Harmonic Table: From elegant simplicity to surprising complexity
Much of this information comes from these sites: C-Thru Music ; Peter Davies' Opal site and the Wikipedia article Harmonic Table keyboard layout. 

When presented with an innovative, totally new device like the Axis 64/49 and the “Harmonic Table” layout. The mind takes a while to sort out the salient new features.

First lets take a look at the key-field and get a feel for the larger features:

First, the basic unit of music - the notes in a octave as they can be grouped on an Axis keyboard. I've coloured the notes of the major scale and marked in the musician's names for the notes. 

At first the notes look a bit daunting to learn, but it's not really too bad, with a good, practical mapping to one's fingers.

Many notes that often be played together are close and can be played with one finger. (I'll talk about playing later, in a separate posting).
 
There is a slightly odd gap of notes (white on this diagram, black on the Axis keyboards) which one normally will want to avoid playing with the others. Whimsically, I call this the Gutter of Dissonance. 

Basic Axii: 

 
The Fifth
This keyboard layout is totally nuts about the 5ths; everything pivots around them. In most music the 5th has a very special relationship with the key (the root note) of the piece. It is in many ways a secondary key center. On this keyboard simply by moving up a key row does this shift. That's all that's needed.  

The replacement of the conventional octave as the dominant musical center (paradigm) in a musician's mind may be the single greatest advantage of this keyboard. It's different, but still profoundly musical. The octave notes are so like the root note that they are boring.

To give just one example: a cool thing one can do on this keyboard is tune the fifth perfectly, rather than temper (flatten) it to fit the octave scale. (if you do this, the octave will be a bit (~2% of a semi-tone) sharp, which is not a grave musical sin, just slightly scandalous, and therefore interesting).

Another, more important example: many pieces of music involve establishing a pattern, then once it is established in the listener's ear, tweaking it in a musical way, so that the harmonics invoked change in simple ways. Thus we musicians keep our listeners interested. The trick is to do this in the aforementioned "musical way". That's the power of this keyboard, as it is wrapped around and deeply embedded in musical theory.

Tip: if you have learned a neat pattern, turn your hand 60 degrees. The same pattern sounds mostly musical and sometimes  interesting. You can repeat this until you have gone all the way around, giving yourself a chance to explore music in a unique way.

The Thirds
Also used everywhere in music is the Major Third and Minor Third intervals. If you learn a pattern and shift it diagonally one note over in any direction you will sound, quite literally, jazzy. That is what jazz musicians do in jazz (caveat alert: this is just one of several tricks used).

What I think will work: shift the hand up, down, sideways etc. for just a moment (one chord, one or two bars), then snap back.

Ridges and gutters:

There are distinct lines of consonance and dissonance (white keys and black keys) in this keyboard. Which makes sense: the consonant notes are grouped together, consequently the dissonant notes are also grouped.

In my diagrams, I've given the "white notes" a unique colour and coloured the black keys white, as this makes the coloured notes stand out better.

Based on my experience the the jammer's Wick-Hayden layout, I strongly recommend that the keys on the Axis be coloured.

Missing Axii

It's often the case that an improvement in one respect comes at an cost in other ways. So it is with the Harmonic Table layout:

The musically important intervals of Octave and Fourth are nearly "Missing In Action". This makes playing some parts of  "normal" music, like inversions, harder. 

Semi-tones and Whole-Tones:
Weirdly, the notes that are close together on a standard piano are stretched out. Fortunately, they can still be reached by the fingers in under the hand without undue movement or stretching, a key requirement for speed and ergonomics. This is partly due to the smaller keys.

Major (and Minor) scales :

The major Scale and it's close cousin, the Minor Scale are shown above, in what I guess is the commonest way to play these scales, showing a numbered ascending sequence on the left side, and the fingers needed on the right.

Note that the distance traveled to play is measured in millimeters and scant centimeters, while on a piano, it is a factor of 10 greater or more. This should translate to a huge improvement in playing speed.

Other observations;
Octave Units :

Around the octave one is playing, is clustered other clusters of octaves, but because an octave unit on the Axis is 6.5 notes wide and less than 2 notes high, the octave units abut in a different way.

The thumb seems to be seldom needed. The thumb has an awkward shape and is awkwardly placed (except to play bass, perhaps) to be used on the smallish hexagonal keys of the board. Note that Jordan Rudess' in his demos seldom uses his thumb. Addendum: Jordan confirms this.

"You are right that it is harder to play the instrument with your 
thumb. I sometimes will use it though for certain musical situations."

One can also "slice up" the notes in other ways. 

Zones: 

Here's another way to group the notes. On the Axis-64, the key-field can be broken up into 3 Zones, each 7 columns wide.