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{{Merge from |Subharmonic |discuss=Talk:Combination tone#Merge discussion |date=January 2012}}
[[Image:Sum and difference tones A220.png|300px|thumb|Sum and difference tones of A220 and unison, just perfect fifth, and octave. Hz marked. {{audio|Sum and difference tones A220.mid|Play}}]]


A '''combination tone''', also called a '''sum tone''' or a '''difference tone''' (also occasionally [[#Resultant tone|resultant tone]]), can be any of at least three similar [[psychoacoustic]] phenomena. When two tones are played simultaneously, a listener can sometimes perceive an additional tone whose [[frequency]] is a sum or difference of the two frequencies. The discovery of some of these phenomena is credited to the violinist [[Giuseppe Tartini]], and so the tones are also called '''Tartini tones'''.


==Explanation==
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One way a difference tone can be heard is when two tones with fairly complete sets of [[harmonics]] make a [[just fifth]]. This can be explained as an example of the [[missing fundamental]] phenomenon.<ref>Beament, James (2001). ''How We Hear Music'', {{page needed|date=October 2010}}. The Boydell Press. ISBN 0-85115-813-7.</ref> If <math>f</math> is the missing [[fundamental frequency]], then <math>2f</math> would be the frequency of the lower tone, and its harmonics would be <math>4f, 6f, 8f,</math> etc. Since a fifth corresponds to a frequency ratio of 2:3, the higher tone and its harmonics would then be <math>3f, 6f, 9f,</math> etc. When both tones are sounded, there are components with frequencies of <math>2f, 3f, 4f, 6f, 8f, 9f,</math> etc. The missing fundamental is heard because so many of these components refer to it.
 
The specific phenomenon that Tartini discovered was physical. Sum and difference tones are thought to be caused sometimes by the [[non-linearity]] of the [[inner ear]]. This causes [[intermodulation distortion]] of the various frequencies which enter the ear. They are [[linear combination|combined linearly]], generating relatively faint components with frequencies equal to the sums and differences of whole multiples of the original frequencies. Any components which are heard are usually lower, with the most commonly heard frequency being just the difference tone, <math>f_2-f_1</math>, though this may be a consequence of the other phenomena. Although much less common, the following frequencies may also be heard:
:<math>2f_1 - f_2, 3f_1 - 2f_2, \ldots, f_1 - k(f_2 - f_1)</math>
 
For a time it was thought that the inner ear was solely responsible whenever a sum or difference tone was heard. However, experiments show evidence that even when using [[headphones]] providing a single [[pure tone]] to each ear separately, listeners may still hear a difference tone{{Citation needed|date=October 2007}}. Since the peculiar, non-linear physics of the ear doesn't come into play in this case, it is thought that this must be a separate, neural phenomenon. Compare [[binaural beats]].
 
[[Heinz Bohlen]] proposed what is now known as the [[Bohlen–Pierce scale]] on the basis of combination tones,<ref>[[Max V. Mathews]] and [[John R. Pierce]] (1989). "The Bohlen–Pierce Scale", p.167. ''Current Directions in Computer Music Research'', Max V. Mathews and John R. Pierce, eds. MIT Press.</ref> as well as the [[833 cents scale]].
 
{{Wide image|Sum and difference tones - chromatic scale.png|600px|{{audio|Sum and difference tones - chromatic scale.mid|Play}}|image width|center|alt=alt text}}
 
==Resultant tone==
In [[pipe organ]]s, a '''resultant tone''' is the sound of a combination of [[organ pipe]]s that allows the listener to perceive a lower [[Pitch (music)|pitch]].<ref>
{{cite book
| title = A Comprehensive Dictionary of Organ Stops: English and foreign, ancient and modern: practical, theoretical, historical, aesthetic, etymological, phonetic
| edition = 2nd
| author = James Ingall Wedgwood
| publisher = G. Schirmer
| year = 1907
| page = 1
| url = http://books.google.com/books?id=he8PAAAAYAAJ&pg=PA1
}}</ref>
This is done by having two pipes, one pipe of the note being played, and another harmonically related, typically at its [[perfect fifth|fifth]], being sounded at the same time. The result is a pitch at a common [[subharmonic]] of the pitches played (one [[octave]] below the first pitch when the second is the fifth, 3:2, two octaves below when the second is the major third, 5:4). This effect is useful especially in the lowest ranks of the pipe organ where cost or space could prohibit having a rank of such low pitch. For example, a 64' pipe would be costly and take up at least 32' of space (if capped) for each pipe.  Using a resultant tone for such low pitches would eliminate the cost and space factor, but would not sound as full as a true 64' pipe.
 
This effect is most often used in the lowest octave of the organ only.
 
==See also==
*[[Harmonic series (music)]]
*[[Missing fundamental]]
*[[Power chord#Analysis]]
*[[Ring modulation]]
 
==References==
{{reflist}}
 
==Further reading==
* Adrianus J. M. Houtsma, Julius L. Goldstein, [http://dspace.mit.edu/bitstream/handle/1721.1/4259/RLE-TR-484-14236636.pdf?sequence=1 "Percepetion of Musical Intervals: Evidence for the Central Origin of the Pitch of Complex Tones"], Massachusetts Institute of Technology, Research Laboratory of Electronics, Technical Report 484, October 1, 1971.
 
==External links==
* [http://www.faculty.ucr.edu/~eschwitz/SchwitzPapers/TitchDemo030417.htm Titchener Difference Tones Training]
* [http://www.patmissin.com/ffaq/q26.html Difference tones on the harmonica]
* [http://www.lifesci.sussex.ac.uk/home/Chris_Darwin/Perception/Lecture_Notes/Hearing4/hearing4.html Pitch Perception Lecture Notes]
* [http://miracle.otago.ac.nz/postgrads/tartini/index.html Tartini computer program.] Uses combination tones for pitch recognition. If certain intervals are played in double-stop, the program can display its Tartini-tone.
*http://www.organstops.org/r/Resultant.html
 
{{Acoustics}}
 
{{DEFAULTSORT:Combination Tone}}
[[Category:Acoustics]]
[[Category:Hearing]]

Latest revision as of 20:50, 1 January 2015


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