26. TONMEISTERTAGUNG – VDT INTERNATIONAL CONVENTION, November 2010
The Physics of E-Guitars:
Vibration – Voltage – Sound wave - Timbre
(Physik der Elektrogitarre)
Manfred Zollner
Hochschule Regensburg, [email protected]
Abstract
The linear tone generation process of an electric guitar can be divided in four
subsystems: Plectrum filter, plucking position filter, pickup position filter, pickup
transfer filter. Nonlinear effects such as bouncing require additional systems.
Cascading these subsystems yields a general system that models the guitars transfer characteristic from string velocity to output voltage.
1. Plectrum filter
Systems theory claims that plucking a string produces a force step function: The plucking
force jumps abruptly from F to zero, corresponding to a 1/f-spectrum. In reality the plucking
force decays gradually, which leads to a "rounded step". As it may take several milliseconds
before the plectrum (or finger nail) has finally left the string, a lowpass filtering of the 1/fspectrum occurs. Bouncing effects between string and plectrum may add comb filtering, so in
total an efficient tone affection is possible simply by modifying the plectrums motion. From
which emanates the expression “It's the finger, not the gear”. However, the plectrum filter is
not the only subsystem in the tone producing process.
2. Plucking position filter
The force step excitation of the string leads to two step waves, traveling in opposite
directions. Each of these waves is reflected at the string ends, e.g. the bridge. Superposition of
all these waves yields the strings place- and time function (Fig. 1). Fig. 1. depicts a string
which is plucked at A. Once the string has left the plectrum, it changes its shape to a
downward moving slant, until the antipole is reached – then it swings back. In the right part of
the figure three time functions are depicted, showing the strings velocity at the marked
positions (a, b, c). As can be seen, each part of the string is either motionless (v = 0), or
moves with constant velocity. Parts closer to the strings center (c) don't move faster, but for a
longer pulse width. By changing the plucking position, the guitarist can modify the shape of
these velocity-squares, and thus the spectrum, which is a superposition of (spectral discrete)
sinc-functions.
In simple models the strings movement is considered to be periodic, so the spectrum is
harmonic, i.e. consists of equidistant lines. But in fact the wave movement is dispersive, with
the high frequency parts traveling much faster than the low frequencies. Thus the spectrum is
stretched, the line spacing increases toward high frequencies. In the time domain, dispersion
means deterioration of the temporal periodicity, which first leads to a kind of superimposed
26. TONMEISTERTAGUNG – VDT INTERNATIONAL CONVENTION, November 2010
ringing. Later the time function completely changes its shape. The degree of inharmonicity
depends on pitch and string diameter, and the relation of core- vs. winding-diameter [1].
A
a
b
c
c
b
a
0
T
2T
Fig. 1: Place functions for different times (left), and 3 velocity time functions of a plucked string [1].
3. Pickup position filter
The typical electromagnetic pickup is a electromechanical transducer, transferring mechanical
energy into electrical energy. The pickup contains one or more permanent magnets, whose
magnetic flux is modulated by the vibrating string. The string approaching the magnetic poles
reduces the magnetic resistance, thus increasing the flux. A coil wound of thin copper wire
(e.g. 10,000 turns) senses these flux variations and produces an electrical voltage (0.1 – 5 V).
The magnetic resistance (reluctance) is mainly determined by the air gap, i.e. the volume
between magnetic pole piece and the neighboring string. Pole piece diameters often range
from 4 – 8 mm, and so does the magnetic effective part of the string, the magnetic aperture.
This means that – neglecting side effects – the strings motion is sampled at a point adjacent to
the pole piece. The law of induction (Faraday/Henry Theorem) postulates that the induced
voltage is proportional to the temporal derivative of the magnetic flux, corresponding to the
derivative of the strings deviation, or in other words to the strings velocity.
Both plucking position and pickup position lead to the multiplication of sinc-functions on the
basic spectrum, with interchangeable results: In the linear model there is no difference between plucking at position A and sensing at position B, or plucking at position B and sensing
at position A.
Fig. 2 depicts a typical velocity DFT-spectrum of a plucked guitar string (Fender Telecaster),
the spectral envelope (dashed blue) shows characteristic comb-filtering (sinc-functions).
90
dB
80
E3
30 ms
70
60
50
40
30
20
0
1
2
3
4
5
6
kHz
7
Fig. 2: Velocity spectrum [1].
Taking a closer look at the strings velocity and the transfer mechanism reveals spatial
movements and nonlinear reluctance, which are published in more detail [1].
