What is the "Harmonic Table": how the C-thru Axis works

How does the Harmonic Table system; the heart of the Axis keyboard, work? 

Harmonics with jammer like scale

To the left is my attempt at showing how harmonics are laid out and inter-relate as explained in the basis of harmony and a jammer sound-byte.

This is a "twice folded scale" whereupon we have notes evenly spaced out, in intervals called a major second, and fold them twice, first by octave, and then again, so that the octave family of harmonics(1, 2, 4, 8, 16, & 32), the green circles, are next to the Perfect Fifth family(3, 6, 12, 24), the Red circles. 

This is the basis of the Wicki-Hayden (Wicki for short) key layout and design of the jammer keyboard. One big advantage is that one can press the two notes most used together in music with a single finger straddling the two.

The jammer has other advantages as well.

Jammer Navigation numbers

But can we do "better", at least in terms of fingers-per-musically-useful notes than the jammer? Yes, we can! The Axis keyboard uses the Harmonic table layout.

The Harmonic table is fairly closely related to the Wicki, as I hope to show (in my addled-geek way). For one thing they are consistently laid out, so one only has to learn one finger pattern to play them instead of 12.
Cast your eyes on the harmonics in the little diagram above. Is there a way to put the blue circles right next to the red and green ones? Yes, but it's not a free lunch. Instead, it's a meal that's reasonably priced, yet with some intriguing spices.

Step 1: Note the Axii of Fifths 

Jammer to Axis Step1

Key to the tranformation is the concept of tonal axii (what's the plural of axis? axes? axises?). You can see that the Wicki layout has a row of keys ascending to the right. This is the Axis of Fifths, with the keys numbered 1, 5, 9, 13 (why do they go up by 4s? 'tis a quirk of musical arithmetic). There is also a flanking set of parallel axii that are up and down in pitch by a relative whole-tone (a.k.a. major second, or 1/6th of an Octave).

We can remove the parallel axiI and keep the vital consistency that a simple-to learn instrument must have.

Step 2: Identify the pesky intervening notes,

Jammer to Axis Step2

They have been colored white.

Step 3: Pull the intervening axii out

Jammer to Axis Step3

Step 4: Re-assemble them: 

Jammer to Axis Step4

Step 5: Add the discarded axii back in

You still need them. On this layout, they have
a place to go that's reasonable for playing.  

Jammer to Axis Step5

Step 6:  Rotate it so the major Fifth Axis is vertical  

Jammer to Axis Step6

And we are done.

This design at first looks bizarre - the octave is way off to the side, and the fifths (and thirds) become the dominant musical principle on this keyboard, instead of the octave. 

But it does have advantages, especially to a musician looking for a new way of viewing music. Just look at the way the harmonics fit together. (!)

That, however, is a discussion all on it's own.

Harmonics with C-Thru like scale rotated

Ken, the Music Science Guy.  

Harmonics with C-Thru like scale