Visual representations of music exist in the form of musical scores.
They exist as graphically-notated scores, such as this example by Cornelius Cardew.
Here is an animation of the graphic score of a work by Gyorgi Ligeti called Artikulation.
Ligeti - Artikulation
by tonicadominante
There are very simple forms of notation that do not require an ability to read music on order to understand. Here is a video example of the Music Animation Machine (created by Stephen Malinowski) that uses piano roll notation and colour to represent the music.
Sound can be visually presented as a waveform or analyzed and presented as a landscape representing spectral changes over time.
For an excellent summary of the history of the visualization of sound, see Symmetry and Harmonics by Joost Rekveld.
Each of these forms of representation were designed for specific purposes, but, with the possible exception of the Musical Animation Machine, they were not designed to replace the listening experience. In recent years, there has been an explosion of music visualization software designed to create visual accompaniment to the music you are playing on your computer. A number of these are quite striking, but are intended to augment the listening experience, and do not attempt to represent the full set of features of the music.