The requirements for this are a little unclear, and it took me a while (long after I'd listened to hundreds of songs from this section) to find a sure fire way to increment the counter.
1. Go to the Top Music twist in Xbox Music.
2. Hit the "All Top Music" button on the left.
3. Select a specific genre, like Pop.
4. Hit the Play button to the left of the genre dropdown.
5. Once the song starts to play, back out to the All Top Music list again. You do not need to listen to the whole song.
6. Select a different genre that you haven't chosen before.
7. Go back to Step 4 and repeat.
You'll need to do this ten times if you're starting from scratch.
(Thanks to VidGamiac for making me realize I'd miswritten a step above!)