On July 31, 2013, Holly Bowling had a profound experience. She was at her 221st Phish concert, when the jam band, playing at the Lake Tahoe Outdoor Arena in Nevada, opened their second set with a 37-minute version of their song “Tweezer.” “I couldn’t get it out of my head,” Bowling, 31, recalls.

But get it out of her head, she did. Bowling, a classically trained pianist with a degree in piano performance from San Francisco State University, started to meticulously transcribe and arrange the heavily improvised number, transforming...