The Physics of E-Guitars: Vibration Voltage Sound wave - Timbre (Physik der Elektrogitarre)

Transcription

1 . TONMEISTERTAGUNG VDT INTERNATIONAL CONVENTION, November The Physics of E-Guitars: Vibration Voltage Sound wave - Timbre (Physik der Elektrogitarre) Manfred Zollner Hochschule Regensburg, manfred.zollner@hs-regensburg.de 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.. 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 /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 /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.. 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. ). Fig.. 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

2 . TONMEISTERTAGUNG VDT INTERNATIONAL CONVENTION, November 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 []. A a b c c b a 0 T T Fig. : Place functions for different times (left), and velocity time functions of a plucked string [].. 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.,000 turns) senses these flux variations and produces an electrical voltage (0. 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 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. depicts a typical velocity DFT-spectrum of a plucked guitar string (Fender Telecaster), the spectral envelope (dashed blue) shows characteristic comb-filtering (sinc-functions) E ms khz 7 Fig. : Velocity spectrum []. Taking a closer look at the strings velocity and the transfer mechanism reveals spatial movements and nonlinear reluctance, which are published in more detail [].

3 . TONMEISTERTAGUNG VDT INTERNATIONAL CONVENTION, November. 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,000 turns of copper wire, according to the specific coil data, produces an inductance of approximately H. This inductance, together with the cable capacitance (0 700 pf), forms a second order lowpass filter with a cutoff frequency of 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 Q- factor 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 []). Fig. depicts typical transfer functions of a Stratocaster-pickup with different loading. Transfer: Strat-7 0 Gain / - pf open - 7 pf Frequency / khz Fig. : Transfer characteristics of a Stratocaster pickup []. 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. depicts a Stratocaster whose

4 . TONMEISTERTAGUNG VDT INTERNATIONAL CONVENTION, November 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. Pickup Pickup khz khz 0 Airborne sound 0 Airborne sound khz khz Fig. : Stratocaster spectrum. Left: Body with ( ) / without (----) box-contact. Right: Neck with/without box-contact []. 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 T, i.e. the time during which the level decays by. For low order partials only minimal dissipation is to be seen, especially for the guitars bass strings.

5 . TONMEISTERTAGUNG VDT INTERNATIONAL CONVENTION, November 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 /s. 0 T /s 0 0 Radiation absorption E A D T /s g b e.5 g b e khz khz Fig. 5: Decay-time T due to direct radiation (left), typical decay-times (right) []. But there are more things between heaven and earth, and more absorbers: If a Gibson ABR-- bridge 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. ). So again we encounter the lemma: it's mainly the hand, not the gear. This holds even more when nonlinearities are considered. T /s G T /s G khz khz Fig. : Decay-Time: ES-5, bridge dislocated (left); Stratocaster without/with capodaster (right) [].. Nonlinear string movements Linearity means proportionality between cause and effect, so nonlinearity is effectively unproportionality. When the string excitation is increased by 0%, the pickup voltage will also increase by 0% 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 rd and th, as st and nd cannot develop their full displacement

6 . TONMEISTERTAGUNG VDT INTERNATIONAL CONVENTION, November amplitude. During 0 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, E on D-string) []. 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. ). The coincidence between calculation and measurement is good, although the Telecaster neck was not new, a fact not considered in the calculation. Measurement 0 Periods Simulation 0 Periods Fig. : Tactigram: Telecaster, E on D-string; distance plucking-point / bridge = cm [].

7 . TONMEISTERTAGUNG VDT INTERNATIONAL CONVENTION, November Fig. 9: Different string positions for a hard plucked string (calculation) [].

8 . TONMEISTERTAGUNG VDT INTERNATIONAL CONVENTION, November When a guitar string is deflected with great force (hit hard), it comes in contact with the last fret (Fig. 9, st 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 th 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: [] Zollner, M.: Physik der Elektrogitarre, (to be downloaded as PDF). The foundations of string movements. The string as a wave guide. String magnetics. The electromagnetic field 5. Electromagnetic pickups. Piezoelectric pickups 7. Neck and body. Psychoacoustics 9. Guitar electrics. Guitar amplifiers. Guitar loudspeakers