View source for Missing fundamental

{{short description|Acoustic phenomenon}}
[[File:Illustration of common periodicity of full spectrum and missing fundamental waveforms.jpg|thumb|400px|The bottom waveform is missing the fundamental frequency, 100 [[hertz]], and the second harmonic, 200 hertz. The periodicity is nevertheless clear when compared to the full-spectrum waveform on top.]]
The pitch being perceived with the first [[harmonic]] being absent in the waveform is called the missing fundamental phenomenon.<ref>{{Cite book |first1=David M.|last1=Howard|first2=J. A. S.|last2=Angus|title=Acoustics and Psychoacoustics Fifth Edition |date=2017 |publisher=Routledge |isbn=9781315716879 |edition=5th |location=New York |publication-date=2017 |pages=123 |language=English}}</ref>

It is established in [[psychoacoustics]] that the auditory system, with its natural tendency to distinguish a tone from another, will persistently assign a pitch to a complex tone given that a sufficient set of harmonics are present in the spectrum.<ref>{{Cite journal |last=Hartmann |first=William |date=December 1996 |title=Pitch, Periodicity, & Auditory Organization |url=https://web.pa.msu.edu/acoustics/tutorial.pdf |journal= The Journal of the Acoustical Society of America|volume=100 |issue=6 |pages=3491–3902 |doi=10.1121/1.417248 |pmid=8969472 |bibcode=1996ASAJ..100.3491H |via=Michigan State University}}</ref>

For example, when a note (that is not a [[pure tone]]) has a [[pitch (music)|pitch]] of 100&nbsp;[[hertz|Hz]], it will consist of frequency components that are integer multiples of that value (e.g. 100, 200, 300, 400, 500.... Hz). However, smaller loudspeakers may not produce low frequencies, so in our example, the 100&nbsp;Hz component may be missing. Nevertheless, a pitch corresponding to the fundamental may still be heard.

== Explanation ==
[[Image:Missing fundamental Fourier series 8.png|thumb|150px|The GCD of the frequency of all harmonics is the fundamental (dashed).]]

A low [[pitch (psychophysics)|pitch]] (also known as the pitch of the missing fundamental or virtual pitch<ref>{{cite web | url=http://jjensen.org/VirtualPitch.html | title=Virtual Pitch Algorithm of Terhardt and Extensions }}</ref>) can sometimes be heard when there is no apparent source or component of that frequency. This perception is due to the brain interpreting repetition patterns that are present.<ref>{{cite book
 |title        = Auditory Neuroscience
 |author       = Jan Schnupp, Israel Nelken and Andrew King
 |publisher    = MIT Press
 |year         = 2011
 |isbn         = 978-0-262-11318-2
 |url          = https://mustelid.physiol.ox.ac.uk/drupal/?q=topics/how-many-repetitions-are-required-produce-pitch
 |access-date  = 2018-08-30
 |archive-url  = https://web.archive.org/web/20120318013518/https://mustelid.physiol.ox.ac.uk/drupal/?q=topics%2Fhow-many-repetitions-are-required-produce-pitch
 |archive-date = 2012-03-18
 |url-status     = dead
}}</ref><ref>
{{cite book
 | title = An Introduction to Phonetics and Phonology
 | author = John Clark, Colin Yallop and Janet Fletcher
 | publisher = Blackwell Publishing
 | year = 2007
 | isbn = 978-1-4051-3083-7
 | url = https://books.google.com/books?id=dX5P5mxtYYIC&q=phantom-fundamental+pitch+perception&pg=PA233}}</ref><ref name=plack>
{{cite book
 | title =Pitch: Neural Coding and Perception
 | author = Christopher J. Plack
 | publisher = Springer
 | year = 2005
 | isbn = 978-0-387-23472-4
 | url = https://books.google.com/books?id=n6VdlK3AQykC&q=Pitch&pg=PP1
 }}</ref>

