This is the Empyrean instrument, upgraded via Trials of the Magians from the Pyf Harp, bought from the Cruor Prospector in Abyssea - Altepa for 20,000 Cruor.
Requires Iron Plate x50, Colorless Soul x75, and Apademak's Horn x75 to fully upgrade.
Because it hasn't been metioned yet. The harp only needs to be on when applying the first round of three songs. After that you can use other instruments.
The "increase song effect duration" is similar to Aoidos' Hongreline +1/+2 and Aoidos' Matinee in that it only affects the duration of buffs. So I guess I can't ditch Nursemaid's after all.
Also, the increase in duration appears to be 25% on buffs (2:00 to 2:30, 2:36 to 3:06).