Editing Shepard tone

{{Short description|Auditory illusion}}
{{Use dmy dates|date=October 2020}}
[[File:Shepard Tones spectrum linear scale.png|thumb|right|350px|A [[spectrogram]] of ascending Shepard tones on a linear frequency scale]]

A '''Shepard tone''', named after [[Roger Shepard]], is a sound consisting of a [[superposition principle|superposition]] of [[sine wave]]s separated by [[octave]]s. When played with the bass [[Pitch (music)|pitch]] of the tone moving upward or downward, it is referred to as the ''Shepard scale''. This creates the [[auditory illusion]] of a tone that seems to continually ascend or descend in pitch, yet which ultimately gets no higher or lower.<ref name="Shepard1964" />

== Construction==
[[File:Shepard tone.jpg|thumb|left|150px|Figure 1: Shepard tones forming a Shepard scale, illustrated in a [[music sequencer|sequencer]]]]
Each square in Figure 1 indicates a tone, with any set of squares in vertical alignment together making one Shepard tone. The color of each square indicates the [[loudness]] of the note, with purple being the quietest and green the loudest. Overlapping notes that play at the same time are exactly one octave apart, and each scale fades in and fades out so that hearing the beginning or end of any given scale is impossible. [[File:Shepard A.ogg|thumb|Shepard tone as of the root note A (A<sub>4</sub> = 440 Hz)]]
[[File:Shepard scale diatonic C.ogg|thumb|Shepard scale, [[Diatonic scale|diatonic]] in [[C major|C Major]], repeated 5 times]]
As a conceptual example of an ascending Shepard scale, the first tone could be an almost inaudible C<sub>4</sub> ([[middle C]]) and a loud C<sub>5</sub> (an octave higher). The next would be a slightly louder C{{music|sharp}}<sub>4</sub> and a slightly quieter C{{music|sharp}}<sub>5</sub>; the next would be a still louder D<sub>4</sub> and a still quieter D<sub>5</sub>. The two frequencies would be equally loud at the middle of the octave (F{{music|sharp}}<sub>4</sub> and F{{music|sharp}}<sub>5</sub>), and the twelfth tone would be a loud B<sub>4</sub> and an almost inaudible B<sub>5</sub> with the addition of an almost inaudible B<sub>3</sub>. The thirteenth tone would then be the same as the first, and the cycle could continue indefinitely. (In other words, each tone consists of two sine waves with frequencies separated by octaves; the intensity of each is e.g. a [[raised cosine]] function of its separation in [[semitone]]s from a peak frequency, which in the above example would be B<sub>4</sub>. According to Shepard, "almost any smooth distribution that tapers off to subthreshold levels at low and high frequencies would have done as well as the cosine curve actually employed."<ref name="Shepard1964">{{cite journal | author-link=Roger N. Shepard | first=Roger N. | last=Shepard | title=Circularity in Judgements of Relative Pitch | journal=Journal of the Acoustical Society of America | volume=36 | issue=12 |date=December 1964 | pages=2346–53 | doi=10.1121/1.1919362 | bibcode=1964ASAJ...36.2346S }}</ref>

The theory behind the illusion was demonstrated during an episode of the BBC's show ''[[Bang Goes the Theory]]'', where the effect was described as "a musical [[barberpole illusion|barber's pole]]".<ref>{{Cite episode
 | title          = Clip from Series 4, Episode 6
 | url            = https://www.bbc.co.uk/programmes/p00gfdg1
 | series         = Bang Goes the Theory
 | network        =BBC
 | date           =18 April 2011
 | quote          = It's like a barber's pole of sound.
 | language       =en
}}</ref>

The scale as described, with discrete steps between each tone, is known as the '''discrete Shepard scale'''. The illusion is more convincing if there is a short time between successive notes ([[staccato]] or [[marcato]] rather than [[legato]] or [[portamento]]).{{citation needed|date=December 2019}}

==Variants==
[[File:Shepard Tone spectrum video visualization.webm|thumb|right|Moving audio and video visualization of a rising Shepard–Risset glissando. See and hear the higher tones as they fade out.]]

