Q. Is one allowed to have a telephone ringtone that’s a segment of a song with musical instruments?

A. In question 104 we mentioned that in regards to the avelus of the Three Weeks, which is usually more stringent than than Sefira period, there are Poiskim who permit listening to music that is not played for the sake of enjoyment or pleasure, as one who employs workers that listen to the radio as they work, or music that is played for a child to calm him to sleep. (Nitey Gavriel , Bein Hametzorim ch. 15, 8-13). Some Poiskim understand the music prohibition as only what conduces to joy and dancing, and permit classical music, played for soothing and comforting the soul. (Shoalin Vedorshim p. 245). However others disagree. See question 1395 above.
Igros Moshe (O.H. 3: 87) permits teaching instrumental music when it is done as a professional duty and not for enjoyment, See Naharos Eisan (p. 252) in regards to permitting singing and playing music for children in school.
Shoishanas Yisroel (Ben Hametzorim 1: 18) mentions that one should change the ringtone if it was recorded for the purpose of joy.
Horav Shlomo Miller’s Shlit’a opinion is that in the onset, one should avoid this type of ringtone the year around. However, once set it does not have to be removed.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a.