<A ID="2263"></A><I>timbre stretching</I>, for reasons we'll develop in this section. There are
|
|
also many other possible ways to extend wavetable synthesis, using for
|
|
instance frequency modulation and waveshaping, but we'll leave them to later
|
|
chapters.
|
|
|
|
<P>
|
|
The central idea of timbre stretching is to reconsider the idea of the wavetable
|
|
oscillator as a mechanism for playing a stored wavetable (or part of one) end
|
|
to end. There is no reason the end of one cycle has to coincide with the
|
|
beginning of another. Instead, we could ask for copies of the waveform to
|
|
be spaced with alternating segments of silence; or, going in the opposite
|
|
direction, the waveform copies could be spaced more closely together so that
|
|
they overlap. The single parameter available in Section
|
|
<A HREF="node27.html#sect2.oscillator">2.1</A>--the frequency--has been heretofore used to control
|
|
two separate aspects of the output: the period at which we start new
|
|
copies of the waveform, and also the length of each individual copy. The
|
|
idea of timbre stretching is to control the two independently.
|
|
|
|
<P>
|
|
Figure <A HREF="#fig02.09">2.9</A> shows the result of playing a wavetable in three ways.
|
|
In each case the output waveform has period 20; in other words, the output frequency
|
|
is <IMG
|
|
WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img181.png"
|
|
ALT="$R/20$"> if <IMG
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img36.png"
|
|
ALT="$R$"> is the output sample rate. In part (a) of the figure, each
|
|
copy of the waveform is played over 20 samples, so that the wave form fits
|
|
exactly into the cycle with no gaps and no overlap. In part (b), although the
|
|
period is still 20, the waveform is compressed into the middle half of the
|
|
period (10 samples); or in other words, the
|
|
<A ID="2267"></A><I>duty cycle</I>--the relative amount of time the waveform fills the
|
|
cycle--equals 50 percent. The remaining 50 percent of the time, the output
|
|
is zero.
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A ID="fig02.09"></A><A ID="2271"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2.9:</STRONG>
|
|
A waveform is played at a period of 20 samples: (a) at 100 percent
|
|
duty cycle; (b) at 50 percent; (c) at 200 percent</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="609" HEIGHT="304" BORDER="0"
|
|
SRC="img221.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig02.09.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
In part (c), the waveform is stretched to 40 samples, and since it is still
|
|
repeated every 20 samples, the waveforms overlap two to one. The duty cycle
|
|
is thus 200 percent.
|
|
|
|
<P>
|
|
Suppose now that the 100 percent duty cycle waveform has a Fourier series
|
|
(Section <A HREF="node14.html#sect1.fourier">1.7</A>) equal to:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{x_{100}}[n] = {a_0} +
|
|
{a_1} \cos \left ( \omega n + {\phi_1} \right ) +
|
|
{a_2} \cos \left ( 2 \omega n + {\phi_2} \right ) + \cdots
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="387" HEIGHT="28" BORDER="0"
|
|
SRC="img222.png"
|
|
ALT="\begin{displaymath}
|
|
{x_{100}}[n] = {a_0} +
|
|
{a_1} \cos \left ( \omega n + {\phi...
|
|
...+
|
|
{a_2} \cos \left ( 2 \omega n + {\phi_2} \right ) + \cdots
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
where <IMG
|
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img27.png"
|
|
ALT="$\omega $"> is the angular frequency (equal to <IMG
|
|
WIDTH="37" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img223.png"
|
|
ALT="$\pi/10$"> in our example since
|
|
the period is 20.) To simplify this example we won't worry about where
|
|
the series must end, and will just let it go on forever.