===Shepard–Risset glissando===
[[Jean-Claude Risset]] subsequently created a version of the scale where the tones glide continuously, and it is appropriately called the '''continuous Risset scale''' or '''Shepard–Risset glissando'''.<ref name="CDM">{{Cite news|url=http://cdm.link/2016/11/jean-claude-risset-reimagined-digital-sound-europe-abroad/|title=Jean-Claude Risset, who reimagined digital synthesis, has died - CDM Create Digital Music|date=2016-11-22|work=CDM Create Digital Music|quote=The sound for which Risset is best known is perhaps the most emblematic of his contributions. Creating a sonic illusion much like M.C. Escher’s optical ones, the Shepherd-Risset glissando / Risset scale, in its present form invented by the French composer, seems to ascend forever.|access-date=2019-12-30|language=en-US}}</ref> When done correctly, the tone appears to rise (or fall) continuously in pitch, yet return to its starting note. Risset has also created a similar effect with rhythm in which tempo seems to increase or decrease endlessly.<ref>{{Cite web|date=12 May 2013|title=Risset rhythm - eternal accelerando|url=http://swiki.hfbk-hamburg.de:8888/MusicTechnology/826}}</ref>
[[File:Risset accelerando beat1 MCLD.ogg|thumb|An example of Risset's accelerating rhythm effect using a [[breakbeat]] loop]]

===Tritone paradox===
{{main|Tritone paradox}}
A sequentially played pair of Shepard tones separated by an [[interval (music)|interval]] of a [[tritone]] (half an octave) produces the [[tritone paradox]]. Shepard had predicted that the two tones would constitute a bistable figure, the auditory equivalent of the [[Necker cube]], that could be heard ascending or descending, but never both at the same time.<ref name="Shepard1964" />
[[File:Paradoxa del tríton.wav|thumb|Sequence of Shepard tones producing the tritone paradox]]
In 1986, [[Diana Deutsch]] discovered that the perception of which tone was higher depended on the absolute frequencies involved and that an individual would usually hear the same pitch as the highest (this is determined by the absolute pitch of the notes).<ref name="Deutsch1986">{{cite journal|last=Deutsch|first=Diana|year=1986|title=A musical paradox|url=http://philomel.com/pdf/MP-1986_3_275-280.pdf|journal=Music Perception|volume=3|issue=3|pages=275–280|doi=10.2307/40285337|jstor=40285337}}</ref> Interestingly, different listeners may perceive the same pattern as being either ascending or descending, depending on the language or dialect of the listener (Deutsch, Henthorn, and Dolson found that native speakers of [[Vietnamese language|Vietnamese]], a [[Tone (linguistics)|tonal]] language, heard the tritone paradox differently from Californians who were native speakers of English).<ref name="Deutsch1992">{{Cite journal 
| last1 = Deutsch | first1 = D. 
| title = Some New Pitch Paradoxes and their Implications 
| doi = 10.1098/rstb.1992.0073 
| journal = Philosophical Transactions of the Royal Society B: Biological Sciences 
| volume = 336 
| issue = 1278 
| pages = 391–397 
| year = 1992 
| pmid = 1354379 
| bibcode = 1992RSPTB.336..391D 
}}</ref><ref>{{Cite journal|last1=Deutsch|first1=Diana|last2=Henthorn|first2=Trevor|last3=Dolson|first3=Mark|date=2004|title=Speech Patterns Heard Early in Life Influence Later Perception of the Tritone Paradox|url=http://dx.doi.org/10.1525/mp.2004.21.3.357|journal=Music Perception|volume=21|issue=3|pages=357–372|doi=10.1525/mp.2004.21.3.357|issn=0730-7829|url-access=subscription}}</ref>

=== Perpetual melody ===
Pedro Patricio observed in 2012 that, by using a Shepard tone as a sound source and applying it to a melody, he could reproduce the illusion of a continuously ascending or descending movement characteristic of the Shepard Scale. Regardless of the tempo and the [[Envelope (music)|envelope]] of the notes, the auditory illusion is effectively maintained. The uncertainty of the scale the Shepard tones pertain allows composers to experiment with deceiving and disconcerting melodies.<ref>Patricio, Pedro. [https://www.researchgate.net/publication/279236493_From_the_Shepard_tone_to_the_perpetual_melody_auditory_illusion From the Shepard tone to the perpetual melody auditory illusion]. Proceedings of the 9th Sound and Music Computing Conference, SMC 2012. 5-10, 2012.</ref>[[File:Melodia Perpètua.wav|thumb|An example of an ascendent perpetual melody|left]]

