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…
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…
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…
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…
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…
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…
Notes are pitched sounds. A given note means little by itself. It could be the tonic of a key, or some member of a scale based on a different tonic. By itself, it generates no tension, resolution or sense of place on the harmonic map. So when I name a note in this blog, I’ll…
When I first learned to play guitar, I would sit on the couch late at night and pluck the low E string, again and again, and just listen to the beauty of the sound as it died away. That low E note is not just a simple vibration. The full length of the string is…
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…
Over the next series of posts, I’m going to explain how the lattice in the Flying Dream video works. Before I do, I want to take time to mention a terrific book. I started investigating just intonation in earnest in early 2011. A couple of months in, my friend Kay Ashley loaned me her copy…