This got interesting.

The pulsing WAS caused by the iPhone. All my songs have started doing it on my iPhone. Even songs that I used these same headphones for in the past that sounded great. It took me a while to notice this new development, because it only is apparent on fairly smooth songs with a slow background.

I turned the phone completely off and back on. And it didn't go away.

So I did a forced phone hardware reset (hold power and front button for half a minute until it forces itself off).

That fixed it. No idea what/why/how, but doing a forced reset fixed the music pulsing issue.