Born in Dresden, Herchet grew up as the son of a driver and a worker in modest circumstances.
From 1962 to 1965 he studied musical composition at the Hochschule für Musik Carl Maria von Weber in Dresden with Johannes Paul Thilman and Manfred Weiss, cello with Clemens Dillner and piano with Ilse Brähmer.
The use of a Franz Kafka text in his composition Interfragmentarium zum werke von franz k. für klavier und alt led to distortions with the Hochschule, whereupon Herchet continued his composition studies at the Hochschule für Musik "Hanns Eisler" in East-Berlin with Rudolf Wagner-Régeny from 1967 to 1969.
After Herchet first worked as an assistant in the book trade and received organ lessons from Gerald Stier and Herbert Collum from 1969 to 1970, he became a master student of Paul Dessau from 1970 to 1974.
His students include Thuon Burtevitz, Michael Flade, Reiko Füting, Johannes Korndörfer, Sergej Newski, Tobias Eduard Schick, Bernhard Schneyer, Theodor Schubach, Karoline Schulz, Johannes Voit, and Lydia Weißgerber.