Editing Combination tone

{{Short description|Perceived tones when real tones are sounded}}
[[Image:Sum and difference tones A220.png|300px|thumb|Combination tones: Unison, just perfect fifth, and octave are played in top row while A220 is sustained in second row, producing third row sum tones and fourth row difference tones. Frequencies are marked in Hz. Midi example contains all four voices for illustration only. {{audio|Sum and difference tones A220.mid|Play}}]]
[[File:Yankee Doodle combination tones.png|300px|thumb|Difference tones (bottom) between "[[Yankee Doodle]]" in F (top) and a [[drone (music)|drone]] on C (middle)<ref>[[Arthur Benade|Benade, Arthur H.]] (2014). ''Horns, Strings, and Harmony'', p.83. Courier, Dover Books on Music. {{ISBN|9780486173597}}.</ref> {{audio|Yankee Doodle combination tones TOP.mid|Play top}}, {{audio|Yankee Doodle combination tones BOTTOM.mid|drone}}, {{audio|Yankee Doodle combination tones BOTH.mid|both}}, {{audio|Yankee Doodle combination tones DIFFERENCE.mid|difference tones}}, or {{audio|Yankee Doodle combination tones ALL.mid|all three}}]]
{{multiple image
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 | header = Sum and difference of frequencies (left) and sum and difference of two pairs of sine waves (right) with frequencies of 1 and 2 (top) and 1 and 3 (bottom)
 | image1 = Sum and difference of frequencies of sine waves 2 and 3.png
 | caption1 = sum and difference of the frequencies
 | image2 = Sum and difference of sine waves 2 and 3.png
 | caption2 = sum and difference of the sine waves
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A '''combination tone''' (also called '''resultant tone''' or '''subjective tone''')<ref name="Brit">"[http://www.britannica.com/science/combination-tone Combination Tone]", ''Britannica.com''. Accessed September 2015.</ref> is a [[psychoacoustic]] phenomenon of an additional tone or tones that are artificially perceived when two real tones are sounded at the same time. Their discovery is credited to the violinist [[Giuseppe Tartini]],<ref name="treccani enciclopedia">{{cite web|title=Tartini, Giuseppe|url=https://www.treccani.it/enciclopedia/giuseppe-tartini/|publisher=[[Enciclopedia Italiana]]|access-date=1 April 2021}}</ref> so they are also called '''Tartini tones'''.

There are two types of combination tones: '''sum tones''' whose [[frequency|frequencies]] are found by adding the frequencies of the real tones, and '''difference tones''' whose frequencies are the difference between the frequencies of the real tones. "Combination tones are heard when two pure tones (i.e., tones produced by simple harmonic sound waves having no overtones), differing in frequency by about 50 cycles per second [[Hertz|[Hertz]]] or more, sound together at sufficient intensity."<ref name="Brit"/>

Combination tones can also be produced electronically by combining two signals in a circuit that has nonlinear distortion, such as an amplifier subject to [[Clipping (signal processing)|clipping]] or a [[ring modulator]].

==Explanation==
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'', p.81-2. 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]].

==Resultant tone==
A '''resultant tone''' <!--(obsolete)<ref>Randel, Don Michael (2003). ''The Harvard Dictionary of Music'', p.192. Harvard. {{ISBN|9780674011632}}.</ref>{{why?|date=February 2018}}-->is "produced when any two loud and sustained musical sounds are heard at the same time."<ref>Maitland, J. A. Fuller; ed. (1909). ''[https://books.google.com/books?id=8gE9AAAAYAAJ&q=resultant+tone Grove's Dictionary of Music and Musicians]'', Volume 4, p.76. Macmillan. {{pre-ISBN}}.</ref>

In [[pipe organ]]s,<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 = https://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 32' pipe would be costly and take up as much as 16' of vertical space (if capped) or more commonly 17-32' (if open-ended) for each pipe.  Using a resultant tone for such low pitches reduces the cost and space factor, but does not sound as full as a true 32' pipe. The effect can be enhanced by using further [[Organ pipe|ranks]] in the [[Harmonic series (music)|harmonic series]] of the desired resultant tone.

This effect is most often used in the lowest octave of the organ only. It can vary from highly effective to disappointing depending on several factors, primarily the skill of the organ voicer, and the acoustics of the room the instrument is installed in.

It is possible to produce a melody with resultant tones from multiple harmonics played by two or more instruments. There is [https://www.youtube.com/watch?v=k7p5tDekMjY an example with seven saxophones].

==See also==
*[[Power chord#Analysis]]
*[[Ring modulation]]
*[[Beat (acoustics)]]

==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.
* Adrian Wehlte, ''Trios for Two'', Practice book with combination tones for two flutes or two recorders – Explanation and examples, Edition Floeno 2020, ISMN 979-0-9000114-2-8

==External links==
* [http://www.faculty.ucr.edu/~eschwitz/SchwitzPapers/TitchDemo030417.htm Titchener Difference Tones Training]
* [https://web.archive.org/web/20061212152545/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.] {{Webarchive|url=https://web.archive.org/web/20130623210638/http://miracle.otago.ac.nz/postgrads/tartini/index.html |date=2013-06-23 }} 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}}

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[[Category:Acoustics]]
[[Category:Hearing]]