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<div class="refentry" title="GENfarey">
<a id="GENfarey"></a>
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<a id="IndexGENfarey" class="indexterm"></a>
<div class="refnamediv">
<h2>
<span class="refentrytitle">GENfarey</span>
</h2>
<p>"farey" —
Fills a table with the Farey Sequence F<sub>n</sub> of the integer n.
</p>
</div>
<div class="refsect1" title="Description">
<a id="idp78636408"></a>
<h2>Description</h2>
<p>
A Farey Sequence F<sub>n</sub> of order n is a list of fractions in their lowest terms between 0 and 1 and in ascending order. Their denominators do not exceed n.
This means a fraction a/b belongs to F<sub>n</sub> if 0 ≤ a ≤ b ≤ n.
The numerator and denominator of each fraction are always coprime.
0 and 1 are included in F<sub>n</sub> as the fractions 0/1 and 1/1.
For example F<sub>5</sub> = {0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1}
Some properties of the Farey Sequence:
</p>
<div class="itemizedlist">
<ul class="itemizedlist" type="disc">
<li class="listitem">
If a/b and c/d are two successive terms of F<sub>n</sub>, then bc - ad = 1.
</li>
<li class="listitem">
If a/b, c/d, e/f are three successive terms of F<sub>n</sub>, then:
c/d = (a+e) / (b+f).
In this case c/d is called the mediant fraction between a/b and e/f.
</li>
<li class="listitem">
If n > 1, then no two successive terms ofF<sub>n</sub> have the same denominator.
</li>
</ul>
</div>
<p>
</p>
<p>
The length of any Farey Sequence F<sub>n</sub> is determined 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="idp78693272"></a>
<h2>Syntax</h2>
<pre class="synopsis"><span class="command"><strong>f</strong></span> # time size "farey" fareynum mode</pre>
</div>
<div class="refsect1" title="Initialization">
<a id="idp78694256"></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 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>).
</p>
<p>
<span class="emphasis"><em>fareynum</em></span> -- the integer n for generating Farey Sequence F<sub>n</sub>
</p>
<p>
<span class="emphasis"><em>mode</em></span> -- integer to trigger a specific
output to be written into the table:
</p>
<div class="itemizedlist">
<ul class="itemizedlist" type="disc">
<li class="listitem">
<p>0 -- outputs floating point numbers representing the
elements of F<sub>n</sub>.</p>
</li>
<li class="listitem">
<p> 1 -- outputs delta values of successive elements of
F<sub>n</sub>, useful for generating note durations for
example.</p>
</li>
<li class="listitem">
<p> 2 -- outputs only the denominators of the integer
ratios, useful for indexing other tables or instruments for
example.</p>
</li>
<li class="listitem">
<p> 3 -- same as mode 2 but with normalised
output.</p>
</li>
<li class="listitem">
<p> 4 -- same as mode 0 but with 1 added to each number,
useful for generating tables for tuning opcodes, for example
<span class="emphasis"><em>cps2pch</em></span>.
</p>
</li>
</ul>
</div>
<p>
</p>
</div>
<div class="refsect1" title="Examples">
<a id="idp78700072"></a>
<h2>Examples</h2>
<p>
</p>
<div class="informalexample"><pre class="programlisting">
<span class="stamnt">f</span>1 0 -23 "farey" 8 0</pre>
Generates generates Farey Sequence F<sub>8</sub>. The table contains all
23 elements of F<sub>8</sub> as floating point numbers.
<pre class="programlisting">
<span class="stamnt">f</span>1 0 -18 "farey" 7 1</pre>
This generates Farey Sequence F<sub>7</sub>. The table contains 18 delta values of F<sub>7</sub>,
i.e. the difference between r<sub>i+1</sub> - r<sub>i</sub>, where r is the ith element of F<sub>n</sub>.
<pre class="programlisting">
<span class="stamnt">f</span>1 0 -43 "farey" 11 2</pre>
This generates Farey Sequence F<sub>11</sub>. The table contains the denominators of all
43 fractions in F<sub>11</sub>.
<pre class="programlisting">
<span class="stamnt">f</span>1 0 -43 "farey" 11 3</pre>
This generates Farey Sequence F<sub>11</sub>. The table contains the denominators of all
43 fractions in F<sub>11</sub>, each of those divided by 11, i.e. normalised.
