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<body class="mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Karplus–Strong_string_synthesis rootpage-Karplus–Strong_string_synthesis skin-vector action-view skin-vector-legacy"><div id="mw-page-base" class="noprint"></div>
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	<h1 id="firstHeading" class="firstHeading" >Karplus–Strong string synthesis</h1>
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		<div id="mw-content-text" lang="en" dir="ltr" class="mw-content-ltr"><div class="mw-parser-output"><table class="box-More_footnotes plainlinks metadata ambox ambox-style ambox-More_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of general <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">references</a>, but it remains largely unverified because <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span>  <small class="date-container"><i>(<span class="date">December 2013</span>)</i></small><small class="hide-when-compact"><i> (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this template message</a>)</i></small></div></td></tr></tbody></table>
<p><b>Karplus–Strong string synthesis</b> is a method of <a href="/wiki/Physical_modelling_synthesis" title="Physical modelling synthesis">physical modelling synthesis</a> that loops a short waveform through a filtered delay line to simulate the sound of a hammered or plucked <a href="/wiki/String_instrument" title="String instrument">string</a> or some types of <a href="/wiki/Percussion_instrument" title="Percussion instrument">percussion</a>. 
</p><p>At first glance, this technique can be viewed as <a href="/wiki/Subtractive_synthesis" title="Subtractive synthesis">subtractive synthesis</a> based on a <a href="/wiki/Feedback_loop" class="mw-redirect" title="Feedback loop">feedback loop</a> similar to that of a <a href="/wiki/Comb_filter" title="Comb filter">comb filter</a> for <a href="/wiki/Z-transform" title="Z-transform">z-transform</a> analysis. However, it can also be viewed as the simplest class of <a href="/wiki/Table-lookup_synthesis" class="mw-redirect" title="Table-lookup synthesis">wavetable</a>-modification algorithms now known as <a href="/wiki/Digital_waveguide_synthesis" title="Digital waveguide synthesis">digital waveguide synthesis</a>, because the delay line acts to store one period of the signal.
</p><p><a href="/w/index.php?title=Alexander_Strong&amp;action=edit&amp;redlink=1" class="new" title="Alexander Strong (page does not exist)">Alexander Strong</a> invented the algorithm, and <a href="/wiki/Kevin_Karplus" title="Kevin Karplus">Kevin Karplus</a> did the first analysis of how it worked.  Together they developed software and hardware implementations of the algorithm, including a custom <a href="/wiki/VLSI" class="mw-redirect" title="VLSI">VLSI</a> chip. They named the algorithm "Digitar" synthesis, as a <a href="/wiki/Portmanteau" title="Portmanteau">portmanteau</a> for "digital guitar".
</p>
<div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#How_it_works"><span class="tocnumber">1</span> <span class="toctext">How it works</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Tuning_the_string"><span class="tocnumber">2</span> <span class="toctext">Tuning the string</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#Refinements_to_the_algorithm"><span class="tocnumber">3</span> <span class="toctext">Refinements to the algorithm</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="#Musical_applications"><span class="tocnumber">4</span> <span class="toctext">Musical applications</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#References"><span class="tocnumber">5</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-6"><a href="#External_links"><span class="tocnumber">6</span> <span class="toctext">External links</span></a></li>
</ul>
</div>

<h2><span class="mw-headline" id="How_it_works">How it works</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karplus%E2%80%93Strong_string_synthesis&amp;action=edit&amp;section=1" title="Edit section: How it works">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<div class="center"><div class="floatnone"><a href="/wiki/File:Karplus-strong-schematic.svg" class="image"><img alt="Karplus-strong-schematic.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Karplus-strong-schematic.svg/519px-Karplus-strong-schematic.svg.png" decoding="async" width="519" height="192" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Karplus-strong-schematic.svg/779px-Karplus-strong-schematic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Karplus-strong-schematic.svg/1038px-Karplus-strong-schematic.svg.png 2x" data-file-width="519" data-file-height="192" /></a></div></div>
