Acoust. Sci. & Tech.

33, 5 (2012) #2012 The Acoustical Society of Japan

PAPER

Humbucking pickup response excited by string vibration

Masahiro Harazono1; , Daichi Kitamura2 and Masashi Nakayama1

1
Department of Electrical and Computer Engineering,
Kagawa National College of Technology,
355 Chokushi-cho, Takamatsu, 761–8058 Japan
2
Graduate School of Information Science, Nara Institute of Science and Technology,
8916–5 Takayama-cho, Ikoma, 630–0192 Japan
( Received 29 June 2011, Accepted for publication 14 March 2012 )

Abstract: As a factor to characterize the sound of an electric guitar, it is thought that a characteristic
of the pickup contributes most. The pickups most often used are classified roughly into single-coil
models and humbucking models. The single-coil pickup is made by winding the thin wires with
several thousand turns of coils around six polarizing pole pieces each corresponding to a string of the
guitar, and the change in the magnetic reluctance owing to the string vibration that causes the change
in the magnetic flux is transformed into an electrical signal. The humbucking pickup is composed of
one magnetic circuit with two single-coil pickups, and made to be in phase electrically and out of
phase magnetically for the purpose of removing circumference magnetic noise. In this paper, the
response of the humbucking pickup excited by a string vibration set up by a real commercial solid
body electric guitar is analyzed, and a simulation result is shown to agree with an actual measured
value with sufficient precision. In addition, the response of the humbucking pickup imitated with two
single-coil pickups is compared with the single-coil pickup and some additional considerations in the
characteristics have been gained through analysis.

Keywords: Electric guitar, Electromagnetic transducer, Humbucking pickup, String vibration,

Negative stiffness

PACS number: 43.38.Dv,43.40.Cw,43.75.Gh, 43.75.Tv [doi:10.1250/ast.33.301]

tone, the characteristic of the pickup that converts the

1. INTRODUCTION string vibration into an electrical signal is thought to have
Approximately 80 years ago, the basic model of the the strongest influence on it.
electric guitar being used at present appeared. Since then, The pickup used for the electric guitar is the electro-
the basic structure of the electric guitar with a pickup magnetic transducer and can be divided roughly into a
in which the pole piece is wound with a coil has not single-coil type or a humbucking type. The former is
changed, and the reduction of noise and the improvement wound with thin coils by several thousand turns around all
of a peripheral device have been treated [1]. For the pole pieces corresponding to the six strings. The latter is
electric guitar with a solid body, the characteristics are made in one magnetic circuit with two single-coil pickups
summarized as follows. The decrement of the string and made to be in phase electrically by connecting the end
vibration is relatively small because the body is made of of the two coils and out of phase magnetically for the
the solid wood. Also, the volume can be adjusted purpose of removing circumference magnetism noise [2].
arbitrarily using an amplifier because the string vibration Generally, it is said that the single-coil pickup is
is changed into an electrical signal by the pickup. In susceptible to noise and feedback, but clean sound is
addition, various sound effects can be added to the provided, and that the humbucking pickup has a large
electrical signal by using an effects unit. Although various output with low noise, and a warmer and more mellow tone
components composing the electric guitar, including the is produced because the middle frequency range compo-
strings, are considered to be factors that characterize the nents are relatively rich.
It has also been sufficiently acknowledged by musi-

e-mail: harazono@t.kagawa-nct.ac.jp cians and guitar lovers that Stratocaster of Fender Com-

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 Acoust. Sci. & Tech. 33, 5 (2012)

String
N S

Terminal

S Magnet N
Fig. 1 Electric guitar with humbucking pickup.

pany, in which a single-coil pickup is used, and Les Paul of Pole piece
Gibson Corporation, in which the humbucking pickup is
used, are representative guitars [3]. It is an established fact
that these electric guitars had a great influence on guitar No. 3
makers as the standard models up to now. However, there String
are few academic studies related to the electric guitar, and
also, the development is carried out mainly through trial
and error based on experience. Therefore, it is thought that
the analysis of the response of the humbucking pickup
acoustically and the comparison with that of the single-coil
pickup are useful for designing future electric guitars.
In this paper, the response of the humbucking pickup
mounted on a real commercial solid-body electric guitar Fig. 2 Diagram of humbucking pickup.
and excited by string vibration is analyzed. It is shown that
a simulation result agrees with the actual measured value
with sufficient precision, and the characteristic of the
imitated humbucking pickup constructed of two single-coil h
pickups is investigated by comparison with the response of
one of two single-coil pickups. l/8 x=l
a1
2. THEORETICAL ANALYSIS x = 0 a2

