  Developing digital sound synthesizers is crucial to the music industry as it
provides a low-cost way to produce high-quality sounds with rich timbres.
Existing traditional synthesizers often require substantial expertise to
determine the overall framework of a synthesizer and the parameters of
submodules. Since expert knowledge is hard to acquire, it hinders the
flexibility to quickly design and tune digital synthesizers for diverse sounds.
In this paper, we propose ``NAS-FM'', which adopts neural architecture search
(NAS) to build a differentiable frequency modulation (FM) synthesizer. Tunable
synthesizers with interpretable controls can be developed automatically from
sounds without any prior expert knowledge and manual operating costs. In
detail, we train a supernet with a specifically designed search space,
including predicting the envelopes of carriers and modulators with different
frequency ratios. An evolutionary search algorithm with adaptive oscillator
size is then developed to find the optimal relationship between oscillators and
the frequency ratio of FM. Extensive experiments on recordings of different
instrument sounds show that our algorithm can build a synthesizer fully
automatically, achieving better results than handcrafted synthesizers. Audio
samples are available at https://nas-fm.github.io/.