26. TONMEISTERTAGUNG – VDT INTERNATIONAL CONVENTION, November 2010
4. Pickup transfer filter
The vibrating string produces a temporally varying magnetic flux, the origin of the induced
voltage. This voltage has to be considered as produced "inside" the pickup, and may not be
confused with the voltage that can be measured at the guitars jack terminal. The induced
source voltage is mapped onto the terminal voltage by the transfer function, which depends on
the network components (inductance, capacitance, resistance). It is understood that 10,000
turns of copper wire, according to the specific coil data, produces an inductance of
approximately 3 H. This inductance, together with the cable capacitance (300 – 700 pF),
forms a second order lowpass filter with a cutoff frequency of 2 – 5 kHz. Underwound
Stratocaster pickups tend towards higher cutoff frequencies, fat P90-coils and Humbucker
pickups towards lower. Potentiometer ratings and the amplifiers input resistance define the Qfactor of the lowpass filter, i.e. the resonance boost. The pickup transfer of a Stratocaster-like
type is fairly simple, but as soon as shielding and focusing parts are used, additional effects
must be considered (eddy-currents, two-point-sampling, inductive coupling [1]). Fig. 3
depicts typical transfer functions of a Stratocaster-pickup with different loading.
Transfer: Strat-72
20
10
en
op
-10
0
45
pF
Gain / dB
0
0
75
-20
pF
-30
-40
.1
.15
.2
.3
.4 .5 .6 .7 .8 .91
1.5 2
Frequency / kHz
3
4
5 6 7 8 910
15
20
Fig. 3: Transfer characteristics of a Stratocaster pickup [1].
5. Damping mechanisms
The above filters specify the time function and spectrum of the initial terminal voltage, but
being time invariant, they do not take damping mechanisms and decay processes into account.
Two mechanical processes dissipate most of the vibration energy: Internal absorption
(irreversible deformation of the string), and radiation (direct radiated sound waves). A third
process has marginal influence on the tone production of an electric guitar, but that process is
believed to be the most important: The vibration of the corpus wood. Many guitarists believe
that an E-guitars body should vibrate as much as possible, but – if the vibration energy has
been guided from the string to the body, the string has lost its energy and its vibration stalled.
No – since for the sake of sustain bridge and nut have to reflect as much energy as possible,
an E-guitars body has only a marginal influence on the electric sound.
There is an easy way to prove this statement: When the guitar corpus is connected to an
external resonator (i.e. a table), the radiated airborne sound changes dramatically, while at the
same time the pickup voltage remains almost unaffected. Fig. 4 depicts a Stratocaster whose
26. TONMEISTERTAGUNG – VDT INTERNATIONAL CONVENTION, November 2010
body is in mechanical contact to a wooden box. While playing an E chord, the guitarist
interrupted this contact by lifting up the guitar by an inch. Thereupon the airborne sound
changed (bottom spectrum) however, the pickup voltage did not.
50
50
Pickup
Pickup
dB
dB
40
40
30
30
20
20
10
10
0
.05
.08 .1
.2
.4 .5
1
2
4
5
kHz
10
80
0
.05
.08 .1
.2
.4 .5
1
2
Airborne sound
5
kHz
10
Airborne sound
dB
dB
70
70
60
60
50
50
40
40
30
.05
4
80
.08 .1
.2
.4 .5
1
2
4
5
kHz
10
30
.05
.08 .1
.2
.4 .5
1
2
4
5
kHz
10
Fig. 4: Stratocaster spectrum. Left: Body with (–––) / without (----) box-contact.
Right: Neck with/without box-contact [1].
When this experiment is repeated with a mechanical contact between neck and wooden box, a
just significant difference is visible in the third octave spectrum of the pickup voltage, that
may or may not become audible – nothing to write home about.
Sustain- and sound-affecting are direct radiation and string-internal dissipation (Fig. 5). These
figures show the decay-time T30, i.e. the time during which the level decays by 30 dB. For low
order partials only minimal dissipation is to be seen, especially for the guitars bass strings.
26. TONMEISTERTAGUNG – VDT INTERNATIONAL CONVENTION, November 2010
The high order partials of the descant strings (g, b, e) suffer in particular both from sound
radiation and from internal dissipation – even with perfect bearings their level decays in the
vicinity of 5 kHz with approximately 15 dB/s.
100
T30 /s
50
T30 /s
Radiation absorption
30
60
g
20
40
15
b
e
30
10
8
20
6
15
A
4
3
D
2
E
10
8
6
g
4
3
1
0.8
0.6
b
e
2
0.4
1.5
1
.08 .1
.15
.2
.3
.4
.6
.8
1
1.5
2
3
4
5
0.2
.08 .1
kHz 8
.15
.2
.3
.4
.6
.8
1
1.5
2
3
4
5 kHz
8
Fig. 5: Decay-time T30 due to direct radiation (left), typical decay-times (right) [1].
