The Mandala of Keys

Let us examine the system of the keys. It is a very worthwhile exercise, and can lead to a broad understanding of the way that the twelve semi-tones that we have divided the octave into relate to each other.
Firstly let us understand that this is our Western system, and that other systems exist. For example, in some Eastern systems of music, up to twenty four partitions of the octave are recognised, including intervals that we refer to as micro-tones being half of a semi-tone.

An octave, being the distance between any given frequency and the doubling of that frequency will naturally contain an infinite amount of potential "notes." We can imagine this if we think of a sound source, such as a variable oscillator or the human voice, being swept continuously through all the frequencies from "f1" to "f2."
Anywhere that we cared to pause the sweeping sound, and listen, we could say was a "note." However, a little thought will lead us to realise that although some instruments are capable of producing all these notes, most are not, and certainly a piano with an infinite amount of notes per octave would be highly impracticable, and mercifully does not concern us here.

So we have arrived, at our system of twelve semi-tones per octave. There are various ideas as to how this evolution took place, but no-one can be really sure. But there does seem to be a recognisable hand, a certain mysterious guidance over the years that has steered it on its way.

We find the recurrence of the mystical numbers twelve, seven and eight. Guided over the many years by the Spirit within him, mankind has overseen the creation of this system. The keyboard would appear to have been behind the logic in the key system, although whether that is possible, historically, I very much doubt.

Consider this if you may, the fact that although the musical alphabet starts with "A" the key system (although a complete cycle, without a real beginning or end) would appear to "start with" "C." The key of "C" being the only all natural key, (no sharps or flats) and that which corresponds to the white notes upon the piano keyboard. I love the mystery, I thrill at the magic.

Let's explore onwards.
We can draw the twelve keys around a circle exactly as the hours appear on a clock face. (This is very appropriate considering how inextricably intertwined are music and time. From the ground floor up, in fact, because we only choose to use, as a rule, periodic frequencies as our sources of sound. That is, frequencies which are consistent over time produce what we recognise as a musically useful sound.)

Let's place "C" at the twelve’ o’clock position.
Let's lay out all the other keys in the hour positions and travelling clockwise put them in the order of increasing quantity of sharps.
Therefore we will have "G" with one sharp at one o’clock, "D" with two sharps at two o’clock and so on.
But wait! if we proceed all the way around the clock with sharp keys, we don't arrive back at "C." Instead we arrive at the key of "B" sharp, with a key signature of twelve sharps!
How on earth can a key have twelve sharps with only seven different notes in the scale? The answer is that some of the sharps are double sharps.
The notes in this scale would be as follows. B sharp, C double sharp, D double sharp, E sharp, F double sharp, G double sharp, A double sharp!
Not the easiest of keys to play in.
Enharmonically the same as C.

Let's try a different order around the clock face.
Let's say that at the six o’clock position, we will have an "interchange" with the flat keys. So, at the six o’clock position we find the key of "F sharp" the note that is halfway between C and its octave.
Let's interchange with the flat keys, and call this "G flat" as well.
Six sharps or six flats.
Now if we proceed round the clock "D flat" with five flats, "A flat", with four flats and so on we will arrive back at "C."

Let's see how we can look further into this chart.
It is sometimes referred to as the cycle of fifths, that is perfect 5ths ascending travelling clockwise, descending travelling anti-clockwise. Really we are looking at an integrated system that is complete, and therefore any direction can only be implied.

I much prefer to call this a Music Mandala. Your philosopher's stone.
Ask any question and it shall be revealed.

Imagine placing a mirror along the axis of "C" to "Fsharp". That is what is actually shown. Everything on one side of this line is reversed and inverted on the other side. "C" and "F sharp" being the two points of integration, "C" has no sharps or flats, "F sharp" or "G flat" have six.

Certain musical patterns can be represented on this mandala, and they can lead to useful insight. For example, by joining the points on the circle in the order of a chromatic scale what shape arrives? A beautiful twelve pointed star.
A whole tone scale produces the image of a hexagon.
Try diminished sevenths and augmented triads, major and minor scales.

Let the shapes speak to you.
Try intervals and their inversions, all will be revealed to you.
Get familiar with this Alice in the Looking Glass world that is the system of music graphically portrayed.
Above all realise that your familiarity of this system, with regards to your knowledge of keys, scales and chords, should become as complete and fluent as possible, for this is the raw material of your musical intellect, here is the syntax, the grammar for your musical creativity.
Take delight in the symmetry of it, and wonder at the mystery of how it all came to be.