What word can I use to describe time-varying spectral content? I have two devices that generate different classes of signals, one simple and one complex. I would like to find a single word or short phrase that describes the increased richness in one class compared to the other. I'm wondering if such a word or phrase exists.

Class #1 is limited to pure tones and colored noise. These sounds are time-invariant, and they are described as having attributes of "frequency and duration," where frequency is taken to mean any number of frequencies together, so long as they are invariant over time.

Class #2 may be any kind of sound. Typically the sounds used are more complex than those in class #1, with time-varying features. They usually have phase and amplitude changes affecting various frequencies over time. Often they contain semantic content such as recognizable real-world sounds or spoken words. This class of sound has been loosely described as having an attribute of "spectral content" as opposed to an attributes of "frequency and duration," but that description doesn't seem to be very good because people may consider spectral content to mean a distribution of time-invariant frequencies.

Perhaps "time-varying spectral content" gets closer? That might be a problem, because the rich class is a superset of the simple class, so there's no requirement for a signal to vary over time. In any event, I'm not sure having time-varying spectral content rather than a frequency and duration really captures the qualitative difference between white noise and a song fragment. Spectro-temporal variations seem to be part of it, and the temporal envelope is certainly important .

Rich sounds would be things like a child laughing, a dog barking, or a human speaking your name, while simple sounds would be things like an 800 Hz sine wave, or white or pink noise.

Is there a pre-existing word or short phrase that meaningfully describes the kind of sonic richness that distinguishes one from the other?
The notions of harmonicity or non-stationary seem connoted.

Stochastic White noise or pink noise, or deterministic pure tones, are (strongly) structured in the spectral domain. So,  strongly structured, or strongly  spectrally-structured. So, in a soft manner, I would  describe the rich class as weakly structured, or weakly spectrally-structured.