A working model, string to speaker
Inside the Electric Guitar
A note starts as a steel string moving a fraction of a millimetre. A magnet and a coil turn that motion into a few hundred millivolts, the guitar's own controls filter it, and the amplifier reshapes it before a speaker moves air.
Every stage below runs a physical model of what is really happening. Pluck the string, change anything, and pluck again. The sound you hear is synthesized from the same equations the charts draw.
Overview
Follow one note
Eight stages, in signal order. Each card shows the same note at that point in the chain. Tap a card to monitor it in the dock below, or open its stage for the detail.
Each scope is scaled to fill its window so you can compare shape. The number underneath is the actual level: millivolts on the guitar side, dB of full scale inside the amp.
Stage 1
The string
A plucked string vibrates at its fundamental and at every whole-number multiple at once. Where you pick decides which of those harmonics start strong and which are missing.
Go deeper: the wave equation and the pluck
A string under tension T with mass per metre μ carries waves at speed v = √(T/μ). Fixed at the fret and the saddle, only waves that fit a whole number of half-wavelengths survive, so the modes are
The last factor is inharmonicity. Real steel is stiff, so high modes run slightly sharp. This model uses B ≈ 10−4, which puts the 20th harmonic about sharp.
A pick pulls the string into a triangle and lets go from rest. Decomposing that triangle into modes, with the pick at fraction β of the length, gives
Two consequences. Amplitudes fall roughly as 1/n², which is why the fundamental dominates. And any harmonic with a node at the pick point, where sin(nπβ) = 0, is not excited at all. Pick at one fifth of the length and the 5th, 10th and 15th harmonics are missing.
Each mode also loses energy at its own rate. Air drag and internal friction grow with frequency, so in this model the decay time is τ = 1 / (0.28 + 3.5×10−7·f²) seconds: about 3.5 s for the fundamental of the A string and a tenth of a second at 5 kHz. That is why a note starts bright and mellows as it rings.
Stage 2
The pickup
A pickup is a magnet wrapped in thousands of turns of fine copper wire. It senses only one short stretch of the string and responds to how fast the string moves there, not how far.
Go deeper: Faraday's law, the comb filter and hum cancelling
The magnet magnetizes the steel string above it. When the string moves, it changes the magnetic flux Φ threading the coil, and Faraday's law says the coil produces a voltage proportional to how fast that flux changes:
Because the output follows velocity, each harmonic gets boosted in proportion to its frequency. The 10th harmonic moves ten times faster than the fundamental for the same displacement. That is why the pickup output in the chart above is brighter than the string motion.
The pickup only sees the string at its own location x. Mode n has the shape sin(nπx/L), so its sensitivity is
where w is the width of the magnetic window, about 10 mm. Harmonics with a node at the pickup vanish, which is the comb pattern in the blue curve. A bridge pickup sits close to the fixed end, where every low mode moves little, so it is thinner and brighter. A neck pickup sits near the middle of the string, where the fundamental moves most.
A humbucker uses two coils about 18 mm apart, wound in opposite directions over magnets of opposite polarity. For the string, the two reversals cancel and the signals add. For hum from a transformer or a light dimmer, only the winding reversal applies, so the hum cancels. The two coils sample two points on the string, which adds its own comb notches and darkens the tone. Their combined inductance of about 4.4 H also pulls the resonance down, as the next stage shows.
Stages 3 and 4
Controls and cable
Electrically, a pickup is a voltage source behind several henries of inductance and a little capacitance. The pots and the cable load it, and together they form a resonant low-pass filter. That resonant peak is a large part of what makes one pickup sound different from another.
Go deeper: the resonant filter and audio-taper pots
The pickup's inductance L and the total capacitance C across it (its own winding capacitance plus the cable) resonate at
Below that frequency the signal passes nearly flat. Near it, the circuit rings and boosts. Above it, output falls at 12 dB per octave. The pots set how tall the peak is: their resistance damps the resonance, so 500 kΩ pots give a taller, brighter peak than 250 kΩ. That is the reason humbucker guitars traditionally use 500 kΩ pots and single-coil guitars 250 kΩ.
The tone control is a variable resistor in series with a capacitor to ground. At 10 its resistance hides the capacitor. As you turn it down, the capacitor shunts more treble, and near 0 it adds so much capacitance that the resonance moves down into the midrange, which gives the nasal "cocked wah" tone.
Both knobs use audio taper pots. Resistance follows a curve close to 10(k/5), so at 5 on the dial the pot is only about 10% of its full value. This model solves the full circuit with complex impedances at every harmonic of the note, so these effects carry straight into the sound.
Stages 5 to 8
The amplifier
The preamp multiplies the signal until its tubes run out of room and clip. A passive tone stack then cuts parts of the spectrum, the power stage drives the speaker and compresses when pushed, and the cabinet filters what is left.
Go deeper: clipping, the tone stack and the cabinet
Preamp. A 12AX7 triode stage has a voltage gain around 60, but the model scales each stage by the Gain knob instead. Its curve is close to linear for small signals and flattens near the supply rails. This model uses an asymmetric soft clipper: the positive half saturates at 1.0 and the negative half at 1.43. Asymmetry adds even harmonics (2nd, 4th), which sound like the same note an octave up and read as warm. Two stages in series, the Drive and Lead channels, clip a signal that has already been clipped, pushing it toward a square wave rich in odd harmonics.
Sustain. Clipping caps the peak, so as the string decays the output level stays nearly constant until the input falls below the clipping point. Watch the Preamp scope on a high-gain preset: the decay flattens out.
Tone stack. The bass, middle and treble knobs are a passive network of three capacitors and four resistors. It can only cut, so even at its flattest it loses 5 to 10 dB, and with every knob at 5 it carves a broad dip in the midrange. This model uses David Yeh's exact third-order transfer function for this circuit, with American (Bassman-style: 250 pF, 20 nF, 20 nF, 56 kΩ) or British (470 pF, 22 nF, 22 nF, 33 kΩ) part values.
Power amp and speaker. The power stage clips symmetrically and later than the preamp, which feels like compression. Presence lifts the top end in the power amp's feedback loop. A 12-inch speaker in a closed cabinet has a low resonance near 100 Hz, a presence bump around 2 to 3 kHz, and rolls off steeply above 5 kHz. That roll-off is why a distorted amp sounds smooth through a cabinet and harsh through hi-fi speakers.