2.1. Analysis Model Fig. 3 String vibration model.

The commercial electric guitar used for the analysis in
this study is shown in Fig. 1. It is a copy product of the Les
Paul model of Gibson Corporation, and the humbucking
pickup is mounted on the front and the rear part. However, The analysis model is shown in Fig. 3. Let l be the
the rear pickup is excluded as the analysis is intended for length between a bridge and the nut and x ¼ a1 ; a2 be the
only front pickup. positions of the two pole pieces of the humbucking pickup
Figure 2 shows the electrical and magnetic principle of with the origin set at the bridge. Also, it is supposed that
the humbucking pickup and the position relation with the the initial condition is set at the height h at the position
string. Two single-coil pickups are located very close to x ¼ l=8.
each other, and a magnetic circuit is composed through the
string by setting the magnet at the bottom center of both 2.2. Vibration Equation
pickups [2]. Therefore, the magnetic flux directions of the When the electromagnetic pickup is used to convert
two pickups against the string are opposite to each other. the string vibration into an electrical signal, it has been
Then, by connecting the end terminals of the two coils, that analyzed in detail that the attraction acts on the string
are wound in same direction, and setting the start terminals vibration via the polarizing magnet of the pickup, and the
to the output terminal, the electrical outputs of the two negative stiffness component reduces the vibration fre-
pickups become in phase. Thus, when the outside magnetic quency of the string depending on the strength and the
flux noise reaches the two pole pieces in phase, the attracting position [4]. It is indicated that the proper
electromotive forces are out of phase and will be canceled simulation result of the string vibration can be provided by
out. linearizing the measured attraction to the static force and

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 M. HARAZONO et al.: HUMBUCKING PICKUP RESPONSE EXCITED BY STRING

the negative stiffness component. Also, when the plural c p

Yðx; pÞ ¼ Yx ð0; pÞ sinh
negative stiffness existed discretely, it was shown that the p c
inharmonicity expressing a nonharmonic property is almost Z x
1 p
the same as the total value of negative stiffnesses that each  gðÞ sinh ðx  Þd
exist independently [5]. Although the flux at two magnetic c 0 c
poles of the humbucking pickup are antiphase, the two j1 
X 
ri 1 p
attractions of the magnet can be divided approximately into þ  i Yðai ; pÞ sinh ðx  ai Þ; ð4Þ
i¼0
p pc c
a static component F0 and the negative stiffness component
Sn yða; tÞ, which is in inverse proportion to the string where a0 ¼ 0, a3 ¼ l, and r0 ¼ 0 ¼ 0. Setting j ¼ 3 and
displacement, and the vibration equation of the model in x ¼ l, and the boundary condition Yðl; pÞ ¼ 0, Yx ð0; pÞ is
Fig. 3 is presented as obtained as
@2 yðx; tÞ @2 yðx; tÞ Zl
p p
 ¼ T Yx ð0; pÞ ¼ gðÞ sinh ðl  Þd
@t2 @x2 p 0 c
X2 ð1Þ c2 sinh l
 ðx  ai ÞfF0i  sni yðai ; tÞg; c
i¼1 p
where  is a linear density and T is the tension of the 2 
X  sinh ðl  ai Þ
ri c
string.   i Yðai ; pÞ : ð5Þ
i¼0
p 2
p
According to the analysis results [4], the static c sinh l
c
displacement affected by the static force is very small
because the attraction is weak compared with the string Substituting this into Eq. (4) with condition i ¼ j, the
tension. However, the negative stiffness reduces the following equation for Yðx; pÞ is given:
vibration frequencies and, as a result, causes beats due to p
the interference between the partial tones whose ampli- sinh ðl  xÞ Z x
c p
tudes are comparatively large. Therefore, the vibration Yðx; pÞ ¼ gðÞ sinh d
p 0 c
waveform and the amplitude change with time. Thus the c sinh l
c
vibration frequencies of the string with the humbucking
pickup will be examined first. p
sinh x Z l
c p
2.3. Characteristic Equation þ gðÞ sinh ðl  Þd
p x c
Equation (1) can be solved using the Laplace trans- c sinh l
c
form. Let Lfyðx; tÞg ¼ Yðx; pÞ and LfYðx; pÞg ¼ Yð; pÞ
represent the Laplace transform for time variance t and p
sinh ðl  xÞ
position variance x, respectively. Setting the initial con- c