{{clear}}

== Examples ==
* [[James Tenney]]'s ''[[For Ann (rising)]]'' consists entirely of a Shepard tone [[glissando]] with gradual modulations.
* A section near the end of [[Karlheinz Stockhausen]]'s ''[[Hymnen]]'' incorporates multiple descending Shepard tone glissandos.<ref>{{cite journal
| last1      = Deutsch
| first1     = Diana
| date       = 2010
| title      = The Paradox of Pitch Circularity
| url        = https://acousticstoday.org/wp-content/uploads/2017/09/Article_1of3_from_ATCODK_6_3.pdf
| journal    = Acoustics Today| volume = 6
| issue = 3
| pages = 8–14
| doi = 10.1121/1.3488670
}}</ref>
* The ending of [[The Beatles]]' "[[I Am the Walrus]]" incorporates a Shepard tone with a chord progression built on ascending and descending lines in the bass and strings that line up to create the auditory illusion.<ref>{{cite web|last=Pollack|first=Alan W.|title=Notes on "I Am The Walrus"|url=https://www.icce.rug.nl/~soundscapes/DATABASES/AWP/iatw.shtml|website=soundscapes.info}}</ref>
* The ending of [[Pink Floyd]]'s "[[Echoes (Pink Floyd song)|Echoes]]" from their 1971 album ''[[Meddle]]'' features an ascending Shepard tone created using a feedback technique involving two tape recorders sharing a single tape, with one set to play and the other to record.<ref>{{cite book|last=Blake|first=Mark|author-link=Mark Blake (writer)|title=Pigs Might Fly: The Inside Story of Pink Floyd|url=https://books.google.com/books?id=fC_BAgAAQBAJ|publisher=Arum Press|year=2011|orig-year=2007|isbn=978-1-781-31519-4|access-date=18 November 2021|archive-date=21 May 2021|archive-url=https://web.archive.org/web/20210521212507/https://books.google.com/books?id=fC_BAgAAQBAJ|url-status=live}}</ref>
* [[Queen (band)|Queen]]'s 1976 album ''[[A Day at the Races (album)|A Day at the Races]]'' opens and closes with a Shepard tone.<ref>{{cite book|title=The Nolan Variations: The Movies, Mysteries, and Marvels of Christopher Nolan|page=172|last=Shone |first=Tom |date=2020 |publisher=Knopf Doubleday|isbn=9780525655329}}</ref>
* In his 1979 book ''[[Gödel, Escher, Bach|Gödel, Escher, Bach: An Eternal Golden Braid]]'', [[Douglas Hofstadter]] explained how Shepard scales could be used on the ''Canon a 2, per tonos'' in [[Johann Sebastian Bach|Bach's]] ''[[Musical Offering]]'' (called the ''Endlessly Rising Canon'' by Hofstadter<ref name="GEB">{{cite book |title=[[Gödel, Escher, Bach: An Eternal Golden Braid]] |last=Hofstadter |first=Douglas |author-link=Douglas Hofstadter |year=1980 |publisher=Penguin Books |isbn=0-14-005579-7 |edition=1st}}</ref>{{rp|10}}) for making the [[modulation (music)|modulation]] end in the same pitch instead of an octave higher.<ref name="GEB" />{{rp|717–719}}
* On [[Yellow Magic Orchestra]]'s 1981 [[Electronic music|electronic]] album ''[[BGM (album)|BGM]]'', the [[Ambient music|ambient]] track "Loom" has "a patiently ascending, two-minute-long" Shepard tone according to ''[[Pitchfork (website)|Pitchfork]]''.<ref name="Yoo2">{{cite web |last=Yoo |first=Noah |author-link=Cafuné |date=7 March 2021 |title=Yellow Magic Orchestra: BGM |url=https://pitchfork.com/reviews/albums/yellow-magic-orchestra-bgm/ |access-date=7 March 2021 |work=[[Pitchfork (website)|Pitchfork]]}}</ref>
* The [[Deep Note]] sound trademark of [[THX]], introduced in 1983, uses a Shepard tone.<ref name="Yoo2" />
* In 1995, Ira Braus argued that the final sequence of [[Franz Liszt]]'s 1885 piano piece ''[[Bagatelle sans tonalité]]'' could be continued to produce a Shepard scale using Hofstadter's technique.<ref>{{cite journal|last=Braus|first=I. |date=1995|title= Retracing one's steps: An overview of pitch circularity and Shepard tones in European music, 1550–1990 |journal= Music Perception|volume= 12 |issue=3 |pages= 323–351|doi=10.2307/40286187 |jstor=40286187 | url=http://www.jstor.com/stable/40286187|url-access=subscription }}</ref>
* In a 1967 [[AT&T]] film by Shepard and E. E. Zajac, a Shepard tone accompanies the ascent of an analogous [[Penrose stair]].<ref>{{cite AV media |first1=Roger N. |last1=Shepard |first2=Edward E. |last2=Zajac |url=http://techchannel.att.com/play-video.cfm/2011/10/10/AT&T-Archives-A-Pair-of-Paradoxes|title=A Pair of Paradoxes |publisher=AT&T Bell Laboratories |year=1967}}</ref>
*In the video game ''[[Super Mario 64]]'' (1996) for the [[Nintendo 64]] console, a piece that plays when the player tries to climb the neverending stairs located in the penultimate room of Peach's Castle incorporates a slightly modified Shepard scale played in the background. This auditory illusion complements the spatial loop effect, seemingly giving the impression that the stairs never end.<ref>{{Cite book|last=Phillips|first=Winifred|url=https://books.google.com/books?id=qebUAgAAQBAJ&pg=PA30|title=A Composer's Guide to Game Music|date=14 February 2014|publisher=MIT Press|isbn=978-0-262-02664-2|language=en}}</ref>
* In [[Godspeed You! Black Emperor]]'s "The Dead Flag Blues" from their 1997 album ''[[F♯ A♯ ∞]]'', a section mainly consisting of [[slide guitar]] is briefly looped into itself to create a downward Shepard tone.
*On their 1998 album ''[[LP5]]'', English electronic duo [[Autechre]] employed a decelerating Risset rhythm for the track "Fold4,Wrap5".
