There have been many recent articles in this journal highlighting simple demonstrations of a wide variety of acoustic phenomena. In introductory physics courses, sound waves and their propagation through air, and resonance in musical instruments, are covered in detail. However, attention is not usually paid to the active role that our ears play in transforming sound waves to create various types of combination tones. Furthermore, many physics textbooks mention that pitch is related to frequency, but do not elaborate on the specifics of the relationship. Teaching the physics behind combination tones allows for an excellent application of physics to biology, and is also an interesting way of exploring the relationship between frequency and pitch. In the first part of the paper, the physics of human hearing and the theory behind combination tones is introduced. In the second part of the paper, demonstrations are outlined and the results are presented.

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,”
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6.
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9.
Supplemental material outlining how the frequencies of piano strings can be used to illustrate the logarithmic relationship between pitch and frequency can be found at TPT Online, https://doi.org/10.1119/1.5098921 , under the Supplemental tab.
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(
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).
11.
Sorge, a German organist, and Tartini, an Italian violinist, have both been noted for their discovery of combination tones
. See
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,
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(
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13.
Helmholtz determined that a series of combination tones can be produced if the auditory system superposes and amplifies the incoming sound in a nonlinear way. See
H.
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,
On the Sensations of Tone
(Ellis translation) (
Dover
,
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14.
Extensive experimental work has been done recently to determine the detailed mechanisms by which the ear produces the distortion products and how the propagation of sound waves correlates with perceived pitch. See
R.
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and
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17.
Sample code and audio files for Demonstrations 1 and 2 are available as a supplement at TPT Online, https://doi.org/10.1119/1.5098921 , under the Supplemental tab.
18.
The frequencies used for each of the notes are for an equal tempered piano and can be found at http://www.sengpielaudio.com/calculator-notenames.htm.
19.
Note that combination tones that are higher in frequency than the stimulus ones are generally inaudible. Combination tones with frequencies below the lowest incoming frequency are the most easily perceived. See Ref. 6.

Supplementary Material

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