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<div class="refentry" title="GEN13">
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<h2>
<span class="refentrytitle">GEN13</span>
</h2>
<p>GEN13 —
Stores a polynomial whose coefficients derive from the Chebyshev polynomials of the first kind.
</p>
</div>
<div class="refsect1" title="Description">
<a id="idp77557920"></a>
<h2>Description</h2>
<p>
Uses Chebyshev coefficients to generate stored polynomial functions which, under waveshaping, can be used to split a sinusoid into harmonic partials having a pre-definable spectrum.
</p>
</div>
<div class="refsect1" title="Syntax">
<a id="idp77558920"></a>
<h2>Syntax</h2>
<pre class="synopsis"><span class="command"><strong>f</strong></span> # time size 13 xint xamp h0 h1 h2 ...</pre>
</div>
<div class="refsect1" title="Initialization">
<a id="idp77569240"></a>
<h2>Initialization</h2>
<p>
<span class="emphasis"><em>size</em></span> -- number of points in the table. Must be a power of 2 or a power-of-2 plus 1 (see <a class="link" href="f.html" title="f Statement (or Function Table Statement)"><em class="citetitle">f statement</em></a>). The normal value is power-of-2 plus 1.
</p>
<p>
<span class="emphasis"><em>xint</em></span> -- provides the left and right values [<span class="emphasis"><em>-xint, +xint</em></span>] of the x interval over which the polynomial is to be drawn. These subroutines both call <a class="link" href="GEN03.html" title="GEN03"><em class="citetitle">GEN03</em></a> to draw their functions; the p5 value here is therefor expanded to a negative-positive p5, p6 pair before <span class="emphasis"><em>GEN03</em></span> is actually called. The normal value is 1.
</p>
<p>
<span class="emphasis"><em>xamp </em></span> -- amplitude scaling factor of the sinusoid input that is expected to produce the following spectrum.
</p>
<p>
<span class="emphasis"><em>h0, h1, h2,</em></span> etc. -- relative strength of partials 0 (DC), 1 (fundamental), 2 ... that will result when a sinusoid of amplitude
</p>
<div class="literallayout">
<p><br />
xamp * int(size/2)/xint<br />
</p>
</div>
<p>
is waveshaped using this function table. These values thus describe a frequency spectrum associated with a particular factor <span class="emphasis"><em>xamp</em></span> of the input signal.
</p>
<p>
<span class="emphasis"><em>GEN13</em></span> is the function generator normally employed in standard waveshaping. It stores a polynomial whose coefficients derive from the Chebyshev polynomials of the first kind, so that a driving sinusoid of strength <span class="emphasis"><em>xamp</em></span> will exhibit the specified spectrum at output. Note that the evolution of this spectrum is generally not linear with varying <span class="emphasis"><em>xamp</em></span>. However, it is bandlimited (the only partials to appear will be those specified at generation time); and the partials will tend to occur and to develop in ascending order (the lower partials dominating at low <span class="emphasis"><em>xamp</em></span>, and the spectral richness increasing for higher values of <span class="emphasis"><em>xamp</em></span>). A negative <span class="emphasis"><em>hn</em></span> value implies a 180 degree phase shift of that partial; the requested full-amplitude spectrum will not be affected by this shift, although the evolution of several of its component partials may be. The pattern +,+,-,-,+,+,... for <span class="emphasis"><em>h0,h1,h2..</em></span>. will minimize the normalization problem for low <span class="emphasis"><em>xamp</em></span> values (see above), but does not necessarily provide the smoothest pattern of evolution.
</p>
</div>
<div class="refsect1" title="Examples">
<a id="idp77610224"></a>
<h2>Examples</h2>
<p>
Here is an example of the GEN13 opcode. It uses the file <a class="ulink" href="examples/gen13.csd" target="_top"><em class="citetitle">gen13.csd</em></a>.
