interpolate

interpolate \in-TUR-puh-layt\, transitive verb:

1. To alter or corrupt (as a book or text) by the insertion of new or foreign matter.
2. To insert (material) into a text or conversation.
3. To insert between other elements or parts.
4. [Mathematics] to estimate a value of (a function) between two known values.
5. To make insertions.

Twenty years earlier, Rodgers was not so pleased when, at the request of the star Belle Baker, Berlin had written a song for her to interpolate into an otherwise all-Rodgers-and-Hart score for the Broadway musical "Betsy."
-- Richard Corliss, "That Old Feeling: A Berlin Bio-pic", Time, December 30, 2001

The staging is by Peter Konwitschny, one of Germany's most progressive directors, and the controversy derives from his decision to interpolate an on-stage disruption that breaks the score at a crucial moment and leads to an additional scene of dialogue.
-- Tim Ashley, "Wagner interrupted", The Guardian, November 23, 2002

A new cover might be designed to replace the original one if a song sold well enough to warrant further printings, however, particularly if the piece had been taken up by a popular performer or interpolated into a show.
-- Charles Hamm, Irving Berlin: Songs from the Melting Pot

To that end, contends Judis, echoing the Progressive-era founder of the New Republic Herbert Croly, we need strong, multifaceted federal institutions that set rules, impose limits, and interpolate themselves into nearly every sphere of society.
-- Chester E. Finn Jr., "The Paradox of American Democracy", Commentary, March 2000

Interpolate comes from the past participle of Latin interpolare, "to polish up, to furbish, to vamp up; hence to falsify," from inter-, "between" + polire, "to polish."