Sticking with the Rhapsody on a theme of Paganini (see last week), people often wonder how what is undoubtedly the most famous bit of the piece, the 18th variation, is derived from Paganini’s theme.

How, in other words, do we get:

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from

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The technical name for what happens is ‘inversion’ and it’s to do with the pitch of the notes, as distinct from the rhythm; here’s how it works:

The first phrase of the Paganini theme you heard above looks like this:

Now, if we invert it i.e. turn it upside down (go down the number of semitones the original went up and, conversely, up when the original went down) we get:

which makes a noise like

[…getting there.]

By the time he’s at his 18th variation Rachmaninoff’s got (by a pretty logical process) from a start in the key of a-minor into the key of D-flat major (5 flats). Let’s transpose our inversion into that key and slow it down a tad so we can hear what’s happening:

[…closer still, but not right.]

You’ll notice that there are two notes that don’t fit in D-flat major (they’re the ones with the extra flats):

Let’s normalise them, so they fit in the key, and see how that works:

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…and now it really does, pitch-wise, sound like the 18th variation. As I explained, inversion is a question of pitch. So, change the time signature into 3 rather than 2, fiddle with the rhythm a bit (like leaving out the first two repeated A-flats and some other notes)

and you get:

[…et voilà!]

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