closed as general reference by Jasper Loy, Barrie England, Andrew Leach, FumbleFingers, tchrist Jan 1 at 21:09
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Extemporaneously means, strictly speaking “without time to prepare”, though it has also acquired a secondary meaning of “without a prepared text or notes”.¹
Improvisation means an act of improvising. To improvise is to “act without foresight”, that is, without seeing in advance what one will do.²
These words are not always synonymous. Imagine a European classical musician asked to perform a set piece without warning. This musician will perform extemporaneously but without improvising. Now imagine a jazz musician asked to perform without warning. This musician will perform extemporaneously and also improvise the music itself.