Extemporization is the name of the act — speaking, playing, or making a plan with no prepared script — and sometimes the name of the thing you made that way: a stretch of talk, a passage of music, a quick fix when the original plan falls apart. It is a formal noun, at home in writing about music, teaching, and public speaking. American spelling uses z; British writing often has extemporisation. Same word, same meaning. Though actually, it is not only “a mess made in a panic.” A skilled pianist’s extemporization can be careful, beautiful work done without a written score. The everyday noun is improvisation, which most people will reach for first.