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At first, of course, it was English composers (Matthew Locke, for example) who provided the songs and atmospheric music that the plays require. Then, after the great 18th-century revolution in taste that made Shakespeare common cultural property, composers of all countries and styles started writing Shakespeare music.
Some of the pieces have been wowsers. Ambroise Thomas wrote a Hamlet with a "happy" ending - Hamlet is alive at the end, and becomes King of Denmark. In Shostakovichs King Lear, one of the Fools tunes turns out to be Jingle Bells. In Rossinis version of Othello, Othello confronts Desdemona to a jaunty little tune that also shows up in The Barber of Seville, which lends the whole thing a certain air of Bugs Bunny and "Whats Opera, Doc?"
There are also some interesting what-ifs. For years, Verdi wrestled with the idea of a King Lear opera - fruitlessly. Mozart had some familiarity with Shakespeare, but it never bore musical results. Its tantalizing, though, to think of a Mozart Twelfth Night or Tempest. But on Shakespeares birthday, lets concentrate not on the obscure, but on some of the best-known classical pieces inspired by his plays:
Wedding March from A Midsummer Nights Dream
Tchaikovsky: Romeo and Juliet
Williams: Fantasia on Greensleeves
West Side Story
Pomp and Circumstance March No. 1