ditions yðx; 0Þ ¼ gðxÞ and y0 ðx; 0Þ ¼ 0, the Laplace trans- p
form of Eq. (1) for t can be obtained as pc sinh l
c
p2 Yðx; pÞ  pgðxÞ ¼ c2 Yxx ðx; pÞ  
X
j1
ri p
X
2     i Yðai ; pÞ sinh ai
ri p c
 ðx  ai Þ  i Yðai ; pÞ ; ð2Þ i¼0
i¼1
p
2
p
where c ¼ T=, ri ¼ F0i =, and i ¼ sni =. sinh x
c
Next, setting LfgðxÞg ¼ GðÞ, carrying the Laplace 
transform for x and setting the boundary condition p
pc sinh l
yð0; tÞ ¼ Yðl; tÞ ¼ 0, the following equation is obtained: c
 2  
1 p X ri p
Yð; pÞ ¼ 2
Yx ð0; pÞ  2 GðÞ   i Yðai ; pÞ sinh ðl  ai Þ: ð6Þ
p c
2  2 i¼j p c
c
2    Namely, the solution is provided separately at three
1X ri sections: 0  x < a1 , a1  x < a2 , and a2  x  l. Because
þ 2  i Yðai ; pÞ eai  : ð3Þ
c i¼1 p Yðx; pÞ is equal at x ¼ a1 in the regions 0  x < a1 and
Then applying the Laplace inverse transform, the next a1  x < a2 , and at x ¼ a2 in the regions a1  x < a2 and
result is provided in the region aj1  x < aj ( j ¼ 1  3): a2  x  l, Yða1 ; pÞ, Yða2 ; pÞ are then provided:

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 Acoust. Sci. & Tech. 33, 5 (2012)

 9 2.0
1 >
>
Yða1 ; pÞ ¼ pD1 ðpÞH2 ðpÞ >
>
WðpÞ >

Inharmonicity [cent]
>
 >
> 1.5
R1 ðpÞH2 ðpÞ R2 ðpÞ >
>
> Humbucking pickup
 þ 2 pD2 ðpÞ  >
>
p p >
>
 >
> 1.0
>
> Single-coil pickup at a2
p p >
>
 sinh a1 sinh ðl  a2 Þ >
=
c c
 ; ð7Þ 0.5
1 >
> Single-coil pickup at a1
Yða2 ; pÞ ¼ pD2 ðpÞH1 ðpÞ >
>
WðpÞ >
>
 >
>
> 0.0
R2 ðpÞH1 ðpÞ R1 ðpÞ >
>
> 5 10 15 20
 þ 1 pD1 ðpÞ  >
>
p p >
> Partial No.
 >
>
p p >
>
>
>
 sinh a1 sinh ðl  a2 Þ ; Fig. 4 Inharmonicity of a string affected by humbucking pickup.
c c

