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<div class="refentry" title="GEN12">
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<h2>
<span class="refentrytitle">GEN12</span>
</h2>
<p>GEN12 —
Generates the log of a modified Bessel function of the second kind.
</p>
</div>
<div class="refsect1" title="Description">
<a id="idp77513000"></a>
<h2>Description</h2>
<p>
This generates the log of a modified Bessel function of the second kind, order 0, suitable for use in amplitude-modulated FM.
</p>
</div>
<div class="refsect1" title="Syntax">
<a id="idp77513888"></a>
<h2>Syntax</h2>
<pre class="synopsis"><span class="command"><strong>f</strong></span> # time size 12 xint</pre>
</div>
<div class="refsect1" title="Initialization">
<a id="idp77517144"></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> -- specifies the <span class="emphasis"><em>x</em></span> interval [0 to <span class="emphasis"><em>+xint</em></span>] over which the function is defined.
</p>
<div class="note" title="Note" style="margin-left: 0.5in; margin-right: 0.5in;">
<table border="0" summary="Note: Note">
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<img alt="[Note]" src="images/note.png" />
</td>
<th align="left">Note</th>
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<p>
</p>
<div class="itemizedlist">
<ul class="itemizedlist" type="disc">
<li class="listitem">
<p>
This subroutine draws the natural log of a modified Bessel function of the second kind, order 0 (commonly written as <span class="emphasis"><em>I</em></span> subscript 0), over the x-interval requested. The call should have rescaling inhibited.
</p>
</li>
<li class="listitem">
<p>
The function is useful as an amplitude scaling factor in cycle-synchronous amplitude-modulated FM. (See Palamin & Palamin, <span class="emphasis"><em>J. Audio Eng. Soc., 36/9</em></span>, Sept. 1988, pp.671-684.) The algorithm is interesting because it permits the normally symmetric FM spectrum to be made asymmetric around a frequency other than the carrier, and is thereby useful for formant positioning. By using a table lookup index of <span class="emphasis"><em>I</em></span>(r - 1/r), where <span class="emphasis"><em>I</em></span> is the FM modulation index and <span class="emphasis"><em>r</em></span> is an exponential parameter affecting partial strengths, the Palamin algorithm becomes relatively efficient, requiring only oscil's, table lookups, and a single <span class="emphasis"><em>exp</em></span> call.
</p>
</li>
</ul>
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<p>
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<div class="refsect1" title="Examples">
<a id="idp77574440"></a>
<h2>Examples</h2>
<p>
Here is an example of the GEN12 opcode. It uses the file <a class="ulink" href="examples/gen12.csd" target="_top"><em class="citetitle">gen12.csd</em></a>.
</p>
<div class="example">
<a id="idp77575384"></a>
<p class="title">
<strong>Example 1048. Example of the GEN12 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 gen12.wav -W ;;; for file output any platform</span>
<span class="csdtag"></CsOptions></span>
<span class="csdtag"><CsInstruments></span>
<span class="comment">;example from the Csound Book, page 87</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="oblock">instr</span> 1
idur <span class="op">=</span> p3
iamp <span class="op">=</span> p4
icarfrq <span class="op">=</span> p5
imodfrq <span class="op">=</span> p6
aenv <span class="opc">expseg</span> .01, idur<span class="op">*</span>.1, iamp, idur<span class="op">*</span>.8, iamp<span class="op">*</span>.75, idur<span class="op">*</span>.1, .01
