 If you liked
Carole & Tuesday
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 ...then you might like
Rock wa Lady no Tashinami deshite
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In both anime the protagonists are young girls making their own music they love. Both anime series have music competitions and some character development.
While Rock wa Lady no Tashinami deshita is more a school-slice of life type of anime with less deep topics coming up, it still has quite a few questions and takes on the lives of children of rich families. Carole & Tuesday on the other hand is set in a more futuristic worl and the girls are more free to spend their time, while having more serious obstacles.