|
|
|
|
<P>
|
|
We would like to relate this to the Fourier series of the other two waveforms
|
|
in the example, in order to show how changing the duty cycle changes the
|
|
timbre of the result. For the 50 percent duty cycle case (calling the
|
|
signal <IMG
|
|
WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img224.png"
|
|
ALT="${x_{50}}[n]$">), we observe that the waveform, if we replicate it out
|
|
of phase by a half period and add the two, gives exactly the original waveform
|
|
at twice the frequency:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{x_{100}}[2n] = {x_{50}}[n] + {x_{50}}[n+{\pi \over \omega}]
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="209" HEIGHT="35" BORDER="0"
|
|
SRC="img225.png"
|
|
ALT="\begin{displaymath}
|
|
{x_{100}}[2n] = {x_{50}}[n] + {x_{50}}[n+{\pi \over \omega}]
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
where <IMG
|
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img27.png"
|
|
ALT="$\omega $"> is the angular frequency (and so <IMG
|
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img226.png"
|
|
ALT="$\pi / \omega$"> is half the
|
|
period) of both signals. So if we denote the Fourier series of <IMG
|
|
WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img224.png"
|
|
ALT="${x_{50}}[n]$">
|
|
as:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{x_{50}}[n] = {b_0} +
|
|
{b_1} \cos \left ( \omega n + {\theta_1} \right ) +
|
|
{b_2} \cos \left ( 2 \omega n + {\theta_2} \right ) + \cdots
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="373" HEIGHT="28" BORDER="0"
|
|
SRC="img227.png"
|
|
ALT="\begin{displaymath}
|
|
{x_{50}}[n] = {b_0} +
|
|
{b_1} \cos \left ( \omega n + {\thet...
|
|
...
|
|
{b_2} \cos \left ( 2 \omega n + {\theta_2} \right ) + \cdots
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
and substitute the Fourier series for all three terms above, we get:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{a_0} +
|
|
{a_1} \cos \left ( 2 \omega n + {\phi_1} \right ) +
|
|
{a_2} \cos \left ( 4 \omega n + {\phi_2} \right ) + \cdots
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="327" HEIGHT="28" BORDER="0"
|
|
SRC="img228.png"
|
|
ALT="\begin{displaymath}
|
|
{a_0} +
|
|
{a_1} \cos \left ( 2 \omega n + {\phi_1} \right ) +
|
|
{a_2} \cos \left ( 4 \omega n + {\phi_2} \right ) + \cdots
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
=
|
|
{b_0} +
|
|
{b_1} \cos \left ( \omega n + {\theta_1} \right ) +
|
|
{b_2} \cos \left ( 2 \omega n + {\theta_2} \right ) + \cdots
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="327" HEIGHT="28" BORDER="0"
|
|
SRC="img229.png"
|
|
ALT="\begin{displaymath}
|
|
=
|
|
{b_0} +
|
|
{b_1} \cos \left ( \omega n + {\theta_1} \right...
|
|
...
|
|
{b_2} \cos \left ( 2 \omega n + {\theta_2} \right ) + \cdots
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
+
|
|
{b_0} +
|
|
{b_1} \cos \left ( \omega n + \pi + {\theta_1} \right ) +
|
|
{b_2} \cos \left ( 2 \omega n + 2 \pi + {\theta_2} \right ) + \cdots
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="389" HEIGHT="28" BORDER="0"
|
|
SRC="img230.png"
|
|
ALT="\begin{displaymath}
|
|
+
|
|
{b_0} +
|
|
{b_1} \cos \left ( \omega n + \pi + {\theta_1} ...
|
|
...\cos \left ( 2 \omega n + 2 \pi + {\theta_2} \right ) + \cdots
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
= 2 {b_0} +
|
|
2 {b_2} \cos \left ( 2 \omega n + {\theta_2} \right ) +
|
|
2 {b_4} \cos \left ( 4 \omega n + {\theta_4} \right ) + \cdots
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="359" HEIGHT="28" BORDER="0"
|
|
SRC="img231.png"
|
|
ALT="\begin{displaymath}
|
|
= 2 {b_0} +
|
|
2 {b_2} \cos \left ( 2 \omega n + {\theta_2} \...
