GEOMETRY:

HYPERCUBES

Ok,

Notice how our three alpha cubes can stack together:

Purple and orange join at red

Orange and green join at Yellow

We can take this stack and fold down the top and bottom cubes:

We simply connect the top and bottom blue notes to make the hypercube:

We did it!

We built the hypercube

The hyperucbe mathematically exists in four dimensions of physical space

Check out this advanced geometry page for detail on that:

BUT

We only need to think of it intuitively as a 3D object

What you see is what you get

So let's look into it:

First things first,

Looking back at the stack, let's identify the perfect 5th intervals:

We thus have four stacks of 5ths here:

Ok, Here's where things get cool

Do you recognise this group of notes?

Can you see what we have?

Try putting them in alphabetical order...

Answer:

The C Major Scale!

In the hypercube the C major scale presents as its own cube:

This is the C major keycube

The hypercube creates four key cubes for each of the four orange keys

This correlates to the Chart from the colour pages:

Look how the grid chords fit onto the hypercube:

If we were specific with the triad notes we would see this:

(revise previous page if unclear)

Each of the four orange key cubes has the same structure as above:

With chords I and vi in the middle, V and iii above, and IV and ii below

The hypercube gives a 3d model of our functional colour chart

We can plot diatonic harmony on the hypercube

Like this 651 in A minor

We can also break out using non diatonic chords

Like this backdoor 2 5 1 in C major:

Check out harmonic functions page to dive into the chord relationships going on here

One last thing

This orange hypercube that we've made only holds four key cubes

The four orange keys: C, Eb, Gb and A

But we need two other hypercubes for the green and purple keys:

Question:

Can you tell which one holds the purple keys vs the green keys?

Hint: the complimentary colour of the keys is on the outside

Answer: The purple keys are in first hypercube (with yellow on the outside) and the green keys are in the next hypercube (with red on the outside)

Alright!

CONGRATULATIONS

You've just build the hypercube

This gives us a foundation for a universe of exploration

but for now we have one question to tackle...

What if we want to change between hypercube keys?

We need the Metaprism...