The Tonic Major Chord

The tonic is the center of the lattice. A drone note on the tonic establishes the center of that particular musical universe. Adding a major third and a perfect fifth (5/4 and 3/2) further reinforces the center and starts to carve out some territory on the map. This is the tonic major chord: In my…

Real Girl

I’ve added a new recording to the Audio page. It’s the first time I’ve consciously written a song using the lattice. The chord progression is especially influenced by how it appears visually. I was moving colored bits of glass around throughout the process, aiming for beauty, tension and resolution. The music tells a small story, of…

The Lattice

In 1739, the great mathematician Leonhard Euler published something he called a Tonnetz, German for “tone network.” It looked like this: Euler’s Tonnetz organizes the notes into a matrix, instead of a scale. Moving down and to the left represents motion by an interval of a fifth (V) in musical space. Down and to the right…

Harmonic Space

Now to relate all this to the lattice in the video. Listening to music is like going on a journey. Most tonal music starts by establishing a center, or basic note, and a basic harmonic framework for the song, such as a major or minor mode. A few melody notes, and a beginning chord, and…

The Major Third

Multiplying a note by 2 creates an octave, and multiplying it by 3 creates a perfect fifth. Multiplying by 5 gives yet another new note, the pure major third.5-1 5/1 is over two octaves above the original note, so you have to reduce it twice (divide by 4) to get it down into the same octave.5-4…

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Octave Reduction

Doubling the frequency of a note certainly changes it. The ear hears a higher-pitched note. But there is something in the essence of the note that does not change, a character that stays consistent through the octaves. This allows a process called octave reduction. When you’re working with notes as ratios, it’s convenient to multiply or…

Between the Keys

I grew up thinking that music was made with a particular set of twelve notes, the ones on the piano keyboard. I had a vague sense that there were other scales in the world, but I thought of them as “more primitive” or perhaps subsets of the 12-tone scale, like that pseudo-Asian music you make…