Semitone Scales Within The Octave
June 30th, 2007The musical octave is divided into 12 semitones or the following musical notes: A, A sharp(B flat), B, C, C sharp(D flat) D, D sharp(E flat), E, F, F sharp(G flat), G, G sharp(A flat). A recent discussion of musical scales with a friend led us to work out something I have thought about a few times before. Where 1 <= n <= 12, take a scale to be defined as a collection of n unique notes within an octave. For example, the common major scale has n=7. Now, my question was, what is the total number of scales which can be derived based on this definition? In other words, how many 1 note scales are there, + how many 2 note scales are there … + how many 12 note scales are there?
Some of these are straightforward. For example, the number of 12 note scales is simply 1, the chromatic scale. The number of 1 note scales is simply 12; there are 12 possible notes to choose from for a 1 note scale. Between 1 and 12 though, things become more complicated. However, it does not take much to realise that this is a problem which can easily be addressed by using resources from the mathematical field known as combinatorics. In particular, we are interested in a formula which deals with combination without repetition; when the order does not matter and each note can be chosen only once. The formula in question is the binomial coefficient or “choose function”:
where n is the number of notes from which you can choose and r is the number to be chosen.
So, plugging in the value 12 for n and 1 to 12 for r, and adding up all 12 results, the answer is obtained. Here is the breakdown:
1 note scales = 12 choose 1 = 12
2 note scales = 12 choose 2 = 66
3 note scales = 12 choose 3 = 220
4 note scales = 12 choose 4 = 495
5 note scales = 12 choose 5 = 792
6 note scales = 12 choose 6 = 924
7 note scales = 12 choose 7 = 792
8 note scales = 12 choose 8 = 495
9 note scales = 12 choose 9 = 220
10 note scales = 12 choose 10 = 66
11 note scales = 12 choose 11 = 12
12 note scales = 12 choose 12 = 1
Therefore, in total there are 4095 such scales. A following question which arises is, how many possible modes are there?
