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For Friends of the Lattice

This is a repository for videos of particular interest to fans of the lattice and just intonation.

In these videos, red represents the bass, yellow the melody, and orange the harmonies. The videos show motion of the notes on a 5-limit lattice, with a few 7-limit notes added.

The notes are named in the following way:

Start with the basic just major scale, 1 thru 7 (1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8)

+ is up a Didymic comma (~22 cents)

– is down a Didymic comma

# is up a chromatic semitone (~70 cents)

b is down a chromatic semitone

7 is down ~49 cents, the difference between the septimal notes and their nearby JI namesakes. For example, a b7 (9:5) is at 1018 cents, and a 7b7 (7:4) is at 969, 49 cents flatter.

Thanks to composer Ben Johnston for +, -, and 7. I haven’t found a use for reciprocal sevens (yet), so I don’t have them on my lattice. If I add them, I’ll use Johnston’s upside-down 7 for the accidental, and they’ll be 49 cents sharp of their namesake.

This system gives every note in the 7-limit lattice a unique name, and defines its exact pitch.


These videos do the best job yet of comparing equal temperament with just intonation.

The piece is example 22.10 in Harmonic Experience. It is a one-minute tour of the lattice, an illustration of how music can move short distances in harmonic space and remain sensible to the ear while being very adventurous. I think it’s a beautiful composition. I got to know it very well in the making of the videos.

The first video features W.A. Mathieu playing the piece on his excellently tuned piano. This is perfect equal temperament.

For the second piece, I retuned the piano to the actual lattice pitches. There are 28 distinct notes in this soundtrack. I used the original track as a template for dynamics and timing, created a set of midi files that were restricted to 12-note regions of the lattice, mixed them down separately and recombined them.

Going back and forth between these videos gives a great feel for what each tuning brings to the party.

The notation is my usual, with letter names instead of numbers. Works well, as long as it’s written in the key of C.


Here is the latest full-song animation. Be Love is a simpler song than Flying Dream, and it illustrates the lattice better. Note the shift in mode between verse and chorus, how it looks on the lattice, and how it feels in the music. I’ve added a green lens to the palette, for the guitar lick.

Here is the original Flying Dream video:

I have illustrated some of the exercises in Harmonic Experience. Here is Exercise 26.7, animated and played in just intonation:

And in ET. Try listening back and forth.

Here’s another exercise, 26-17, in JI:

And here it is in ET:

I also got curious about a Mozart passage, discussed at length on pages 104-105. Here it is in just intonation, with the suggested spelling (melody starts on a b3):

The visual logic is as dead-on as the musical logic. Beauty! It’s in slow motion, to let the harmonies sink in a bit.

Now the most sensible interpretation (melody starts on a #2) of Mozart’s original spelling. It changes the whole effect, and visually it’s almost comical. It seems clear to me that this could not have been what Mozart was hearing.

Finally, here’s the same passage in equal temperament. How does it feel?


  1. dude…this is cool. Mathieu is my Bible

  2. Amen bro!

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