Introduction
So this lesson has absolutely nothing to do with Jazz, and is just a little bit of fun with maths. I wanted to answer the questions: How Many Scales Are There? Let’s start with some semantics.
- In Western Music, there are 12 notes in an octave
- A Scale is just a way of combining some subset of these 12 notes
- Some scales exist and are widely used (Major, minor, Blues, etc.) while others may be just hypothetical and not yet existent. But in theory, any combination of notes CAN be a scale.
- We all know that Major and minor scales have 7 notes, but there’s no reason why scales need only have 7 notes. Some scale have eight notes (diminished), some have six (wholetone), some have five (pentatonic).
- You could even conceivably have a 1 note scale.
So, let’s say you can have anywhere from a 1 note scale to a 12 note scale, with the scale that contains all 12 notes called, of course, the Chromatic Scale.
So How Many Scales are There?
For the moment let’s ignore ‘modes’. That is, let’s ignore the fact that if you start the same scale on a different root note and play all the same notes, you get a different scale. Let’s first assume that if the notes are the same, the root note doesn’t matter. So assume C Major and A natural minor are the same scale, again, for the moment. Let’s also call this a ‘unique scale’.
Now, you can use a mathematical formula called a ‘Combination’ or a ‘Binomial Coefficient’ to calculate the number of unique scales that exist in music. This formula is shown below:
n = 12 (there are 12 notes in an octave)
k = the number of notes in a scale
We can use this formula to calculate how many unique scales are possible – based on the number of notes in that scale.
For example, there are:
- 12 One note scales → 12! / (1! (12 – 1)! ) = 12
- 66 Two note scales → 12! / (2! (12 – 2)! ) = 66
and so on… - 1 Twelve note scale → 12! / (12! (12 – 12)! ) = 1
We can then sum all these up to find that there are 4,095 different possible unique scales.
A shortcut formula can be used to get the same answer:
But, as I said before, this doesn’t take into account the fact that if you play the same notes from a different root note, this is technically a new scale.
- C chromatic vs D chromatic
- C Major vs A natural minor
- C Ionian vs D Dorian vs E Phrygian vs etc…
There can be as many possible root notes as notes in the scale. So all we have to do is multiply our ‘Combination’ for each ‘No. of notes’ line (below) with the ‘No. of notes’ in the scale.
This tells us that there are 24,576 different possible scales in music.
| A | B | C |
| No. of Notes | Combination | A x B |
| 1 | 12 | 12 |
| 2 | 66 | 132 |
| 3 | 220 | 660 |
| 4 | 495 | 1,980 |
| 5 | 792 | 3,960 |
| 6 | 924 | 5,544 |
| 7 | 792 | 5,544 |
| 8 | 495 | 3,960 |
| 9 | 220 | 1,980 |
| 10 | 66 | 660 |
| 11 | 12 | 132 |
| 12 | 1 | 12 |
| Total | 4,095 | 24,576 |
