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<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
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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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* with significant contributions from:
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
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<HTML>
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<HEAD>
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<TITLE>Sampling</TITLE>
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<BR>
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<B> Next:</B> <A NAME="tex2html961"
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HREF="node29.html">Enveloping samplers</A>
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<B> Up:</B> <A NAME="tex2html955"
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HREF="node26.html">Wavetables and samplers</A>
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<B> Previous:</B> <A NAME="tex2html949"
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HREF="node27.html">The Wavetable Oscillator</A>
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<B> <A NAME="tex2html957"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html959"
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HREF="node201.html">Index</A></B>
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<BR>
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<BR>
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<!--End of Navigation Panel-->
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<H1><A NAME="SECTION00620000000000000000"></A>
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<A NAME="sect2.sampling"></A>
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<BR>
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Sampling
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||||
</H1>
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<P>
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``Sampling"
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<A NAME="2203"></A>
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is nothing more than recording a live signal into a wavetable, and then later
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||||
playing it out again. (In commercial samplers the entire wavetable is
|
||||
usually called a ``sample" but to avoid confusion we'll only use the word
|
||||
``sample" here to mean a single number in an audio signal.)
|
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|
||||
<P>
|
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At its simplest, a sampler is simply a wavetable oscillator, as was shown in
|
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Figure <A HREF="node27.html#fig02.03">2.3</A>. However, in the earlier discussion we imagined playing
|
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the oscillator back at a frequency high enough to be perceived as a pitch, at
|
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least 30 Hertz or so. In the case of sampling, the frequency is usually lower
|
||||
than 30 Hertz, and so the period, at least 1/30 second and perhaps much more, is
|
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long enough that you can hear the individual cycles as separate events.
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|
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<P>
|
||||
Going back to Figure <A HREF="node26.html#fig02.02">2.2</A>, suppose that instead of 40 points the
|
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wavetable <IMG
|
||||
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img80.png"
|
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ALT="$x[n]$"> is a one-second recording, at an original
|
||||
sample rate of 44100, so that it has 44100 points; and let
|
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<IMG
|
||||
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> in part (b) of the figure have a period of 22050 samples. This
|
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corresponds to a frequency of 2 Hertz. But what we hear is not a pitched sound
|
||||
at 2 cycles per second (that's too slow to hear as a pitch) but rather, we
|
||||
hear the original recording <IMG
|
||||
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img80.png"
|
||||
ALT="$x[n]$"> played back repeatedly at double speed. We've
|
||||
just reinvented the sampler.
|
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|
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<P>
|
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In general, if we assume the sample rate <IMG
|
||||
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
|
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ALT="$R$"> of the recording is the same as
|
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the output sample rate, if the wavetable has <IMG
|
||||
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
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SRC="img3.png"
|
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ALT="$N$"> samples, and if we index
|
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it with a sawtooth wave of period <IMG
|
||||
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
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SRC="img86.png"
|
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ALT="$M$">,
|
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the sample is sped up or slowed down by
|
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a factor of <IMG
|
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WIDTH="40" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img191.png"
|
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ALT="$N/M$">, equal to <IMG
|
||||
WIDTH="47" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img192.png"
|
||||
ALT="$N f / R$"> if <IMG
|
||||
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img112.png"
|
||||
ALT="$f$"> is the frequency in Hertz of the
|
||||
sawtooth. If we denote the transposition factor by <IMG
|
||||
WIDTH="9" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img82.png"
|
||||
ALT="$t$"> (so that, for instance,
|
||||
<IMG
|
||||
WIDTH="54" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img193.png"
|
||||
ALT="$t=3/2$"> means transposing upward a perfect fifth), and if we denote the
|
||||
transposition in half-steps by <IMG
|
||||
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img194.png"
|
||||
ALT="$h$">, then we get the
|
||||
<A NAME="2206"></A>Transposition Formulas for Looping Wavetables:
|
||||
<A NAME="sect2.transpositionformula"></A>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
t = {N / M} = {{N f} / R}
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="130" HEIGHT="28" BORDER="0"
|
||||
SRC="img195.png"
|
||||
ALT="\begin{displaymath}
|
||||
t = {N / M} = {{N f} / R}
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
h = 12 \, {\log _ 2} \left ( {N \over M} \right ) =
|
||||
12 \, {\log _ 2} \left ( {N f \over R} \right )
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="241" HEIGHT="45" BORDER="0"
|
||||
SRC="img196.png"
|
||||
ALT="\begin{displaymath}
|
||||
h = 12 \, {\log _ 2} \left ( {N \over M} \right ) =
|
||||
12 \, {\log _ 2} \left ( {N f \over R} \right )
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
Frequently the desired transposition in half-steps (<IMG
|
||||
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img194.png"
|
||||
ALT="$h$">) is known and the
|
||||
formula must be solved for either <IMG
|
||||
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img112.png"
|
||||
ALT="$f$"> or <IMG
|
||||
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img3.png"
|
||||
ALT="$N$">:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
f = {{2^{h/12} R} \over N}
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="79" HEIGHT="42" BORDER="0"
|
||||
SRC="img197.png"
|
||||
ALT="\begin{displaymath}
|
||||
f = {{2^{h/12} R} \over N}
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
N = {{2^{h/12} R} \over f}
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="86" HEIGHT="45" BORDER="0"
|
||||
SRC="img198.png"
|
||||
ALT="\begin{displaymath}
|
||||
N = {{2^{h/12} R} \over f}
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
|
||||
<P>
|
||||
So far we have used a sawtooth as the input wave <IMG
|
||||
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img199.png"
|
||||
ALT="$y[t]$">, but, as suggested in
|
||||
parts (d) and (e) of Figure <A HREF="node26.html#fig02.02">2.2</A>, we could use anything we like as an
|
||||
input signal. In general, the transposition may be time dependent and
|
||||
is controlled by the rate of change of the input signal.
