A Harmonic Journey: ET and JI Compared

The Harmonic Lattice can be viewed as a map of harmonic space. Music moves in harmonic space, just as it moves in melodic space (the world of scales and keyboards). The two spaces are very different from each other.

In melodic space, such as a piano keyboard, when two notes are close together, it means they are close in pitch.

In a harmonic space, such as the lattice, when two notes are close together, it means they are harmonically related.

“Harmonically related” means that one note can be converted into the other note by multiplying and dividing by small whole numbers. A note vibrating at 100 cycles per second is closely related to a note at 300 cycles per second. In melodic space, these two notes are far apart, but in harmonic terms they are right next door to each other — they harmonize.

In my video, Flying Dream, I animated the movement of one of my songs on the lattice. Now I’ve animated a composition of W. A. Mathieu’s.

Mathieu is the author of Harmonic Experience, an astonishing book that takes music back to its origins in resonance and pure harmony, and then uses the lattice concept to bring that harmonic understanding forward into the world of equal temperament. For me, the book opened the study of music like a flower.

The lattice, and my stop-motion animations, have given me a sort of musical oscilloscope. Instead of the music being some sort of black box, I can see inside it, get a visual image of what is going on harmonically. The new tool has made songwriting, improvising and arranging much easier.

I’ve animated Example 22.10 from the book. It’s intended to be an illustration of unambiguous harmony — the chord progression moves by short distances on the lattice, so it is clear to the eye and ear where you are. I think it’s a beautiful piece of music in its own right, a one-minute tour of a huge area of the lattice. It uses 28 different notes!

There are two versions of the video. The first one, in ET, has a soundtrack of Allaudin Mathieu playing the piece on his beautifully tuned piano. This is perfect equal temperament. It uses twelve notes to approximate the twenty-eight notes that the piece visits.

For the second one, in JI, I retuned the piano to the actual pitches of the lattice notes. Now, magically, the piano has all 28 notes. There is a whole new dimension to the music. In the JI version, I feel:

• Slight vertigo when the music moves quickly
• Satisfaction when a spread-out (tense) pattern collapses to a compact (resolved) one
• A great sense of homecoming at the end
• Stronger consonance and dissonance than in the ET version.

The four voices, from lowest to highest, are red, green, orange and yellow. It’s fun to follow one voice at a time.

This lattice is notated differently. It’s my usual system, but with letters instead of numbers. C is the tonic.