FM sideband amplitudes are set by Bessel functions of the first kind indexed by sideband order and modulation index

In FM/PM synthesis the signal expands as an infinite sum of sidebands: s(t) = Σ Jk(I)·sin(2π(fC + k·fM)t), where Jk(I) is the Bessel function of the first kind of order k evaluated at the modulation index I. Each integer k is one sideband (the carrier is k=0), and its amplitude is Jk(I) — not a linear or simple function of I. Bessel functions are oscillatory: as I increases, J0 first decreases, goes negative (a phase inversion), then oscillates around zero, while higher-order functions peak at progressively larger I. Because they cross zero, individual partials can momentarily vanish and then return as I changes — the carrier itself disappears at Bessel zeros (J0(I)=0 at I≈2.4, 5.5, 8.6…). So sweeping the modulation index does not simply add ever-more harmonics at rising amplitude; spectral evolution is non-monotonic, with components growing and fading in turn. The full amplitude-vs-order-vs-I surface is a complete, frequency-independent map of FM timbre.

Examples

J0(0)=1 (carrier only at I=0). J1(2.4)≈0, so the first sideband disappears near I≈2.4; J0=0 at I≈2.4 and 5.5 give characteristic hollow timbres. DX7 brass patches exploit J1/J2 growth around I=2–5. Ramping I from 0 to 20 makes partials appear, peak, and fade in turn, giving evolving timbres.

Assessment

Explain why increasing the FM index does not simply add more harmonics at monotonically rising amplitudes, and what happens to the carrier as I increases past 2. What does Jk(I)=0 imply about the k-th partial, and name one sound-design consequence of sweeping I through a Bessel zero.