Music Theory Tree

INTRODUCING THE MTT TORUS OF FIFTHS

And now for something… completely different. Or… perhaps, seemingly so.

This is my best attempt so far to distill down as much relevant music theory as possible, into the simplest format—a grid system. It may not be as pretty as a Music Theory Tree, but it has the potential to help music theory click for you. I've personally found it very useful, and so have students of mine.

What does this chart show? Let’s break it down, part by part.

THE CIRCLE OF FIFTHS/FOURTHS

First of all, the contents in this chart have been taught for hundreds if not thousands of years, in one form or another. The MTT Torus of Fifths is simply a reimagining of what may be the most well-known and widely-used diagrams in all of Western music theory–the legendary Circle of Fifths (sometimes called the Circle of Fourths).

If you’ve studied the Circle of Fifths, this alternate configuration may be immediately useful and intuitive to you. It reveals things about key structures that aren’t immediately apparent from looking at the Circle of Fifths, because it takes what’s implicit in the Circle of Fifths, and makes it explicit. It’s like opening the book instead of reading the cover.

THE COLUMNS

The grid is laid out such that all like-pitch-classes (which you may know as notes), share a column. There are 12 columns, one for each pitch class. All ‘C’s’ are in the same column, for example. B♯ is enharmonically equivalent to C, so it shares a column. Enharmonic equivalence refers to musical elements that sound the same, but are written differently, based on context. For more information on pitch classes, I recommend my essay on the Chromatic Scale, found here (INSERT)

THE ROWS

The rows are diatonic systems, commonly called “musical keys”. There are major keys and minor keys. In total, there are 30 keys in Western tonal harmony–15 major keys, and their 15 relative minor keys. Some might argue that there are only 12 or 24 keys, and in a sense there are, because some keys are enharmonically equivalent to others. These are keys whose scales sound the same when played, but are written differently. They contain all of the same pitch classes. For example; C Major and B♯ Major are enharmonically equivalent keys. To clarify the argument, one might say that there are 12 or 24 “playable keys” and 15 or 30 “common theoretical keys”. Though one can imagine more keys, the rules of key signatures dictate otherwise. More on this topic another time.

This chart is meant to help teach music theory, so it shows all common theoretical keys. If you’re curious why one might choose to write in a given key or its enharmonic equivalent, follow along for future posts. It’s quite a fascinating space of ideas, and it’s not as arbitrary as one might first think.

THE LEFT LEGEND

The legend on the far left shows relative major and minor pairings. The relative minor of C Major is A Minor, for example. It also shows the number of pitch classes with sharps or flats in that row.

SCALE DEGREES

The small numbers above and below the pitch classes are the scale degrees of the major and minor scales, respectively. So, ‘C’ in the key of C Major has a ‘1’ above it, because it’s the first degree of the C Major Scale. It has a ‘♭3’ below it, because it corresponds to scale degree flat-3 in C Major's relative minor scale, A Minor. Conversely, ‘A’ has a ‘6’ above it, because it corresponds to scale degree 6 in the C Major Scale. However, A is scale degree 1 in the A Minor Scale, so it has a 1 below it.

ORGANIZING THE ROWS

Without getting too deeply into the theory in this post, it suffices to say that the rows are arranged such that their first scale degrees are perfect fifth intervals apart. This is the interval that spans between any given scale degree 1, and its scale degree 5.

So, the key of C Major (or C Major Scale) occupies the middle row. Scale degree 5 of ‘C’ is ‘G’. Therefore, the key of G Major occupies the row below the key of C Major.

Scale degree 5 of ‘F’ is ‘C’, so the key of C Major is found in the row just below the key of F Major.

THE COLOURS OF TRIADS

Each of the pitch classes that have pink backgrounds are the roots of major triads in that row. Notice that ‘C’ in the C Major row has pink behind it. This is because the triad whose root is ‘C’ in the key of C Major is the C Major Triad.