*Austrian composer [[Georg Friedrich Haas]] incorporates Shepard tones at various points in his orchestral piece ''in vain'' (2000/02).<ref>{{Cite journal|last=Hutchinson|first=Mark|title=Stairways in the Dark: Sound, Syntax and the Sublime in Haas's in Vain|date=April 2019|url=https://www.cambridge.org/core/journals/tempo/article/abs/stairways-in-the-dark-sound-syntax-and-the-sublime-in-haass-in-vain/BD3AF1FBE848F2D9F5247AE33C9685C5|journal=Tempo|volume=73|issue=288|pages=7–25|doi=10.1017/S0040298218000943|s2cid=151161376|issn=0040-2982|url-access=subscription}}</ref>
* Christopher Nolan said in an interview that the soundtrack of his 2006 film ''The Prestige'' (composed by David Julyan) explores the potential of Shepard tones as a fundamental basis for compositions.<ref>{{Cite web|last=Guerrasio|first=Jason|title=Christopher Nolan explains the biggest challenges in making his latest movie 'Dunkirk' into an 'intimate epic'|url=https://www.businessinsider.com/christopher-nolan-dunkirk-interview-2017-7|access-date=2020-11-14|website=Business Insider}}</ref> This is fully realised in his 2017 film ''[[Dunkirk (2017 film)|Dunkirk]] where'' a Shepard tone is used to create the illusion of an ever increasing moment of intensity across intertwined storylines.<ref>{{cite web |last=Haubursin |first=Christopher |date=26 July 2017 |title=The sound illusion that makes Dunkirk so intense |url=https://www.vox.com/videos/2017/7/26/16033868/dunkirk-soundtrack-shepard-tone |website=Vox}}</ref>
* In [[Stephin Merritt]]'s 2007 song "Man of a Million Faces", composed for [[NPR]]'s "Project Song", the Shepard tone is a key aspect.<ref>{{cite AV media |date= 4 November 2007 |title= Stephin Merritt: Two Days, 'A Million Faces' |medium=video |url=https://www.npr.org/2007/11/04/15859351/stephin-merritt-two-days-a-million-faces |access-date=9 October 2015 |publisher=NPR |quote='It turns out I was thinking about a Shepard tone, the illusion of ever-ascending pitches.'}}</ref>
* In the 2008 film ''[[The Dark Knight]]'' and its 2012 followup ''[[The Dark Knight Rises]]'', a Shepard tone was used to create the sound of the Batpod, a motorcycle that the filmmakers did not want to change gear and tone abruptly but to accelerate constantly.<ref>{{cite news |title='The Dark Knight' sound effects |first=Richard |last=King |url=https://www.latimes.com/archives/la-xpm-2009-feb-04-en-lightsknight4-story.html |newspaper=Los Angeles Times |date=4 February 2009 }}</ref>
* The 2009 [[progressive house]] song "[[Leave the World Behind (song)|Leave the World Behind]]" by [[Swedish House Mafia]] features a Shepard tone in the form of an ongoing "riser" to build up the tension throughout the track.<ref>{{cite AV media|url=https://www.youtube.com/watch?v=xG4AhAnyqvE?t=1m14s|title=Axwell, Ingrosso, Angello, Laidback Luke ft. Deborah Cox - Leave The World Behind (Original)|via=YouTube}}</ref>
* The [[non-Euclidean]] video game [[HyperRogue]] uses a Shepard tone in its music for the land "R'Lyeh" and its subland "Temple of Cthulhu". Since the latter is an infinite sequence of concentric [[horocycle]]s, the music conveys the feeling of the player continually descending, but never getting any closer to the center.
*In [[Lucrecia Martel]]'s feature film ''Zama'' (2017), there is extensive use of the Shepard tone creating a "loud and shreechy soundscape, in order to achieve closeness to the viewer", according to the director.<ref>{{Cite magazine|last1=Gemünden|first1=Gerd|last2=Spitta|first2=Silvia|date=2018-06-01|title='I Was Never Afraid': An Interview with Lucrecia Martel|url=https://online.ucpress.edu/fq/article-abstract/71/4/33/42163/I-Was-Never-Afraid-An-Interview-with-Lucrecia?redirectedFrom=fulltext|magazine=Film Quarterly|language=en|volume=71|issue=4|pages=33–40|doi=10.1525/fq.2018.71.4.33|issn=0015-1386}}</ref>
* The 2018 track "[[Always Ascending (song)|Always Ascending]]" by [[Franz Ferdinand (band)|Franz Ferdinand]] from the album of the same name features a rising Shepard tone throughout the song. The video for the song echoes the effect, with the camera apparently rising continually throughout.<ref>{{cite web|url=https://www.telegraph.co.uk/music/what-to-listen-to/franz-ferdinand-still-operating-elevated-plateau-always-ascending/|title=Franz Ferdinand are still operating on an elevated plateau – Always Ascending, review |last=McCormick|first=Neil|website=The Telegraph|date=9 February 2018}}</ref>
*In [[Sumio Kobayashi]]'s piano work "Unreal Rain", the Shepard tone is entirely used.<ref>{{Citation|title=Sumio Kobayashi "Unreal Rain" (Japan)|url=https://www.youtube.com/watch?v=wXjiC8IRC3o| archive-url=https://ghostarchive.org/varchive/youtube/20211211/wXjiC8IRC3o| archive-date=2021-12-11 | url-status=live|language=en|access-date=2021-10-15}}{{cbignore}}</ref>{{clarify|date=November 2023}}
*In the song "[[Fear_Inoculum_(song)|Fear Inoculum]]", Tool drummer [[Danny Carey]] introduces the track with the Shepard tone.
*The track "Neuron Activator" from the [[Cruelty Squad]] soundtrack uses a constantly repeating Shepard tone, in line with the intentionally crude and semi-[[Dadaist]] nature of the game's soundtrack.

