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Today we’re going to go over some basic music theory: the 12-tone matrix. Then we’ll expand on how to use the idea of retrograde and inversion in everyday tonal composition.

In the above video, I discussed the use of permutations on a theme, but let’s just get some basic definitions out of the way:

*Prime* – the main theme

*Retrograde* – the main theme backwards

*Inversion* – a mirror of the main theme

*Retrograde-Inversion* – the inversion backwards

To take a simple example, let’s choose a brief prime theme:

C – D – E – G

The retrograde of prime is:

G – D – E – C

The inversion of prime is:

C – Bb – Ab – F

I got to this point by taking the intervals in prime and reversing them. C – D is whole step up, so I went a whole step down (C – Bb). The same is true with D – E (which became a whole step down, or Bb – Ab). The final interval E – G is a minor third ascent, so I made a minor third descent (Ab – F). It’s just that easy!

The retrograde-inversion of prime is:

F – Ab – Bb – C

It’s very simple to do, but these aren’t all of the permutations possible. Let’s transpose our prime theme up a perfect fourth:

F – G – A – C

We’ll call the above theme *Prime 5*, since it’s 5 half steps above the original prime theme. Its inversion will become *Inversion 5*, its retrograde will become *Retrograde 5*, and its inverted retrograde will become *Retrograde-Inversion 5*.

For short we’ll be using P-5, I-5, R-5, and RI-5 to refer to Prime 5, Inversion 5, Retrograde 5, and Retrograde-Inversion 5, respectively.

That makes R-5:

C – A – G – F

I-5 is:

F – Eb – Db – Bb

RI-5 is:

Bb – Db – Eb – F

In atonal, or twelve-tone music it’s common for music students to start off with a 12×12 table and write out all of the permutations. In reality, only P-0 (the original prime theme) and I-0 (the inversion of the original theme) need to be worked out. R-0 and RI-0 will be immediately apparent, and then P-1 through P-11 and I-1 through I-11 become simple transpositions.

I cheated and used the wonderful matrix calculator tool from ComposersTools.com (no relation) to create this matrix based on a twelve-tone theme:

The numbers along the top and left hand sides of the matrix denote how many half steps above the starting pitch (C) the new prime/inversion themes are. Each matrix will be slightly different.

Now in the above Youtube video, I discuss how to compose using these different permutations as starting points in developing additional themes. However, I take a diatonic approach. Returning to our original prime theme:

C – D – E – G

I can take the intervals and move them throughout the C major scale. Since I’m moving within a key, there are only 7 different starting pitches, so I’ll ignore the half step business and label them all 0-6. I’ll also substitute P, R, I, and RI with dP, dR, dI, and dRI to denote the fact that they are diatonic and not chromatic:

I chose C major as the key center since it seems the most fitting, but I could have easily chosen G or F major, since they both contain C and E naturals; moving along the circle of fourths, both Bb and D major would be out of the running.

If I choose a minor key, it gets a bit more complex with the introduction of melodic and harmonic minor. Let’s look at the same example in the key of F melodic minor:

I can get really into the minor mode by using melodic minor to ascend and natural minor to descend:

Note that the diatonic retrograde and diatonic retrograde-inversion are no longer true mirrors of their respective tone rows.

I can also use this same technique with partial scales (like pentatonic patterns), ethnic scales, diminished themes, whole tone themes, blues scale themes, or chromatic themes.

The above examples are also great ways to introduce motivic development into improvised solos. It doesn’t take much practice to plane simple motifs through various key centers.

Let’s take a look at how we can get further permutations out of our prime theme. This time, let’s move the theme throughout the diatonic key centers and see what we come up with:

Suddenly we get the introduction of new accidentals. Each of these new themes (the Em, Am, and Bdim themes) offers their own set of inversions and transpositions.

Further more, we can take the original prime theme and move it through all of the major and minor keys for even more permutations:

This approach is one of the oldest and most basic. Let’s take a look at the first half of J.S. Bach’s Prelude BWV 846 to see how the master uses permutations to create theme and variation:

I’ve decided to ignore the left hand for simplicity’s sake. The downbeat of every bar functions as the bass line, and the second attack establishes the harmony. In that respect, only the right hand is melody (although to any ear, harmony and melody are virtually the same in this work). I’ve also color coded the themes to help make a point of when and where Bach alters his prime theme.

This may seem like a very elementary way to look at a work such is this –one with colorful harmonic progressions, but it really is just a theme and variation work. All of the variation comes from melody and harmony; there is no rhythmic variation.

Even a very young composer can grasp the simplicity and beauty of writing using permutation as a basis for variations on a theme. I encourage you all to try writing your own work.

I’ll close today with a simple etude I wrote for a young piano student I had. The first 8 measures are simply diatonic permutations of the theme:

Enjoy, and be sure to keep an eye out for new videos, blog posts, and scores in our online store!

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