|
||||
|
||||
<P>
|
||||
The
|
||||
transposition multiple <IMG
|
||||
WIDTH="9" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img82.png"
|
||||
ALT="$t$"> and the transposition in half-steps <IMG
|
||||
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img194.png"
|
||||
ALT="$h$"> are then
|
||||
given by the
|
||||
<A NAME="2217"></A>Momentary Transposition Formulas for Wavetables:
|
||||
<A NAME="sect2.momentaryformula"></A>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
t[n] = \left | y[n] - y[n-1] \right |
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="153" HEIGHT="28" BORDER="0"
|
||||
SRC="img200.png"
|
||||
ALT="\begin{displaymath}
|
||||
t[n] = \left \vert y[n] - y[n-1] \right \vert
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
h[n] = 12 {{\log_2} \left | y[n] - y[n-1] \right |}
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="202" HEIGHT="28" BORDER="0"
|
||||
SRC="img201.png"
|
||||
ALT="\begin{displaymath}
|
||||
h[n] = 12 {{\log_2} \left \vert y[n] - y[n-1] \right \vert}
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
(Here the enclosing bars ``<IMG
|
||||
WIDTH="7" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img202.png"
|
||||
ALT="$\vert$">" mean absolute value.)
|
||||
For example, if
|
||||
<IMG
|
||||
WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img203.png"
|
||||
ALT="$y[n] = n$">, then <IMG
|
||||
WIDTH="78" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img204.png"
|
||||
ALT="$z[n] = x[n]$"> so we hear the wavetable at its original pitch,
|
||||
and this is what the formula predicts since, in that case,
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
y[n]-y[n-1] = 1
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="129" HEIGHT="28" BORDER="0"
|
||||
SRC="img205.png"
|
||||
ALT="\begin{displaymath}
|
||||
y[n]-y[n-1] = 1
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
On the other hand, if <IMG
|
||||
WIDTH="69" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img206.png"
|
||||
ALT="$y[n] = 2n$">, then the wavetable is transposed up an
|
||||
octave, consistent with
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
y[n]-y[n-1] = 2
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="129" HEIGHT="28" BORDER="0"
|
||||
SRC="img207.png"
|
||||
ALT="\begin{displaymath}
|
||||
y[n]-y[n-1] = 2
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
If values of <IMG
|
||||
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$y[n]$"> are
|
||||
decreasing with <IMG
|
||||
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img75.png"
|
||||
ALT="$n$">, you hear the sample backward, but the transposition
|
||||
formula still gives a positive multiplier. This all agrees with the
|
||||
earlier Transposition Formula for Looping Wavetables; if a sawtooth ranges
|
||||
from <IMG
|
||||
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img179.png"
|
||||
ALT="$0$"> to <IMG
|
||||
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img3.png"
|
||||
ALT="$N$">, <IMG
|
||||
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img112.png"
|
||||
ALT="$f$"> times per second,
|
||||
the difference of successive samples is just <IMG
|
||||
WIDTH="47" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img192.png"
|
||||
ALT="$N f / R$">--except at the sample
|
||||
at the beginning of each new cycle.
|
||||
|
||||
<P>
|
||||
It's well known that transposing a recording also transposes its timbre--this
|
||||
is the ``chipmunk" effect. Not only are any periodicities (such as might
|
||||
give rise to pitch) transposed, but so are the frequencies of
|
||||
the overtones. Some timbres, notably those of vocal sounds, have characteristic
|
||||
frequency ranges in which overtones are stronger than other nearby ones.
