The behavior of a unison pair of piano strings coupled by the soundboard bridge, when one string has localized anisotropy in the reactive part of the bridge admittance for a given partial frequency, can be investigated using a theoretical matrix description. The anisotropy can cause what in piano tuning terminology is referred to as “false beating” in a partial of the single string. A mathematical model can be used to illustrate how “mistunings” between the strings of the unison (measured when the strings are sounding in isolation from each other) may theoretically arise as a consequence of the normal practice in piano tuning, of eliminating or reducing audible beating in the unison when both strings are sounding. “False beats” in a single string partial can be “inherited” by a partial of the coupled unison’s spectrum, and mistunings between the strings can eliminate or reduce the appearance of this inheritance.

Topics
Musical instruments, Mathematical modeling
1.
R. E.
Kirk
, “
Tuning preferences for piano unison groups
,”
J. Acoust. Soc. Am.
31
,
1644
–
1648
(
1959
).
Google Scholar
Crossref
Search ADS
 
2.
G.
Weinreich
, “
Coupled piano strings
,”
J. Acoust. Soc. Am.
62
,
1474
–
1484
(
1977
).
Google Scholar
Crossref
Search ADS
 
3.
See, for example, W. Braid White, Piano Tuning and Allied Arts (1917, 14th reprint 1972), p. 106. The description includes some unsupported assumptions about the physics of falseness.
4.
In older texts this is sometimes called a “false wave.” See, for example, J. Cree Fischer, Piano Tuning/Regulating and Repairing (1907), reprinted as Piano Tuning. A Simple and Accurate Method for Amateurs (Dover, New York, 1975), pp. 160–161.
5.
The component partials of the decay tone (in an approximately harmonic frequency series) can with training be heard in practice, as discrete audible tones.
6.
The idea that false beats can be “neutralized” by tuning the false string “slightly off” from the other strings in the group was suggested by W. Braid White, Piano Tuning and Allied Arts (1917, 14th reprint 1972), p. 106.
7.
T. C.
Hundley
,
H.
Benioff
, and
D. W.
Martin
, “
Factors contributing to the multiple rate of piano tone decay
,”
J. Acoust. Soc. Am.
64
(
5
),
1303
–
1309
(
1978
). See, in particular, p. 1306.
Google Scholar
Crossref
Search ADS
 
8.
G. Weinreich, in Ref. 2, p. 1479.
9.
O. H.
Schuck
and
R. W.
Young
,
J. Acoust. Soc. Am.
15
(
1
),
1
–
11
(
1943
).
Google Scholar
Crossref
Search ADS
 
10.
S. K.
Wolf
and
W.
Sette
,
J. Acoust. Soc. Am.
6
,
160
–
168
(
1935
).
Google Scholar
Crossref
Search ADS
 
11.
“Horizontal” and “vertical” refer, respectively, to the direction parallel and perpendicular to the surface of a grand piano soundboard.
12.
The grand piano arrangement is adopted as a standard reference.
13.
H.
 
Tanaka
,
K.
 
Mizutani
, and
K.
 
Nagai
, “
Experimental analysis of two-dimensional vibration of a piano string measured with an optical device
,”
J. Acoust. Soc. Am.
 
105
(
2
),
1181
(
1999
),
Google Scholar
Crossref
Search ADS
 
H.
Tanaka
,
K.
Mizutani
, and
K.
Nagai
, and “
Two-dimensional motion of a single piano string
,”
J. Acoust. Soc. Am.
,
100
(
4
),
2843
(
1996
).
Google Scholar
Crossref
Search ADS
 
14.
See P. M. Morse, Vibration and Sound (New York, 1948), Sec. III, p. 13.
15.
T. C.
Hundley
,
H.
Benioff
, and
D. W.
Martin
, “
Factors contributing to the multiple rate of piano tone decay
,”
J. Acoust. Soc. Am.
64
(
5
),
1303
–
1309
(
1978
).
Google Scholar
Crossref
Search ADS
 
16.
G. Weinreich, in Ref. 2, p. 1480.
17.
G. Weinreich, in Ref. 2, p. 1480.
18.
G. Weinreich, in Ref. 2, p. 1481.
19.
G. Weinreich, in Ref. 2, p. 1482.
20.
From the soundboard bridge, the string angles downward towards the hitch pin on the iron frame. This angle is the “downbearing angle.” The downbearing angle keeps the string firmly on the bridge surface. The bridge is “crowned” with a two-dimensional arch supporting the total downbearing force of the strings against it. Thus, the entire soundboard bridge and strings system acts as an elastic, acoustical structure.
21.
G. Weinreich, in Ref. 2, pp. 1474–1475.
22.
G. Weinreich, in Ref. 2, p. 1475.
This content is only available via PDF.
© 2004 Acoustical Society of America.
2004
Acoustical Society of America
You do not currently have access to this content.