It was once thought that this effect was because the missing fundamental was replaced by distortions introduced by the physics of the ear. However, experiments subsequently showed that when a noise was added that would have masked these distortions had they been present, listeners still heard a pitch corresponding to the missing fundamental, as reported by [[J. C. R. Licklider]] in 1954.<ref>{{cite book | title = Music and Connectionism | author = Peter M. Todd and D. Gareth Loy | publisher = MIT Press | year = 1991 | isbn = 978-0-262-20081-3 | url = https://books.google.com/books?id=NxycaQH6PeoC&q=missing-fundamental+pitch+perception+distortion+noise+masked&pg=PA86}}</ref> It is now widely accepted that the brain processes the information present in the overtones to calculate the fundamental frequency. The precise way in which it does so is still a matter of debate, but the processing seems to be based on an [[autocorrelation]] involving the timing of neural impulses in the auditory nerve.<ref>{{cite journal|last=Cariani|first=P.A.|author2=Delgutte, B.|title=Neural Correlates of the Pitch of Complex Tones. I. Pitch and Pitch Salience|journal=Journal of Neurophysiology|date=September 1996|volume=76|issue=3|pages=1698–1716|pmid=8890286|url=http://www.brainmusic.org/MBB91%20Webpage/Pitch_II_Cariani.pdf|accessdate=13 November 2012|doi=10.1152/jn.1996.76.3.1698}}</ref> However, it has long been noted that any neural mechanisms which may accomplish a delay (a necessary operation of a true autocorrelation) have not been found.<ref name=plack/> At least one model shows a temporal delay to be unnecessary to produce an autocorrelation model of pitch perception, appealing to [[Phase shift#Phase shift|phase shifts]] between [[Cochlea|cochlear filters]];<ref>{{cite journal|last=de Cheveigné|first=A.|author2=Pressnitzer, D.|title=The case of the missing delay lines: Synthetic delays obtained by cross-channel phase interaction|journal=Journal of the Acoustical Society of America|date=June 2006|volume=119|issue=6|pages=3908–3918|pmid=16838534|url=http://audition.ens.fr/dp/pdfs/cheveigne-2006-synthetic_delay.pdf|accessdate=13 November 2012|bibcode = 2006ASAJ..119.3908D |doi = 10.1121/1.2195291 }}</ref> however, earlier work has shown that certain sounds with a prominent peak in their autocorrelation function do not elicit a corresponding pitch percept,<ref>{{cite journal|last=Kaernbach|first=C.|author2=Demany, L.|title=Psychophysical evidence against the autocorrelation theory of auditory temporal processing|journal=Journal of the Acoustical Society of America|date=October 1998|volume=104|issue=4|pages=2298–2306|pmid=10491694|bibcode = 1998ASAJ..104.2298K |doi = 10.1121/1.423742 |s2cid=18133681}}</ref><ref>{{cite journal|last=Pressnitzer|first=D. |author2=de Cheveigné, A. |author3=Winter, I.M.|title=Perceptual pitch shift for sounds with similar waveform autocorrelation|journal=Acoustics Research Letters Online|date=January 2002|volume=3|issue=1|pages=1–6|doi=10.1121/1.1416671|s2cid=123182480 |doi-access=free}}</ref> and that certain sounds without a peak in their autocorrelation function nevertheless elicit a pitch.<ref>{{cite journal|last=Burns|first=E.M.|author2=Viemeister, N.F.|title=Nonspectral pitch|journal=Journal of the Acoustical Society of America|date=October 1976|volume=60|issue=4|pages=863–869|bibcode = 1976ASAJ...60..863B |doi = 10.1121/1.381166 }}</ref><ref>{{cite journal|last=Fitzgerald|first=M.B.|author2=Wright, B.|title=A perceptual learning investigation of the pitch elicited by amplitude-modulated noise|journal=Journal of the Acoustical Society of America|date=December 2005|volume=118|issue=6|pages=3794–3803|pmid=16419824|bibcode = 2005ASAJ..118.3794F |doi = 10.1121/1.2074687 }}</ref> Autocorrelation can thus be considered, at best, an incomplete model.

