Sukay performs Music of the Andes.

They have had three major group formations. The first, under the musical direction of Edmond Badoux (now of Chaskinakuy), featured his then wife, Quentin, as well as Javier Canelas and Gonzalo Vargas (who would go on as a founding member of Inkuyo). This formation of the group released two albums, "Music of the Andes" in 1978, recorded in Quebec, and released on Flying Fish records; and "Pacha Siku" in 1979, recorded in Berkeley, CA, which was released on the Aural Tradition label, in association with the Vancouver Folk Music Festival. This group formation performed together until around 1985. This group formation focused on traditional Andean Village music.

The second major formation of the group started in 1985, when Edmond, Gonzalo, and Javier left the group, Carlos Crespo and Omar Sepulveda joined the group. This group formation focused on more contemporary mestizo music from the Andes, and released four albums on Flying Fish records, all recorded in San Francisco, CA - They were "Socavon" in 1985, "Tutayay" in 1986, "Mama Luna" in 1987, and "Huayrasan" in 1988. The group occasionally featured other musicians, such as Mario Lino Perez, who was a guest artist on "Tutayay". Carlos Crespo contributed most of the group's original compositions during this period. Additionally, they released "Sukay Instrumental," a compilation of instrumental pieces from this period, later in 1988. "Sukay Instrumental" was the group's first release on CD, though Flying Fish later re-released several of the group's recordings on CD. "Sukay Instrumental" has a second release after Flying Fish was acquired by Rounder Records, and the second release had a completely different track listing. This was done without the permission of Sukay. Rounder has also continued selling Sukay albums from 1988 and earlier without paying royalties to the group, maintaining that they were clearing out existing back-stock.

In 1988, Eddy Navia of Savia Andina started touring with the group, and shortly thereafter Carlos Crespo left the group, founding the group "Aymura". Omar Sepulveda had been recording with the group, but had long since stopped touring with them, and had joined Gonzalo Vargas as a member of "Inkuyo". Shortly after joining, Navia became the musical director of Sukay, and in 1990, they formed their own record label, Sukay World Music, and released the recording "Cumbre (The Summit)". This third major formation of the group included Enrique Corea, and Alcides Meija, also of Savia Andina. Carlos Hilario was touring with the group at this point, and also recorded on "Cumbre." This formation of the group focused more on the contemporary compositions of Eddy Navia, and largely abandoned the group's traditional roots. In 1991, they released "Return of the Inca" featuring David Grisman on two songs. Since then, they released several other albums, including "Navidad Andina," "Encuentros," and "Love songs of the Andes." At some point after "Return of the Inca," Enrique Corea stopped recording and touring with the group, and Eddy Navia's son, Gabriel, started recording and touring with them. The group's most recent album was released in 1996, and since then, they've been operating the restaurant "Peña Pachamama" in San Francisco, where they perform regularly. As of 2006, all of the group's recordings are out-of-print, though some continue to be available online.

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