Timbre is through frequency response, as it will alter the source signal and color the original timbre as tuned by the mixing and mastering engineer.
All those other characteristics can be measured by a battery of tests including square waves and DIRAC pulses by observing the delta between the original digital signal and the recorded sound from the transducer or a reconversion through an ADC if you are testing components upstream of the transducer.
"Sound stage" is a more contentious topic because that depends on a number of factors that contribute to modifying the source signal's spatial cues as implemented by the recording engineer (binaural recording techniques for instance) or the mixing/mastering engineer (through software plug-ins playing with panning, reverb, EQ, phase differentials, etc) in such a way that renders a universal analysis practically impossible for HPs, let alone IEMs. These tracks are mixed on studio monitors usually, so spatial cues are building off a HRTF and HRIR that are not translatable exactly to the audience, and mastering those tracks further muddies the water by adding frequency exciters and distortion to ensure performance on systems that are not up to spec, especially speaker systems that don't have a proper subwoofer and room treatment. The measurement methodology for this is as yet unknown because of the psychoacoustic nature of sound imaging, although we can get close by mimicing the spatial cues observed through a binaural recording setup.
I find this topic of 'qualitative' vs 'quantitative' aspects of sound really interesting. I would disagree that timbre is solely a function of frequency response. For me, it goes beyond driver tuning to include more subtle qualities of driver behaviour over time, such as attack, decay, transient control and resonant properties which I feel is more driver composition and housing/ventilation. For example, I have heard a single BA tuned with what I consider to be a 'tonal balance' that on paper should produce a convincing realism and capture all the 'information' but the resolution, note density and contours of the attack and decay are unnatural and/or not sufficiently revealing. This makes distinguishing instrumental character more difficult. So, it's not just about the information but how that information is conveyed.
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