Those with powder blue behind them are the roots of minor triads, such as the A Minor Triad in the key of C Major.

The remaining pitch classes with light purple backgrounds are the roots of diminished triads. In the key of C Major, it's only the B Diminished Triad.

MOVING BETWEEN KEYS

If you’re first starting to study key structures and signatures, the following is a simple way to make sense of the pattern (which may seem arbitrary, at first).

Beginning from the C Major Scale row, notice that the number of sharps increases by one every time we move down one row in the torus. The number of flats increases by one every time we move up one row in the torus.

Why does this happen?

One short answer is that major scales are imbalanced, asymmetrical, and contain a dissonant interval called a tritone. Every time we move from one row to another row, we’re displacing the tritone to a new location. The dissonance and location of the tritone plays a crucial role in dictating where the most consonant tone in the scale is. This most consonant tone in a major key is called the Tonic, or scale degree 1 in its major scale.

The tritone in the C Major Scale spans between ‘F’ and ‘B’. This is a crucially important detail. There's a lot to unpack with this, but for now, just remember this as a brute fact.

The basic instructions for how to displace the tritone follow. Don’t be afraid to ask for help if you would like assistance with understanding this. It’s a bit complex, but learnable.

MOVING TO THE KEY BELOW - ADDING SHARPS

To move to the key below a given key, we find scale degree 4 in our starting major scale, and we sharpen its corresponding pitch class. For example, scale degree 4 in the C Major Scale is ‘F’. We then sharpen ‘F’ to become ‘F♯’. The letter previously attributed to scale degree 4 of the C Major has altered to become scale degree 7 of the G Major Scale. The letter ‘F’ comes along for the ride, but the pitch is now a semitone higher. ‘F♯’ is the “Leading Tone” in the key of G Major, or scale degree 7 in the G Major Scale. Its triad is a diminished triad, which includes that dissonant tritone I mentioned before. The tritone in the G Major Scale spans between ‘F♯’ and ‘C’. So, we’ve successfully displaced the tritone that previously spanned from ‘F’ to ’B’.

I won’t dig too much more into detail about the relevance of the tritone in major scales in this post, because videos with audio examples will do a much better job explaining this. If you want to learn more about the theoretical side of the tritone interval, I have an in-depth video on my page where I teach you how to grow a C Major Music Theory Tree from scratch. I discuss the construction of a B Diminished Triad in that video. There are accompanying exercise sheets for you to download if you like.

MOVING TO THE KEY ABOVE - ADDING FLATS

Let’s finish this short lesson by learning how to move up a row. To do so, we find degree 7 in the major scale and flatten it. In C Major Scale, scale degree 7 is ‘B’. So, we flatten ‘B’ to ‘B♭’. ‘B♭’ is scale degree 4 in the F Major Scale. So, the letter ‘B’, previously attributed to scale degree 7 of ‘C’, has been flattened a semitone to become ‘B♭’, and is now scale degree 4 of ‘F’. What happened to the tritone between ‘F’ and ‘B’ in the C Major Scale? It has displaced to become the the interval between ‘E’, the Leading Tone of ‘F’, and ‘B♭’. Mission accomplished.

CONCLUSION (FOR NOW)

Through repeating these processes, sharpening scale degree 4, and flattening scale degree 7, we can construct every major scale in the torus. This usually takes practice and repetition to make sense of. If you don’t understand the first time, don’t feel alone or discouraged. Read this post multiple times carefully, if necessary. Look closely at the MTT Torus of Fifths and diagrams of the Circle of Fifths. Search for instructional videos online about “changing keys using the Circle of Fifths”, and read through Wikipedia, and other articles. If you want to get these concepts totally down, please get the Music Theory Tree Exercises. They are specifically crafted to help you memorize these patterns.

If you’re curious and would like to learn more (super relatable), follow me. I’ve been studying music theory for a long time, and I have A LOT more to share with you. I’m a person who genuinely wants to help you.

Let’s reimagine music theory.

Be well,

Steve