In After Effects, ever since the now defunct Motion Math, you can synchronize any parameter with any other parameter, so a range of audio effects can be tied to graphic properties to create visual music.
Visual music has a long history in media like color organs, film and abstract animation, light shows, CGI, installation art, and even cave art. Abstract animation is not just eye candy, but often attempts to communicate or stimulate synesthesia or mystical states. While forecasts for an expanded or synaesthetic cinema (PDF) haven't quite come to fruition, motion graphics has. The term seems to have been coined or at least popularized by visual music artist John Whitney, who pioneered motion control cameras and the slit scan technique (showing it to Trumbull and Kubrick) -- and in 1960 named his company Motion Graphics, Inc. SIGGRAPH has a peek at some of his movies.
Later, Vice featured Whitney too. Here's some of his work, along with Lapis from his brother James Whitney.
In addition to the Center for Visual Music, mentioned up top, another specific resource for visual music is The iotaCenter. Also noteworthy is the work of William Moritz, who was tireless in documenting early work by abstract animation artists at CalArts and elsewhere and filmmakers like Oscar Fischinger. The Moritz article "Abstract Film and Color Music" in The Spiritual In Art: Abstract Painting 1890-1985 is quite good. Online, there's a good roundup of color organs and such in Colour and Sound: Visual Music by Maura McDonnell.
Greg Jalbert, author of Bliss Paint (for Mac, RIP) collected info on some artists at Tonecolor some time ago -- including the rarely mentioned "Raster Masters" who used to provide incredible live visuals at concerts in the Bay Area. Some say that the genesis of Trips Festival and rock music light shows was the Vortex Concerts at the Morrison Planetarium in San Francisco, organized by visual artist Jordan Belson and composer Henry Jacobs in 1957–59.
Motion Graphic Design and Fine Art Animation: Principles and Practice by Jon Krasner. Another excellent source, which includes streaming video, is "A Critical History of Computer Graphics and Animation " by Wayne Carlson.
Update: Motionographer posted some background info on Larry Cuba's 1977 computer graphics Death Star animation for Star Wars -- which was commented on by The iotaCenter's Cuba himself.
With "A" for effort and exposure, Jakob Trollback's visual music experiment with the song "Moonlight in Glory" by David Byrne and Brian Eno was shown at the 2007 TED Conference. More interesting stuff can be found at the Center for Visual Music and Cuba's iotaCenter, and in Internet resources like Motionographer, a recent AEP post Motion graphics 1961 (pictured right), resources for DJs, VJs, and others (including the aging Tonecolor), and summaries on video art (or searches) and artists like Bill Viola and Steven Beck (UC lecture QuickTime movie).
Even these links will just scratch the surface of related activities now happening in motion graphics, VJing, and visualization.
After Effects and the Velvet Revolution and in the nice history of motion graphics by Matt Frantz.
As noted, Motion Graphic Design and Fine Art Animation: Principles and Practice by Jon Krasner (Focal Press, 2004) has a great history of motion and film with clips, but see also Brand Identity for Television: with Knobs On, by Martin Lambie-Nairn, (1997 Phaidon) which discusses and illustrates the design process for short television spots.
A Composition of the “Things Themselves”: Visual music in practice by Maura McDonnell is a nice summary with bibliography. She also runs Visual Music, a blog that documents artists, filmmakers, composers, musicians, video artists, and events -- since 2005.Visual Music: Synaesthesia in Art and Music Since 1900, which has the same title as the museum show, and The Visual Music Film by Aimee Mollaghan (2015).
And let's not forget Cymatics, which was discussed in another post here on AEP.