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<H1><A NAME="SECTION001190000000000000000"></A>
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<A NAME="sect7.pitchshift"></A>
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<BR>
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Pitch shifting
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</H1>
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<P>
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A favorite use of variable delay lines is to alter the pitch of an incoming
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sound using the Doppler effect. It may be desirable to alter the pitch
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variably (randomly or periodically, for example), or else
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to maintain a fixed musical interval of transposition over a length of time.
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<P>
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Returning to Figure <A HREF="node113.html#fig07.17">7.17</A>, we see that with a single variable
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delay line we can maintain any desired pitch shift for a limited
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interval of time, but if we wish to sustain a fixed transposition we will always
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eventually land outside the diagonal strip of admissible delay times. In
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the simplest scenario, we simply vary the transposition up and down so as to remain
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in the strip.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig07.19"></A><A NAME="8151"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.19:</STRONG>
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Vibrato using a variable delay line. Since the pitch shift
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alternates between upward and downward, it is possible to maintain it without
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drifting outside the strip of admissible delay.</CAPTION>
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<TR><TD><IMG
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WIDTH="380" HEIGHT="344" BORDER="0"
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SRC="img799.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.19.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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This works, for example, if we wish to apply vibrato
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to a sound as shown in Figure <A HREF="#fig07.19">7.19</A>.
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Here the delay function is
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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d[n] = {d_0} + a \cos(\omega n)
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\end{displaymath}
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-->
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<IMG
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WIDTH="145" HEIGHT="28" BORDER="0"
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SRC="img800.png"
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ALT="\begin{displaymath}
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d[n] = {d_0} + a \cos(\omega n)
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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where <IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img801.png"
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ALT="$d_0$"> is the average delay, <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$"> is the amplitude of variation about
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the average delay, and <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $"> is an angular frequency. The Momentary
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Transposition (Page <A HREF="node113.html#eq.momentarydel"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>), is approximately
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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t = 1 + a \omega \cos(\omega n - \pi/2)
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\end{displaymath}
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-->
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<IMG
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WIDTH="174" HEIGHT="28" BORDER="0"
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SRC="img802.png"
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ALT="\begin{displaymath}
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t = 1 + a \omega \cos(\omega n - \pi/2)
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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This ranges in value between <IMG
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WIDTH="49" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img803.png"
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ALT="$1 - a \omega$"> and <IMG
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WIDTH="49" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img804.png"
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ALT="$1 + a \omega$">.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig07.20"></A><A NAME="8159"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.20:</STRONG>
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Piecewise linear delay functions to maintain a constant transposition
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(except at the points of discontinuity). The outputs are enveloped as suggested
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by the bars above each point, to smooth the output at the points of discontinuity
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in delay time.</CAPTION>
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<TR><TD><IMG
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WIDTH="325" HEIGHT="346" BORDER="0"
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SRC="img805.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.20.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, on the other hand, that we wish to maintain a constant transposition
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over a longer interval of time. In this case we can't maintain the
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transposition forever, but it is still possible to maintain it over fixed
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intervals of time broken by discontinuous changes, as shown in Figure
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<A HREF="#fig07.20">7.20</A>.
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The delay time is the output of a suitably normalized sawtooth function, and
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the output of the variable delay line is enveloped as shown in the figure
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to avoid discontinuities.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig07.21"></A><A NAME="8165"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.21:</STRONG>
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Using a variable delay line as a pitch shifter. The sawtooth wave
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creates a smoothly increasing or decreasing delay time. The output of the
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delay line is enveloped to avoid discontinuities.
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Another copy of the same diagram should run 180 degrees (<IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $"> radians) out
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of phase with this one.</CAPTION>
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<TR><TD><IMG
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WIDTH="281" HEIGHT="403" BORDER="0"
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SRC="img806.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.21.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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This is accomplished as shown in Figure <A HREF="#fig07.21">7.21</A>.
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The output of the sawtooth generator is used in two ways. First it is adjusted
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to run between the bounds <IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img807.png"
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ALT="${d_0}$"> and <IMG
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WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img808.png"
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ALT="${d_0}+s$">, and this adjusted sawtooth
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controls the delay time, in samples. The initial delay <IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img801.png"
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ALT="$d_0$"> should be at
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least enough to make the variable delay feasible; for four-point interpolation
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it must be at least one sample. Larger values of <IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img801.png"
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ALT="$d_0$"> add a constant,
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additional delay to the output; this is usually offered as a control in a
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pitch shifter since it is essentially free. The quantity <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img208.png"
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ALT="$s$"> is sometimes
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called the
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<A NAME="8171"></A><I>window size</I>. It corresponds roughly to the sample length in a
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looping sampler (Section <A HREF="node28.html#sect2.sampling">2.2</A>).
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<P>
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The sawtooth output is also used to envelope the output in exactly the same
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way as in the enveloped wavetable sampler of Figure <A HREF="node29.html#fig02.07">2.7</A>
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(Page <A HREF="node29.html#fig02.07"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>). The
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envelope is zero at the points where the sawtooth wraps around, and in
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between, rises smoothly to a maximum value of 1 (for unit gain).
