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Posted by on Nov 15, 2012 in The Lattice | 4 comments

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 shows movement by a major third (III).

The lattice has been rediscovered and redrawn many times over the years. One of my favorites is the Duodenarium of Alexander Ellis, which showed up in his appendix to Helmholtz’s pioneering book, On the Sensations of Tone, in the late 1800’s.

Now we’re talkin’! C is at the center. The fifths go up and down, and thirds from left to right, leading to a square grid.

One of W. A. Mathieu’s innovations in Harmonic Experience is to slant the axes and make them line up with the musical staff:

Seriously, if this blog interests you, please get a copy of this book. I have no stake in you doing this, except that I believe the more broadly understood this man’s work is, the more great music will be made.

I’ve been messing around with the lattice for a year and a half now, and I’ve morphed it into a form that suits my own musical work.

Further slanting the thirds axis to 60 degrees makes it a hexagonal lattice, and for me the relationships between the notes become more intuitive. The major chord is now, appropriately, a stable-looking triangle. And a new axis appears, northwest to southeast: movement by minor thirds. I follow Mathieu’s example and show this one with a dotted line, because it isn’t a direct move: the minor third is a third down and a fifth up, a compound move on the lattice — a major (sorry) insight into the nature of the minor third. Much more on that one later.

Japanese mathematician Shohé Tanaka drew a hexagonal tone lattice in the 1800’s. I haven’t been able to find a picture.

Movement to the right represents multiplication by 3, that is, up a fifth. Up and to the right means you’ve multiplied by 5, up a major third. Left means division by 3, down a fifth. Down left is division by 5, down a major third. The tonic, 1, is at the center (below left of center in this portion). The grid goes out to infinity. This is the region encompassed by Flying Dream, which in fact covers most of the territory I’ve found useful so far, a major reason I chose that song for the video.

Next: The Tonic Major Chord

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Posted by on Nov 14, 2012 in The Lattice | 0 comments

Harmonic Space

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 you have some idea of the space in which the journey will be occurring. Strauss’ Also Sprach Zarathustra (of 2001 fame) is a great example. The famous opening section, called “Sunrise,” gives an extremely clear sense of home. You know exactly where you are, sonically.

By the way, it’s fun to hum this while using an electric toothbrush.

The piece goes on to travel away from this home, and back again, many times. The journey takes place in a space of some sort, an auditory environment.

But what might this space look like? One way to visualize music is staff notation:

It’s beautiful, and if I know how to read it, it will tell me what the music sounds like. It doesn’t do such a good job of showing me why music sounds the way it does. Neither staff notation, nor the 12-tone scale, gives me a particularly clear idea of how music works. Why would this be restful and sonorous:


While this, though beautiful in a different way, has tremendous tension? Sounds like the villain (or the cat) is about to pop out and scare you.


Okay, okay, here’s the resolution:major


If I know a lot about music theory, I can interpret the notation and come up with explanations. The second example is an augmented chord, and yes it sounds like that. But why, Mom, wh-wh-why?

Next: The Lattice

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Posted by on Nov 11, 2012 in Just Intonation, The Lattice, The Notes | 0 comments

Notes As Ratios

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 usually be referring to a ratio, the relationship between the note and a reference note — the tonic, or the root of a chord, or another note in the harmony or melody.

Ratios are fractions. The first number is divided by the second number to give the value of the ratio.

If the tonic is, say, 100 Hz, then another 100 Hz note is related to the tonic by the ratio 1/1. This is the interval of a unison, two identical notes.

Each note name on the lattice represents a unique ratio, relative to the tonic. The 1, at the center, stands for 1/1.

Next: Octave Reduction

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Posted by on Nov 8, 2012 in Background, Just Intonation, The Lattice | 0 comments

Harmonic Experience

Over the next series of posts, I’m going to explain how the lattice in the Flying Dream video works. Before I do, I want to take time to mention a terrific book.

I started investigating just intonation in earnest in early 2011. A couple of months in, my friend Kay Ashley loaned me her copy of Harmonic Experience, by W. A. Mathieu. Thank you so much, Kay!

I spent a few weeks with Kay’s copy and very soon knew I had to have my own. I devoured the book almost daily for at least a year. I still pull it out often, lug it to a cafe for browsing over breakfast, do bibliomancy with it if I’m stuck creatively, take it on vacations.

Harmonic Experience is the only music theory book I’ve read so far that actually increases my understanding of music, rather than obfuscating it. It’s huge, which could be intimidating. But I found it to be immediately accessible and entertaining. Mathieu has a great, light sense of humor. The concepts are introduced at a beautiful pace. And the ideas he presents are enlightening. “Aha” experiences abound.

Much of what I’ll present in this blog is heavily influenced and inspired by Mathieu. The lattice itself goes back to Euler in the 1700’s, but Mathieu expands on the idea enormously, arranging it so it corresponds to traditional musical staff notation, using it as a means to understand equal temperament, harmony, melody, chord progressions, world music, and much more.

Mathieu uses the term “positional analysis” to describe his system. For me, positional analysis opens the black box. It shows what’s happening in there. When my music is informed by the lattice, it makes more sense. I have more control over the effect it has on me and my audience. And it’s way more fun, because I know more about what I’m doing and why, rather than flailing around finding good sounds by instinct. And when I do compose by instinct (which is essential), I understand better why it sounds good, and can expand on my inspirations in a rewarding way.

‘Nuff said! If music theory has been frustrating for you in the past, as it has been for me, I can’t recommend this book highly enough.

Next: Between the Keys

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Posted by on Nov 7, 2012 in Just Intonation, The Lattice | 0 comments

The Tonic

The heart of the lattice is the note called 1. This note is the tonic.

Almost all the music you hear — pop, rock, classical — has one note that is at the center, a master note against which all other notes are measured. That note is the tonic. It’s the Do of Do Re Mi. When you call a scale “G Major,” or say that a song is in the key of G, the G is the tonic.

A single note means little by itself. But when it’s considered in relation to the tonic, it acquires meaning. The examples in yesterday’s post show how a note changes character when played against different tonics.

The tonic establishes the framework for the rest of the notes in a piece. It’s the anvil on which the music is forged.

The tonic can be any note. When you tune your guitar by the campfire, without a tuner, just tuning it to itself, you’ve chosen a reference frame that will make perfect sense, regardless of whether it’s the same frame as a piano or orchestra back home. You can happily play great music in the key of G-and-a-half, if you’re playing solo.

Once you’ve established the tonic, the rest of the notes are tuned, and named, relative to that note. The tonic is the center, the Big Bang of that particular musical universe. The rest of the structure comes from the interplay between the tonic and small, whole numbers — mainly 2, 3, 5 and 7.

The tonic is Home. The lattice shows how music is a journey, away from home and back again, through different lands, each with its own scenery and feeling.

Next: Harmonic Experience

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Posted by on Oct 28, 2012 in Just Intonation, Recordings, The Lattice | 9 comments

Flying Dream on YouTube

This may have been the most intense art project of my life. Some time early in 2012, I got it into my head to create a stop-motion animation of my song Flying Dream, moving in harmonic space. I’ve spent the past five months working like crazy on it. The song is carefully arranged using the lattice of fifths and thirds, sung and played in just intonation, and animated using colored lenses, rice paper and a flashlight.

If it doesn’t play smoothly, don’t hesitate to lower the resolution. It survives scaling well, and the timing is crucial.

This video encompasses almost everything I’ve learned in the past couple of years. I will be writing much more about it. Meanwhile, here are a couple of external links about the basic subjects:

just intonation

lattice (music)

Next: Back Story


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