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The "Nine Presentations" refers to the sequential offerings made during a grand sacrifice—such as those to Heaven or Confucius—where music is played during each stage of the rite, from the welcoming of the spirits to their departure.
Historically, the "Four Barbarians" (Siyi) represented the peoples surrounding ancient China. Including their music in state rituals symbolized the universal reach of the Emperor’s harmony and the submission of all corners of the world to ritual order.