</p>
<div class="example">
<a id="idp77611208"></a>
<p class="title">
<strong>Example 1049. Example of the GEN13 opcode.</strong>
</p>
<div class="example-contents">
<p>See the sections <a class="link" href="UsingRealTime.html" title="Real-Time Audio"><em class="citetitle">Real-time Audio</em></a> and <a class="link" href="CommandFlags.html" title="Csound command line"><em class="citetitle">Command Line Flags</em></a> for more information on using command line flags.</p>
<pre class="programlisting">
<span class="csdtag"><CsoundSynthesizer></span>
<span class="csdtag"><CsOptions></span>
<span class="comment">; Select audio/midi flags here according to platform</span>
-odac <span class="comment">;;;realtime audio out</span>
<span class="comment">;-iadc ;;;uncomment -iadc if realtime audio input is needed too</span>
<span class="comment">; For Non-realtime ouput leave only the line below:</span>
<span class="comment">; -o gen13.wav -W ;;; for file output any platform</span>
<span class="csdtag"></CsOptions></span>
<span class="csdtag"><CsInstruments></span>
<span class="ohdr">sr</span> <span class="op">=</span> 44100
<span class="ohdr">ksmps</span> <span class="op">=</span> 32
<span class="ohdr">nchnls</span> <span class="op">=</span> 2
<span class="ohdr">0dbfs</span> <span class="op">=</span> 1
<span class="comment">;example by Russell Pinkston - Univ. of Texas (but slightly modified)</span>
gisine <span class="ohdr">ftgen</span> 0, 0, 16384, 10, 1 <span class="comment">;sine wave</span>
<span class="oblock">instr</span> 1
ihertz <span class="op">=</span> <span class="opc">cpspch</span>(p4)
ipkamp <span class="op">=</span> p5
iwsfn <span class="op">=</span> p6 <span class="comment">;waveshaping function</span>
inmfn <span class="op">=</span> p7 <span class="comment">;normalization function</span>
agate <span class="opc">linen</span> 1, .01, p3, .1 <span class="comment">;overall amp envelope</span>
kctrl <span class="opc">linen</span> .99, 2, p3, 2 <span class="comment">;waveshaping index control</span>
aindex <span class="opc">poscil</span> kctrl<span class="op">/</span>2, ihertz, gisine <span class="comment">;sine wave to be distorted</span>
asignal <span class="opc">tablei</span> .5<span class="op">+</span>aindex, iwsfn, 1 <span class="comment">;waveshaping</span>
knormal <span class="opc">tablei</span> kctrl, inmfn, 1 <span class="comment">;amplitude normalization</span>
asig <span class="op">=</span> asignal<span class="op">*</span>knormal<span class="op">*</span>ipkamp<span class="op">*</span>agate
<span class="opc">outs</span> asig, asig
<span class="oblock">endin</span>
<span class="csdtag"></CsInstruments></span>
<span class="csdtag"><CsScore></span>
<span class="comment">; This proves the statement in Dodge (p. 147) that Chebyshev polynomials</span>
<span class="comment">; of order K have "only the kth harmonic." This is only true when the </span>
<span class="comment">; waveshaping index is at the maximum - i.e., when the entire transfer</span>
<span class="comment">; function is being accessed. RP.</span>
<span class="comment">;--------------------------------------------------------------------------------------------------------------------------------------------</span>
<span class="comment">; quasi sawtooth transfer function: </span>
<span class="comment">; h0 h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h13 h14 h15 h16 h17 h18 h19 h20</span>
<span class="stamnt">f</span>1 0 513 13 1 1 0 100 -50 -33 25 20 -16.7 -14.2 12.5 11.1 -10 -9.09 8.333 7.69 -7.14 -6.67 6.25 5.88 -5.55 -5.26 5
<span class="stamnt">f</span>2 0 257 4 1 1 <span class="comment">; normalizing function with midpoint bipolar offset</span>
<span class="comment">; st dur pch amp wsfn nmfn</span>
<span class="stamnt">i</span>1 0 4 6.00 .7 1 2
<span class="stamnt">i</span>1 4 . 7.00 .
<span class="stamnt">i</span>1 8 . 8.00 .
<span class="comment">;-------------------------------------------------------------------------------------------------------------------------------------------- </span>
<span class="comment">; quasi square wave transfer function: </span>
<span class="comment">; h0 h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h13 h14 h15 h16 h17 h18 h19</span>
<span class="stamnt">f</span>3 0 513 13 1 1 0 100 0 -33 0 20 0 -14.2 0 11.1 0 -9.09 0 7.69 0 -6.67 0 5.88 0 -5.26
<span class="stamnt">f</span>4 0 257 4 3 1 <span class="comment">; normalizing function with midpoint bipolar offset</span>
<span class="comment">; st dur pch amp wsfn nmfn</span>
<span class="stamnt">i</span>1 16 4 6.00 .7 3 4
<span class="stamnt">i</span>1 20 . 7.00 .
<span class="stamnt">i</span>1 24 . 8.00 .