<p><br />
</p>
<ol><li>A short excitation waveform (of length L samples) is generated. In the original algorithm, this was a burst of <a href="/wiki/White_noise" title="White noise">white noise</a>, but it can also include any <a href="/wiki/Wideband" title="Wideband">wideband</a> signal, such as a rapid <a href="/wiki/Sine_wave" title="Sine wave">sine wave</a> <a href="/wiki/Chirp" title="Chirp">chirp</a> or frequency sweep, or a single cycle of a <a href="/wiki/Sawtooth_wave" title="Sawtooth wave">sawtooth wave</a> or <a href="/wiki/Square_wave" title="Square wave">square wave</a>.</li>
<li>This excitation is output and simultaneously fed back into a <a href="/wiki/Analog_delay_line" title="Analog delay line">delay line</a> L samples long.</li>
<li>The output of the delay line is fed through a <a href="/wiki/Audio_filter" title="Audio filter">filter</a>.  The <a href="/wiki/Gain_(electronics)" title="Gain (electronics)">gain</a> of the filter must be less than 1 at all frequencies, to maintain a stable <a href="/wiki/Positive_feedback" title="Positive feedback">positive feedback</a> loop. The filter can be a first-order lowpass filter (as pictured).  In the original algorithm, the filter consisted of averaging two adjacent samples, a particularly simple filter that can be implemented without a multiplier, requiring only shift and add operations. The filter characteristics are crucial in determining the harmonic structure of the decaying tone.</li>
<li>The filtered output is simultaneously mixed back into the output and fed back into the delay line.</li></ol>
<h2><span class="mw-headline" id="Tuning_the_string">Tuning the string</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karplus%E2%80%93Strong_string_synthesis&amp;action=edit&amp;section=2" title="Edit section: Tuning the string">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>The <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental frequency</a> (specifically, the lowest nonzero resonant frequency) of the resulting signal is the lowest frequency at which the unwrapped phase response of the delay and filter in cascade is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle -2\pi }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mo>&#x2212;<!-- − --></mo>
        <mn>2</mn>
        <mi>&#x03C0;<!-- π --></mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle -2\pi }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6565511f35e075d5f1054dcc82d91eb7548081d4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:4.303ex; height:2.343ex;" alt="-2\pi "/></span>.  The required <a href="/wiki/Phase_delay" class="mw-redirect" title="Phase delay">phase delay</a> <i>D</i> for a given fundamental frequency <i>F</i><sub>0</sub> is therefore calculated according to <i>D</i> = <i>F</i><sub><i>s</i></sub>/<i>F</i><sub>0</sub> where <i>F</i><sub><i>s</i></sub> is the sampling frequency.
</p><p>The length of any digital delay line is a whole-number multiple of the sampling period.  In order to obtain a <a href="/w/index.php?title=Fractional_delay&amp;action=edit&amp;redlink=1" class="new" title="Fractional delay (page does not exist)">fractional delay</a>, <a href="/wiki/Interpolation" title="Interpolation">interpolating filters</a> are used with parameters selected to obtain an appropriate phase delay at the fundamental frequency.  Either <a href="/wiki/Infinite_impulse_response" title="Infinite impulse response">IIR</a> or <a href="/wiki/Finite_Impulse_Response" class="mw-redirect" title="Finite Impulse Response">FIR</a> filters may be used, but FIR have the advantage that transients are suppressed if the fractional delay is changed over time. The most elementary fractional delay is the <a href="/wiki/Linear_interpolation" title="Linear interpolation">linear interpolation</a> between two samples (e.g., <i>s</i>(4.2) = 0.8<i>s</i>(4) + 0.2<i>s</i>(5)).  If the phase delay varies with frequency, <a href="/wiki/Harmonic" title="Harmonic">harmonics</a> may be sharpened or flattened relative to the fundamental frequency.  The original algorithm used equal weighting on two adjacent samples, as this can be achieved without multiplication hardware, allowing extremely cheap implementations.
</p><p><a href="/wiki/Z-transform" title="Z-transform">Z-transform</a> analysis can be used to get the pitches and decay times of the harmonics more precisely, as explained in the 1983 paper that introduced the algorithm.
</p><p>A demonstration of the Karplus-Strong algorithm can be heard in the following <a href="/wiki/Vorbis" title="Vorbis">Vorbis</a> file.  The algorithm used a loop gain of 0.98 with increasingly attenuating first order lowpass filters.  The pitch of the note was A2, or 220&#160;Hz.
