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<title>Spectral centroid</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Spectral centroid</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>The <b>spectral centroid</b> is a measure used in <a href="Digital_signal_processing" title="Digital signal processing">digital signal processing</a> to characterise a <a href="Spectrum" title="Spectrum">spectrum</a>. It indicates where the <a href="Center_of_mass" title="Center of mass">center of mass</a> of the spectrum is located. Perceptually, it has a robust connection with the impression of <a href="Brightness_(sound)" class="mw-redirect" title="Brightness (sound)">brightness of a sound</a>.<sup id="cite_ref-greygordon78_1-0" class="reference"><a href="#cite_note-greygordon78-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It is sometimes called center of spectral mass.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Calculation">Calculation</h2></div>

<p>It is calculated as the <a href="Weighted_mean" class="mw-redirect" title="Weighted mean">weighted mean</a> of the frequencies present in the signal, determined using a <a href="Fourier_transform" title="Fourier transform">Fourier transform</a>, with their magnitudes as the weights:<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<mtext>centroid</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\text{centroid}}={\frac {\sum _{n=0}^{N-1}f(n)x(n)}{\sum _{n=0}^{N-1}x(n)}},}</annotation>

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</math></span><img src="./970a886599d78a6e96c9452aa53c8dbc20425d29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.506ex; height:7.509ex;" alt="{\displaystyle {\text{centroid}}={\frac {\sum _{n=0}^{N-1}f(n)x(n)}{\sum _{n=0}^{N-1}x(n)}},}" loading="lazy"></span></dd></dl>

<p>where <i>x</i>(<i>n</i>) represents the weighted frequency value, or magnitude, of <a href="Histogram" title="Histogram">bin</a> number <i>n</i>, and <i>f</i>(<i>n</i>) represents the center frequency of that bin.

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<div class="mw-heading mw-heading2"><h2 id="Alternative_usage">Alternative usage</h2></div>

<p>Some people use "spectral centroid" to refer to the <a href="Median" title="Median">median</a> of the spectrum. This is a <i>different</i> statistic, the difference being essentially the same as the difference between the unweighted median and <a href="Mean" title="Mean">mean</a> statistics. Since both are <a href="Average" title="Average">measures of central tendency</a>, in some situations they will exhibit some similarity of behaviour. But since typical audio spectra are not <a href="Normal_distribution" title="Normal distribution">normally distributed</a>, the two measures will often give strongly different values. Grey and Gordon in 1978 found the mean a better fit than the median.<sup id="cite_ref-greygordon78_1-1" class="reference"><a href="#cite_note-greygordon78-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>Because the spectral centroid is a good predictor of the "brightness" of a sound,<sup id="cite_ref-greygordon78_1-2" class="reference"><a href="#cite_note-greygordon78-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> it is widely used in digital audio and music processing as an automatic measure of musical <a href="Timbre" title="Timbre">timbre</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFGreyGordon1978" class="citation journal cs1">Grey, John M.; Gordon, John W. (1978). "Perceptual effects of spectral modifications on musical timbres". <i>The Journal of the Acoustical Society of America</i>. <b>63</b> (5). Acoustical Society of America (ASA): <span class="nowrap">1493–</span>1500. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978ASAJ...63.1493G">1978ASAJ...63.1493G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1121%2F1.381843">10.1121/1.381843</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0001-4966">0001-4966</a>.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFPulavartiMaguireYuenHarrison2022" class="citation journal cs1">Pulavarti, Surya V. S. R. K.; Maguire, Jack B.; Yuen, Shirley; Harrison, Joseph S.; Griffin, Jermel; Premkumar, Lakshmanane; Esposito, Edward A.; Makhatadze, George I.; Garcia, Angel E.; Weiss, Thomas M.; Snell, Edward H. (2022-02-17). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9281400">"From Protein Design to the Energy Landscape of a Cold Unfolding Protein"</a>. <i>The Journal of Physical Chemistry B</i>. <b>126</b> (6): <span class="nowrap">1212–</span>1231. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Facs.jpcb.1c10750">10.1021/acs.jpcb.1c10750</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1520-6106">1520-6106</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9281400">9281400</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/35128921">35128921</a>.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://recherche.ircam.fr/equipes/analyse-synthese/peeters/ARTICLES/Peeters_2003_cuidadoaudiofeatures.pdf">A Large Set of Audio Features for Sound Description</a>. Technical report published by <a href="IRCAM" title="IRCAM">IRCAM</a> in 2003. Section 6.1.1 describes the spectral centroid.</span>

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<cite id="CITEREFSchubertWolfeTarnopolsky2004" class="citation conference cs1">Schubert, Emery; Wolfe, Joe; Tarnopolsky, Alex (2004). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110810004531/http://icmpc8.umn.edu/proceedings/ICMPC8/PDF/AUTHOR/MP040215.PDF">"Spectral centroid and timbre in complex, multiple instrumental textures"</a> <span class="cs1-format">(PDF)</span>. <i>Proceedings of the 8th International Conference on Music Perception & Cognition, North Western University, Illinois</i>. <a rel="nofollow" class="external text" href="http://www.icmpc8.umn.edu/index_all.htm">International Conference on Music Perception & Cognition</a>. Lipscomb, S.D.; Ashley, R.; Gjerdingen, R. O.; Webster, P. (Eds.). Sydney, Australia: School of Music and Music Education; School of Physics, University of New South Wales. Archived from <a rel="nofollow" class="external text" href="http://icmpc8.umn.edu/proceedings/ICMPC8/PDF/AUTHOR/MP040215.PDF">the original</a> <span class="cs1-format">(PDF)</span> on 2011-08-10.</cite></span>

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