Music on Nine Days

I understand the Halacha regarding the prohibition of listening to singing accompanied by instruments during the nine days before Tisha B’Av. But what I don’t understand is why it is permissable to listen and enjoy singing not accompanied by instruments. If listening to singing can be pleasurable, what is the logic of permitting it during the nine days?

There is no prohibition to enjoy during the nine days. There are a set of mourning practices our sages instituted in addition to those which evolved over the generations, all in order to actively remember the destruction of Yerushalaim and feel the lack of our temple. Among the mourning practices is refraining from music. Some authorities understand "music" to include any type of music, including that mentioned in your question. Others limit the prohibition to music only, whereas non instrumental recordings are permissible especially when the songs are in accordance with the spirit of these days. I think that if you feel uncomfortable with the lenient approach you might as well follow the strict one.