We present a novel decomposition for sinusoidal audio coding using amplitude modulation of sinusoids via a linear combination of arbitrary basis vectors. The proposed method, which incorporates a perceptual distortion measure, is based on a relaxation of a non-linear least squares minimization. It offers benefits in the modeling of transients in audio signals. We compare the decomposition to constant-amplitude sinusoidal coding using rate-distortion curves and listening tests. Both indicate that, at the same bit-rate, perceptually significant improvements can be achieved using the proposed decomposition.