The pitch of the missing fundamental, usually at the [[greatest common divisor]] of the frequencies present,<ref>{{cite journal|last=Schwartz|first=D.A.|author2=Purves, D.|title=Pitch is determined by naturally occurring periodic sounds|journal=Hearing Research|date=May 2004|volume=194|issue=1–2|pages=31–46|doi=10.1016/j.heares.2004.01.019|pmid=15276674|s2cid=40608136|url=http://www.purveslab.net/publications/schwarrtz_purves_pitch.pdf|archive-url=https://web.archive.org/web/20121208222727/http://www.purveslab.net/publications/schwarrtz_purves_pitch.pdf|url-status=dead|archive-date=2012-12-08|accessdate=4 September 2012}}</ref> is not, however, always perceived. Research conducted at [[Heidelberg University]] shows that, under narrow stimulus conditions with a small number of harmonics, the general population can be divided into those who perceive missing fundamentals, and those who primarily hear the overtones instead.<ref>{{cite journal |last=Schneider |first=P. |author2=Sluming, V. |author3=Roberts, N. |author4=Scherg, M. |author5=Goebel, R. |author6=Specht, H. |author7=Dosch, H.G. |author8=Bleeck, S. |author9=Stippich, C. |author10=Rupp, A. |title=Structural and functional asymmetry of lateral Heschl's gyrus reflects pitch perception preference |journal=Nature Neuroscience |date=August 2005 |volume=8 |issue=9 |pages=1241–1247 |doi=10.1038/nn1530 |url=http://lumiere.ens.fr/Audition/P2web/eval2006/ALG_nnRupp.pdf |pmid=16116442 |s2cid=16010412 |access-date=2012-07-22 |archive-date=2017-08-09 |archive-url=https://web.archive.org/web/20170809032034/http://lumiere.ens.fr/Audition/P2web/eval2006/ALG_nnRupp.pdf |url-status=dead }}</ref> This was done by asking subjects to judge the direction of motion (up or down) of two complexes in [[Melody|succession]]. The authors used structural [[MRI]] and [[Magnetoencephalography|MEG]] to show that the preference for missing fundamental hearing correlated with left-hemisphere lateralization of pitch perception, where the preference for spectral hearing correlated with right-hemisphere lateralization, and those who exhibited the latter preference tended to be musicians.

In ''Parsing the Spectral Envelope: Toward a General Theory of Vocal Tone Color'' (2016) by [https://www.ianhowellcountertenor.com/ Ian Howell], He wrote that although not everyone can hear the missing fundamentals, noticing them can be taught and learned.<ref>Howell, I. (2017). ''Parsing the Spectral Envelope: Toward a General Theory of Vocal Tone Color''[Doctoral Thesis, New England Conservatory of Music]''.'' https://www.nats.org/_Library/So_You_Want_To_Sing_Book_Series/HOWELL-Parsing-the-spectral-envelope-PROQUEST-FINAL.pdf</ref> [[Robert Ladd (linguist)|D. Robert Ladd]] et al. have a related study that claims that most people can switch from listening for the pitch from the harmonics that are evident to finding these pitches spectrally. <ref>{{Cite journal |last=Ladd |first=Robert |date=2013 |title=Patterns of Individual Differences in the Perception of Missing Fundamental Tones |url=https://supp.apa.org/psycarticles/supplemental/a0031261/a0031261_supp.html |journal=Journal of Experimental Psychology |volume=39 |issue=5 |pages=1386–1397 |doi=10.1037/a0031261 |pmid=23398251 |hdl=11858/00-001M-0000-0010-247B-4 |via=Pubmed|hdl-access=free }}</ref>

==Examples==
[[File:Timpani missing fundamental.png|thumb|Timpani bodies modify modes of vibration to match harmonics.<ref name="H&A"/> Red: Harmonics of perceived pitch. Dark blue: Prominent modes of vibration.<!--Dashed light blue: Quieted modes of vibration.--> {{audio|Timpani versus harp middle C.mid|Play C0 harp-timpano-harp}}]]

