Interpolation refers to different things in classical music and hip hop music.
Interpolation (also known as replayed), especially in 20th-century music and later, is an abrupt change of musical elements, with the (almost immediate) resumption of the main theme or idea. Pieces that are cited as featuring interpolation, among other techniques, are Music for Brass Quintet by Gunther Schuller and Threnody to the Victims of Hiroshima by Krzysztof Penderecki, both 1960-61.
This device is commonly used to extend what would normally be a regular phrase into an irregular and extended phrase. Such expansion by interpolation is achieved by the addition of extra music in the middle of a phrase (commonly through the use of sequence). A clear example exists in the second movement of Mozart's Piano Sonata No. 10, K.330.
Formerly, in the sung portions of the Mass, such as the introit or kyrie, it was permissible, especially during the medieval period, to amplify a liturgical formula by interpolating a farse (from Medieval Latin farsa, forcemeat),[clarification needed] also called trope. This might consist of an explanatory phrase or verse, usually in the form of an addition or paraphrase, often in the vernacular.
In the classical suite, consisting strictly of the allemande, courante, saraband and gigue, composers often interpolated a gavotte, bourrée, minuet, musette or passepied.
In hip hop music, interpolation refers to using a melody--or portions of a melody (often with modified lyrics)--from a previously recorded song but re-recording the melody instead of sampling it. The method is often used when the original artist or label declines to license the actual sample, since sampling already released music is subject to compulsory licenses.
Example: "Ghetto Supastar" by Pras features a hook sung by Mýa that was originally written for the song "Islands in the Stream" by Kenny Rogers and Dolly Parton. The song "I'll Be Missing You" by Faith Evans and P. Diddy is an interpolation of "Every Breath You Take" by the Police. Frank Ocean's "White Ferrari" interpolates the Beatles' "Here, There and Everywhere." Trippie Redd also uses a method of interpolation throughout the hook in his song "Dark Knight Dummo" (feat. Travis Scott), which is reminiscent of Trippie's feature in XXXTentacion's F**k Love.