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The level of my musical education leaves much to be desired, so I apologize in advance for maybe amateurish question. Several times I've encountered a situation when a composer repeats some piece in a song, keeping the overall shape of the melody but applying a small variation to it. It looks like an improvisation where some notes have remained unchanged, while the others have been shifted randomly or according to some rule unfamiliar to me.

Here's an example of what I mean. It is a music composed by Laurent Boutonnat to the Gourmandises (2000) album by Alizée, reissued as an instrumental album in 2019:

Alizée - Abracadabra (Instrumental Version)

In this example, please pay attention to the following fragments:

2:13 - 2:29 (main phrase),

2:31 - 2:47 (the variation).

Is there a term for such a technique (I think improvisation is too broad for this)? Are there common rules for this? I've found a similar question here and also found the 'Variation' article on Wikipedia, but they are too broad and I cannot figure out which parts of them apply to the above example and whether they are relevant at all.

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I think variation is the term to use. Ground bass is a specific kind of variation where the bass part repeats as the melody on top changes. While that description seems apt for the track you linked, ground bass suggests Baroque music, like Bach or Handel. Some may think it an inappropriate term for that reason, this isn't 18th century music.

In the track the main thing I hear is the first iteration melody is on the beat and the second is syncopated so it's most noticeably a rhythmic variation.

There aren't really rules about making variations other than keeping some aspect of the theme identifiable in the variations. Usually the basic harmonic pattern is maintained. Also, each variation tends to keep to one simple idea.

There is sort of a convention to gradually use shorter rhythmic values in the variations. Like eighth notes for a variation before sixteenth in a subsequent variation. Changing from major to minor mode toward the end of the set of variations is common. Putting one of the variations in a new meter is pretty common too.

But, again, that's about music from the 18th century and the focus was mostly on melodic decoration of a basic musical structure. You don't necessarily need to do that in pop music. Variation can be viewed as a general concept. Repetition with variation is a kind of musical axiom. In pop music that variation often comes from changes in instrumentation and dynamics.

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    Thank you for your answer. It really sheds light on the matter. Searching for the term Ground bass, I also found the more general term 'Ostinato'. – John McClane Jun 26 at 23:47
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I think what you're talking about is what's called a motif, or leitmotif, which is frequently used in film and video game scores. It's a theme (often a short melody) that suggests the arrival of a familiar, or otherwise preconceived, appearance, but through a musical conduit.

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