i1 <span class="op">=</span> p7<span class="op">*</span>imodfrq <span class="comment">;p7=modulation index start</span>
i2 <span class="op">=</span> p8<span class="op">*</span>imodfrq <span class="comment">;p8=modulation index end</span>
adev <span class="opc">line</span> i1, idur, i2 <span class="comment">;modulation frequency</span>
aindex <span class="opc">line</span> p7, idur, p8 <span class="comment">;modulation index</span>
ar <span class="opc">linseg</span> 1, .1, p9, p3<span class="op">-</span>.2, p10, .1, 1 <span class="comment">; r value envelope: p9-p10 =exp. partial strength parameter start and end</span>
amp1 <span class="op">=</span> (aindex<span class="op">*</span>(ar<span class="op">+</span>(1<span class="op">/</span>ar)))<span class="op">/</span>2
afmod <span class="opc">oscili</span> amp1, imodfrq, 1 <span class="comment">;FM modulator (sine)</span>
atab <span class="op">=</span> (aindex<span class="op">*</span>(ar<span class="op">-</span>(1<span class="op">/</span>ar)))<span class="op">/</span>2 <span class="comment">;index to table</span>
alook <span class="opc">tablei</span> atab, 37 <span class="comment">;table lookup to GEN12</span>
aamod <span class="opc">oscili</span> atab, adev, 2 <span class="comment">;am modulator (cosine)</span>
aamod <span class="op">=</span> (<span class="opc">exp</span>(alook<span class="op">+</span>aamod))<span class="op">*</span>aenv
acar <span class="opc">oscili</span> aamod, afmod<span class="op">+</span>icarfrq, 1 <span class="comment">;AFM (carrier)</span>
asig <span class="opc">balance</span> acar, aenv
<span class="opc">outs</span> asig, asig
<span class="oblock">endin</span>
<span class="csdtag"></CsInstruments></span>
<span class="csdtag"><CsScore></span>
<span class="stamnt">f</span> 1 0 8192 10 1
<span class="stamnt">f</span> 2 0 8192 9 1 1 90
<span class="stamnt">f</span>37 0 1024 -12 40 <span class="comment">;Bessel function-defined from 0 to 40</span>
<span class="stamnt">i</span> 1 0 2 .2 800 800 1 6 .1 2
<span class="stamnt">i</span> 1 + . . 1900 147 8 1 4 .2
<span class="stamnt">i</span> 1 . . . 1100 380 2 9 .5 2
<span class="stamnt">i</span> 1 . 10 . 100 100 11 3 .2 5
<span class="stamnt">s</span>
<span class="stamnt">i</span> 1 0 1 .1 200 100 1 6 .1 2
<span class="stamnt">i</span> 1 + . < < < < < < <
<span class="stamnt">i</span> 1 + . . < < < < < <
<span class="stamnt">i</span> 1 + . . < < < < < <
<span class="stamnt">i</span> 1 + . . < < < < < <
<span class="stamnt">i</span> 1 + . . < < < < < <
<span class="stamnt">i</span> 1 + . . < < < < < <
<span class="stamnt">i</span> 1 + 10 .2 800 800 9 1 .9 6
<span class="stamnt">s</span>
<span class="stamnt">i</span> 1 0 11 .25 50 51 1 6 .1 2
<span class="stamnt">i</span> 1 1 9 .05 700 401 1 6 .1 2
<span class="stamnt">i</span> 1 2 8 . 900 147 8 1 4 .2
<span class="stamnt">i</span> 1 3 7 . 1100 381 2 9 .5 2
<span class="stamnt">i</span> 1 4 6 . 200 102 11 3 .2 5
<span class="stamnt">i</span> 1 5 6 . 800 803 9 1 .9 6
<span class="stamnt">e</span>
<span class="csdtag"></CsScore></span>
<span class="csdtag"></CsoundSynthesizer></span>
</pre>
</div>
</div>
<p><br class="example-break" />
<span>This is the diagram of the waveform of the GEN12 routines, as used in the example:</span>
</p>
<div class="mediaobject">
<img src="images/gen12.png" alt="f 37 0 1024 -12 40 - Bessel function-defined from 0 to 40" />
<div class="caption">
<p>f 37 0 1024 -12 40 - Bessel function-defined from 0 to 40</p>
</div>
</div>
<p>
</p>
</div>
<div class="refsect1" title="Credits">
<a id="idp77579880"></a>
<h2>Credits</h2>
<p>
Example is, with minor modifications, taken from The Csound Book (page 87).
</p>
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