Sinusoidal neural networks have been shown effective as implicit neural representations (INRs) of low-dimensional signals, due to their smoothness and high representation capacity. However, initializing and training them remain empirical tasks which lack on deeper understanding to guide the learning process. To fill this gap, our work introduces a theoretical framework that explains the capacity property of sinusoidal networks and offers robust control mechanisms for initialization and training. Our analysis is based on a novel amplitude-phase expansion of the sinusoidal multilayer perceptron, showing how its layer compositions produce a large number of new frequencies expressed as integer combinations of the input frequencies. This relationship can be directly used to initialize the input neurons, as a form of spectral sampling, and to bound the network's spectrum while training. Our method, referred to as TUNER (TUNing sinusoidal nEtwoRks), greatly improves the stability and convergence of sinusoidal INR training, leading to detailed reconstructions, while preventing overfitting.