When using FM synthesis for melodic sounds, how does the modulation index affect the timbre and harmonic content of the resulting sound?

In FM (Frequency Modulation) synthesis, the modulation index directly controls the amount of frequency deviation applied to the carrier oscillator by the modulator oscillator. The modulation index significantly affects the timbre and harmonic content of the resulting sound. FM synthesis works by using one oscillator (the modulator) to modulate the frequency of another oscillator (the carrier). The modulation index determines the depth or intensity of this frequency modulation. A low modulation index results in subtle changes to the carrier oscillator's frequency, creating a sound with relatively few additional harmonics. The resulting timbre will be closer to the original waveform of the carrier. As the modulation index increases, the frequency of the carrier oscillator is modulated more intensely, resulting in a greater number of sidebands (new frequencies) being generated. These sidebands create a more complex and harmonically rich sound. With higher modulation index values, the timbre becomes increasingly bright, metallic, and complex, introducing more dissonant and non-harmonic frequencies. Extremely high modulation index values can lead to very harsh and noisy sounds. The modulation index provides the primary means of controlling the timbre and harmonic complexity in FM synthesis. By carefully adjusting the modulation index, it is possible to create a wide range of sounds, from subtle and delicate tones to aggressive and distorted textures. The ratio between the carrier and modulator frequencies also influences the resulting harmonic content, but the modulation index determines the overall intensity of the FM effect.