We have considered the vibration of various types of pinned guitar strings and have investigated the deviation of the partials from integer multiples of the string's fundamental vibration frequency. We measured the inharmonicity parameter B and compared it to a direct calculation based on a model equation. We generally found very good agreement between the two determinations of B for monofilament strings, but perhaps not surprisingly, we find rather poor agreement for wound strings. Furthermore, we show that the methodology used to carry out this experiment can easily serve as the basis for an upper division physics laboratory on physical acoustics including a more thorough investigation of the classical wave equation in a real-world application.
REFERENCES
1.
P.
Perov
,
W.
Johnson
, and
N.
Perova-Mello
, “
The physics of guitar string vibrations
,” Am. J. Phys.
84
, 38
–43
(2016
).2.
N. H.
Fletcher
and
T. D.
Rossing
, The Physics of Musical Instruments
(
Springer-Verlag
,
New York
, 1998
).3.
R. E.
Berg
and
D. G.
Stork
, The Physics of Sound
(
Pearson/Prentice Hall
,
New Jersey
, 2005
), p. 341
.4.
Lord
Rayleigh
, The Theory of Sound
(
MacMillan Publishers
,
London
, 1894
), Vol.
1
, pp. 298
–301
.5.
6.
H.
Fletcher
, “
Normal vibration frequencies of a stiff piano string
,” J. Acoust. Soc. Am.
36
, 203
–209
(1964
).7.
8.
P.
Chumnantas
,
C. A.
Greated
, and
R.
Parks
, “
Inharmonicity of nonuniform overwound strings
,” J. Phys. IV
4
, C5-649
(1994
).9.
A.
Galembo
,
A.
Askenfelt
,
L. L.
Cuddy
, and
F. A.
Russo
, “
Perceptual relevance of inharmonicity and spectral envelope in the piano bass range
,” Acta Acust. United Acust.
90
(3
), 528
–536
(2004
).10.
J. P.
Dalmount
and
S.
Maugeais
, “
Piano strings with reduced inharmonicity
,” Acta Acust. Acust.
105
(1
), 714
–717
(2019
).11.
J. A.
Kemp
, “
On inharmonicity in bass guitar strings with application to tapered and lumped constructions
,” SN Appl. Sci.
2
, 636
–648
(2020
).12.
H.
Järveläinen
and
M.
Karjalainen
, “
Perceptibility of inharmonicity in the acoustic guitar
,” Acta Acust. United Acust.
92
(5
), 842
–847
(2006
).13.
A. J.
Houtsma
, “
Inharmonicity of wound guitar strings
,” J. Acoust. Soc. Am.
71
(S1
), S9
(1982
).14.
A piezo is a device that measures the mechanical stress an object undergoes and converts the amplitude of that stress into an electrical output. See, e.g., https://en.wikipedia.org/wiki/Pickup_(music_technology) and references therein.
15.
An electric guitar pickup works on the principle of magnetic induction. It consists of a small coil of wire wrapped around a small permanent magnet. The permanent magnet magnetizes the steel guitar string. As the string vibrates, it creates a changing magnetic field that is picked up by the wire coil wrapped around the permanent magnet. The strength of the induced voltage in the coil is directly proportional to the string's vibrational motion. See, e.g., https://en.wikipedia.org/wiki/Pickup_(music_technology) and references therein.
16.
N. G.
Horton
and
T. R.
Moore
, “
Modeling the magnetic pickup of an electric guitar
,” Am. J. Phys.
77
, 144
–150
(2009
).17.
In the work of Fletcher on piano strings (Ref. 6), it was not possible to determine the tension in the strings directly. Instead, the tension was inferred via
by measuring its fundamental frequency ( f0) directly, its length L, and calculating
using the density of steel
. Likewise, a direct determination of Y was not carried out, rather (in the case of monofilament strings) the value of Y = 19.5 × 1011 dyne/cm2 (195 GPA) for steel was used. Fletcher does not state where he obtained these particular values for steel or why he chose them.
18.
N.
Lynch-Aird
and
J.
Woodhouse
, “
Mechanical properties of nylon harp strings
,” Materials
10
(5
), 497
–529
(2017
).19.
Scarlett 2i2 audio interface
, <https://focusrite.com/en/usb-audio-interface/scarlett/scarlett-2i2>.20.
REAPER digital audio workstation
, <https://www.reaper.fm/>.21.
Origin-data analysis and graphical software package
, <https://www.originlab.com/>.22.
S. B.
Whitfield
and
K. B.
Flesch
, “
An experimental analysis of a vibrating string using high-speed photography
,” Am. J. Phys.
82
, 102
–109
(2014
).23.
J. C.
Brown
, “
Time dependent behavior of strings using Fourier analysis
,” Am. J. Phys.
54
, 125
–130
(1986
).© 2022 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2022
Author(s)
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