</p>
<table role="presentation" class="mbox-small noprint" style="background-color:#f9f9f9;border:1px solid #aaa;color:#000;">
<tbody><tr>
<td class="mbox-image"><div class="center"><div class="floatnone"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/50px-Gnome-mime-sound-openclipart.svg.png" decoding="async" width="50" height="50" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/75px-Gnome-mime-sound-openclipart.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/100px-Gnome-mime-sound-openclipart.svg.png 2x" data-file-width="160" data-file-height="160" /></div></div></td>
<td class="mbox-text plainlist" style="line-height:1.1em"><div class="haudio">
<div style="padding:4px 0"><a href="/wiki/File:Karplus-strong-A2.ogg" title="File:Karplus-strong-A2.ogg">Karplus-Strong #1</a></div>
<div><div class="mediaContainer" style="width:220px"><audio id="mwe_player_0" controls="" preload="none" style="width:220px" class="kskin" data-durationhint="11.337868480726" data-startoffset="0" data-mwtitle="Karplus-strong-A2.ogg" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/2/28/Karplus-strong-A2.ogg" type="audio/ogg; codecs=&quot;vorbis&quot;" data-title="Original Ogg file (63 kbps)" data-shorttitle="Ogg source" data-width="0" data-height="0" data-bandwidth="63245" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/2/28/Karplus-strong-A2.ogg/Karplus-strong-A2.ogg.mp3" type="audio/mpeg" data-title="MP3" data-shorttitle="MP3" data-transcodekey="mp3" data-width="0" data-height="0" data-bandwidth="69640" /></audio></div></div>
<div class="description" style="padding:2px 0 0 0"><i>F</i><sub>1</sub> = 220Hz</div></div></td></tr>
<tr><td colspan="2" class="mbox-text" style="line-height:1.1em"><hr /><i class="selfreference">Problems playing this file? See <a href="/wiki/Help:Media" title="Help:Media">media help</a>.</i></td></tr>
</tbody></table>
<p>Holding the period (= length of the delay line) constant produces vibrations similar to those of a string or bell.  Increasing the period sharply after the transient input produces drum-like sounds.
</p>
<h2><span class="mw-headline" id="Refinements_to_the_algorithm">Refinements to the algorithm</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karplus%E2%80%93Strong_string_synthesis&amp;action=edit&amp;section=3" title="Edit section: Refinements to the algorithm">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p><a href="/w/index.php?title=Alex_Strong&amp;action=edit&amp;redlink=1" class="new" title="Alex Strong (page does not exist)">Alex Strong</a> and <a href="/wiki/Kevin_Karplus" title="Kevin Karplus">Kevin Karplus</a> realized that the Karplus-Strong algorithm was physically analogous to a sampling of the transversal wave on a string instrument, with the filter in the feedback loop representing the total string losses over one period. <a href="/wiki/Julius_O._Smith_III" class="mw-redirect" title="Julius O. Smith III">Julius O. Smith III</a> <a rel="nofollow" class="external autonumber" href="http://ccrma.stanford.edu/~jos/">[1]</a> and others 
generalized the algorithm to <a href="/wiki/Digital_waveguide_synthesis" title="Digital waveguide synthesis">digital waveguide synthesis</a>, which could also be used to model acoustic waves in tubes and on drum membranes. The first set of extensions and generalizations was presented in a paper in 1982 at the International Computer Music Conference in Venice, Italy, and published in more detail in 1983 in Computer Music Journal in an article entitled "Extensions of the Karplus Strong Plucked String Algorithm," by David A. Jaffe and Julius O. Smith.<sup id="cite_ref-JaffeSmith1983_1-0" class="reference"><a href="#cite_note-JaffeSmith1983-1">&#91;1&#93;</a></sup>
</p><p><a href="/w/index.php?title=Alex_Strong&amp;action=edit&amp;redlink=1" class="new" title="Alex Strong (page does not exist)">Alex Strong</a> developed a superior <a href="/wiki/Table-lookup_synthesis" class="mw-redirect" title="Table-lookup synthesis">wavetable</a>-modification method for plucked-string synthesis, but only published it as a patent.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">&#91;2&#93;</a></sup><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">&#91;<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Which of these patents? (July 2019)">clarification needed</span></a></i>&#93;</sup>
</p>
<h2><span class="mw-headline" id="Musical_applications">Musical applications</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karplus%E2%80%93Strong_string_synthesis&amp;action=edit&amp;section=4" title="Edit section: Musical applications">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>The first musical use of the algorithm was in the work <i>May All Your Children Be Acrobats</i> written in 1981 by <a href="/wiki/David_A._Jaffe" title="David A. Jaffe">David A. Jaffe</a>, and scored for eight guitars, mezzo-soprano and computer-generated stereo tape, with a text based on <a href="/wiki/Carl_Sandburg" title="Carl Sandburg">Carl Sandburg</a>'s <i>The People, Yes</i>. Jaffe continued to explore the musical and technical possibilities of the algorithm in <i>Silicon Valley Breakdown</i>, for computer-generated plucked strings (1982), as well as in later works such as <i>Telegram to the President, 1984</i> for string quartet and tape, and <i>Grass</i> for female chorus and tape (1987).
