What does it mean when a song 'interpolates' a melody from another song? 16

I love the explanation of interpolation. “We ask the license holder if we can use their song, they so no, so we just do it anyway without paying.”

To be clear, interpolation DOES STILL REQUIRE an agreement made with the original songwriters. They absolutely still get a PUBLISHING ROYALTY on the new song. Though the original writers are not always listed as songwriters on the new song, as you saw with the first Dua Lipa example. It depends on the deal that's made. Sometimes quite a lot of the pub from the new song is given to the writers of the interpolated song depending on how it's used.
Mind you this is ONLY from the publishing royalty. As opposed to sampling which requires you to give MASTER royalties to owners of the original recording. Interpolating allows the label or artist to retain full ownership of the master of the new song.

In journalism, interpolation is when the writer or editor adds/removes part of a quotation. It's usually done to clarify an oddly spoken sentence, or to cut out a part of a sentence that's unnecessary or confusing. To indicate an interpolation, you put an ellipses (...) where you removed words, and you put added words [inside brackets]. In journalism ethics, it's perfectly acceptable to use interpolation when quoting someone, but you're supposed to ensure your interpolations don't change the meaning or intention of the speaker's original entire quote.

I too remember hearing “Break My Heart”, having no idea about what interpolation was, and asking myself: “Where have I heard this before?”
And it seems interpolation is more common than one might think - Tchaikovsky’s “1812 Overture” interpolates the opening of the French national anthem (“La Marseillaise”), for instance.
However, what I’ve noticed is that from songs I enjoy that interpolate others, it’s a lot harder to notice - these musicians took another’s work, but they made it their own and changed it in such a way where it’s really hard to catch the similarity, even when trying to find it. For instance, apparently the Beatles’ classic “All You Need is Love” also interpolates the opening of “La Marseillaise”, but I was completely unaware until I looked it up. Most people who are at least familiar with the Beatles should know they do start singing “She Loves You” and “Yesterday” at the end.
Further still, Billy Joel rather famously interpolated the second movement of Beethoven’s “Pathétique Sonata” into the chorus of his 1983 song, “This Night”. Not only did Joel give Beethoven a songwriting credit for the song, but I recall listening to both at separate times for a years, and it wasn’t until I could put a name to the Beethoven composition that it finally clicked that Billy had worked it into his song, 180 years later.
I don’t think interpolation is necessarily a bad thing, but it needs to be done well enough to the point that the average listener doesn’t pick up on it. If they read into it and have any brains, they can then appreciate the genius of turning something already out there into something entirely new but familiar. As for Dua Lipa, she (and the other 800 writers of that song - what’s up with that?) didn’t make any kind of changes to the flow of INXS’s riff at all…

What kind of person redefines theft as interpolation?
Rick you’re the best.

Hey Rick, music teacher here! Love your stuff. From an education perspective interpolation comes from a system called motivic development which describes how a melody can be changed. Interpolation implies notes are added, the order is switched around, and in essence a new melody is created. Much like the 1990s version of ‘polyphonic ringtones’ these people have no idea what the word means! Would love to explain in chat with you :)

Remember Pink Floyd's 1971 album Meddle? The track "Fearless (Interpolating "You'll Never Walk Alone)" does credit Rodgers and Hammerstein for use of the melody sung or chanted by the fans of Liverpool Football Club. The crowd noise heard at the beginning and coda of the song is actually credited as an interpolation way back when!
Side note: This was how the song title was printed on the gatefold of the US release that I bought back in the mid eighties but the reissue leaves off the parentheses and listed as simply “Fearless”.

The Ten song “Dawn Star” from the 2000 album “Babylon” reused (probably by accident) the riff of Deep Purple’s “Burn”. Back then, the label insisted it be removed from the record only three weeks before the album release. Later though, it was included as a bonus track on the Japanese remaster of “Babylon”, and for everyone else, it’s available as part of Ten’s “Opera Omnia”. And I guess if all these other artists get away with a much more public “interpolation”, that’s just fair enough.

The earliest case of interpolation I recall (only just learned the term, though) is Madonna re-using the organ hook from ABBA’s “Gimme, Gimme, Gimme (A Man After Midnight)” on one of her songs, the name of which escapes me at the moment.

Interpolate has a VERY clear mathematical definition: to complete an incomplete data set based on surrounding data. For example: 2, 4, 6, __, 10. What is the missing number? Interpolation would say that 8 is the answer.
How this word got used in the context of "borrowing" melodies is beyond me. If they had said that they used tune A to INSPIRE tune B, then sure, maybe. But there is no sense where the first tune is incomplete and the second tune completes it. Just theft.

We were taught this term, interpolation, when I was in music school at IU in the late 1980s. It refers to way phrases could be extended, especially in the sung portions of a mass during the medieval period. I also heard composers use the term casually to mean, "a melody from another existing piece used as counterpoint to a new melody", a little like what David Bowie and Bing Crosby did with "Carol of the Drum". I not sure I ever heard the term used like that again after college, and I was completely unfamiliar with the term being used in popular music, and with this new meaning. Wild.

Vanilla Ice should have "INTERPOLATED" that Queen bass riff...

These modern artists and writers should try "interpolating" some integrity.

"Some Say" by Nea is an "interpolation" of "Blue" by Eiffel 65 -- that's where I heard the word first. The worst part is that since they seem to be getting away with it, there will be more of it.

A dillusional word, "interpolation". Great short and on point as always. Thank you Rick.

Interpolation has been around for decades, if you interpolate a song you have the credit the songwriters, no lawyers get involved unless you don't. A lot of 90s music is built on interpolations, especially late 90s hiphop and eurodance. It's not a new concept.

If you want some funny examples from the 90s, Pras' global megahit Ghetto Supastar from 1998 interpolates Dolly Parton's Islands in the Stream with Kenny Rodgers, and Vengaboy's song Boom Boom Boom Boom which was a number one hit worldwide in 1998 interpolates Abba's Lay All Your Love on Me.