Seeing as Penn very specifically cautions applicants to be discerning in their submission of supplementary materials, I’m torn as to whether I should include an additional rec from my band teacher. It’s already written- I used it in an application for a summer program. It is concise, very effusive, insightful, and kind, and beautifully written. (My teacher showed it to me even though I (verbally) waived my right to see it). Should I include this?
One supplementary recommendation, especially one that is complementary to your academic recommendations, is fine. Your are following the “discerning” criteria since the supplementary recommendation is brief and it shows a side of you that the other recommendations do not.
As @fogcity said, if the recommendation shows a side of you that the other recommendations don’t, and if it highly praises you, you should send it. I sent an external recommendation from a previous employer that I’m sure helped me get into Wharton.