Processing HarmonicTable: Part 1

Earlier this year while reading Harmonic Experience by W. A. Mathieu, I was introduced to the concept of lattices to represent tones, chords and keys. These lattices can be used to represent the basics of music composition in a visual way that makes more sense than standard scales on staffs. Here is an example:

ChordLattice

The lattice is effectively several staffs of music stacked on top of each other, so that the note can be displayed horizontally and vertically. Starting from any note in the lattice, moving horizontally (diagonally right or left), jumps by a 5th (for example C to G, or G to D). Moving vertically jumps by thirds (C to E, or G to B). Since nearly all music is the construction of 5ths in 3rds, this system makes it easy to create patterns visually for chords, cadences, scales etc, that will always look the same no matter where you start on the lattice. In the example above, you can see that Major chords are always point up triangles, Minor chords are the opposite, point down. Until recently this system didn’t translate directly to an instrument. Any chord on a keyboard will have similar but different fingerings and patterns depending on the starting note, so one still has to memorize a ton of different patterns for one scale, chord, or modulation.

A few weeks into reading Harmonic Experience, providence saw fit to lead me to the C-Thru AXiS controller. The AXiS is a Harmonic Table based midi controller that creates a table much like the lattice in Harmonic Experience, separating notes not in semi-tones like a keyboad, but by 5ths and 3rds (and of course many other inter-relationships based on this). Here is the layout of the AXiS: natural_keyboard If you look the relationships between the keys become clear. They are reversed from Mathieu’s lattice, 3rds are horizontal (diagonally), and 5ths are always straight up. To play a Major triad on the AXiS, just play that triangle pattern using starting at any key. C Major will have the same pattern as Bb Major, etc. There are of course drawbacks to this system (playing inversions, etc), but for the most part it accomplishes the goal of simplifying the muscle memory aspects of playing music so the user can concentrate on composition and performance. you learn the pattern for a scale, chord or mode only once, then you may modulate it anywhere without the need to retrain your fingers.

The problem with AXiS is the price tag. Its around $1700 for one from what I can tell, which is a pretty steep entrance fee for a device I may not be able to get used to. They are in the process of creating a cheaper smaller version, but I want to try it now. The solution was to build one in processing. I’ve only just started, but combining the midi reference classes I’ve been working on along with (eventually) a touch screen, I think I should be able to pull something off that works well enough for me to test this thing out. I started by looking for ways to draw regular hexagons, and came across this site. With a some basic trigonometry and a lot of cut-and-try with the vertex functions, I came up with this function:
int length=30;
float a = length/2;
float b = sin(radians(60))*length;
float c = length;
public void drawHex(){
beginShape();
vertex(0,b);
vertex(a,0);
vertex(a+c,0);
vertex(2*c,b);
vertex(a+c,2*b);
vertex(a+c,2*b);
vertex(a,2*b);
vertex(0,b);
endShape();
}
This will construct a regular hexagon based on the length of one side. After that I used a series of translation matrices to draw them all over the screen:
public void draw(){
rowNumber=0;
setNoteNumber(rowNumber);
for (int j=2; j<20; j++){
pushMatrix();
translate(0+space, (height-(j*(b+space/2)+1)));
drawHex(getNextNote());
for (int i=1; i <12; i++){
translate(space+(2*a)+(2*c), 0);
drawHex(getNextNote());
}
popMatrix();
j++;
rowNumber++;
setNoteNumber(rowNumber);
pushMatrix();
translate(a+c+1.5f*space, (height-(j*(b+space/2))));
drawHex(getNextNote());
for (int i=1; i <12; i++){
translate(space+(2*a)+(2*c), 0);
drawHex(getNextNote());
}
popMatrix();
rowNumber++;
setNoteNumber(rowNumber);
}
}
I predefined the number of horizontal hexagons to 12 for each row, this means that STRAIGHT horizontally across a row I will have a perfect chromatic scale. The number of vertical keys I simply copied from the AXiS.After predefining the number of columns and rows, it meant that I could construct the size of the screen based on the length of one side of the hexagon. I also included a variable that allows me to declare an amount of fixed space in between the keys (in case my fingers are too big for the key’s surface). One variable, named length, can be changed to create a bigger surface with larger keys. After implementing some simple code to write the note name into the key as well, I was finished with the layout. Here is a snapshot of it directly out of processing: Virtual Harmonic Table Its of course much bigger. With the key surface complete now all I have to do is map the key locations to the midi note they’re associated with, and use some on press functions to trigger them. After that I just need to get access to a touch screen, anyone want to donate one? Here is the preliminary code to draw the hexagons, as well as the code required to map midi notes: HarmonicTable NoteReference In┬áPart 2 I’ll have have midi functionality implemented and some testing done. In Part 3 I’ll have it implemented with a touchscreen, most likely a laptop, at that point I’ll make the rest of the code available.

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