</p><p>The patent was licensed first to Mattel Electronics, which failed as a company before any product using the algorithm was developed, then to a startup company founded by some of the laid-off Mattel executives.  They never got sufficient funding to finish development, and so never brought a product to market either.  Eventually Yamaha licensed the patent, as part of the Sondius package of patents from Stanford.  It is unknown whether any hardware using the algorithm was ever sold, though many software implementations (which did not pay any license fees to the inventors) have been released.
</p><p>While they may not adhere strictly to the algorithm, many hardware components for modular systems have been commercially produced that invoke the basic principles of Karplus-Strong Synthesis: using an inverted, scaled control system for very small time values in a filtered delay line to create playable notes in the Western Tempered tuning system, controlled with volt per octave tracking or MIDI data. The Inventors were not specifically credited, though the term "Karplus-Strong Synthesis" is referenced in some of the manuals. 
</p><p>Hardware components capable of Karplus-Strong style synthesis include the Moog Clusterflux 108M, Mutable Instruments Elements, 4ms Company Dual Looping Delay, 2HP Pluck, Make Noise Mimeophon and the Arturia MicroFreak.
</p>
<h2><span class="mw-headline" id="References">References</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karplus%E2%80%93Strong_string_synthesis&amp;action=edit&amp;section=5" title="Edit section: References">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<dl><dt>Citations</dt></dl>
<style data-mw-deduplicate="TemplateStyles:r1011085734">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-JaffeSmith1983-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-JaffeSmith1983_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJaffeSmith1983">Jaffe &amp; Smith 1983</a></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r999302996">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://patents.google.com/?inventor=Alexander+R.+Strong">"inventor:(Alexander R. Strong)"</a>. <i>Google Patents</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-07-17</span></span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=unknown&amp;rft.jtitle=Google+Patents&amp;rft.atitle=inventor%3A%28Alexander+R.+Strong%29&amp;rft_id=https%3A%2F%2Fpatents.google.com%2F%3Finventor%3DAlexander%2BR.%2BStrong&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarplus%E2%80%93Strong+string+synthesis" class="Z3988"></span></span>
</li>
</ol></div></div>
<dl><dt>Bibliography</dt></dl>
<ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFKarplusStrong1983" class="citation journal cs1">Karplus, Kevin; Strong, Alex (1983). "Digital Synthesis of Plucked String and Drum Timbres". <i><a href="/wiki/Computer_Music_Journal" title="Computer Music Journal">Computer Music Journal</a></i>. MIT Press. <b>7</b> (2): 43–55. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3680062">10.2307/3680062</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="//www.jstor.org/stable/3680062">3680062</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Computer+Music+Journal&amp;rft.atitle=Digital+Synthesis+of+Plucked+String+and+Drum+Timbres&amp;rft.volume=7&amp;rft.issue=2&amp;rft.pages=43-55&amp;rft.date=1983&amp;rft_id=info%3Adoi%2F10.2307%2F3680062&amp;rft_id=%2F%2Fwww.jstor.org%2Fstable%2F3680062%23id-name%3DJSTOR&amp;rft.aulast=Karplus&amp;rft.aufirst=Kevin&amp;rft.au=Strong%2C+Alex&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarplus%E2%80%93Strong+string+synthesis" class="Z3988"></span><span class="cs1-maint citation-comment">CS1 maint: ref=harv (<a href="/wiki/Category:CS1_maint:_ref%3Dharv" title="Category:CS1 maint: ref=harv">link</a>)</span></li></ul>
<ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFJaffeSmith1983" class="citation journal cs1">Jaffe, David A.; Smith, Julius O. (1983). <a rel="nofollow" class="external text" href="http://static1.1.sqspcdn.com/static/f/1195073/15927045/1326030818550/Jaffe-Smith-Extensions-CMJ-1983.pdf?token=jBAicC7FH9EIwp3LPLGs7NE%2BOjo%3D">"Extensions of the Karplus-Strong Plucked String Algorithm"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Computer_Music_Journal" title="Computer Music Journal">Computer Music Journal</a></i>. MIT Press. <b>7</b> (2): 56–69. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3680063">10.2307/3680063</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="//www.jstor.org/stable/3680063">3680063</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Computer+Music+Journal&amp;rft.atitle=Extensions+of+the+Karplus-Strong+Plucked+String+Algorithm&amp;rft.volume=7&amp;rft.issue=2&amp;rft.pages=56-69&amp;rft.date=1983&amp;rft_id=info%3Adoi%2F10.2307%2F3680063&amp;rft_id=%2F%2Fwww.jstor.org%2Fstable%2F3680063%23id-name%3DJSTOR&amp;rft.aulast=Jaffe&amp;rft.aufirst=David+A.&amp;rft.au=Smith%2C+Julius+O.&amp;rft_id=http%3A%2F%2Fstatic1.1.sqspcdn.com%2Fstatic%2Ff%2F1195073%2F15927045%2F1326030818550%2FJaffe-Smith-Extensions-CMJ-1983.pdf%3Ftoken%3DjBAicC7FH9EIwp3LPLGs7NE%252BOjo%253D&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarplus%E2%80%93Strong+string+synthesis" class="Z3988"></span><span class="cs1-maint citation-comment">CS1 maint: ref=harv (<a href="/wiki/Category:CS1_maint:_ref%3Dharv" title="Category:CS1 maint: ref=harv">link</a>)</span></li></ul>
<ul><li><span class="citation patent" id="CITEREFUS_application_4649783"><a rel="nofollow" class="external text" href="https://worldwide.espacenet.com/textdoc?DB=EPODOC&amp;IDX=US4649783">US&#32;application 4649783</a>,&#32;Alexander R. Strong, Kevin J. Karplus,&#32;"<a rel="nofollow" class="external text" href="https://patents.google.com/patent/US4649783A">Wavetable Modification Instrument and Method for Generating Musical Sound</a>",&#32;published 1987-03-17</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Apatent&amp;rft.applnumber=4649783&amp;rft.cc=US&amp;rft.title=%5Bhttps%3A%2F%2Fpatents.google.com%2Fpatent%2FUS4649783A+Wavetable+Modification+Instrument+and+Method+for+Generating+Musical+Sound%5D&amp;rft.inventor=Alexander+R.+Strong%2C+Kevin+J.+Karplus&amp;rft.appldate=1984-05-24&amp;rft.pubdate=1987-03-17"><span style="display: none;">&#160;</span></span></li></ul>
<ul><li><span class="citation patent" id="CITEREFUS_application_4622877"><a rel="nofollow" class="external text" href="https://worldwide.espacenet.com/textdoc?DB=EPODOC&amp;IDX=US4622877">US&#32;application 4622877</a>,&#32;Alexander R. Strong,&#32;"<a rel="nofollow" class="external text" href="https://patents.google.com/patent/US4622877A">Independently controlled wavetable-modification instrument and method for generating musical sound</a>",&#32;published 1986-11-18</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Apatent&amp;rft.applnumber=4622877&amp;rft.cc=US&amp;rft.title=%5Bhttps%3A%2F%2Fpatents.google.com%2Fpatent%2FUS4622877A+Independently+controlled+wavetable-modification+instrument+and+method+for+generating+musical+sound%5D&amp;rft.inventor=Alexander+R.+Strong&amp;rft.pubdate=1986-11-18"><span style="display: none;">&#160;</span></span></li></ul>
<ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFMoore1990" class="citation book cs1">Moore, F. Richard (1990). <i>Elements of Computer Music</i>. Upper Saddle River: Prentice-Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-13-252552-6" title="Special:BookSources/0-13-252552-6"><bdi>0-13-252552-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Elements+of+Computer+Music&amp;rft.place=Upper+Saddle+River&amp;rft.pub=Prentice-Hall&amp;rft.date=1990&amp;rft.isbn=0-13-252552-6&amp;rft.aulast=Moore&amp;rft.aufirst=F.+Richard&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarplus%E2%80%93Strong+string+synthesis" class="Z3988"></span><span class="cs1-maint citation-comment">CS1 maint: ref=harv (<a href="/wiki/Category:CS1_maint:_ref%3Dharv" title="Category:CS1 maint: ref=harv">link</a>)</span></li></ul>
<h2><span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karplus%E2%80%93Strong_string_synthesis&amp;action=edit&amp;section=6" title="Edit section: External links">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li><a rel="nofollow" class="external text" href="http://ccrma.stanford.edu/~jos/pasp/Karplus_Strong_Algorithm.html">The Karplus-Strong Algorithm</a></li>
<li><a rel="nofollow" class="external text" href="http://ccrma.stanford.edu/~jos/Mohonk05/Karplus_Strong_Algorithm.html">Sound Examples</a></li>
<li><a rel="nofollow" class="external text" href="http://www.freesound.org/browse/tags/karplus-strong/">More sound examples under CC license</a></li>
