## The Compass Points

There are two basic directions on the lattice: multiplication and division. If I start with a note, and then multiply it by 3, or 5, or 7, I will get a harmony note with overtonal energy. Such a note is in the natural overtone series of the original note. Overtonal energy is stable, restful, it…

## The Minor Third

Here’s an interesting and perhaps misunderstood note. It’s a compound move on the lattice: down a third and up a fifth. Or up a fifth and down a third, it doesn’t matter what order. So the ratio is 3/5, or 6/5, octave reduced. The note is the minor third. I call it b3. It lives a…

## Reciprocal Thirds

There are four basic moves available on the lattice of fifths and thirds. They are: Up a fifth (x3) Down a fifth (÷3) Up a major third (x5) Down a major third (÷5) Each of these moves has its own harmonic flavor, and they can be combined to create new flavors. The major scale only…

## 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…

## 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…

## 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…

## Notes and Intervals

A note, in music, is a sound with a particular pitch. Pitch is frequency, measured in cycles per second, or Hertz (Hz). The faster the vibration, the higher the pitch. A vibration, at, say, 220 Hz, all by itself is a note by that general definition. But the note doesn’t acquire its distinct personality until it’s…