where
frequency can be shown as !n ¼ 2 fn . The string used here
WðpÞ ¼ H1 ðpÞH2 ðpÞ
is the 3rd string of the G3 tone with a basic frequency of
p p
 1 2 sinh2 a1 sinh2 ðl  a2 Þ: ð8Þ 195.997718 Hz. The data is given as follows
c c
String length l ¼ 0:6305 [m]
For i ¼ 1 and 2, Hi ðpÞ, Di ðpÞ, and Ri ðpÞ are represented as
Linear Density  ¼ 0:9721  103 [kg/m]
p
Hi ðpÞ ¼ pc sinh l Tension T ¼ 59:36 [N/m]
c
p p Initial height at l=8 h ¼ 1  103 [m]
 i sinh ðl  ai Þ sinh ai ð9Þ
c c
Z ai Positions of pickup a1 ¼ 0:1365 [m]
p p
Di ðpÞ ¼ sinh ðl  ai Þ gðÞ sinh d
c 0 c a2 ¼ 0:1535 [m]
Zl
p p Negative stiffness sn1 ¼ 0:8 [N/m]
þ sinh ai gðÞ sinh ðl  Þd ð10Þ
c ai c sn2 ¼ 1:3 [N/m]:
p p Here, the ratio of two negative stiffnesses are determined
Ri ðpÞ ¼ r1 sinh a1 sinh ðl  ai Þ
c c from the measured flux densities at the pole piece of two
p p pickups, because the attraction is proportional to the square
 r2 sinh ðl  a2 Þ sinh ai : ð11Þ
c c of the magnetic flux, and both negative stiffnesses are
Substituting Yða1 ; pÞ and Yða2 ; pÞ into Eq. (6), the final estimated by referring to the calculated nonharmonic
image equation can be obtained. frequencies and measured beat frequency.
Namely, Eq. (8) is the characteristic equation with
the humbucking pickup, and also, H1 ðpÞ and H2 ðpÞ are 3.2. Vibration Displacement
characteristic equations respectively when the two single- The influence of the negative stiffness on the amplitude
coil pickup are each set up alone. of the string is very small [4]. Because it can be thought
that the influence of damping on the vibration frequency is
3. ANALYSIS AND MEASUREMENT also small, the vibration displacement of the string can be
RESULTS assumed to be
3.1. Numerical Data and Inharmonicity X
1
n
The inharmonicity n is defined as yðx; tÞ ¼ An en t sin x  cos !n t; ð13Þ
n¼1
l
fn
n ¼ 1200 log2 [cent]; ð12Þ where An is the amplitude of the nth partial tone when the
n f1
initial displacement of the string is gðxÞ, and is approx-
where f1 is the fundamental frequency and fn is the nth imately given as follows [6]:
partial tone frequency. The inharmonicity of the vibration Z
2 l 2n f1
frequencies calculated by setting Eq. (8) to zero is An ¼ gðxÞ sin dx: ð14Þ
l 0 c
indicated in Fig. 4. Because the solution of the character-
istic equation becomes a purely imaginary number, by In addition, n is the damping coefficient and shall be
setting p ¼ j!, the relation with the nth partial tone determined using experimentally obtained values as

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 M. HARAZONO et al.: HUMBUCKING PICKUP RESPONSE EXCITED BY STRING

