Musical training is frequently associated with benefits to linguistic abilities, and recent focus has been placed on possible benefits of bilingualism to lifelong executive functions; however, the neural mechanisms for such effects are unclear. The aim of this study was to gain better understanding of the whole-brain functional effects of music and second-language training that could support such previously observed cognitive transfer effects. We conducted a 28-day longitudinal study of monolingual English-speaking 4- to 6-year-old children randomly selected to receive daily music or French language training, excluding weekends. Children completed passive EEG music note and French vowel auditory oddball detection tasks before and after training. Brain signal complexity was measured on source waveforms at multiple temporal scales as an index of neural information processing and network communication load. Comparing pretraining with posttraining, musical training was associated with increased EEG complexity at coarse temporal scales during the music and French vowel tasks in widely distributed cortical regions. Conversely, very minimal decreases in complexity at fine scales and trends toward coarse-scale increases were displayed after French training during the tasks. Spectral analysis failed to distinguish between training types and found overall theta (3.5–7.5 Hz) power increases after all training forms, with spatially fewer decreases in power at higher frequencies (>10 Hz). These findings demonstrate that musical training increased diversity of brain network states to support domain-specific music skill acquisition and music-to-language transfer effects.