  Frequency modulation (FM) and phase modulation (PM) are well-known synthesis
methods, which have been deployed widely in musical instruments. More recently,
some synthesisers have implemented direct forms of FM (as opposed to PM),
allowing, at least as part of their design, for higher-order modulation
topologies. However, such implementations are affected by well-known
difficulties that arise in the modulation of frequency, which are normally
solved by the use of PM. In this article, we analyse these problems and using a
direct comparison with PM, we put forward a solution for the direct application
of FM in higher-order modulation arrangements. We begin by reviewing the theory
of first-order FM, contrasting it to PM. We then proceed to develop a
formulation of second-order FM which is equivalent to the issue-free PM
synthesis, and present a closed-form expression for the evaluation of the
second-order FM spectrum. We then extend the principle to higher-order
topologies, by advancing the concept of an FM operator, analogous to the one
used in PM instrument designs. From this we demonstrate that feedback FM is
also a practical possibility. Finally, we complement the paper by giving a
reference implementation in C++.
