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Which fugal techniques did Beethoven use in the Hammerklavier Piano Sonata?

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In the finale of his Piano Sonata in B-flat Major, Opus 106, commonly known as the Hammerklavier, Ludwig van Beethoven employed several sophisticated fugal techniques to push the form to new extremes. These include:

  • Stretti: The use of overlapping entrances of the fugue subject.1
  • Inversion: Moving the melody in the opposite direction.1
  • Augmentation: Lengthening the note values of the subject.1
  • Cancrizans: A seldom-used technique where the fugue subject is written backward, note for note, often referred to as "crablike" motion.1

Additionally, Beethoven structured this fugue by dividing it into distinct sections that feature changes in key, meter, and tempo.1