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

Comments

[Taylor Livingston]

Well, I should point out that I've never used the C-Thru keyboard, so I have no real-world experience. I was just speculating. I have always experienced intangible conecpts like sound in a very spacial, visual way. My spacial concept of pitch is somewhat at odds with this layout, but that's just my personal thing of course. 

As above, I like the sort of "stacked linear" (jargon I just made up) approach of the bass guitar. Horizontally, notes are linear, sequential semitones. Vertically, you have 4ths or 5ths. I like the way semitones are right next to each other, because that's how "see" notes in my head. But having that 4th/5th jump obviously makes chordal playing easier. I'm making myself a keyboard based on this.

[MusicScienceGuy]

I believe it is becoming popular because there is finally a solid promise of a  affordable, hexagonal-array, velocity-sensitive keyboard, the "holy grail" of some of us alternative keyboard keeners.

I'll only really believe it when I see it, but still, I'm hopeful enough to have spent the time on this analysis. I'm also hoping for more substance, details and time-lines to come forth at the winter NAMM in 2 weeks.

Thanks for the info and the comments.

Ken.

[Grant Muller]

Interesting that this is suddenly becoming so popular right now! I actually created a mouse controlled replica of the the C-thru Axis (a harmonic table keyboard) using processing. It can currently control any midi device, I’m working on setting it up as a touchscreen application and adding some additional features. You can find it here:

http://www.grantmuller.com/processing-harmonictable-part-2/

There is a a book called Harmonic Experience by Walter Mathieu that explores this subject, and uses it as the basis of a composition system. He refers to it as a lattice, and its laid out differently, but the system is basically the same. Very Very good book.

@Taylor: I can't say that I totally agree with you. I don't feel like the setup tells me what to play at all, but simply that I can play whatever I want anywhere I want (meaning modulating to any key without any change in fingering). I will agree that some scales (and chord inversions) are basically impossible, but as a composition tool this setup has been invaluable to me.

[Taylor Livingston]

"Axes" is the plural of axis.

It's a nice layout, but to me it seems designed for people who don't know anything about music and want everything done for them, and only want to play certain kinds of music (mainly pop). There is absolutely nothing wrong with that, of course, and I think music should not be for an elite crowd. I am entirely self-taught myself (well, taught by the internet, but I've never had a lesson or music class). Music definitely should be available to everyone.

But, for someone who does know something about music, and wants to write their own, this layout kind of defines what you can play on it, and makes anything else harder. Just looking at it, trying to play an ascending diatonic scale looks rather awkward. The 2nd, 4th, 6th, and octave are over in a whole other neighborhood. Personally, the appeal of 1-finger chords is not powerful to me. I can make that happen quite easily on any keyboard with MIDI, and even the lowly Casio keyboards we had as kids can do that.

The architecture of the harmonic table seems to tell me what to play, and I don't really like that. Of all "isomorphic" layouts I've seen or used, I actually most prefer that of my bass guitar. Same fingering in all keys, any chord type is the same no matter where it's played, and there are no notes that have been shunned because the designer didn't think they were useful.