Playing an octave up halves ß, so the Modulator amplitude must double per octave to keep the timbre constant

Because ß = carrier deviation ÷ modulator frequency, and both carrier and modulator track the keyboard so their harmonic relationship holds, raising pitch by an octave doubles the modulator frequency and therefore halves ß. A halved ß means a narrower, simpler, duller spectrum — so a naive FM patch sounds bright low on the keyboard and thin up high. To keep the timbre constant across the keyboard, the Modulator amplitude must scale with the modulator frequency, doubling for every octave. Achieving this precisely is almost impossible with drifting analogue components, which is the main reason commercial FM is implemented digitally, where the arithmetic is exact.

Examples

Analogue fix: route the modulator level through keyboard-tracking scaling so its amplitude doubles per octave. Digital FM operators do this scaling automatically, so a DX7 patch keeps its brightness up and down the keyboard.

Assessment

A simple FM patch is bright in the bass and dull in the treble. Explain the ß-based reason, and state the correction needed to make the timbre consistent across the keyboard.