In my last post, I proposed a simple way to graph tonal gravity against the octave. Overtonal notes, generated by multiplying, are restful, stable — they have positive polarity, pulling toward the center. Reciprocal notes, generated by division, are restless, unstable — they push. I call this negative polarity. Mixed-polarity notes have both, and I’ve…
Chords and other collections of notes have consistent, recognizable shapes on the lattice. A major chord is a triangle sitting on its base, a minor chord is a triangle on its point. Yesterday’s post has videos showing these chords. In the songs I know and write, the next most common chords after major and minor…
I just recorded a new song, and it’s a perfect example of how equal temperament and just intonation can get along together. Here’s the cut: Reading this blog might give you the impression that I’m “against” equal temperament and “for” just intonation, or untempered music. True, discovering untempered music has been like sailing to…
The picture to the right shows the lattice of fifths and thirds, a map of untempered harmonic space, extending to infinity. The map shows how to venture farther and farther from home, the tonic, and stay perfectly in tune no matter how far you roam. Every single note in this infinite matrix is tuned to a different…
The lattice goes on forever in every direction. It starts with the tonic, the 1, the Big Bang of the musical universe. Multiplying and dividing the tonic by 3 generates the horizontal axis. This is the familiar circle of fifths, although in just intonation, it doesn’t quite come out exact. If you multiply by 3,…
One more comma shows up in the central portion of the lattice. In equal temperament, three major thirds adds up to an octave. The major third is an interval of four piano keys (out of 12) or 400 cents. Three of them is 1200 cents, exactly an octave. In just intonation, this is not the…
When I started exploring the extended lattice beyond the central 12 notes, the first note that was really new to me was the 10/9 major second, also called the minor or lesser whole tone. Now I call it the 2-. The lattice extends forever in all directions. When you continue multiplying and dividing, generating new notes beyond the boundaries…
Why are the rings of Saturn so beautiful? The photo to the right was taken by the Cassini spacecraft when the sun was behind the planet, and backlighting the rings and the edge of the atmosphere. A type of solar eclipse never before seen by humans! The rings are composed of millions of small particles, mostly ice, orbiting Saturn….
Yesterday, I described a simple way to hear, and more importantly, feel, the difference between equal temperament and just intonation, by singing Frère Jacques over an open G chord with ET major thirds, and then over a G chord that has only roots and fifths in it. The second half of the experiment is called singing over…