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Is there any insight or documenation on this? I think it's interesting that you can tell straight away from the sound if the music was made by Capcom or Konami, even if the composers were completely different across several games from the respective developers. Did they all use the same internally developes tools with preset instruments or something, or was it a separate coder responsible for implementing the music from the composers, who remained the same for every title?

Capcom music usually sounded technically simpler than Konami. There's also two distinct "eras" of Capcom NES music with different styles. Their 80's sound was usually harsher with less dynamics in the tones, and there was always this tiny gap between each note. This changed some time in the early 90's as the note cutoffs were eliminated, the tones generally decayed in volume more and there were the occasional tremolo-style notes.

Konami of course made use of DPCM which Capcom didn't (is there any Capcom game at all that uses DPCM?), and they had more tricks like using duty cycle changes in the transient/attacks to get more complex plucky timbres.

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Well for a start I'd imagine they had their own company sound drivers, that will be an integral part of the "sound" of those games.  Even similar instrument patch setups would behave differently depending on the code.