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<div class="refentry" title="fareyleni">
<a id="fareyleni"></a>
<div class="titlepage"></div>
<a id="IndexFareyleni" class="indexterm"></a>
<div class="refnamediv">
<h2>
<span class="refentrytitle">fareyleni</span>
</h2>
<p>fareyleni —
returns the length of a Farey Sequence.
</p>
</div>
<div class="refsect1" title="Description">
<a id="idp19859504"></a>
<h2>Description</h2>
<p>
This opcode can be used in conjunction with <a class="link" href="GENfarey.html" title="GENfarey"><em class="citetitle">GENfarey</em></a>.
It calculates the length of Farey Sequence F<sub>n</sub>. Its length is given by:
|F<sub>n</sub>| = 1 + SUM over n phi(m)
where phi(m) is Euler's totient function, which gives the number of integers ≤ m that are coprime to m.
</p>
<p>
Some values for the length of F<sub>n</sub> given n:
</p>
<div class="informaltable">
<table border="1">
<colgroup>
<col />
<col />
</colgroup>
<thead>
<tr>
<th>n</th>
<th>F<sub>n</sub></th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>2</td>
</tr>
<tr>
<td>2</td>
<td>3</td>
</tr>
<tr>
<td>3</td>
<td>5</td>
</tr>
<tr>
<td>4</td>
<td>7</td>
</tr>
<tr>
<td>5</td>
<td>11</td>
</tr>
<tr>
<td>6</td>
<td>13</td>
</tr>
<tr>
<td>7</td>
<td>19</td>
</tr>
<tr>
<td>8</td>
<td>23</td>
</tr>
<tr>
<td>9</td>
<td>29</td>
</tr>
<tr>
<td>10</td>
<td>33</td>
</tr>
<tr>
<td>11</td>
<td>43</td>
</tr>
<tr>
<td>12</td>
<td>47</td>
</tr>
<tr>
<td>13</td>
<td>59</td>
</tr>
<tr>
<td>14</td>
<td>65</td>
</tr>
<tr>
<td>15</td>
<td>73</td>
</tr>
<tr>
<td>16</td>
<td>81</td>
</tr>
<tr>
<td>17</td>
<td>97</td>
</tr>
<tr>
<td>18</td>
<td>103</td>
</tr>
<tr>
<td>19</td>
<td>121</td>
</tr>
<tr>
<td>20</td>
<td>129</td>
</tr>
</tbody>
</table>
</div>
<p>
</p>
</div>
<div class="refsect1" title="Syntax">
<a id="idp19923464"></a>
<h2>Syntax</h2>
<pre class="synopsis">ifl <span class="command"><strong>fareyleni</strong></span> ifn</pre>
</div>
<div class="refsect1" title="Initialisation">
<a id="idp19924552"></a>
<h2>Initialisation</h2>
<p>
The length of the identified Farey sequence is returned.
</p>
<p>
<span class="emphasis"><em>ifn</em></span> -- Integer identifying the sequence.
</p>
</div>
<div class="refsect1" title="Examples">
<a id="idp19925808"></a>
<h2>Examples</h2>
<p>
Here is an example of the fareyleni opcode. It uses the file <a class="ulink" href="examples/fareyleni.csd" target="_top"><em class="citetitle">fareyleni.csd</em></a>.
</p>
<div class="example">
<a id="idp19926808"></a>
<p class="title">
<strong>Example 223. Example of the fareyleni 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 for RT audio input as well </span>
<span class="comment">; For Non-realtime ouput leave only the line below:</span>
<span class="comment">; -o fareyleni.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">; initialise integer n for Farey Sequence F_8</span>
gifarn <span class="opc">init</span> 8
<span class="comment">; calculate length of F_8, it should return 23 for |F_8|</span>
gires <span class="opc">fareyleni</span> gifarn
<span class="comment">; convert to negative number for the GEN routine because</span>
<span class="comment">; the table length won't be a power of 2 </span>
<span class="comment">; (The length of a Farey Sequence of n > 1 is always an odd number)</span>
gilen <span class="opc">init</span> gires <span class="op">*</span> <span class="op">-</span>1
<span class="comment">; create F_8 with GENfarey, the negative table number prevents </span>
<span class="comment">; unnecessary normalisation (F_8 IS already normalised)</span>
<span class="comment">; n mode: </span>
gifarey <span class="ohdr">ftgen</span> 100, 0, gilen, "farey", gifarn, 4
<span class="comment">; if mode=4 then 1 is added to each element of F_n.</span>
<span class="comment">; Useful for creating just tuning tables that can be read by the cps2pch opcode.</span>
<span class="oblock">instr</span> 1
<span class="comment">; the very last element of F_n is not needed in the case of tuning tables</span>
ires <span class="op">=</span> gires <span class="op">-</span> 1
<span class="comment">; read out and print to file</span>
kndx <span class="opc">init</span> 0
<span class="octrl">if</span> (kndx <span class="op"><</span> ires) then
kelem <span class="opc">tab</span> kndx, gifarey
<span class="opc">fprintks</span> "farey8_tuning.txt", "<span class="op">%</span>2.6f\\n", kelem
kndx <span class="op">=</span> kndx<span class="op">+</span>1
<span class="octrl">endif</span>
<span class="oblock">endin</span>
<span class="oblock">instr</span> 2
ip <span class="opc">cps2pch</span> p4, <span class="op">-</span>100
asig <span class="opc">poscil</span> .5, ip, 1
aenv <span class="opc">linseg</span> 0, 0.1, 1, p3<span class="op">-</span>0.2, 1, 0.1, 0
<span class="opc">outs</span> asig <span class="op">*</span> aenv, asig <span class="op">*</span> aenv
<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="comment">;sine wave</span>
<span class="stamnt">i</span>1 0 .1
<span class="stamnt">i</span>2 1 .5 8.00
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + . >
<span class="stamnt">i</span>2 + 1 8.22
<span class="stamnt">e</span>
<span class="csdtag"></CsScore></span>
<span class="csdtag"></CsoundSynthesizer></span>
</pre>
</div>
</div>
<p><br class="example-break" />
</p>
</div>
<div class="refsect1" title="Credits">
<a id="idp19929088"></a>
<h2>Credits</h2>
<p>
</p>
<table border="0" summary="Simple list" class="simplelist">
<tr>
<td>Author: Georg Boenn </td>
</tr>
<tr>
<td>University of Glamorgan, UK</td>
</tr>
</table>
<p>
</p>
<p>New in Csound version 5.13</p>
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