Rinck was born in Elgersburg (in present-day Thuringia), and died in Darmstadt, aged 76.He studied with Johann Christian Kittel (1732–1809), (a pupil of Johann Sebastian Bach), and eventually became Kantor at the music school in Darmstadt, where he was also a court organist from 1813. He composed prolifically, and an organ primer of his enjoyed wide popularity.
Among his works is a set of Variations on ‘Ah! vous dirai-je, Maman’, Opus 90, published by Simrock in 1828. It is based on a tune made familiar by Mozart (K. 265) (and generally associated with the words Twinkle Twinkle Little Star).

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