How many melodies are there? | plus.maths.org
The equivalent of a writer staring at a blank page, wondering how to fill it, is a composer staring at the 88 black and white notes on a piano wondering how to compose a melody that's never been heard before. How can one possibly take the eight notes of a standard scale and write a brand new melody when so many great melodies have already been written? Perhaps they've all been taken!
So, to counter the fear of there being no new melodies, I thought it would be interesting to examine the number of melodies available to a composer looking at his blank stave to see how many there potentially are.
We will tackle this problem by starting with the simplest possible melody — one consisting of two notes — and then building up the melody length one note at a time until we see a pattern that can be turned into a formula.
Now, you might be wondering why combinations like G# - F and E – A are not included. This is because we are only interested in relative pitch, not absolute notes. For the purposes of this exercise the melody C – C is identical to D – D or G – G as they are all unison melodies (i.e., they have 0 as their pitch difference). That's why we won't count the unison melody C' – C' — unison was already covered in the first table by C – C.
