The spectral component of timbre has a lot in common with harmony, in that both musical features are defined by relationships between a number of simultaneous frequencies. The spectrum of an individual instrument is much more likely to be full of harmonically-related frequencies than music is, due to the physics involved in most instruments production of sound. Some instruments, like bells, gongs, and cymbals have some non-harmonic components, but they are in the minority. So unlike the representation of intervals and chords, spectral representation can usually be simplified into static shapes. This is because most single notes on an instrument, having only frequency components related to the fundamental pitch of the tone, has a periodic waveform (one that repeats). As noted earlier in the discussion of lissajous curves, if you use a sine wave to move a point horizontally and then use another sine wave of the same frequency delayed by 1/4 of its wavelength to move that point vertically, the point draws a circle. If, instead of a pure sine-wave, the deflections of the point are controlled by more complex waves reflecting richer sound spectra, instead of a circle, the point draws other shapes, predominantly triangles, bow-ties, squares, crosses, and other simple and iconic shapes. As the spectrum gets "brighter" (i.e. higher frequencies are emphasized), the shape gets sharper points at vertices, and as the spectrum gets "darker" (i.e. lower frequencies are emphasized), the shape softens towards a circle.
Here is an interactive example. Press '0' for sine wave. Press '1' for sawtooth wave. Press '2' for square wave. Press '3' for a brass tone. Press 's' to darken the spectrum by reducing higher harmonics. This also simplifies the shape. Click and drag on the bars at the bottom to change the amount of each harmonic.
If this sort of shape representation were used in representing timbre, it could either be used iconically, wherein certain shapes are understood to represent certain instruments, or it could be used algorithmically, where the actual timbre of a note on an instrument is used to define the shape. This second approach offers more expressive potential. Theoretically, the nuances of performance that impact timbre could be represented as fluid variations on the basic shape. To capture the true nuance of an existing performance however, this would require isolating the sound of each note of each instrument in order to properly construct that note's shape. In many cases this would not be possible. In the simpler scenario, the music is supplied as a MIDI file, and the MIDI controller value changes and the changes in "velocity" (how hard or soft the instrument was played) could modulate the timbral shape in some useful manner.
Stephen Malinowski experimented with simplified shape representations for instruments in an attempt to add more musical information to his Music Animation Machine. He seems, in his experiments to have used shapes very similar to the ones that the above process generates. You can see an example in the second half of this video. (There was obviously a scanning problem with the transfer of this video from tape, but you can still gather something from the example I think) In this example, it is clear that one can quickly generate too much visual information when representing even a portion of a piece of music's qualities.
Colour is also a promising candidate for timbral expression. Colour words are sometimes used to describe timbres (dark, bright, warm, cool) although they seem to be used more to express a general sense of the relative weights of higher and lower harmonics rather than more detailed spectral information. Although colour can mix as harmonics in a sound spectrum mix, the result of mixing colours is less expressive since the component colours average out to a single undifferentiated colour.
Another possibility for visually expressing timbre is texture. Texture and wave-shape seem to have a natural affinity. A two dimensional texture with colour might be even more effective than a gray texture, however that depends on what else colour might be representing. Texture allows groups of different colours to combine without losing information (as happens when colours are simply mixed). An additional possibility might be to introduce 3-dimensional textures (using a true 3-d systems with polarized-glasses). If the depth dimension could be used, it might be able to coexist with other dimensions of visual expression without substantially interfering with it.