Every pipe organ in every concert hall and church in the world is essentially built to order and, of course, immobile. Each is also usually conceived with a specific repertory in mind. Important touring organists must adjust to each instrument and locale, which can be fascinating, frustrating or both.

Like most organists, Cameron Carpenter, the 32-year-old superstar, who combines a punkish persona with tremendous skills and an admirable way of connecting with audiences, has spent a lot of time playing on vastly different instruments here and there. On Sunday at Alice Tully Hall, he realized a fantasy with the world premiere of his International Touring Organ, made by Marshall & Ogletree, organ builders in Needham, Mass., who, as Mr. Carpenter has said, are redefining the digital organ as an instrument of artistic significance.

This one was designed specifically for Mr. Carpenter to be his own portable instrument. In building it, Marshall & Ogletree sampled sounds from a wide range of traditional pipe organs, including some of Mr. Carpenter’s favorite instruments around the world.

The International Touring Organ is not your typical, compact digital instrument, as the audience for Mr. Carpenter’s Sunday afternoon program (the first of two that day) found out upon entering the hall. In the rear of the stage was a row of nine cabinet-like boxes, each with four vertical speakers. Behind that row were four large additional speakers.