See the Pri Megadim A.A. on O.C. 551:2. That is the basic Makor for our Minhag not to listen to music. See the Sh’arim Metzuyanim B’halacha on the Kitzur 122:2 (with the Kuntros Acharon) and 126.
The Beis Yosef in O.C. 560 discusses the Halachah about music all year round. The Minhag almost everywhere is to be Meikil for “Shiros V’Sishbachos” (560:3). The Beis Yosef also mentions that to be Mezarez Bim’lachah or to remove depression is okay. The Poskim bring those two ideas down in several places.
See the Tzitz Eliezer quoted above (15:33) and the Igros Moshe that he quotes (I think O.C. 1:166 or so). But, as I said above, those two base their Issur of a capella for Sefirah off of the idea that you don’t listen to any music all year round. I didn’t see the Igros today, but I don’t think either of them mentioned the issue of being M’zarez Bim’lacha either way. Still, since neither explicitly disagrees with the Beis Yosef there is no reason to assume that they do.
Chacham: From the way the TZ”E Paskened like R’ Moshe for Sefirah I got the feeling that Os 1 was a Limud Z’chus.