The short answer is – they did. Mute was founded by a gentleman by the I didn’t mean to push all your buttons I was just looking for mute shirt name of Daniel Miller in 1978. He was a musician who performed and recorded under the name of Normal Essence and founded Mute Records as a means of distribution for his material through Rough Trade record store (store) group that year. Depeche Mode sought to contract with a brand for several months after its establishment and drew attention from Phonogram and other major labels, but after it was approached by Miller in September 1980 (at a the concert they were playing in Camden Town, North London) and seeking his support for electronic / synth based music, chose Mute subscriptions instead.
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Their first single My Dream was just a small hit, but the second – New New Life – broke UK top 20 and saw them appear on the I didn’t mean to push all your buttons I was just looking for mute shirt prestigious BBC music show on top of The Pops punches. Neither the band nor the brand looked back from that point onwards, with Mute remaining one of the leading independent people on the market for the next two decades (adding Yazoo, Nick Cave & The Bad Seeds and Erasure  while the band has risen to worldwide stars. Mute was later acquired by the EMI team in 2002, but since 2009, it has once again become an independent brand (thanks to EMI’s financial conflict, and the subsequent breakup).