response (0–4.05 s), the 50 ms-long waveform for every 1 s

n ¼ 0 þ s !n þ v !2n ; ð15Þ
is shown, which approximately corresponds to ten periods.
where 0 ¼ 0:6295, s ¼ 0:296  105 , and  ¼ 2:585  The low-pass filter that cuts off the frequency of 6.25 kHz
108 . However, it was assumed that 2 ¼ 0:35889, because has been applied because there was high-frequency noise in
it greatly deviated from the Eq. (15) near the 2nd partial the measuring signal. It is confirmed that, in addition to the
tone. In addition, the vibration frequency generally changes decrease of amplitude due to damping, the waveforms vary
with damping, but such change shall be ignored because with time because of the existing inharmonicity indicated
the amount of change here is extremely small. in Fig. 4.
The data was saved by confirming that the same wave
3.3. Electromotive Force of Pickup pattern was provided through several trials because the
The response of the single-coil pickup excited by a initial condition was given by picking the string up as
string is analyzed in detail as a case of the effect of the height h at x ¼ l=8 in two fingers. Although the initial
single attraction described above [6,7]. Here, for each condition has an angle with prominent sharpness at x ¼ l=8
single-coil pickup constructing the humbucking pickup, theoretically, it is confirmed that roundness appears to the
assuming the magnetomotive force Ui (i ¼ 1; 2), the wave pattern in the first stage, because the realization is
compound reluctance Ri , the turn number of the coil Ni , difficult and the high-frequency components are lacked.
the magnetic flux
i , the current of the coil Ii , the voltage However, subsequently, it is recognized that the simulation
added to the coil Ei and the load impedance Zi , the results are in good agreement with the measured results.
fundamental equation of each single-coil pickup is repre- Figure 6 shows the envelopes of the two results by
sented as narrowing a time scale from 0 to 4 s. The beats of a long
9 period are generated by the interference between the par-
Ui ¼ Ri
i þ Ni Ii =
tial tones because the inharmonicity occurs under the
d
i : ð16Þ
Ei ¼ Zi Ii  Ni ; influence of negative stiffness; it is recognized that the
dt calculation results are in good agreement with the measure-
Also, assuming that the voltage source and the load are ment results.
not connected, and that only electromotive force will be
treated, only the first term shall be a target. The reluctance
4. CHARACTERISTICS OF HUMBUCKING
is divided into the static component and the dynamical
PICKUP
component, that is, Ri ¼ RSi þ i yðxi ; tÞ [6]. Thus, because 4.1. Response of Imitated Humbucking Pickup
of
i ¼ Ui =Ri , the electromotive force Emi can be obtained In this section, the characteristic of the response of the
as imitated humbucking pickup, constructed of two single-
d
i Ni U i  @yðx; tÞ coil pickups on the guitar mentioned in the previous
Emi ¼ Ni ¼ 2
 : ð17Þ section, is compared with the one of two single-coil
dt fRSi þ yðx; tÞg @t
pickups to clarify the property of the humbucking pickup.
Then the output of the humbucking pickup is given as the The imitated humbucking pickup is shown in Fig. 7.
sum of the two single-coil pickup outputs that calculated The setting condition is the same as for the original
from Eq. (17) for i ¼ 1; 2. humbucking type described in chapter 3, and the single-
Because the electric guitar treated here is a product coil pickup is set at midway between the two pickups, that
already on the market, the number of turns of the coil as is x ¼ 0:145. The negative stiffnesses of both of the two
well as the magneto-motive force of the humbucking single-coil pickups used here are 2.1 N/m.
pickup are unknown, and it also is difficult to establish the The vibration frequencies and the inharmonicity of
value of the reluctance. Then, those values are obtained by both single-coil and imitated humbucking pickups are
reverse calculation using the simulation result as follows: shown in Table 1 and Fig. 8, respectively. It is confirmed
N1 ¼ N2 ¼ 8000 that inharmonicity of the humbucking pickup becomes
U1 ¼ U2 ¼ 355:442 [A] approximately 2 times that of the single-coil pickup, as
described above.
RS1 ¼ RS2 ¼ 1:72903  1010 [A/Wb]
The measured responses of the two types of pickup are
1 ¼ 2 ¼ 1:97752  1012 [A/Wb/m]: shown in Fig. 9. The amplitude ratio of both pickups is
about 1:2, and the beats occur strongly in the humbucking
3.4. Comparisons with Experimental Results pickup compared with the single-coil pickup. It can be
On the basis of the above-mentioned analysis, the explained that strong beats are generated by the interfer-
calculated outputs of the humbucking pickup are shown ence between the low partial tones because the partial tone
in Fig. 5 with the measured result. To confirm the long frequencies of the humbucking pickup having two negative

305
 Acoust. Sci. & Tech. 33, 5 (2012)

0.05 0.05

0.00 0.00

-0.05 -0.05

0.00 0.01 0.02 0.03 0.04 0.05 0.00 0.01 0.02 0.03 0.04 0.05
Time [s] Time [s]
0.05
Response [V]

0.05

0.00 0.00

-0.05
-0.05
1.00 1.01 1.02 1.03 1.04 1.05 1.00 1.01 1.02 1.03 1.04 1.05
Time [s] Time [s]
0.03 0.03
0.00 0.00
-0.03 -0.03
2.00 2.01 2.02 2.03 2.04 2.05 2.00 2.01 2.02 2.03 2.04 2.05
Time [s] Time [s]
0.02 0.02
0.00 0.00
-0.02 -0.02
3.00 3.01 3.02 3.03 3.04 3.05 3.00 3.01 3.02 3.03 3.04 3.05
Time [s] Time [s]
(a) Measured (b) Calculated

Fig. 5 Calculated and measured humbucking pickup responses excited by string vibration.

