Calculating the Spectrum of Electric Guitar Tones

In this handout we show how to calculate the relative amplitudes of all string harmonics in the pickup signal from a solid body electric guitar, as a function of pickup position and the location and width of the plectrum (pick).

Provisions also are made for fingered as well as open string notes, and for a variety of pickup designs including humbucking (both normal and "out of phase" modified), single coil, and velocity as well as displacement sensitive.

Boundary conditions: both ends of string fixed (at nut and bridge)

Initial conditions: initial velocity is zero at every point on the string; initial position of the string is defined by three straight segments as shown below

Parameters

L -- open string length

x -- distance of center of pickup from bridge

p -- distance of center of plectrum from bridge

w -- width of plectrum in contact with string

m -- number of semitones (half steps) above the open string pitch [i.e., the number of the fret being fingered] open string corresponds to m = 0

We will be calculating Y_n, the relative displacement amplitude of the n^th harmonic mode of the string at the pickup. [The fundamental (harmonic mode 1) amplitude is defined to be 1.0]

Y_n will equal the relative amount of the n^th mode excited by plucking, multiplied by the relative amount of the n^th mode detected by the pickup.

For an arbitrarily narrow pick on an open string:

This is the best equation to examine in order to get a sense of the symmetrical interaction between plucking position and pickup position in affecting the output.

Several elaborations to this equation are needed to model typical "real world" playing situations accurately:

(1) For a pick of significant width, we replace by .

(2) For a fingered note, we replace L by (assuming an equal-tempered scale and no change in anything but length).

With both these refinements, our equation becomes:

where ,

,

,

and .

Two further elaborations may be relevant, depending on the particular guitar being modeled, and are included as options in our GUITAR computer modeling program:

(3) For humbucking pickups (pairs of coils separated by a distance d, typically 6 mm)

we replace sin(nr) by

and sin(r) by .

[Some players have modified the wiring of their humbucking pickups to achieve an interesting effect. That situation is easily modeled by changing the signs of the second terms in each of the above substitutions, i.e.

replace sin(nr) by

and sin(r) by .

(4) Finally, for a velocity sensitive pickup (rather than displacement sensitive), the pickup's signal will be proportional to nY_n. [Including time dependence, the relative velocity amplitude may be written

(where is the first partial derivative with respect to time). Then

(where is the (angular) frequency of the n^th mode).] So for such pickups we use the same equations as for Y_n but with n rather than n² in the denominator.