|
|
... {b_4} \cos \left ( 4 \omega n + {\theta_4} \right ) + \cdots
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
and so
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{a_0} = 2{b_0}, \,\, {a_1} = 2{b_2}, \,\, {a_2} = 2{b_4}
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="199" HEIGHT="27" BORDER="0"
|
|
SRC="img232.png"
|
|
ALT="\begin{displaymath}
|
|
{a_0} = 2{b_0}, \,\, {a_1} = 2{b_2}, \,\, {a_2} = 2{b_4}
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
and so on: the even partials of <IMG
|
|
WIDTH="26" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img233.png"
|
|
ALT="$x_{50}$">, at least, are obtained by stretching
|
|
the partials of <IMG
|
|
WIDTH="32" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img234.png"
|
|
ALT="$x_{100}$"> out twice as far. (We don't yet know about the odd
|
|
partials of <IMG
|
|
WIDTH="26" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img233.png"
|
|
ALT="$x_{50}$">, and these might be in line with the even ones or not,
|
|
depending on factors we can't control yet. Suffice it to say for the moment,
|
|
that if the waveform connects smoothly with the horizontal axis at both
|
|
ends, the odd partials will act globally like the even ones. To make this
|
|
more exact we'll need Fourier analysis, which is developed in Chapter 9.)
|
|
|
|
<P>
|
|
Similarly, <IMG
|
|
WIDTH="32" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img234.png"
|
|
ALT="$x_{100}$"> and <IMG
|
|
WIDTH="32" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img235.png"
|
|
ALT="$x_{200}$"> are related in exactly the same way:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{x_{200}}[2n] = {x_{100}}[n] + {x_{100}}[n+{\pi \over \omega}]
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="222" HEIGHT="35" BORDER="0"
|
|
SRC="img236.png"
|
|
ALT="\begin{displaymath}
|
|
{x_{200}}[2n] = {x_{100}}[n] + {x_{100}}[n+{\pi \over \omega}]
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
so that, if the amplitudes of the fourier series of <IMG
|
|
WIDTH="32" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img235.png"
|
|
ALT="$x_{200}$"> are denoted
|
|
by <IMG
|
|
WIDTH="17" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img237.png"
|
|
ALT="$c_0$">, <IMG
|
|
WIDTH="17" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img238.png"
|
|
ALT="$c_1$">, <IMG
|
|
WIDTH="22" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img239.png"
|
|
ALT="$\ldots$">, we get:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{c_0} = 2{a_0}, {c_1} = 2{a_2}, {c_2} = 2{a_4}, \ldots
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="213" HEIGHT="27" BORDER="0"
|
|
SRC="img240.png"
|
|
ALT="\begin{displaymath}
|
|
{c_0} = 2{a_0}, {c_1} = 2{a_2}, {c_2} = 2{a_4}, \ldots
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
so that the partials of <IMG
|
|
WIDTH="32" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img235.png"
|
|
ALT="$x_{200}$"> are those of <IMG
|
|
WIDTH="32" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img234.png"
|
|
ALT="$x_{100}$"> shrunk, by half,
|
|
to the left.
|
|
|
|
<P>
|
|
We see that squeezing the waveform by a factor of 2 has the effect of
|
|
stretching the Fourier series out by two, and on the other hand stretching the
|
|
waveform by a factor of two squeezes the Fourier series by two. By the same
|
|
sort of argument, in general it turns out that stretching the waveform by a
|
|
factor of any positive number <IMG
|
|
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img112.png"
|
|
ALT="$f$"> squeezes the overtones, in frequency, by the
|
|
reciprocal <IMG
|
|
WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img241.png"
|
|
ALT="$1/f$">--at least approximately, and the approximation is at least
|
|
fairly good if the waveform "behaves well" at its ends.
|
|
(As we'll see later, the waveform can always be forced to behave at least
|
|
reasonably well by enveloping it as in Figure <A HREF="node29.html#fig02.07">2.7</A>.)
|
|
|
|
<P>
|
|
Figure <A HREF="#fig02.10">2.10</A> shows the spectra of the three waveforms--or in
|
|
other words the one waveform at three duty cycles--of Figure
|
|
<A HREF="#fig02.09">2.9</A>. The figure emphasizes the relationship between the three
|
|
spectra by drawing curves through each, which, on inspection, turn out to be
|
|
the same curve, only stretched differently; as the duty cycle goes up, the
|
|
curve is both compressed to the left (the frequencies all drop) and amplified
|
|
(stretched upward).
