replace '<A NAME=' with '<A ID='
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node14.html
60
node14.html
@@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<BODY >
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<!--Navigation Panel-->
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<A NAME="tex2html749"
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<A ID="tex2html749"
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HREF="node15.html">
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<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
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SRC="next.png"></A>
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<A NAME="tex2html743"
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<A ID="tex2html743"
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HREF="node7.html">
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<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
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SRC="up.png"></A>
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<A NAME="tex2html737"
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<A ID="tex2html737"
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HREF="node13.html">
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<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
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SRC="prev.png"></A>
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<A NAME="tex2html745"
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<A ID="tex2html745"
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HREF="node4.html">
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<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
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SRC="contents.png"></A>
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<A NAME="tex2html747"
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<A ID="tex2html747"
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HREF="node201.html">
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<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
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SRC="index.png"></A>
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<BR>
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<B> Next:</B> <A NAME="tex2html750"
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<B> Next:</B> <A ID="tex2html750"
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HREF="node15.html">About the Software Examples</A>
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<B> Up:</B> <A NAME="tex2html744"
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<B> Up:</B> <A ID="tex2html744"
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HREF="node7.html">Sinusoids, amplitude and frequency</A>
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<B> Previous:</B> <A NAME="tex2html738"
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<B> Previous:</B> <A ID="tex2html738"
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HREF="node13.html">Superposing Signals</A>
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<B> <A NAME="tex2html746"
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<B> <A ID="tex2html746"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html748"
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<B> <A ID="tex2html748"
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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="SECTION00570000000000000000"></A>
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<A NAME="sect1.fourier"></A>
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<H1><A ID="SECTION00570000000000000000"></A>
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<A ID="sect1.fourier"></A>
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<BR>
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Periodic Signals
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</H1>
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@@ -105,7 +105,7 @@ the smallest <IMG
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WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img135.png"
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ALT="$\tau$"> if any at which a signal repeats is called the signal's
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<A NAME="1175"></A><I>period</I>.
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<A ID="1175"></A><I>period</I>.
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In discussing periods of digital audio signals, we quickly run into the
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difficulty of describing signals whose "period" isn't an integer, so that the
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equation above doesn't make sense. For now we'll effectively
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@@ -149,7 +149,7 @@ with frequencies <!-- MATH
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SRC="img138.png"
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ALT="$2 \pi / \omega$"> samples.
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Such a sum is called a <I>Fourier Series</I>:
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<A NAME="eq-fourierseries"></A>
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<A ID="eq-fourierseries"></A>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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@@ -184,12 +184,12 @@ multiples of <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 $">. They are called the
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<A NAME="1187"></A><I>harmonics</I>
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<A ID="1187"></A><I>harmonics</I>
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of <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">, which in turn is called the
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<A NAME="1189"></A><I>fundamental</I>. In terms of pitch, the harmonics
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<A ID="1189"></A><I>fundamental</I>. In terms of pitch, the harmonics
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<!-- MATH
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$\omega, 2 \omega, \ldots$
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-->
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@@ -205,7 +205,7 @@ pitches of the same scale; the third and sixth miss only by 2 cents and the
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fifth misses by 14.
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<P>
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<A NAME="r-intervals"></A>Put another way, the frequency ratio 3:2 (a perfect fifth in Western
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<A ID="r-intervals"></A>Put another way, the frequency ratio 3:2 (a perfect fifth in Western
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terminology) is almost exactly
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seven half-steps, 4:3 (a perfect fourth) is just as near to five half-steps,
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and the ratios 5:4 and 6:5 (perfect major and minor thirds) are fairly close to
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@@ -221,7 +221,7 @@ by the three component sinusoids.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig01.08"></A><A NAME="1196"></A>
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<DIV ALIGN="CENTER"><A ID="fig01.08"></A><A ID="1196"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.8:</STRONG>
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A Fourier series, showing three sinusoids and their sum. The
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@@ -242,7 +242,7 @@ block diagram for doing this.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig01.09"></A><A NAME="1202"></A>
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<DIV ALIGN="CENTER"><A ID="fig01.09"></A><A ID="1202"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.9:</STRONG>
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Using many oscillators to synthesize a waveform with desired harmonic
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@@ -256,43 +256,43 @@ amplitudes.</CAPTION>
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<P>
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This is an example of
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<A NAME="1205"></A><I>additive synthesis</I>; more generally the term can be applied to networks
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<A ID="1205"></A><I>additive synthesis</I>; more generally the term can be applied to networks
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in which the frequencies of the oscillators are independently controllable.
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The early days of computer music rang with the sound of additive synthesis.
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<P>
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<HR>
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<!--Navigation Panel-->
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<A NAME="tex2html749"
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<A ID="tex2html749"
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HREF="node15.html">
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<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
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SRC="next.png"></A>
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<A NAME="tex2html743"
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<A ID="tex2html743"
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HREF="node7.html">
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<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
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SRC="up.png"></A>
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<A NAME="tex2html737"
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<A ID="tex2html737"
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HREF="node13.html">
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<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
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SRC="prev.png"></A>
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<A NAME="tex2html745"
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<A ID="tex2html745"
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HREF="node4.html">
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<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
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SRC="contents.png"></A>
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<A NAME="tex2html747"
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<A ID="tex2html747"
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HREF="node201.html">
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<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
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SRC="index.png"></A>
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<BR>
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<B> Next:</B> <A NAME="tex2html750"
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<B> Next:</B> <A ID="tex2html750"
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HREF="node15.html">About the Software Examples</A>
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<B> Up:</B> <A NAME="tex2html744"
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<B> Up:</B> <A ID="tex2html744"
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HREF="node7.html">Sinusoids, amplitude and frequency</A>
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<B> Previous:</B> <A NAME="tex2html738"
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<B> Previous:</B> <A ID="tex2html738"
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HREF="node13.html">Superposing Signals</A>
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<B> <A NAME="tex2html746"
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<B> <A ID="tex2html746"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html748"
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<B> <A ID="tex2html748"
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HREF="node201.html">Index</A></B>
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<!--End of Navigation Panel-->
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<ADDRESS>
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