Hyve Touch Synthesizer - Tonnetz keyboard

I'm exploring the Hyve's upper keyboard. This is not your familiar black/white piano.
Apart from it not having any moving parts, it's arranged in a lattice structure.
This is a network representing tonal space "first described by the mathematician Leonhard Euler in 1739.Various visual representations of the Tonnetz can be used to show traditional harmonic relationships in European classical music".

It's all about chords and harmonies. Each note is harmonically related to its adjacent notes.
Straight up is a perfect 5th (a interval spanning 7 semitones), up to the right is a major 3rd (a interval of 4 semitones), and up to the left is a minor 3rd (interval of 3 semitones).

So the way it's arranged is if you play any note in a straight line from top to bottom, or bottom to top, it will play in perfect 5ths.
If you play notes going up to the right you will have a augmented pattern (it will go up or down in Major 3rds).
And playing notes to the left will have a diminished pattern (Minor thirds).

This layout of neighbouring fifths and thirds also makes it easy to form major and minor seventh, ninth, 11th and 13th chords.

It seems that all the most important scales — major, minor, chromatic, whole‑tone, diminished, blues, etc — have logical and distinctive patterns that basically climb rightwards and up. 

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How it forms chords is really interesting.

I'm starting by looking at how it groups the Major Triads.
These are the most common  chords and are built by adding the third and fifth notes in the scale above a starting note (root). For example, in C major, the triad built on C contains:

• C (the root)
• E (the third note above C; often called just "the third")
• G (the fifth note above C; often called just "the fifth")

The 3 major white-white-white triads are: C Major, F Major, and G Major.

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C major: C E G

  F Major : F A C

 G Major: G B D

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Minor Triads

 In C minor, the triad on C is built the same way:

• C (the root)
• E♭ (the third note above C; often called just "the third")
• G (the fifth note above C; often called just "the fifth")

This is called the C minor triad.

C Minor Triad : C Ef G

C-sharp Minor Triad : C# E G#

 D Minor Triad : D F A

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As discussed earlier, this layout of neighbouring fifths and thirds also makes it easy to form major and minor seventh, ninth, 11th and 13th chords.

First the 7th & 9th chords.

The C Major 7th Chord is C E G B

The C Major 9th chord is C E G B D

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The C major 11th chord is C E G B D F

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The C major 13th chord (drawn in light green) is C E G B D F A

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The minor chords now.

Below are just random examples.

The G-minor 7th chord interval is: G A# D F (G Bf D F)

The E-minor 9th chord interval is: E F# G B D

The C minor 13 chord contains C, E♭ , G, B♭, D, F and A♭ (C D# G A# D F G#) as on the diagram below: