Note permutations

The equivalent of a writer staring at a blank page, wondering how to fill it, is a composer staring at 88 black and white notes.* The composer is wondering how to arrange them into a pleasing sequence when so many great melodies have already been written – perhaps they’ve all been taken!

(*Actually, a more accurate analogy would be a writer staring at the 26 letters wondering how to arrange those, but strangely that’s not how the saying goes.)

So, to counter the fear of there being no new melodies, I thought it would be interesting to examine the permutations available to a composer writing a one note melody.

AND IT TURNS OUT THAT THEY ARE MANIFOLD!!!

1. Two note melodies.
How many two note melodies can be written within an octave?
Easy, they are:

C -> C C’ -> C
C -> C# C’ -> C#
C -> D C’ -> D
C -> D# C’ -> D#
C -> E C’ -> E
C -> F C’ -> F
C -> F# C’ -> F#
C -> G C’ -> G
C -> G# C’ -> G#
C -> A C’ -> A
C -> A# C’ -> A#
C -> B C’ -> B
C -> C’ C’ -> C’

Notice how the final C’ – C’ is struck because it’s a duplicate of C-C. Although the notes are different, the ‘melody’ is the same, and I’m interested in relative pitches here, not absolute.

=25

Twenty-five exciting melodies!

2. Three Note melodies

Imagine my surprise to discover that there are 469 ways of arranging a three note sequence! Quite a lot more than the 25 previously. This number was arrived at using the following rules:

1. All melodies must be fully contained within an octave (C-C’), as very few melodies jump more than an octave in one leap.
2. All chromatic notes can be used; I haven’t restricted to any particular key. (No accidentals would make for dull music!)

3. Duplicates are excluded as we’re interested in melody patterns, not absolute notes. So for example C,G,G, is the same as D,A,A as they are the same melody, but the latter is played a tone higher.

Here’s a grid showing how the three note permutations work. [previous grid was wrong – watch this space!)

It turns out the best way to calculate the number of permutations is to calculate the total number of permutations for the number of notes and sequence length, and then to subtract the number of duplicates. (Remember the ‘unison’ melody from the first step. There is an increasing number of duplicates as the note sequence length increases). Have a look at the grid to see this in action.

This basically gives us a nice easy geometric formula:
(N^S) – (N-1)^S

– where N = number of notes in scale = 13
– where S = number of notes in sequence

Plugging in values of S from 2 (a two note sequence) to 10 (a ten note sequence) gives the following table:

 

Notes in Scale (N) Melody length (S)

N^S

(N-1)^S (N^S)-(N-1)^S Easier to read format:

13

2

169

144

25

25

13

3

2197

1728

469

469

13

4

28561

20736

7825

7,825

13

5

371293

248832

122461

122,461

13

6

4826809

2985984

1840825

1.84m

13

7

62748517

35831808

26916709

26.9m

13

8

815730721

429981696

385749025

385m

13

9

10604499373

5159780352

5444719021

5.4bn

13

10

137858491849

61917364224

75941127625

75bn

And this doesn’t even begin to take into account the fact that the same sequence can be played in a multitude of different rhythms, and never mind the therefore near infinity provided by harmonisation, orchestration, tempo, or heavens above – bringing in a new counter melody!Which gets to a fantastically large number!

So, I think the message is: there is no excuse for writers’  block!

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