<pre class="programlisting">
<span class="stamnt">f</span>1 0 -18 "farey" 7 4</pre>
This generates Farey Sequence F<sub>7</sub>. The table contains all fractions of F<sub>7</sub>,
same as mode 0, but this time '1' is added to each table element.
</div>
<p>
</p>
<p>
</p>
<div class="example">
<a id="idp78707744"></a>
<p class="title">
<strong>Example 1076. A simple example of the GENfarey routine.</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="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>10
<span class="ohdr">nchnls</span><span class="op">=</span>1
<span class="oblock">instr</span> 4
kndx <span class="opc">init</span> 0 <span class="comment">; read out elements of F_8 one by one and print to file</span>
<span class="octrl">if</span> (kndx <span class="op"><</span> 23) then
kelem <span class="opc">tab</span> kndx, 1
<span class="opc">fprintks</span> "farey8table.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="csdtag"></CsInstruments></span>
<span class="csdtag"><CsScore></span>
<span class="comment">; initialise integer for Farey Sequence F_8</span>
<span class="stamnt">f</span>1 0 -23 "<span class="stamnt">f</span>arey" 8 0
<span class="comment">; if mode=0 then the table stores all elements of the Farey Sequence</span>
<span class="comment">; as fractions in the range [0,1]</span>
<span class="stamnt">i</span>4 0 1
<span class="stamnt">e</span>
<span class="csdtag"></CsScore></span>
<span class="csdtag"></CsoundSynthesizer></span>
</pre>
</div>
</div>
<p><br class="example-break" />
</p>
<p>
Here is a complete example of the GENfarey routine. It uses the files <a class="ulink" href="examples/genfarey-2.csd" target="_top"><em class="citetitle">genfarey-2.csd</em></a>.
</p>
<div class="example">
<a id="idp78710616"></a>
<p class="title">
<strong>Example 1077. Another example of the GENfarey routine.</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 genfarey.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">; GENfarey creates table gidelta. </span>
<span class="comment">; The table contains the delta values of Farey Sequence 7 (p5=7).</span>
<span class="comment">; They are used as Inter Onset Intervals (IOIs) or event durations.</span>
<span class="comment">; If p6 is set to 1 for IOI output then the length of the table (p3=-18) is -(|F_7| - 1)</span>
<span class="comment">; Remember that a negative sign is for non-power-of-2 table lengths.</span>
<span class="comment">; The negative sign in front of the GEN number prevents post-normalisation of its values.</span>
gidelta <span class="ohdr">ftgen</span> 0,0,<span class="op">-</span>18,"farey",7,1
<span class="comment">; Use GENfarey with p6 set to 2 to generate the denominators of fractions of F_7 </span>
<span class="comment">; this is used in this example as factors to create a series of pitches:</span>
gimult <span class="ohdr">ftgen</span> 0,0,<span class="op">-</span>18,"farey",7,2
<span class="comment">;-------- loop and trigger instrument 901 using a Farey Sequence polyrhythm</span>
<span class="oblock">instr</span> 1
kindx <span class="opc">init</span> 0
kindx2 <span class="opc">init</span> 0
ktrigger <span class="opc">init</span> 0
ktime_unit <span class="opc">init</span> p6
kstart <span class="opc">init</span> p4
kloop <span class="opc">init</span> p5
kinitndx <span class="opc">init</span> 0
kfn_times <span class="opc">init</span> gidelta
knote <span class="opc">init</span> 60
kbasenote <span class="opc">init</span> p8
ifundam <span class="opc">init</span> p7
ktrigger <span class="opc">seqtime</span> ktime_unit, kstart, kloop, kinitndx, kfn_times
<span class="octrl">if</span> (ktrigger <span class="op">></span> 0 ) then
kpitch <span class="op">=</span> <span class="opc">cpspch</span>(ifundam)
kmult <span class="opc">tab</span> kindx2, gimult
kpitch <span class="op">=</span> kpitch <span class="op">*</span> kmult
knote <span class="op">=</span> kbasenote <span class="op">+</span> kmult
