Historically, painting and music have interacted in many different ways: see, e.g., Peter Vergo's two volumes, That Divine Order and The Music of Painting, which outline this history from Ancient China and Greece, through the middle ages and fin de siecle Vienna, to modern abstract art and John Cage. The interactions go both ways: there are many paintings inspired by music, and many musical compositions inspired by paintings; here I'll say something apropos the former, since many of my own artworks belong to this category.
There are numerous different paths which a musically-inclined or -inspired painter can take. One can try to paint in a manner reminiscent of the art of musical composition; this was the approach, in some sense or other, of numerous painters including Nicolas Poussin, J. M. Whistler, Paul Klee and Wassily Kandinsky. One can try to capture the effect of music visually; Oskar Kokoschka did this in a series of sketches of a woman listening to various pieces of music.* One can paint pictures inspired by specific pieces of music; examples include the Brahmsphantasie of Max Klinger and Kandinsky's designs for a stage production based on Mussorgsky's Pictures at an Exhibition.** And there are probably many other methods than these.
I will write more about some of these examples, and others, in the future. My own approach to painting music is most similar to the last on the list: I paint specific pieces of music. Sometimes I listen to music while I work, and in these cases my brush strokes are often directly responsive to its dynamics and rhythm: I suppose the end result could be seen as a kind of two-dimensional dance. But I am far more interested in conveying the meaning of a piece of music, as I understand it, than in transposing any of its superficial features into pictorial form. So at other times I merely hold in mind the overall impression of a piece of music, or what I regard as its overall meaning, in just the same way as I might hold in mind the thoughts to be conveyed in a piece of writing when I want to communicate in words. Listening to the music while working can in these cases prove more of a distraction than an aid. I view the painting as a success if, when I look at the finished product, it seems to express some of the same meaning as the music on which it was based. Likewise, I regard a written paragraph as more or less successful to the extent that I feel it conveys the thoughts I want to communicate through it.
This procedure might seem farfetched. A portrait usually resembles the person it is a portrait of; the same goes for landscapes and other figurative paintings. We can tell what they are paintings of just by looking at them. Surely, therefore, if a painting represents a piece of music then anyone who is familiar with the music ought to be able to recognise it in the painting. But this is not usually the case. The painting will instead be a success if someone who already knows what piece of music it refers to, and is familiar with that piece of music, sees it as conveying some of the same thoughts and feelings as the music.
But the procedure has pedigree: in itself it is inspired by music, because it is essentially the same as that used by composers of non-imitative programmatic and incidental music. For example, no one will guess from listening to Liszt's Die Ideale that it refers to Schiller's poem of the same name; but once we are aware that it is based on that poem, we can evaluate whether or not it does so successfully. We don't need explicit criteria for this kind of evaluation: we only need to have understood the poem, and to be capable of understanding the music. Likewise, to recognize that the Imperial March theme is a suitable accompaniment for Darth Vader but not for Obi-Wan Kenobi, we need to understand the two characters and the music, and see that the significance of Darth Vader corresponds better to the significance of the music.
This kind of visual or musical reference is necessarily less determinate than in the case of, e.g., a portrait, where the subject can usually be identified on the basis of the painting alone. No purely abstract art will be able to represent anything determinatively, that is, in such a way as to allow a viewer or listener to determine its subject, without simply encoding what it represents. The contemporary artist Jack Ox, for example, has painted the works of several composers. Her aim is to translate various important aspects of the music into extremely long, segmented paintings that can be decoded by a viewer who has understood her method. (These works are effectively musical scores, although they would not be very useful as such, since they leave out a lot of information relevant to musical performances.) Where they do not employ codes of this kind, however, we must rely on the verbal promptings of artists—the titles they give their works, or the explanations they provide—to indicate what specific subjects, if any, their abstract works are about.
Is this a problem? Only if we assume that the full significance of a painting or a piece of music should be contained on the canvas or in the notes themselves. But why should we set such arbitrary limits? If we are interested in the expression and communication of meaning, there is no reason to do so. If we allow the titles, explanations and descriptions of artists to count as integral parts of their works conceived as communicative expressions, then abstract arts are on the same footing, in their capacity to convey meaning by referring to particular subjects, as the most literal descriptions and depictions. And they enjoy the distinct advantage of not being constrained, in doing so, to imitate or resemble, in some identifiable respects, the features of their subjects; they are free to exploit the much greater expressive potential of purely imaginative forms.
Furthermore, unlike figurative or imitative arts, abstract arts are thereby also permitted to refer directly to other abstract entities—whether artworks or ideas or feelings—and to convey their meanings too. Since there is no requirement to resemble or imitate, there is no requirement to stick to subjects than can be resembled or imitated in other mediums. A figurative painter who wants to paint something abstract—say, the concept of charity—must resort to allegory; an abstract painter can try to convey the significance of charity directly, by depicting forms that—to the painter, and, he or she hopes, the viewer—have the same significance as the concept, and then say "This is a painting of charity." Whether or not it is a painting of charity, or indeed a good painting of charity, cannot be decided or verified with reference to explicit criteria; all we can do is try to understand it as such, and see for ourselves. The same goes for a painting of a piece of music.
*The composer H. E. Apostel later composed a set of piano variations inspired by Kokoschka's drawings, adding another layer to this musical-visual interaction: music to sketches then back to music.
**Which in turn was inspired by the pictures of Viktor Hartmann. Paint to music then back to paint.