Just wondering. It was a little funky and strange. (It was supposed to be) but it also worked. Someone had to write it. I couldn't tell from the credits.
The songs in question, "Cantina Band 1 & 2," were written by John Williams and performed by the London Symphony Orchestra.
Williams was the composer for all the tracks in the movie.
I think this is the one you are looking for:
This is by the artists: Figrin D'an and the Modal Nodes
In-Universe we now have canon confirmation from the short story Not For Nothing that the music was an original composition by Bith composer and bandleader Figrin D’an and played in the cantina by his band, Figrin D’an and the Modal Nodes.
The reason should be obvious: Figrin D’an is the best composer and bandleader you will ever find this side of the galaxy. We knew that if we left, we’d never be able to find another leader quite like him. When we’re getting thrown into shackles or washing sand out of our eyes, it’s hard to appreciate him. When we’re playing, nothing in all the worlds is better.
As we played another one of D’an’s newest songs (captivity had been strangely inspirational to him, I have to admit), I thought about those droids and the stormtroopers that searched for them. I thought about a possible reward. And then I decided that if that human could do us a favor without us knowing him at all, we could do those other humans a favor and not report them. We were safe now, and could keep playing.