<span class="comment">;-------------------------------------------------------------------------------------------------------------------------------------------- </span>
<span class="comment">; quasi triangle wave transfer function: </span>
<span class="comment">; h0 h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h13 h14 h15 h16 h17 h18 h19</span>
<span class="stamnt">f</span>5 0 513 13 1 1 0 100 0 -11.11 0 4 0 -2.04 0 1.23 0 -.826 0 .59 0 -.444 0 .346 0 -.277
<span class="stamnt">f</span>6 0 257 4 5 1 <span class="comment">; normalizing function with midpoint bipolar offset</span>
<span class="comment">; st dur pch amp wsfn nmfn</span>
<span class="stamnt">i</span>1 32 4 6.00 .7 5 6
<span class="stamnt">i</span>1 36 . 7.00 .
<span class="stamnt">i</span>1 40 . 8.00 .
<span class="comment">;--------------------------------------------------------------------------------------------------------------------------------------------</span>
<span class="comment">; transfer function1: h0 h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h13 h14 h15 h16</span>
<span class="stamnt">f</span>7 0 513 13 1 1 0 1 -.8 0 .6 0 0 0 .4 0 0 0 0 .1 -.2 -.3 .5
<span class="stamnt">f</span>8 0 257 4 7 1 <span class="comment">; normalizing function with midpoint bipolar offset</span>
<span class="comment">; st dur pch amp wsfn nmfn</span>
<span class="stamnt">i</span>1 48 4 5.00 .7 7 8
<span class="stamnt">i</span>1 52 . 6.00 .
<span class="stamnt">i</span>1 56 . 7.00 .
<span class="comment">;--------------------------------------------------------------------------------------------------------------------------------------------</span>
<span class="comment">;=========================================================================;</span>
<span class="comment">; This demonstrates the use of high partials, sometimes without a ;</span>
<span class="comment">; fundamental, to get quasi-inharmonic spectra from waveshaping. ;</span>
<span class="comment">;=========================================================================;</span>
<span class="comment">; transfer function2: h0 h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h13 h14 h15 h16</span>
<span class="stamnt">f</span>9 0 513 13 1 1 0 0 0 -.1 0 .3 0 -.5 0 .7 0 -.9 0 1 0 -1 0
<span class="stamnt">f</span>10 0 257 4 9 1 <span class="comment">; normalizing function with midpoint bipolar offset</span>
<span class="comment">; st dur pch amp wsfn nmfn</span>
<span class="stamnt">i</span>1 64 4 5.00 .7 9 10
<span class="stamnt">i</span>1 68 . 6.00 .
<span class="stamnt">i</span>1 72 . 7.00 .
<span class="comment">;--------------------------------------------------------------------------------------------------------------------------------------------</span>
<span class="comment">; transfer function3: h0 h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h13 h14 h15 h16 h17 h18 h19 h17 h18 h19 h20</span>
<span class="stamnt">f</span>11 0 513 13 1 1 0 0 0 0 0 0 0 -1 0 1 0 0 -.1 0 .1 0 -.2 .3 0 -.7 0 .2 0 -.1
<span class="stamnt">f</span>12 0 257 4 11 1 <span class="comment">; normalizing function with midpoint bipolar offset</span>
<span class="comment">; st dur pch amp wsfn nmfn</span>
<span class="stamnt">i</span>1 80 4 5.00 .7 11 12
<span class="stamnt">i</span>1 84 . 5.06 .
<span class="stamnt">i</span>1 88 . 6.00 .
<span class="comment">;--------------------------------------------------------------------------------------------------------------------------------------------</span>
<span class="comment">;=========================================================================;</span>
<span class="comment">; split a sinusoid into 3 odd-harmonic partials of relative strength 5:3:1</span>
<span class="comment">;=========================================================================;</span>
<span class="comment">;--------------------------------------------------------------------------------------------------------------------------------------------</span>
<span class="comment">; transfer function4: h0 h1 h2 h3 h4 h5</span>
<span class="stamnt">f</span>13 0 513 13 1 1 0 5 0 3 0 1
<span class="stamnt">f</span>14 0 257 4 13 1 <span class="comment">; normalizing function with midpoint bipolar offset</span>
<span class="comment">; st dur pch amp wsfn nmfn</span>
<span class="stamnt">i</span>1 96 4 5.00 .7 13 14
<span class="stamnt">i</span>1 100 . 5.06 .
<span class="stamnt">i</span>1 104 . 6.00 .
