- Because it is played via a keyboard, the organ can allow for more musical complexity. For brass and woodwind instruments, you're limited to a single note. String instruments can play as many notes as the number of strings they have, but that still pales in comparison to keyboard instruments because keyboard instruments can simultaneously play chords and a melody but string instruments cannot. Additionally, they can play a melody with a bass accompaniment or simultaneously play multiple parts, as in a fugue. As far as playing multiple parts on a single instrument at once, strings are better than brass or woodwinds, but the organ is still greater.
- Organs have a longer sustain than most instruments. The length of the notes it plays isn't bound by the player's physical limitations, unlike brass or woodwind instruments, which can hold a note only as long as the player has breath left. Or string instruments, where the string's vibration eventually ends.
- Because of the different stops, organs can sound a bit like other instruments. The mellotron is often cited as the first synthesizer, but organs have been doing the same sort of thing for hundreds of years. Different stops are meant to sound like strings or horns. Different playing techniques can alter the sound of other instruments to some degree (like strings playing pizzicato), but I wouldn't say that they're as drastic as the changes the organ undergoes when playing with different stops.
If you want to get the most complexity out of a single instrument, organ is the one to play.
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