Lots of people who should be, but are not interested in Western classical music would probably have their brain tickled more if they saw the "visual pattern" appeal of many interacting musical voices. As I was searching YouTube, I came across the videos of a guy who uses an animation program to visualize the melodic and harmonic aspects of several pieces. These aspects are implicit in sheet music, but you have to be trained to read musical notation, whereas his animations are incredibly intuitive. A trained musician or music lover might be able to hear all of the different voices and their interactions, but in order to reel in those who aren't already aficionados, something less demanding has to do the job.
For example, one reason why classical music snobs (of whom I'm not one) deplore much popular music, including film music, is that it lacks the richness that only comes from interacting voices, as opposed to the single line of melody, perhaps backed up by slavish accompaniment, that dominates most popular music. Since those who don't prefer classical music don't get what the "richness of interactions" argument is about -- which is understandable, given how vague a lot of the explanations are -- here are some visuals to the rescue. Let's start off with a simple example of counterpoint (a common method of producing interacting voices), namely a round:
The four lines are going in their own particular directions, but at each point in time they harmonize or produce agreeable sounds. However, the "particular directions" are really the same pattern, only shifted a little to the right as each "new" voice is added. As the lines go by, you can actually see this "same voice but shifted rightward" idea on the screen at once: watch the 1st and 3rd lines or the 2nd and 4th lines. So what we have is a single quirky pattern, then several copies of that pattern, and then shifting these copies to dovetail with the original and with each other. In other words, it's like a tessellation (some examples). Although popular among geeks, M.C. Escher prints ain't exactly the summit of visual art for at least three reasons: first, they don't appear to express much. And they are easily copied -- you can make your own tessellations with "how-to" websites on the internet. Furthermore, it's hard for them to sustain the audience's interest for very long after the initial appeal of "Wow, look at that repeating pattern!" has waned. Still, much more interesting than comic book or billboard art.
Let's have a look at more complicated examples from Baroque counterpoint, in which the two voices aren't just independent shifts of each other but are distinct patterns:
Scarlatti, Sonata K. 455
Bach, Toccata & Fugue in D Minor
Off the top of my head I can't think of a good art or math analogy for what the difference is between these and the round , but consider a dancing / martial arts analogy. Fill an otherwise empty room with people, so that each one is about three feet from their neighbors (let's say the room has a chessboard grid on the floor, with each person in one square). It would be pretty hard for each person to swing their fists without at least someone (probably lots of them) getting injured. The round's basic solution to this is to have everyone do the exact same thing -- say, stick your arms out in an "on the cross" pose, and spin your body in place -- but in such a way that the parts dovetail (i.e., in the opposite clock direction as your neighbors; try this with pens or toothpicks if that's not clear). So, when my neighbors have their arms pointed at me, mine are pointed perpendicularly to theirs, and by the time my arms are pointed at them, they have turned their arms to be perpendicular to mine. From the ceiling, it would look like a bunch of gears were turning, each gear with two teeth (a person's arms). No injuries! Cool -- but still, a pretty boring sight.
The voices in the Baroque counterpoint, especially the fugue, are like capoeira dancers, each of whom performs his own complicated dance moves, and who are doing so in close proximity to each other, but who have coreographed the dance so that they never hit each other. That's damn hard, and therefore it wouldn't look boring at all. Pedestrians could watch all day. Now, imagine four people doing this in close proximity, for nearly an hour and without repeating that much, and then you get a hint of the genius that produced The Art of the Fugue.
Even in the realm of popular music, some compositions are animated by more counterpoint than others. To provide two examples, consider the bass playing of Paul Simonon of The Clash vs Dee Dee Ramone of The Ramones. Or take the video game music in Castlevania 2 vs Gradius. The former parts sound richer than the latter parts since the accompanying voices don't merely "prop up" the main voice but instead do their own thing. Audiences readily recognize which music is richer than the other, so the potential is there for a better popular appreciation of what Western classical music has to offer. And with the more widespread use of visually intuitive aids such as the animation program used to generate the videos above, more potential art music lovers will become actual art music lovers.
 Here is my attempt. The idea is that each shape does something unique, but that they all blend together continuously -- leaving no gaps. The "no gaps" condition isn't so hard to meet with a tessellation, but it gets tricky when you use different, irregular shapes. For example, imagine trying to make a circle and square dovetail -- the aesthetically unpleasing, "pull it out of your ass" solution would be to inscribe one inside the other, and fill in the gaps with shapes that were tailored (stipulated) to fit them. That strategy is for Procrustes. I can't think of a good example on-the-fly, again since it's so hard, but it would look like three or four wacky 3-D shapes that meet together in a gapless knot.