It is a human voice, insists McGregor at the start. She, like Yma Sumac, is a coloratura soprano, and the sounds she conjures up with her voice — on behalf of her alter ego Sumac — are so astounding it’s good to be reminded.

Growling, purring, trilling, McGregor swoops — seemingly effortlessly, and with pinpoint precision — from a low baritone to a register so high her voice becomes a crystalline whistle.

Sumac’s extraordinary vocal range was rumoured to extend across five octaves (an opera singer usually manages three) and McGregor replicates this with such astonishing skill that during her rendition of “Chuncho”, a song which echoes Sumac’s childhood spent singing to birds and trees among the Peruvian mountains, you could have heard a pin drop.

The charm of this performance lies in the interweaving of Sumac’s stories with McGregor’s own. The Cabaret Festival artistic director tells how she came to be fascinated by Sumac, recounting how she connected with the Peruvian singer, who died in 2008, through her long-time assistant and companion who lives in LA, where the star spent her final years.

Managing her legacy, the assistant had arranged a sale of Sumac’s effects, and McGregor bought the last remaining box. To reveal what she does with its contents would spoil the surprise, but it’s a device McGregor uses with great effect to invoke the spirit of her hero.

Moving from Peru to New York in 1946, Sumac toured the States with her husband Moisés Vivanco and cousin Cholita Rivero as the Inca Taqui Trio. During this time, Sumac’s life resembled, as McGregor points out, that of a Peruvian soap opera, with Vivanco fathering twins with her assistant, he and Sumac marrying and divorcing twice over, and tensions leading at one point to a public brawl.

Meanwhile, Sumac was signed to Capitol Records in 1950 and became the best-selling artist on its books, outselling even Bing Crosby. Despite this, and despite samples of her music being used more recently in commercials for Coors and Kahlua, in a German techno track and in The Black Eyed Peas’ “Hands Up”, her name is little known. Most of the earnings went to Vivanco and her producers.

McGregor unpacks Sumac from the myth Hollywood buried her under — the story that she was an Incan princess, for example — and from the generic Latin-American exoticism that post-war America hungered for.

Behind the dazzle and mystery was a woman, she says, who was denied her own agency. She was more than just a novelty act. Like her flamboyant jewellery, Sumac was brilliant and multi-faceted.

McGregor’s impressive tribute returns some of that agency to her. This is an unforgettable show. I suspect tickets will be selling fast, so be quick.

Yma Sumac — The Peruvian Songbird is at the Dunstan Playhouse until June 16. See more Adelaide Cabaret Festival coverage here.