The notes get more exotic as you move outward from the center. The ninth is quite consonant, but not nearly as consonant as the fifth. (Consonance and dissonance are descriptions of feelings; they are part of the flavor of an interval, and I don’t think the last word has been written on them yet. I’ll be…
Multiplying the tonic by 2, 3 and 5 creates the octave, fifth and third respectively. The ear hears these intervals very well. We can easily sing them. Each one has a feel, a sort of harmonic flavor, that makes a fifth a fifth and a third a third. It turns out that the ear can…
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…