|
||||
Such frequency ranges are also transposed, and this is is heard as
|
||||
a timbre change. In language that will be made more precise in Section
|
||||
<A HREF="node76.html#sect5-spectra">5.1</A>,
|
||||
we say that the
|
||||
<A NAME="2221"></A><I>spectral envelope</I> is transposed along with the pitch or
|
||||
pitches.
|
||||
|
||||
<P>
|
||||
In both this and the preceding section, we have considered playing
|
||||
wavetables periodically. In Section <A HREF="node27.html#sect2.oscillator">2.1</A> the playback
|
||||
repeated quickly enough that the repetition gives rise to a pitch, say between
|
||||
30 and 4000 times per second, roughly the range of a piano. In the current
|
||||
section we assumed a wavetable one second long, and in this case
|
||||
``reasonable" transposition factors (less than four octaves up) would give rise
|
||||
to a rate of repetition below 30, usually much lower, and going down as low as
|
||||
we wish.
|
||||
|
||||
<P>
|
||||
The number 30 is significant for another reason: it is roughly the maximum
|
||||
number of separate events the ear can discern per second; for instance, 30
|
||||
vocal phonemes, or melodic notes, or attacks of a snare drum are about the most
|
||||
we can hope to crowd into a second before our ability to distinguish them
|
||||
breaks down.
|
||||
|
||||
<P>
|
||||
A continuum exists between samplers and wavetable oscillators, in that the
|
||||
patch of Figure <A HREF="node27.html#fig02.03">2.3</A> can either be regarded as a sampler (if the
|
||||
frequency of repetition is less than about 20 Hertz) or as a wavetable oscillator
|
||||
(if the frequency is greater than about 40 Hertz). It is possible to move
|
||||
continuously between the two regimes.
|
||||
Furthermore, it is not necessary to play an entire wavetable in a loop; with a bit
|
||||
more arithmetic we can choose sub-segments of the wavetable, and these can change
|
||||
in length and location continuously as the wavetable is played.
|
||||
|
||||
<P>
|
||||
The practice of playing many small segments of a wavetable in rapid succession
|
||||
is often called
|
||||
<A NAME="2225"></A><I>granular synthesis</I>.
|
||||
For much more discussion of the possibilities, see [<A
|
||||
HREF="node202.html#r-roads01">Roa01</A>].
|
||||
|
||||
<P>
|
||||
Figure <A HREF="#fig02.05">2.5</A> shows how to build a very simple looping sampler. In the
|
||||
figure, if the frequency is <IMG
|
||||
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img112.png"
|
||||
ALT="$f$"> and the segment size in samples is <IMG
|
||||
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img208.png"
|
||||
ALT="$s$">,
|
||||
the output transposition factor is given by <IMG
|
||||
WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img209.png"
|
||||
ALT="$t = fs/R$">, where R is the sample
|
||||
rate at which the wavetable was recorded (which need not equal the sample rate
|
||||
the block diagram is working at.) In practice, this equation must usually
|
||||
be solved for either <IMG
|
||||
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img112.png"
|
||||
ALT="$f$"> or <IMG
|
||||
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img208.png"
|
||||
ALT="$s$"> to attain a desired transposition.
|
||||
|
||||
<P>
|
||||
In the figure, a sawtooth oscillator controls the location of wavetable lookup,
|
||||
but the lower and upper values of the sawtooth aren't statically
|
||||
specified as they were in Figure <A HREF="node27.html#fig02.03">2.3</A>; rather, the sawtooth
|
||||
oscillator simply ranges from 0 to 1 in value and the range is adjusted
|
||||
to select a desired segment of samples in the wavetable.