<span class="stamnt">e</span>
<span class="csdtag"></CsScore></span>
<span class="csdtag"></CsoundSynthesizer></span>
</pre>
</div>
</div>
<p><br class="example-break" />
<span>These are the diagrams of the waveforms of the GEN13 routines, as used in the example:</span>
</p>
<div class="mediaobject">
<img src="images/gen13_1.png" alt="f1 0 513 13 1 1 0 100 -50 -33 25 20 -16.7 -14.2 12.5 11.1 -10 -9.09 8.333 7.69 -7.14 -6.67 6.25 5.88 -5.55 -5.26 5 - quasi sawtooth transfer function" />
<div class="caption">
<p>f1 0 513 13 1 1 0 100 -50 -33 25 20 -16.7 -14.2 12.5 11.1 -10 -9.09 8.333 7.69 -7.14 -6.67 6.25 5.88 -5.55 -5.26 5 - quasi sawtooth transfer function</p>
</div>
</div>
<p>
</p>
<div class="mediaobject">
<img src="images/gen13_2.png" alt="f3 0 513 13 1 1 0 100 0 -33 0 20 0 -14.2 0 11.1 0 -9.09 0 7.69 0 -6.67 0 5.88 0 -5.26 - quasi square wave transfer function" />
<div class="caption">
<p>f3 0 513 13 1 1 0 100 0 -33 0 20 0 -14.2 0 11.1 0 -9.09 0 7.69 0 -6.67 0 5.88 0 -5.26 - quasi square wave transfer function</p>
</div>
</div>
<p>
</p>
<div class="mediaobject">
<img src="images/gen13_3.png" alt="f5 0 513 13 1 1 0 100 0 -11.11 0 4 0 -2.04 0 1.23 0 -.826 0 .59 0 -.444 0 .346 0 -.277 - quasi triangle wave transfer function" />
<div class="caption">
<p>f5 0 513 13 1 1 0 100 0 -11.11 0 4 0 -2.04 0 1.23 0 -.826 0 .59 0 -.444 0 .346 0 -.277 - quasi triangle wave transfer function</p>
</div>
</div>
<p>
</p>
<div class="mediaobject">
<img src="images/gen13_4.png" alt="f7 0 513 13 1 1 0 1 -.8 0 .6 0 0 0 .4 0 0 0 0 .1 -.2 -.3 .5 - transfer function 1" />
<div class="caption">
<p>f7 0 513 13 1 1 0 1 -.8 0 .6 0 0 0 .4 0 0 0 0 .1 -.2 -.3 .5 - transfer function 1</p>
</div>
</div>
<p>
</p>
<div class="mediaobject">
<img src="images/gen13_5.png" alt="f9 0 513 13 1 1 0 0 0 -.1 0 .3 0 -.5 0 .7 0 -.9 0 1 0 -1 0 - transfer function 2" />
<div class="caption">
<p>f9 0 513 13 1 1 0 0 0 -.1 0 .3 0 -.5 0 .7 0 -.9 0 1 0 -1 0 - transfer function 2</p>
</div>
</div>
<p>
</p>
<div class="mediaobject">
<img src="images/gen13_6.png" alt="f11 0 513 13 1 1 0 0 0 0 0 0 0 -1 0 1 0 0 -.1 0 .1 0 -.2 .3 0 -.7 0 .2 0 -.1 - transfer function 3" />
<div class="caption">
<p>f11 0 513 13 1 1 0 0 0 0 0 0 0 -1 0 1 0 0 -.1 0 .1 0 -.2 .3 0 -.7 0 .2 0 -.1 - transfer function 3</p>
</div>
</div>
<p>
</p>
<div class="mediaobject">
<img src="images/gen13_7.png" alt="f13 0 513 13 1 1 0 5 0 3 0 1 - split a sinusoid into 3 odd-harmonic partials of relative strength 5:3:1" />
<div class="caption">
<p>f13 0 513 13 1 1 0 5 0 3 0 1 - split a sinusoid into 3 odd-harmonic partials of relative strength 5:3:1</p>
</div>
</div>
<p>
</p>
</div>
<div class="refsect1" title="See Also">
<a id="idp77626760"></a>
<h2>See Also</h2>
<p>
<a class="link" href="GEN03.html" title="GEN03"><em class="citetitle">GEN03</em></a>,
<a class="link" href="GEN14.html" title="GEN14"><em class="citetitle">GEN14</em></a>, and
<a class="link" href="GEN15.html" title="GEN15"><em class="citetitle">GEN15</em></a>.
</p>
<p> Information about the Chebyshev polynomials on Wikipedia: <a class="ulink" href="http://en.wikipedia.org/wiki/Chebyshev_polynomials" target="_top"><em class="citetitle">http://en.wikipedia.org/wiki/Chebyshev_polynomials</em></a></p>
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