Q: Syncing iPhone with other iTunes library

What if iPhone was synced with computer that broke? How can I sync it with new computer/iTunes without losing content previously synced with other iTunes? I have a different situation. My iPhone 4 (iOS 6) was synced with my Mac mini. I've made clean OS X reinstallation. And now, of course I cannot sync that iPhone with fresh iTunes library without losing contet which was already synced previously with other iTunes library (on the same mini!). It all should be easy but it is not. I sometimes think I should choose some android device. I don't wanna steal anything, I don't wanna sync purchased music or movies with another computer. I just would like to safe voice memos. Any idea?