Defining two index breakpoints (I1, I2) and an envelope shape controls the full spectral trajectory of an FM note

Chowning’s practical implementation specifies the modulation index as a time-varying function by setting two index values, I1 and I2, plus an envelope shape. The deviation at any moment is computed as deviation = I1 × m (when the envelope is at 0) up to I2 × m (when the envelope is at 1). By choosing different envelope shapes and the I1/I2 pair, the designer fully controls the spectral arc of the note: wide bandwidth at the attack (high I) narrowing to a simpler tone (low I), or the reverse (woodwind behavior). This two-breakpoint system is the prototype for the operator envelope generators on the DX7, which extend it to six breakpoints per operator but follow the same logical structure.

Examples

Brass: I1=0, I2=5, envelope rises and falls with amplitude → spectrum richest at loudest point. Woodwind: I1=4, I2=2, index inversely proportional to amplitude → spectrum richest at onset, simplifies as tone sustains.

Assessment

Design the I1, I2, and envelope shape for a plucked string patch: rich attack that decays to a pure tone. State your choices and justify each.