[[Timpani]] produce [[inharmonic]] overtones, but are constructed and tuned to produce near-harmonic overtones to an implied missing fundamental. Hit in the usual way (half to three-quarters the distance from the center to the rim), the fundamental note of a timpani is very weak in relation to its second through fifth "harmonic" overtones.<ref name="H&A">{{cite book |title=Acoustics and Psychoacoustics |last=Howard |first=David M. |author2=Jamie Angus |year=2006 |publisher=Focal Press |isbn=978-0-240-51995-1 |pages=200–3 |url=https://books.google.com/books?id=Yyk5nAIJcBcC }}</ref> A timpani might be tuned to produce sound most strongly at 200, 302, 398, and 488&nbsp;Hz, for instance, implying a missing fundamental at 100&nbsp;Hz (though the actual dampened fundamental is 170&nbsp;Hz).<ref>[http://www.physics.mcgill.ca/~guymoore/ph224/notes/lecture26.pdf McGill University. Physics Department. Guy D. Moore. ''Lecture 26: Percussion''] {{Webarchive|url=https://web.archive.org/web/20150924072753/http://www.physics.mcgill.ca/~guymoore/ph224/notes/lecture26.pdf |date=2015-09-24 }}. "The sequence 1; 1:51; 1:99; 2:44; 2:89 is almost 1; 1:5; 2; 2:5; 3 which is the harmonic series of
a missing fundamental."<!--1x100x2=200--></ref>

A [[violin]]'s lowest air and body resonances generally fall between 250&nbsp;Hz and 300&nbsp;Hz. The fundamental frequency of the open [[Scientific pitch notation|G3]] string is below 200&nbsp;Hz in modern tunings as well as [[concert pitch#Pitch inflation|most historical tunings]], so the lowest notes of a violin have an attenuated fundamental, although listeners seldom notice this.{{Citation needed|date=March 2021}}

Most common [[telephones]] cannot reproduce sounds lower than 300&nbsp;Hz, but a male voice has a fundamental frequency of approximately 150&nbsp;Hz. Because of the missing fundamental effect, the fundamental frequencies of male voices are still perceived as their pitches over the telephone.<ref name="Mather 2006">{{cite book|last=Mather |first=George |title=Foundations of perception |publisher=Taylor & Francis |year=2006 |page=125 |isbn=978-0-86377-835-3 |url=https://books.google.com/books?id=LYA9faq3lt4C&pg=PA125 |accessdate=May 11, 2010 }}</ref>{{Update inline|?=yes|reason=Telephones have changed a lot since 2006, so it should be verified whether this example still holds.|date=July 2020}}

The missing fundamental phenomenon is used electronically by some pro audio manufacturers to allow sound systems to seem to produce notes that are lower in pitch than they are capable of reproducing.<ref>[http://www.wavescaraudio.com/htmls/maxxbass.htm Waves Car Audio. ''MaxxBass Bass Enhancement Technology'']</ref> In a hardware [[effects unit]] or a software plugin, a [[Audio crossover|crossover filter]] is set at a low frequency above which the sound system is capable of safely reproducing tones. Musical signal content above the high-pass part of the crossover filter is sent to the main output which is amplified by the sound system. Low frequency content below the low-pass part of the crossover filter is sent to a circuit where harmonics are synthesized above the low notes. The newly created harmonics are mixed back into the main output to create a perception of the filtered-out low notes.<ref>{{Cite patent |country= US |number= 5930373 |title= Method and system for enhancing quality of sound signal}}</ref> Using a device with this synthetic process can reduce complaints from low frequency noise carrying through walls and it can be employed to reduce low frequency content in loud music that might otherwise vibrate and damage breakable valuables.<ref>{{Cite web |url=http://srforums.prosoundweb.com/index.php/m/335893/72/ |title=ProSoundWeb. LAB: The Classic Live Audio Board. ''Re: maxxbass'' posts by Doug Fowler June 28-29, 2008. |access-date=2008-09-03 |archive-url=https://web.archive.org/web/20110521194507/http://srforums.prosoundweb.com/index.php/m/335893/72/ |archive-date=2011-05-21 |url-status=dead }}</ref>

Some [[pipe organ]]s make use of this phenomenon as a [[resultant tone]], which allows relatively smaller bass pipes to produce very low-pitched sounds.

