The brightness of sound is an important concept of the brass instruments and is related to the nonlinearity : "shock waves certainly correspond to a dramatic amount of high frequencies in the radiated sound: a bright ‘‘metallic’’ sound." (Hirschberg et al, JASA 99 (3), March 1996).
For more details, see publications from A.Hirschberg or J.Gilbert for examples. Maybe the book of N.H. Fletcher, T.D. Rossing, "The physics of musical instruments" can be useful.
In a more general view, brightness relates with the high frequency content of sound, regardless of its structure. Useful to compare sounds rather than describing absolute values; useful to compare sounds at perceptive level, it also relates with the "color" of sound, e.g. white noise is brighter than pink noise, and pink noise is brighter than brown noise. With harmonic sounds, e.g. musical instruments, it relates mainly with the harmonic composition. However with musical instruments your perception can modified by both pitch and harmonic structure. Depending on the structure of your sounds you may use several different measures .... spectral centroid, the level of octave bands, the number and level of harmonics ....
have you seen this wiki ?
https://en.wikipedia.org/wiki/Brightness#Brightness_of_sounds
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