Equal temperament divides the octave into twelve equal intervals, each a frequency ratio of the twelfth root of two. This is a compromise. The natural harmonic series produces intervals at simple integer ratios — a perfect fifth is 3:2, a major third is 5:4. Equal temperament approximates these with irrational numbers. The major third is 13.7 cents sharp. The fifth is 2 cents flat. These errors are small enough that most listeners cannot detect them in clean tones. Equal temperament works because its compromises stay below the perceptual threshold.
Mullin and Leinweber at the University of Adelaide (arXiv:2504.04919, 2025) show that amplifier distortion makes the compromises audible. The mechanism is not subjective — it is arithmetic.
A clean amplifier is approximately linear: the output is proportional to the input. A distorted amplifier is nonlinear. Its transfer function includes higher-order terms — squared, cubed, and beyond. When two frequencies enter a nonlinear system, the higher-order terms generate new frequencies: sums, differences, harmonics, and intermodulation products. The amplifier computes with the input frequencies, producing outputs that didn't exist in the original signal.
In just intonation, a power chord (root and fifth at 3:2) passing through cubic distortion generates a new tone at 5:2 — a major third. The distortion fills in the chord. The generated frequency sits exactly where the harmonic series predicts. The amplifier adds consonance.
In equal temperament, the same power chord has a fifth at 1.4983... instead of 1.5. The distortion still generates cross-terms, but the generated frequencies are slightly wrong — they don't align with the harmonics of the root. Those misaligned tones beat against the harmonics that are present, producing audible roughness. The 13.7-cent error on the major third, inaudible in a clean signal, becomes a dissonant artifact when the amplifier generates the third from the power chord's two notes.
The structural point: a linear system transmits its input faithfully. Whatever compromises exist in the input remain at their original magnitude. A nonlinear system generates new outputs from the relationships between inputs. If those relationships are approximate — as they are in equal temperament — the nonlinear system exposes the approximation by computing with it. The cross-terms are the amplifier doing math with slightly wrong numbers.
This explains a practice that predates the physics. Rock and metal guitarists have tuned by ear rather than by electronic tuner since the 1960s. Eddie Van Halen flattened his major thirds by feel. Blues players bent notes toward just intervals instinctively. The physics was unknown to them, but the consequence was audible: distortion rewards accurate ratios and punishes approximations. The players were optimizing for a nonlinear regime using a linear-era tuning system, and their ears told them the system was wrong before anyone wrote down why.
The general pattern: an approximation that works in a linear regime fails in a nonlinear one, because nonlinearity generates new information from the errors the linear system silently passes through. The compromise isn't eliminated when you move to the nonlinear regime — it is multiplied.