==See also==
{{Portal|Psychology|Physics|Music}}
* [[Chorus (audio effect)]]
* [[Deep Note]]
* [[Flanging]]
* [[Interference (wave propagation)]]
* [[Phaser (effect)]]
* [[Pitch circularity]]
* [[Sound effect]]
* [[Strange loop]]

== References ==
{{Reflist|30em}}
<!-- Dead note "<name>": {{cite journal | first=Risset | last=J. C. | title=Pitch and rhythm paradoxes: comments on "Auditory paradox based on fractal waveform" | journal=Journal of the Acoustical Society of America | volume=80 | issue=3 | year=1986 | pages=961–962 | pmid=3760341 | doi = 10.1121/1.393919}} -->

== External links ==
{{Commons category|Shepard tone}}
* [http://www.bbc.co.uk/programmes/p00gfdg1 BBC science show, Bang Goes the Theory, explains the Shepard Tone]
* [http://www.exploratorium.edu/exhibits/highest_note/ex.about.fr.html Demonstration of discrete Shepard tone (requires Macromedia Shockwave)]
* [http://www.netalive.org/tinkering/shepard-effect/ Visualization of the Shepard Effect using Java]
* [https://www.youtube.com/watch?v=PCs1lckF5vI A demonstration of a rising Shepard Scale as a ball bounces endlessly up a Penrose staircase] ([https://www.youtube.com/watch?v=boJD_gTLavA and down])
* [http://codepen.io/yukulele/pen/PPgxbG Shepard tone Keyboard] on [[CodePen]]
*[https://www.cs.ubc.ca/labs/imager/contributions/flinn/Illusions/TT/tt.html Tritone paradox example (requires Java)]
*[https://sites.google.com/site/pp2007pt/music Pedro Patricio's compositions formed around the auditory illusion of the perpetual melody (''Perpetual Melody - contrasting moments'', 1-7)]
*[https://www.youtube.com/watch?v=P8RgeUa7_WI An E5 Shinkansen Bullet Train Departing Tokyo Station plays the sound of a Shepard Tone possibly due to how many cars and systems have to activate]

{{Auditory illusions}}
{{Music psychology}}
{{Music technology}}

{{DEFAULTSORT:Shepard Tone}}
[[Category:Auditory illusions]]