==Audio processing applications==

This very concept of "missing fundamental" being reproduced based on the overtones in the tone has been used to create the illusion of bass in sound systems that are not capable of such bass. In mid-1999, Meir Shashoua of [[Tel Aviv]], co-founder of [[Waves Audio]], patented an algorithm to create the sense of the missing fundamental by synthesizing higher harmonics.<ref>{{US patent|5930373}}</ref> Waves Audio released the MaxxBass [[Plug-in (computing)|plug-in]] to allow computer users to apply the synthesized harmonics to their audio files. Later, Waves Audio produced small [[subwoofer]]s that relied on the missing fundamental concept to give the illusion of low bass.<ref name="Maximum PC 2004">{{cite journal|last=Norem |first=Josh |date=May 2004 |title=MaxxBass MiniWoofer |journal=Maximum PC |page=78 |issn=1522-4279 |url=https://books.google.com/books?id=cAIAAAAAMBAJ&q=%22missing+fundamental%22&pg=PA78 |accessdate=May 11, 2010 }}</ref> Both products processed certain overtones selectively to help small loudspeakers, ones which could not reproduce low-frequency components, to sound as if they were capable of low bass. Both products included a [[high-pass filter]] which greatly attenuated all the low frequency tones that were expected to be beyond the capabilities of the target sound system.<ref name="Bundschuh 2004">{{cite web |url=http://www.maxx.com/objects/PDF/Loudspeaker_U-MaxxBass_with_High_BL.pdf |title=MaxxBass Applications for Small, Full Range Loudspeakers |last=Bundschuh |first=Paul |date=April 15–17, 2004 |website=Loudspeaker University |publisher=Waves Audio |accessdate=May 11, 2010 |location=Nashua, New Hampshire |archive-date=July 14, 2011 |archive-url=https://web.archive.org/web/20110714055045/http://www.maxx.com/objects/PDF/Loudspeaker_U-MaxxBass_with_High_BL.pdf |url-status=dead }}</ref> One example of a popular song that was recorded with MaxxBass processing is "[[Lady Marmalade]]", the 2001 Grammy award-winning version sung by [[Christina Aguilera]], [[Lil' Kim]], [[Mýa]], and [[Pink (singer)|Pink]], produced by [[Missy Elliott]].<ref name="Bundschuh 2004"/>

Other software and hardware companies have developed their own versions of missing fundamental-based bass augmentation products. The poor bass reproduction of [[Headphones|earbuds]] has been identified as a possible target for such processing.<ref name="Arora 2006">{{cite journal |last=Arora |first=Manish |author2=Seongcheol Jang |author3=Hangil Moon  |date=September 2006 |title=Low Complexity Virtual Bass Enhancement Algorithm For Portable Multimedia Device |journal=AES Conference |url=http://www.aes.org/e-lib/browse.cfm?elib=13868 |accessdate=May 11, 2010 }}</ref> Many computer sound systems are not capable of low bass, and songs offered to consumers via computer have been identified as ones that may benefit from augmented bass harmonics processing.<ref name="Houghton 2007">{{cite journal|last=Houghton |first=Matt |date=April 2007 |title=Better Bass: The Complete Guide To Recording, Mixing & Monitoring The Low End |journal=Sound on Sound |url=http://www.soundonsound.com/sos/apr07/articles/betterbass.htm |accessdate=May 11, 2010 }}</ref>

==See also==
* [[Psychoacoustics]]
* [[Subharmonic]]

==References==
{{Reflist}}

==External links==
* [http://www.ihear.com/Pitch/paradoxical.html Pitch Paradoxical] {{Webarchive|url=https://web.archive.org/web/20030813182102/http://www.ihear.com/Pitch/paradoxical.html |date=2003-08-13 }}
* [http://www.nature.com/neuro/journal/v8/n9/abs/nn1530.html Structural and functional asymmetry of lateral Heschl's gyrus reflects pitch perception preference] – abstract of the Heidelberg research, as published in ''Nature Neuroscience'' 8, 1241–1247 (2005); downloading the full article requires payment
* [https://web.archive.org/web/20131213012614/http://www.hydrogenaudio.org/forums/lofiversion/index.php/t40690.html How do you hear tones?] – discussion forum thread about the Heidelberg research, with a link to a sound file used in the research so that readers can determine whether they are fundamental or overtone hearers

{{Acoustics}}
{{Auditory illusions}}

[[Category:Psychoacoustics]]
[[Category:Waves]]