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 view, the tonic major helps the ear grab onto the center, by adding two notes that point directly at it. The ear has more information to work with.
The mind has amazing real time mathematical ability. Maybe a more accurate way to say this is that the mind has an amazing ability to quickly analyze and predict physical phenomena. The physical phenomena can be described by math. I don’t think the mind is working with arithmetic calculations at blinding speed, like a computer. It’s more of a massively parallel, holistic analog processor, that achieves a similar result.
Willie Mays used to catch fly balls with his back to the plate. Here’s a famous one:
Mays watches the ball start its flight, calculates the parabola it will follow (fine tuned by the conditions that day), and sets out at top speed for the spot, 400+ feet deep in center field, where he knows it’s going to land. He doesn’t (can’t!) look at the ball until it’s almost upon him. Marvelous.
So the ear hears a note, another one at 3x the frequency (remember octaves don’t count, 3/2 works like 3/1 in this regard), and another one at 5x. All three notes are direct signposts, pointing exactly at the tonic. Here we are, says the mind.
This may be why the equal-tempered major third gives me that slight queasy feeling. The tonic is the tonic, all right, but that equal-tempered third doesn’t point right at it! It’s close enough that the ear correctly identifies it, but it’s actually pointing at a note about 1% sharp of the tonic, and something sounds subtly off, like day-old sushi.
Here it is again: pure third, ET third, pure third. The middle note, the ET third, has a ratio of about 5.04/4.
I’ve added a new recording to the Audio page. It’s the first time I’ve consciously written a song using the lattice. The chord progression is especially influenced by how it appears visually. I was moving colored bits of glass around throughout the process, aiming for beauty, tension and resolution. The music tells a small story, of a journey around the map.
The lattice seems to create a connection between the auditory and the visual. The forms and movements are beautiful, like chess moves are beautiful. The auditory beauty tracks somehow with the visual beauty. How it looks can be used to predict how it will sound.
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.
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:
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
Now we have four notes: 1/1, 5/4, 3/2 and 2/1 — enough for a scale.1-3-5-8
This scale is contained in the chord of nature, and it pops up all over the place. A clear example is the bugle.
Bugles have no valves or keys. So how can you play more than one note on one?
A bugle is a long tube full of air, curved so it fits in a small space. The player’s lips get the air column vibrating, and by changing the tightness of her lips, the player can coax the air column into vibrating along its whole length, or get it to break up into sections, just like the jump rope in the Chord of Nature demonstration.
1) Isn’t it strange that when you multiply by 3 you get a fifth and when you multiply by 5 you get a third? The note names come from their position in a seven-tone scale. Here’s how our new scale fits with the standard do-re-mi. The notes we’ve explored are played louder to set them apart. five notes in do re mi
The 5/4 note pops up third in the scale and the 3/2 note comes up fifth. It’s just a confusing coincidence, based on our fondness for seven-tone scales.
2) Here’s a sneak preview of why I’m going to all this trouble. The equal-tempered major third that we’ve been hearing all these years is not tuned to the 5/4 ratio. It’s tuned sharp, by almost 1%. This isn’t enough to make the note sound obviously sour, but it’s certainly enough to change the feel of it.
Try listening to the following example a few times, and pay attention to how you feel while listening. JI3 vs ET3
The first note you hear is the tonic with a pure major third. The second note is with an equal tempered major third. Then it goes back to the pure 5/4 note. The pitch difference is small, but I perceive an uneasiness, almost a queasiness about the equal-tempered version. Do you hear a difference, and if so, how does it feel to you?
Doubling the frequency of a note certainly changes it. The ear hears a higher-pitched note. But there is something in the essence of the note that does not change, a character that stays consistent through the octaves.
This allows a process called octave reduction. When you’re working with notes as ratios, it’s convenient to multiply or divide the raw ratio by 2, as many times as is necessary to bring it into the same octave as the tonic.