0.10 0.10

0.05 0.05
Response [V]

0.00 0.00

-0.05 -0.05

-0.10 -0.10
0 1 2 3 4 0 1 2 3 4
Time [s] Time [s]
(a) Measured (b) Calculated

Fig. 6 Calculated and measured humbucking pickup responses from 0 to 4 s.

stiffnesses are reduced much more than that of the single- coil and humbucking pickups are shown in Fig. 10. The
coil pickup. partial tone amplitudes determined by the initial condition
expressed in Eq. (14) are indicated in Fig. 11. It is
4.2. Partial Tones Constitution of Pickup understood that, when paying attention to the low partial
The spectrograms of the responses of both the single- tones, the output of the second partial tone is the largest,

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 M. HARAZONO et al.: HUMBUCKING PICKUP RESPONSE EXCITED BY STRING

the humbucking pickup, and only the 15th partial tone is

recognized from among higher partial tones than 12th.
For example, in the 13th partial tone, because there is a
nodal point in the center of the humbucking pickup, the
output is not at all seen in the single-coil pickup or in the
humbucking pickup. It has been reported in detail that the
response of the pickup had a nonlinear property of against
the displacement of the string, and that many distortion
components were generated [7]. The 12th partial tone is not
seen in the output, although there is no nodal point. It is
thought that the component has been canceled out with the
Fig. 7 Imitated humbucking pickup.
distortion components regenerated in out of phase by the
low partial tones of relatively large amplitudes. On the
Table 1 Partial frequencies of a string affected by contrary, the 7th partial tone that is originally a small
negative stiffness. displacement is generated for a relatively long time in the
Partial No. Single-coil Humbucking humbucking pickup response. Such differences are why the
sound of the single-coil pickup is clean, or why the sound
1 195.997718 195.997718
2 392.165539 392.340553
of humbucking pickup has warmer and more mellow tone
3 588.474262 588.959882 because the middle frequency range is relatively empha-
4 784.759971 785.526836 sized.
5 980.940317 981.89154 When watching the 5th partial tone of the humbucking
6 1,177.086507 1,178.198492
pickup, it is confirmed that the beats occur because the
7 1,373.285987 1,374.602086
8 1,569.520792 1,571.059840 weak points are seen about every 0.7 s. It is considered
9 1,765.723032 1,767.461602 analytically that the various frequency components, such as
10 1,961.888136 1,963.811866 f2 þ f3 , 2 f1 þ f3 , and 3 f1 þ f2 , are generated as intermo-
dulation distortion of the low partial tones with compara-
tively large amplitudes. They are slightly different from
4.0 the value of f5 , and as a result, they generate the beats as
3.5 interference between the partial tone. The values of f5 
Inharmonicity [cent]

3.0 2 f1  f3 and f5  3 f1  f2 become 1.25 and 1.597 Hz

Imitated humbucking pickup
by calculating each frequency using Eq. (8), and it is
2.5
presumed that the beats seen in measurement results
2.0
occurred because plural beats acted at the same time.
1.5
1.0
Single-coil pickup 4.3. Examination of SNR of Pickup
0.5
Figure 12 illustrates the signal-noise ratios of both
0.0 single-coil and imitated humbucking pickups, that are
5 10 15 20
Partial No. calculated from the RMS value for 0.05 s at every moment,
where the signal and the noise are the low-pass-filtered and
Fig. 8 Inharmonicity of a string affected by imitated high-pass-filtered pickup responses with the cut-off fre-
humbucking pickup and single-coil pickup. quency 6.25 kHz, respectively. Although, the difference of
the SNR at the early parts between the single-coil pickup
and the humbucking pickup is only 2.8 dB because the
because the loop position is consistent with the positions numerous highly partial tones exist in the single-coil
of both pickup types, and the output of the pickup is pickup, and subsequently, the SNR of the humbucking
proportional to the velocity indicated in Eq. (17). In pickup becomes greater than that of the single-coil pickup
addition, for the humbucking pickup, the output shows a nearly 5 dB.
large value at from the 1st to 3rd partial tones because the
responses of both pickups are added. Also, when paying
5. CONCLUSIONS
attention to the partial tone constitution for approximately The responses of the humbucking pickup mounted on a
1 s in the initial stage, the difference from the case of the commercial electric guitar were analyzed by linearizing the
single-coil is marked because some partial tones shall be attraction to the static component and the negative stiffness
canceled out, being in inverse phase at two pole piece of component. It has been shown that the analytical solution