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A ID="fig02.10"></A><A ID="2511"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2.10:</STRONG>
|
|
The Fourier series magnitudes for the waveforms shown in Figure
|
|
<A HREF="#fig02.09">2.9</A>. The horizontal axis is the harmonic number. We only "hear"
|
|
the coefficients for integer harmonic numbers; the continuous curves are the
|
|
"ideal" contours.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="474" HEIGHT="317" BORDER="0"
|
|
SRC="img242.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig02.10.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
The continuous curves have a very simple interpretation. Imagine squeezing the
|
|
waveform into some tiny duty cycle, say 1 percent. The contour will be
|
|
stretched by a factor of 100. Working backward, this would allow us to
|
|
interpolate between each pair of consecutive points of the 100 percent duty
|
|
cycle contour (the original one) with 99 new ones. Already in the figure the
|
|
50 percent duty cycle trace defines the curve with twice the resolution of
|
|
the original one. In the limit, as the duty cycle gets arbitrarily small, the
|
|
spectrum is filled in more and more densely; and the limit is the "true"
|
|
spectrum of the waveform.
|
|
|
|
<P>
|
|
This "true" spectrum is only audible at suitably low duty cycles, though. The
|
|
200 percent duty cycle example actually misses the peak in the ideal
|
|
(continuous) spectrum because the peak falls below the first harmonic. In
|
|
general, higher duty cycles sample the ideal curve at lower resolutions.
|
|
|
|
<P>
|
|
Timbre stretching is an extremely powerful technique for generating
|
|
sounds with systematically variable spectra. Combined with the possibilities of
|
|
mixtures of waveforms (Section <A HREF="node27.html#sect2.oscillator">2.1</A>) and of snatching
|
|
endlessly variable waveforms from recorded samples (Section
|
|
<A HREF="node28.html#sect2.sampling">2.2</A>), it is possible to generate all sorts of sounds.
|
|
For example, the block diagram of Figure <A HREF="node29.html#fig02.07">2.7</A> gives us a
|
|
way to to grab and stretch timbres from a recorded wavetable. When the
|
|
"frequency" parameter <IMG
|
|
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img112.png"
|
|
ALT="$f$"> is high enough to be audible as a pitch, the
|
|
"size"
|
|
parameter <IMG
|
|
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img208.png"
|
|
ALT="$s$"> can be thought of as controlling timbre stretch, via the
|
|
formula <IMG
|
|
WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img209.png"
|
|
ALT="$t = fs/R$"> from Section <A HREF="node28.html#sect2.sampling">2.2</A>, where we now
|
|
reinterpret <IMG
|
|
WIDTH="9" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img82.png"
|
|
ALT="$t$"> as the factor by which the timbre is to be stretched.
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html988"
|
|
HREF="node31.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A ID="tex2html982"
|
|
HREF="node26.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A ID="tex2html976"
|
|
HREF="node29.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A ID="tex2html984"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A ID="tex2html986"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A ID="tex2html989"
|
|
HREF="node31.html">Interpolation</A>
|
|
<B> Up:</B> <A ID="tex2html983"
|
|
HREF="node26.html">Wavetables and samplers</A>
|
|
<B> Previous:</B> <A ID="tex2html977"
|
|
HREF="node29.html">Enveloping samplers</A>
|
|
<B> <A ID="tex2html985"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A ID="tex2html987"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|
|
|