Think of linear systems as showing the results as independent signals but nonlinear sensing as a mixer with by-products. From Laplace transforms and Fourier analysis we get mixing, think of it as multiplying in the time domain and adding and subtracting in the frequency domain. I defer the math to my colleagues explanations or any book on Laplace transforms of a mixer output in the time domain or the analysis by Fourier.
So adding in time domain does not affect spectral content in spite of baseband apparent beat frequencies of the difference between spectral content of pure f1 and pure f2. Now our brains do a Fourier Analysis when we hear music instantaneously because of parallel processing of the signal. THe difference in beats of similar notes is an apparent result of nulling the signals then the phases cancel on the carrier rather than an intermodulation from mixing in a non-linear detector. Although our ears are also non-linear (See Fletcher-Munson curves) and logarithmic, they are pure linear systems with an adaptive detective of parallel nerve sensors and not actually a logarithmic detector in the passband which is non-linear. It is a logarithm detector in the Fourier transform of every continuous frequency.
Imagine a million Op Amp log detectors in parallel (See Analog computers) as our brains doing linear Fourier transforms with a 120 dB dynamic range. This may not match your application, but it shows how complex out brain's function with parallel processing and clearly distinguishing a pair of notes instantly.
I wish I could multiply as well as my calculator or do spectrogram of what I hear. Musicians can do this visually in their mind by hearing the sounds in a rather interesting way and have auditory memory like a strip chart with a "spectrogram" of Fourier content. The interest part is there is no conscious transform in our mind and we cannot imagine it in the time domain other than the envelope unless we digitize it onto music charts.
Engineers prefer Spectrum Analyzers. Many free tools on the web including Audacity's free Audio recording studio with a built in software Spectrum Analyzer.
I hope this helped even if I didn't have to use math.
I trust it raises more questions than you asked and answered at least one you intended. That's what learning is all about.
Thanks for listening ;)