|
||||
|
||||
<P>
|
||||
It might be desirable to specify the segment's location <IMG
|
||||
WIDTH="8" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img210.png"
|
||||
ALT="$l$"> either as its
|
||||
left-hand edge (its lower bound) or else as the segment's midpoint; in either
|
||||
case we specify the length <IMG
|
||||
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img208.png"
|
||||
ALT="$s$"> as a separate parameter. In the first
|
||||
case, we start by
|
||||
multiplying the sawtooth by <IMG
|
||||
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img208.png"
|
||||
ALT="$s$">, so that it then ranges from <IMG
|
||||
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img179.png"
|
||||
ALT="$0$"> to <IMG
|
||||
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img208.png"
|
||||
ALT="$s$">; then
|
||||
we add <IMG
|
||||
WIDTH="8" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img210.png"
|
||||
ALT="$l$"> so that it now ranges from <IMG
|
||||
WIDTH="8" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img210.png"
|
||||
ALT="$l$"> to <IMG
|
||||
WIDTH="35" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img211.png"
|
||||
ALT="$l+s$">. In order to specify the
|
||||
location as the segment's midpoint, we first subtract <IMG
|
||||
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img98.png"
|
||||
ALT="$1/2$"> from the sawtooth
|
||||
(so that it ranges from <IMG
|
||||
WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img212.png"
|
||||
ALT="$-1/2$"> to <IMG
|
||||
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img98.png"
|
||||
ALT="$1/2$">), and then as before multiply by <IMG
|
||||
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img208.png"
|
||||
ALT="$s$">
|
||||
(so that it now ranges from <IMG
|
||||
WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img213.png"
|
||||
ALT="$-s/2$"> to <IMG
|
||||
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img214.png"
|
||||
ALT="$s/2$">) and add <IMG
|
||||
WIDTH="8" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img210.png"
|
||||
ALT="$l$"> to give a range from
|
||||
<IMG
|
||||
WIDTH="51" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img215.png"
|
||||
ALT="$l-s/2$"> to <IMG
|
||||
WIDTH="51" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img216.png"
|
||||
ALT="$l+s/2$">.
|
||||
|
||||
<P>
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="fig02.05"></A><A NAME="2232"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2.5:</STRONG>
|
||||
A simple looping sampler, as yet with no amplitude control.
|
||||
There are inputs to control the frequency and the segment size and location.
|
||||
The ``-" operation is included if we wish the segment location to be specified
|
||||
as the segment's midpoint; otherwise we specify the location of the left
|
||||
end of the segment.</CAPTION>
|
||||
<TR><TD><IMG
|
||||
WIDTH="320" HEIGHT="398" BORDER="0"
|
||||
SRC="img217.png"
|
||||
ALT="\begin{figure}\psfig{file=figs/fig02.05.ps}\end{figure}"></TD></TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
|
||||
<P>
|
||||
In the looping sampler,
|
||||
we will need to worry about maintaining continuity between the beginning and the
|
||||
end of segments of the wavetable; we'll take this up in the next section.
|
||||
|
||||
<P>
|
||||
A further detail is that, if the segment size and location are changing
|
||||
with time (they might be digital audio signals themselves, for instance),
|
||||
they will affect the transposition factor, and the pitch or timbre of the
|
||||
output signal might waver up and down as a result. The simplest way to
|
||||
avoid this problem is to synchronize changes in the values of <IMG
|
||||
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img208.png"
|
||||
ALT="$s$"> and <IMG
|
||||
WIDTH="8" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img210.png"
|
||||
ALT="$l$">
|
||||
with the regular discontinuities of the sawtooth; since the signal jumps
|
||||
discontinuously there, the transposition is not really defined there anyway,
|
||||
and, if you are enveloping to hide the discontinuity, the effects of changes
|
||||
in <IMG
|
||||
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img208.png"
|
||||
ALT="$s$"> and <IMG
|
||||
WIDTH="8" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img210.png"
|
||||
ALT="$l$"> are hidden as well.
|
||||
|
||||
<P>
|
||||
<HR>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html960"
|
||||
HREF="node29.html">
|
||||
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
||||
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
|
||||
<A NAME="tex2html954"
|
||||
HREF="node26.html">
|
||||
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
||||
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
|
||||
<A NAME="tex2html948"
|
||||
HREF="node27.html">
|
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<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
||||
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
|
||||
<A NAME="tex2html956"
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||||
HREF="node4.html">
|
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<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
||||
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
|
||||
<A NAME="tex2html958"
|
||||
HREF="node201.html">
|
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<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
||||
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
|
||||
<BR>
|
||||
<B> Next:</B> <A NAME="tex2html961"
|
||||
HREF="node29.html">Enveloping samplers</A>
|
||||
<B> Up:</B> <A NAME="tex2html955"
|
||||
HREF="node26.html">Wavetables and samplers</A>
|
||||
<B> Previous:</B> <A NAME="tex2html949"
|
||||
HREF="node27.html">The Wavetable Oscillator</A>
|
||||
<B> <A NAME="tex2html957"
|
||||
HREF="node4.html">Contents</A></B>
|
||||
<B> <A NAME="tex2html959"
|
||||
HREF="node201.html">Index</A></B>
|
||||
<!--End of Navigation Panel-->
|
||||
<ADDRESS>
|
||||
Miller Puckette
|
||||
2006-12-30
|
||||
</ADDRESS>
|
||||
</BODY>
|
||||
</HTML>
|
||||
Reference in New Issue
Block a user