Permutations

When we look at permutations, we are concerning ourselves with the different combinations a phrase can be played in. It might not seem immediately obvious to even try this but there is good reason to, it can reveal some phrases that you might have missed and perhaps force you to play something slightly out of your reach.

Let's take for example, the Major7 chord.
The Major7 chord contains the intervals 1-3-5-7
If we use basic counting, we can figure out that there are a total of 24 combinations for this set of notes. (4x3x2x1)

If this were a 5 note set such as a Major9, the amount of permutations available to us would be 120. (5x4x3x2x1)

Knowing this, we can begin inverting and modifying our phrases from a simple upwards arpeggiation of a Major7 chord :

1 3 5 7

To a more outside phrase starting on a sharp dissonance (the major 7th) and resolving to the I :

7 3 5 1

Depending on the genre or your style, you may choose to emphasise different tones. Jazz generally tends to emphasise the strongest tones of the chord on the main beats, so we could say we're playing quarter notes and emphasis would be on the 1st and 3rd beat.

1st 2nd 3rd 4th
1 7 3 5

Notice here that on the first beat (The strongest beat), we are playing the root note. The second and fourth beats are the weaker beats so we try and insert the most dissonant/redundant tones in, so for the 2nd beat there is a seventh and for the 4th there is a 5th. Although a perfect 5th is not a dissonance, it is less useful to the chord than the major third and so was placed here.
For the third beat (The second strongest beat) we have used the third, the third is considered an important tone to keep as it defines the type of chord (major or minor) and so should not be omitted.

(And for all you jazzers, I am well aware you guys like to emphasise the 2nd and 4th beat but for the sake of simplicity we have gone with the traditional approach.)

Here we have stressed the most important tones of the chord, on the most important beats.

Now aside from re-ordering, what else can we do?
Thankfully this device has come a long way and has some standardized names for some types of permutation, these are listed below :

Prime - If the permutation is 'prime', there has been no modification made to the original set of notes.

Diatonic Inversion - There are two types of inversion, diatonic inversion, and true inversion. We will discuss both for clarity. Diatonic inversion is playing the phrase upside-down but adhering to the scale it is in, for example:

Let's say we have the simple phrase of C-E-G-A and we are in the key of C.

We can see that this is a C major triad with an A on top (Cmaj6 if you like), now if we were to convert this to intervals, we would have;

1 3 5 6
C E G A

To invert this diatonically we count backwards from the first tone we start on. So instead of going up a third from C, we go down a third diatonically from C, this would give us;
C - A
because A is a minor third down from C.
Once we have worked out this part, the rest will fall naturally into place and the whole phrase becomes.

1 6 4 3
C A F E

...The keen eyed amongst you may have noticed something. All of these degrees follow basic inversion principles!
Let's take another look

1 3 5 6
C E G A

Has become

1 6 4 3
C A F E

So, a 3rd has become a 6th. A 5th has become a 4th and a 6th has become a 3rd!
If you know your interval inversion, then diatonic inversion should be a slightly easier process for you now.

Now we will take a look at True inversion...

True Inversion

True inversion is exactly the same as diatonic inversion except that is non-diatonic. Using the same example as above will easily show what I mean.

Original Set

1 3 5 6
C E G A

Diatonic inversion

1 6 4 3
C A F E

With true inversion, instead of sticking to the key, we concern ourselves more with the intervallic nature of the tones and strictly adhere to them.
So in the original set:

C to E = An upward major third leap

In order to make this 'true' we leap down from C but stick exactly to the intervals used originally so:
C to Ab = A downward major third leap

Following this strict inversion, our new set of true inverted notes becomes:

1 b6 4 b3
C Ab F Eb

I suggest practicing this on paper before you try and apply it as it can become quite the mind-bender and you can easily get it wrong upon your first few tries.

Retrograde - A retrograde permutation is just the same phrase but played backwards and diatonically so:

C-E-G would become G-E-C, just as A-F-G#-E would become E-G#-F-A... Simples!

Retrograde Inversion - Retrograde inversion is just the idea of joining inversion and retrograde together. Therefore a phrase is now played backwards (retrograde) and upsidedown (inversion).

this concept doesn't really need explaining as it is just a combination of the above and once you have mastered those concepts then you can easily apply this.

It should be noted: A retrograde inversion can again, be true or diatonic.

This has been a brief yet exhaustive look at permutations and as you can see, there are many possibilities for this lesson and I urge you to try them yourself, you don't have to possess a PHD in combinatorics.

And as a final note, remember moderation:

With one chord (Major7) we can see the possibilities and potential of these devices , with an entire scale or indeed all 12 tones, it begins to get unwieldy. A traditional major scale would have over 5,000 different variants!