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<P>
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If the frequency of the sawtooth wave is <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$"> (in cycles per second), then
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its value sweeps from 0 to 1 every <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img809.png"
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ALT="$R/f$"> samples (where <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$"> is the sample rate).
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The difference between successive values is thus <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img810.png"
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ALT="$f/R$">. If we let <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$">
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denote the output of the sawtooth oscillator, then
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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x[n+1] - x[n] = {f \over R}
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\end{displaymath}
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-->
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<IMG
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WIDTH="136" HEIGHT="39" BORDER="0"
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SRC="img811.png"
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ALT="\begin{displaymath}
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x[n+1] - x[n] = {f \over R}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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(except at the wraparound points). If we adjust the output range of the
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wavetable oscillator to the value <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img208.png"
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ALT="$s$"> (as is done in the figure) we get a
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new slope:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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s \cdot x[n+1] - s \cdot x[n] = {{sf} \over R}
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\end{displaymath}
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-->
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<IMG
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WIDTH="180" HEIGHT="39" BORDER="0"
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SRC="img812.png"
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ALT="\begin{displaymath}
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s \cdot x[n+1] - s \cdot x[n] = {{sf} \over R}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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Adding the constant <IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img801.png"
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ALT="$d_0$"> has no effect on this slope. The Momentary
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Transposition (Page <A HREF="node113.html#eq.momentarydel"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>) is then:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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t = 1 - {{sf} \over R}
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\end{displaymath}
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-->
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<IMG
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WIDTH="74" HEIGHT="39" BORDER="0"
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SRC="img813.png"
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ALT="\begin{displaymath}
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t = 1 - {{sf} \over R}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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To complete the design of the pitch shifter we must add the other copy halfway
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out of phase. This gives rise to a delay reading pattern as shown in
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Figure <A HREF="#fig07.22">7.22</A>.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig07.22"></A><A NAME="8183"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.22:</STRONG>
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The pitch shifter's delay reading pattern using two delay lines,
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so that one is at maximum amplitude exactly when the other is switching.</CAPTION>
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<TR><TD><IMG
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WIDTH="326" HEIGHT="343" BORDER="0"
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SRC="img814.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.22.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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The pitch shifter can transpose either upward (using negative sawtooth
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frequencies, as in the figure) or downward, using positive ones. Pitch shift is
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usually controlled by changing <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$"> with <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img208.png"
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ALT="$s$"> fixed. To get a
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desired transposition interval <IMG
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WIDTH="9" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img82.png"
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ALT="$t$">, set
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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f = {{(t - 1) R} \over s}
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\end{displaymath}
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-->
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<IMG
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WIDTH="89" HEIGHT="40" BORDER="0"
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SRC="img815.png"
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ALT="\begin{displaymath}
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f = {{(t - 1) R} \over s}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<A NAME="sect7.psformula"></A>The window size <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img208.png"
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ALT="$s$"> should be chosen small enough, if possible, so that the
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two delayed copies
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(<IMG
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WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img214.png"
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ALT="$s/2$"> samples apart) do not sound as distinct echoes. However, very small
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values of <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img208.png"
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ALT="$s$"> will force <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$"> upward; values of <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$"> greater than about 5 Hertz
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result
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in very audible modulation. So if very large transpositions are required, the
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value of <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img208.png"
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ALT="$s$"> may need to be increased. Typical values range from 30 to 100
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milliseconds (about <IMG
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WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img816.png"
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ALT="$R/30$"> to <IMG
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WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img817.png"
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ALT="$R/10$"> samples).
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<P>
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Although the frequency may be changed at will, even discontinuously, <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img208.png"
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ALT="$s$"> must
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be changed more carefully. A possible solution is to mute the output while
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changing <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img208.png"
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ALT="$s$"> discontinuously; alternatively, <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img208.png"
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ALT="$s$"> may be ramped continuously but
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this causes hard-to-control Doppler shifts.
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<P>
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A good choice of envelope is one half cycle of a sinusoid. If we assume
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on average that the two delay outputs are uncorrelated (Page
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<A HREF="node13.html#eq-meanpowersum"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>), the signal power from the two delay lines, after
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enveloping, will add to a constant (since the sum of squares of the two
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envelopes is one).
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<P>
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Many variations exist on this pitch shifting algorithm. One
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classic variant uses a single delay line, with no enveloping at
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all. In this situation it is necessary to choose the point at
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which the delay time jumps, and the point it jumps to, so that the output
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stays continuous. For example, one could find a point where the output signal
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passes through zero (a "zero crossing") and jump discontinuously to another one.
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Using only one delay line has the advantage that the signal output sounds
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more "present". A disadvantage is that, since
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the delay time is a function of input signal value, the output is no longer
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a linear function of the input, so non-periodic inputs can give rise to
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Miller Puckette
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2006-12-30
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