Each playable character has an instrument. When you hit an enemy, that character plays a brief melody part on their instrument. If you combo another hit, another melody part will play, and another one if you time a 3rd hit. Then it just loops these melody parts (creating a melody), until you reach 16 hits or mess up.
Each time you are presented with the choice menu, where you can choose what to do next, the game randomizes the next tune for each character. By reverseengineering it, I found a code that sets the first party member's tune:
0201DB2A:XXXX, where XXXX is a number from 0 (first) to 5 (last). Any number above 5 will play sounds that don't belong there, which can lead to some interesting results, specially with Duster (since he's the last on the list, and from there on, are completely unrelated sounds)! If the code doesn't work for you, try setting the character as the only one in your party (using the debug room).
Bonus: use the 'always 16 hit' code,
A986DFE18532B3F6
7750E1A03FACA8E8
368FBAB54691351A
to listen to the tunes more thoroughly.

I tried to search the address for the other party members' instruments, but I couldn't find it. If it's just the standard party Lucas, Kumatora, Duster, Boney I can find the address for Kumatora and Boney (Duster's doesn't change, remember), but if the order is switched, even if it's Kumatora trading places with Duster, the addresses go wild. I couldn't figure out a pattern on such a subtle change in the party, much less adding characters like Salsa or Claus. So, at least for now, I can only provide the address for the first character, which seems to be the same address all the time.

Curiously, due to a glitch, Duster's tune isn't randomized like the rest it's always 0. Duster has 6 tunes like everybody else, and with this code, the unused tunes can also be heard. The folks behind the English patch said they don't plan on fixing this on a new release.

I used the Debug room to change characters and recorded the 3 notes of each tune of each character with Zombeat. AFAIK, these melodies where made in 120BPM, which is Zombeat's tempo.

Lucas' keytar: 0:00
Kumatora's electric guitar: 0:13
Duster's bass: 0:28
Boney's bark: 0:42
Claus' sitar: 0:56
Flint's saxophone: 1:11
Salsa's synthesizer: 1:26 (or cuíca, as pointed out by TheSaxophoneStar)

At the time I made this video, I forgot about the Fake Bat and the Fake Frying Pan for Lucas and Kumatora, respectively; those two have their own instrument. I made a video about those afterwards:    • Fake Bat and Fake Frying Pan tunes  ...  

Update: Jeffman patched Duster's missing songs! http://forum.starmen.net/forum/Games/...