Are there any standard exercises where you play a chord progression with one hand while playing a scale with the other hand?
I'm adding the usual I–IV–I–V–I at the end of each scale (forcing myself to identify the actual chord name as I play it), but that doesn't give me any practice with the other chords (let alone any important 7th chords, where I'm very weak). Nor does it exercise the practice of chords on one hand with melody on the other hand.
I could come up with my own chord progression, but it would be nice to have something with pedagogical value rather my meager attempts at cobbling together some chords that just rotate through the possibilities in a vaguely meaningful way. I would imagine someone has put together something using common cadences or turn-arounds that are commonly encountered, etc.
After 15 years, I've finally dusted off my old piano book and am re-learning to play the piano (and my prior experience is just a single semester undergraduate course!) I was quickly able to get back up to speed with the simple five-finger position exercises/songs from my old piano book. To force myself to move beyond these simple/comfortable pieces, I started playing some familiar christmas carols and hymns. I then got distracted for a few months learning about chord progressions, chord inversions, roman numeral notation, etc. and I had some fun transcribing several carols and hymns (which are written for 4 part vocal harmony - currently rather difficult for me to play) into the more simple right-hand melody + left-hand chord progression (great way to force myself to learn LilyPond).
It's clear that the only way to improve my technique is to put some serious time and effort into scales. I've spent the past few weeks working on scales and I'm making great progress. However, I would like to use my time practicing scales to also practice the important chord shapes and progressions in each key as well as to learn the chord progressions in terms of the actual chord names (I've been relying on roman numerals to just play chord shapes, ignoring what the chords actually are).