<li><a rel="nofollow" class="external text" href="http://lab.andre-michelle.com/karplus-strong-guitar">Flash application implementing a guitar</a></li>
<li><a rel="nofollow" class="external text" href="http://amid.fish/javascript-karplus-strong">A HTML5 port of the above application</a></li>
<li><a rel="nofollow" class="external text" href="http://www.jaffe.com">David A. Jaffe's music, including sound examples</a></li></ul>
<div role="navigation" class="navbox" aria-labelledby="Sound_synthesis_types" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r992953826">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}.mw-parser-output .infobox .navbar{font-size:100%}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Sound_synthesis_types" title="Template:Sound synthesis types"><abbr title="View this template" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Sound_synthesis_types" title="Template talk:Sound synthesis types"><abbr title="Discuss this template" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">t</abbr></a></li><li class="nv-edit"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Template:Sound_synthesis_types&amp;action=edit"><abbr title="Edit this template" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">e</abbr></a></li></ul></div><div id="Sound_synthesis_types" style="font-size:114%;margin:0 4em"><a href="/wiki/Synthesizer" title="Synthesizer">Sound synthesis</a> types</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div id="*_Frequency_modulation_*_Linear_arithmetic_*_Phase_distortion_*_Scanned_*_Subtractive_*_Additive_*_Distortion">
<ul><li><a href="/wiki/Frequency_modulation_synthesis" title="Frequency modulation synthesis">Frequency modulation</a></li>
<li><a href="/wiki/Linear_arithmetic_synthesis" title="Linear arithmetic synthesis">Linear arithmetic</a></li>
<li><a href="/wiki/Phase_distortion_synthesis" title="Phase distortion synthesis">Phase distortion</a></li>
<li><a href="/wiki/Scanned_synthesis" title="Scanned synthesis">Scanned</a></li>
<li><a href="/wiki/Subtractive_synthesis" title="Subtractive synthesis">Subtractive</a></li>
<li><a href="/wiki/Additive_synthesis" title="Additive synthesis">Additive</a></li>
<li><a href="/wiki/Distortion_synthesis" title="Distortion synthesis">Distortion</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Sample-based_synthesis" title="Sample-based synthesis">Sample-based</a> or <a href="/wiki/Sampler_(musical_instrument)" title="Sampler (musical instrument)">Sampler</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Wavetable_synthesis" title="Wavetable synthesis">Wavetable</a></li>
<li><a href="/wiki/Granular_synthesis" title="Granular synthesis">Granular</a></li>
<li><a href="/wiki/Vector_synthesis" title="Vector synthesis">Vector</a></li>
<li><a href="/wiki/Concatenative_synthesis" title="Concatenative synthesis">Concatenative</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Physical_modelling_synthesis" title="Physical modelling synthesis">Physical modelling</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Banded_waveguide_synthesis" title="Banded waveguide synthesis">Banded waveguide</a></li>
<li><a href="/wiki/Digital_waveguide_synthesis" title="Digital waveguide synthesis">Digital waveguide</a></li>
<li><a href="/wiki/Direct_digital_synthesizer" class="mw-redirect" title="Direct digital synthesizer">Direct digital</a></li>
<li><a href="/wiki/Formant_synthesis" class="mw-redirect" title="Formant synthesis">Formant</a></li>
<li><a class="mw-selflink selflink">Karplus–Strong string</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Analog_synthesizer" title="Analog synthesizer">Analog synthesizer</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Graphical_sound" title="Graphical sound">Graphical sound</a></li>
<li><a href="/wiki/Modular_synthesizer" title="Modular synthesizer">Modular</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Digital_synthesizer" title="Digital synthesizer">Digital synthesizer</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Analog_modeling_synthesizer" title="Analog modeling synthesizer">Analog modeling</a></li>
<li><a href="/wiki/Scanned_synthesis" title="Scanned synthesis">Scanned synthesis</a></li>
<li><a href="/wiki/Software_synthesizer" title="Software synthesizer">Software synthesizer</a></li></ul>
</div></td></tr></tbody></table></div>
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