<span class="opc">event</span> "i", 901, 0, .4, .10, kpitch, kpitch <span class="op">*</span> .9, 0.4, 5, .75, .8, 1.0, .15, .0, .125, .125, .25, .5, 1.0, .0, .0, .0, .0, .125, .25, .25, .25, knote
kindx <span class="op">=</span> kindx <span class="op">+</span> 1
kindx <span class="op">=</span> kindx <span class="op">%</span> kloop
kindx2 <span class="op">=</span> kindx2 <span class="op">+</span> 1
kindx2 <span class="op">=</span> kindx2 <span class="op">%</span> kloop
<span class="octrl">endif</span>
<span class="oblock">endin</span>
<span class="comment">;------ basic 2 Operators FM algorithm ----------------</span>
<span class="oblock">instr</span> 901
inotedur <span class="op">=</span> p3
imaxamp <span class="op">=</span> p4 <span class="comment">;ampdb(p4)</span>
icarrfreq <span class="op">=</span> p5
imodfreq <span class="op">=</span> p6
ilowndx <span class="op">=</span> p7
indxdiff <span class="op">=</span> p8<span class="op">-</span>p7
knote <span class="op">=</span> p27
aampenv <span class="opc">linseg</span> p9, p14<span class="op">*</span>p3, p10, p15<span class="op">*</span>p3, p11, p16<span class="op">*</span>p3, p12, p17<span class="op">*</span>p3, p13
adevenv <span class="opc">linseg</span> p18, p23<span class="op">*</span>p3, p19, p24<span class="op">*</span>p3, p20, p25<span class="op">*</span>p3, p21, p26<span class="op">*</span>p3, p22
amodosc <span class="opc">oscili</span> (ilowndx<span class="op">+</span>indxdiff<span class="op">*</span>adevenv)<span class="op">*</span>imodfreq, imodfreq, 10
acarosc <span class="opc">oscili</span> imaxamp<span class="op">*</span>aampenv, icarrfreq<span class="op">+</span>amodosc, 10
<span class="opc">outs</span> acarosc, acarosc
<span class="oblock">endin</span>
<span class="csdtag"></CsInstruments></span>
<span class="csdtag"><CsScore></span>
<span class="stamnt">f</span>10 0 4096 10 1 <span class="comment">;sine wave </span>
<span class="comment">; p4 kstart := index offset into the Farey Sequence</span>
<span class="comment">; p5 kloop := end index into Farey Seq.</span>
<span class="comment">; p6 timefac := time in seconds for one loop to complete</span>
<span class="comment">; p7 fundam := fundamental of the FM instrument</span>
<span class="comment">; p8 basenote:= root pitch of the midi voice output</span>
<span class="comment">; note that pitch structures of the midi file output are not equivalent to the</span>
<span class="comment">; ones used for the FM real-time synthesis.</span>
<span class="comment">; start dur kstart kloop timefac fundam. basenote</span>
<span class="stamnt">i</span>1 0.0 44 0 18 2 6.05 60
<span class="stamnt">i</span>1 4 30 0 18 3 7.05 72
<span class="stamnt">i</span>1 34 12 9 18 3 7.05 72
<span class="stamnt">i</span>1 10 12 0 18 1.5 8 84
<span class="stamnt">i</span>1 22 12 0 9 1.5 8 84
<span class="stamnt">i</span>1 15 16 0 18 1 5 48
<span class="stamnt">i</span>1 22 20 5 17 1.7 4 36
<span class="stamnt">i</span>1 46 20 3 11 2.5 7.04 71
<span class="stamnt">i</span>1 51 20 5 13 2.5 7.06 72
<span class="stamnt">i</span>1 73.5 1.5 11 18 1.5 5.05 48
<span class="stamnt">i</span>1 75 1 12 18 1 6.03 58
<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 GENfarey routines, as used in the example:</span>
</p>
<div class="mediaobject">
<img src="images/genfarey-2_1.png" alt="gidelta ftgen 100,0,-18,"farey",7,1 - delta values of Farey Sequence 7" />
<div class="caption">
<p>gidelta ftgen 100,0,-18,"farey",7,1 - delta values of Farey Sequence 7</p>
</div>
</div>
<p>
</p>
<div class="mediaobject">
<img src="images/genfarey-2_2.png" alt="gimult ftgen 101,0,-18,"farey",7,2 - generate the denominators of fractions of F_7" />
<div class="caption">
<p>gimult ftgen 101,0,-18,"farey",7,2 - generate the denominators of fractions of F_7 </p>
</div>
</div>
<p>
</p>
</div>
<div class="refsect1" title="Credits">
<a id="idp78716824"></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</td>
</tr>
<tr>
<td>2010</td>
</tr>
</table>
<p>
</p>
<p>New in Csound version 5.13</p>
</div>
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