307
 Acoust. Sci. & Tech. 33, 5 (2012)

0.3
0.1 0.2

Response [V]

Response [V]
0.1
0.0 0.0
-0.1
-0.1 -0.2
-0.3
0 1 2 3 4 0 1 2 3 4
Time [s] Time [s]
(a) Single-coil pickup (b) Imitated humbucking pickup

Fig. 9 Envelopes of pickup responses from 0 to 4 s.

4 4

Frequency [kHz]
Frequency [kHz]

3 3

2 2

1 1

0 0
0 1 2 3 4 0 1 2 3 4
Time [s] Time [s]
(a) Single-coil pickup (b) Imitated humbucking pickup

Fig. 10 Spectrograms of pickup responses from 0 to 4 s.

1.0 80
0.8 Partial No.1

60 Imitated humbucking
Amplitude [mm]

0.6
SNR (dB)

0.4 3 2
4 6 5 40
0.2 7

0.0
Single-coil
Pole piece 20
-0.2 position a a 1 2

-0.4
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0
Position [m] 0 1 2 3 4
(a) Amplitudes of partial tones from 1st to 7th Time [s]
20
10 12 11 Fig. 12 SNR of single coil and Imitated humbucking
Partial No.9 13
14 pickup response.
10
Amplitude [mm]

15

0
was provided by solving the vibration equation using the
-10 Laplace transform and dividing the reluctance of the
Pole piece
position a a
magnetic circuit of the pickup and the string into static and
-3 1 2

-20x10 dynamical components, and that the result was in good

0.10 0.12 0.14 0.16 0.18 0.20
Position [m] agreement with the measurement results. Additionally, the
(b) Amplitudes of partial tones from 9th to 15th characteristics of both pickup types expressed subjectively
were indicated objectively using the spectrum structure, by
Fig. 11 Initial condition and amplitudes of partials. analyzing the responses of the single-coil pickup and the

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 M. HARAZONO et al.: HUMBUCKING PICKUP RESPONSE EXCITED BY STRING

imitated humbucking pickup composing two single-coil Pap/Com edition, New York, 2009).
[3] D. Hunter, The Electric Guitar Sourcebook (Backbeat Books;
pickups. Also, the generating the beats in the response was
Pap/Com edition, San Francisco, 2006), pp. 37–51.
clarified by analyzing the partial tone frequencies. Fur- [4] M. Harazono, M. Tomioka, K. Nakamura and Y. Tomita, ‘‘A
thermore, the SNR at all moments for both pickups were string vibration affected by concentrated negative stiffness,’’
calculated, and it was found that the humbucking pickup J. Acoust. Soc. Jpn. (J), 36, 615–623 (1980) (in Japanese).
clearly contributes to noise reduction. [5] M. Harazono, ‘‘A string vibration affected by discrete negative
stiffness,’’ J. Acoust. Soc. Jpn. (J), 44, 187–193 (1988) (in
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[6] M. Harazono, ‘‘Electromagnetic pickup response excited by a
[1] T. Evans, Guitars: Music, History, Construction and Players string vibration,’’ J. Acoust. Soc. Jpn. (E), 10, 23–29 (1989).
from the Renaissance to Rock (Paddington Press, New York & [7] M. Harazono, ‘‘Beats of partials of electromagnetic transducer
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[2] D. Hunter, The Guitar Pickups Handbook (Backbeat Books; 45, 101–106 (1989) (in Japanese).

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