But there are more things between heaven and earth, and more absorbers: If a Gibson ABR-1bridge is dislocated within the boundaries of mechanical tolerance, or if the guitarist’s hand
touches the neck (not unusual), or a capodaster is mounted, the decay time can be reduced by
a factor of four, or even more (Fig. 6). So again we encounter the lemma: “it's mainly the
hand, not the gear”. This holds even more when nonlinearities are considered.
50
T30 /s
30
50
T30 /s
30
G3
20
15
10
8
10
8
6
6
4
3
4
3
2
2
1
0.8
0.6
1
0.8
0.6
0.4
0.4
0.2
.08 .1
.15
.2
.3
.4
.6
.8
1
1.5
2
3
4
5 kHz
G3
20
15
8
0.2
.08 .1
.15
.2
.3
.4
.6
.8
1
1.5
2
3
4
5 kHz
8
Fig. 6: Decay-Time: ES-335, bridge dislocated (left); Stratocaster without/with capodaster (right) [1].
6. Nonlinear string movements
Linearity means proportionality between cause and effect, so nonlinearity is effectively
“unproportionality”. When the string excitation is increased by 100%, the pickup voltage will
also increase by 100% if the system is linear. This happens when the exciting force is low.
Hitting the string hard yields a nonlinear behavior, and a pickup voltage that cannot be
predicted from the standpoint of low excitation. Fig. 7 depicts voltage spectrograms of a
Telecaster. In both cases the D-string was plucked at the same position, with the same bridgepickup, the only parameter that varied was the plucking force. Whereas for gentle plucking
the partials decay more or less regularly, hard plucking generates an unpredictable behavior.
The strongest partials are 3rd and 4th, as 1st and 2nd cannot develop their full displacement
26. TONMEISTERTAGUNG – VDT INTERNATIONAL CONVENTION, November 2010
amplitude. During 0 – 150 ms a strong attack pattern occurs which is due to string bouncing
(snapping): The string repeatedly is in contact with the frets, even with the "unused" high
order frets.
Fig. 7: Spectrograms of a gently (left) and hard (right) plucked string (Telecaster, E3 on D-string) [1].
String/fret-contacts depend on subtle fret heights, and this is one of the reasons for interindividual guitar sounds. To document the occurrence of string bouncing, a logic analyzer
was connected to the frets of a Telecaster (Fig. 8). The coincidence between calculation and
measurement is good, although the Telecaster neck was not new, a fact not considered in the
calculation.
Measurement
0
1
2
3
Periods
4
1
2
3
Periods
4
Simulation
0
Fig. 8: Tactigram: Telecaster, E3 on D-string; distance plucking-point / bridge = 12 cm [1].
26. TONMEISTERTAGUNG – VDT INTERNATIONAL CONVENTION, November 2010
1
2.Bund
9
2.Bund
Steg
Steg
2
10
2.Bund
2.Bund
Steg
Steg
3
11
2.Bund
2.Bund
Steg
Steg
4
12
2.Bund
2.Bund
Steg
Steg
5
13
2.Bund
2.Bund
Steg
Steg
6
14
2.Bund
2.Bund
Steg
Steg
7
15
2.Bund
2.Bund
Steg
Steg
8
16
2.Bund
2.Bund
Steg
Fig. 9: Different string positions for a hard plucked string (calculation) [1].
Steg
26. TONMEISTERTAGUNG – VDT INTERNATIONAL CONVENTION, November 2010
When a guitar string is deflected with great force (hit hard), it comes in contact with the last
fret (Fig. 9, 1st picture). Once the string has left the plectrum (following pictures) it leaves the
frets, but soon snaps back to them. The consequence is a specific snapping sound, which is
formed by all frets – even the "unused" ones. So even if a guitarist thinks he will never play
beyond his guitars 15th fret – the tone will be affected by the surface qualities of these higher
order frets.
Summarized: The sound of a solid E-guitar depends on the player, the plucking and pickup
position, the pickup, the bridge and the frets. The neck with its unavoidable eigenmodes will
contribute to an extent that (in contrary to E-basses) is inferior in most cases, but the corpus
wood is largely insignificant – unless it were made of insulating material (…which it never
is). As long as an E-guitar is crafted according to the rules, the most sound affecting parts are
the pickups.
Reference:
[1] Zollner, M.: Physik der Elektrogitarre, http://homepages.hs-regensburg.de/~elektrogitarre
(to be downloaded as PDF)
1. The foundations of string movements
2. The string as a wave guide
3. String magnetics
4. The electromagnetic field
5. Electromagnetic pickups
6. Piezoelectric pickups
7. Neck and body
8. Psychoacoustics
9. Guitar electrics
10. Guitar amplifiers
11. Guitar loudspeakers