There are existing projects that allow the Uno and other devices using the ATmega8u2 to operate as a USB MIDI HID, such as MocoLUFA. Unfortunately, as far as I am aware, these require uploading new firmware that isn't compatible with programming via the Arduino UI.
Are there any similar projects for the ATmega32u4 (Leonardo, Micro)?
Is it possible to use an Arduino as a USB MIDI HID, while still being able to program it as normal?