Graphic music visualizer without after effects?
A free alternative for After Effects is Jashaka. It runs on every plattform and has some decent features. While it wont have the comfort of after effects for creating such an effect its certainly possible.
If you just don't want to spend all that much but still willing to pay a small price you can go for Apple Motion which costs 50$ and is a fairly powerful tool for animation and motion graphics for the price.
https://www.apple.com/final-cut-pro/motion/
Also here a tutorial on how make something react to music in Apple Motion: http://www.larryjordan.biz/motion-5-make-audio-move-video/
I personally have not worked in detail with both software so I can not give you guidance for actually creating the animation, expect for this tutorial.