Modes. They seem to be difficult for many musicians to get the hang of. Fortunately, I’ve found that the easiest way to understand them and to explain them lies in a nice bit of maths.

A quick note on Cycles

Cycles are a neat piece of mathematics, lying as a part of permutations within the realms of combinatorics and algebra. Despite its simple nature, it has very powerful applications. Let me give you a short introduction.

A permutation of a set of numbers is a function that does something to the order of those numbers. For example, suppose you have the numbers {1,2,3}. There is a function that does this to the numbers:

1 becomes 3
3 becomes 1
2 becomes 2

This is a description of a specific permutation. However, it is a bit long winded and messy. Therefore we use cycle notation to make this function easier to understand. We can write this permutation as: (1,3)(2)

The (1,3) means that this function makes 1 into 3, then back into 1. This repeats ad infinitum. The 2 just stays the same, and so it is in its own set of brackets.

If we perform this function a few times, we can see it at work:
1. 1,2,3
2. 3,2,1
3. 1,2,3
4. 3,2,1
etc.

The key thing about this notation is that we can also write the cycle (1,3) as (3,1), because they both do the same things, i.e. they represent the same permutation. Equally, the cycle (1,2,3) is the same as (2,3,1), which is also the same as (3,1,2). We are just shifting the elements by taking the last number inside the brackets and putting it to the beginning, shifting all the other numbers one place to the right. 123123123123 is the same thing as 231231231231 which is the same as 312312312312.

As you can see from the above: for a cycle of length 2, there are 2 different ways to write the same cycle. For a cycle of length 3, there are 3 different ways to write the same cycle. Hence, for a cycle of length n, there are n different ways to write the same cycle.

Back to the Music

Hopefully, you are familiar with the major scale – possibly the first thing you are ever taught how to play on any instrument, or in musical theory in general.

The major scale is based on the formula: Tone, Tone, Semitone, Tone, Tone, Tone, Semitone; or in abbreviated form: T,T,S,T,T,T,S. This describes the length of the interval between two notes in the scale. The first note in the major scale is a whole tone away from the second (which equates to 2 keys away on a keyboard, or two frets away on a guitar). But the third note in the scale is only half a tone away from the fourth.

Here is the major scale in cycle form: (T,T,S,T,T,T,S). Remember I said that a cycle of length n can be written in n ways? This means that our major scale cycle of length 7 can be written in 7 different ways. Let’s do this by taking the first element in the cycle and moving it to the end each time.

Congratulations! You’ve just learnt all the modes! Don’t believe me? Here are all 7 ways to write the major scale cycle:

1. (T,T,S,T,T,T,S) Ionian Mode (Major scale)
2. (T,S,T,T,T,S,T) Dorian Mode
3. (S,T,T,T,S,T,T) Phrygian Mode
4. (T,T,T,S,T,T,S) Lydian Mode
5. (T,T,S,T,T,S,T) Mixolydian Mode
6. (T,S,T,T,S,T,T) Aeolian Mode (Minor scale)
7. (S,T,T,S,T,T,T) Locrian Mode

As you can see, one more shift from the final cycle will take you back to the first one – back to the major scale.

It’s a neat and easy way to remember how to play all of the modes (also known as diatonic scales). Looking at them this way also enables you to clearly pick out all of the patterns and relationships between each mode.

A useful observation is that if we take the C-Major scale: (C,D,E,F,G,A,B) and shift it one position, we get (D,E,F,G,A,B,C), which is the D-Dorian. Note that they contain exactly the same notes as each other, but in a different order. This carries on for the rest of the modes – they consist of the same notes, but shifted, so that they are in a different order.

This epiphany made the fuzzy world of modes very clear to me, and I hope it does for you too. Have fun creating mathematical music!