Posted by on Jul 20, 2013 in Consonance, Just Intonation, The Lattice | 0 comments

## Mixolydian Matchsticks

In yesterday’s post I mentioned matchstick harmony.

This concept is from Mathieu’s book Harmonic Experience, which I’ve discussed a lot on this blog.

Matchstick harmony is governed by a rule: It’s easiest for the ear to follow harmonies that move short distances on the lattice.

Imagine that the lines of the lattice are matchsticks. The triangles that they make are triads, major ones pointing up and minors pointing down.

If you move by as few matchsticks as possible when going from triad to triad, you will generate a chord progression that “makes sense” to the ear.

Here’s a rather artificial matchstick chord progression in Mixolydian mode. All I do is flip from each triangle to the one that borders it. It isn’t great music, but it shows how moving small distances on the lattice can draw the ear to a distant spot and bring it back again.

Actually this progression does drag the ear along rather fast. The roots move by major thirds (solid lines) and minor thirds (broken lines), which are not the shortest distances on the lattice. I like those equilateral triangles — they make visualizing easier for me — but if I wanted to accurately show harmonic distance, the horizontal lines, showing movement by fifths, should be the shortest, and the broken lines, the minor thirds, should be the longest, with the major thirds in between.

Progressions that move left or right, by fifths, are easier yet to follow.

Posted by on Mar 4, 2013 in Just Intonation, The Lattice | 2 comments

## The Infinite Lattice

The lattice goes on forever in every direction.

It starts with the tonic, the 1, the Big Bang of the musical universe.

Multiplying and dividing the tonic by 3 generates the horizontal axis. This is the familiar circle of fifths, although in just intonation, it doesn’t quite come out exact. If you multiply by 3, twelve times, you run through all of the scale degrees and land back on the tonic, 19 octaves up … almost. Three to the twelfth power, octave reduced until it’s back in the original octave, comes out to about 24 cents, not zero. Equal temperament flattens this out by subtracting two cents from every fifth. Very handy.

Multiplying and dividing by 5 generates the vertical axis. The two together create a plane, a map of harmonic space. Tonal music, that is, music that is organized around a key center or tonic, can be viewed as a journey on this map.

In the center of the map, there is a lovely pattern of twelve notes that form a chromatic scale.

Each of these notes has a cousin, four fifths down and a third up, that is tuned almost the same. It’s 22 cents flat of the original note, a distance of a Didymic comma. Thus there is another major second, the 2-, just outside the 12-note pattern.

Doing the same thing for all twelve notes creates another chromatic scale to the west. It’s the same scale, 22 cents flat of the original. It works the other way too — there is another block of notes to the east, same scale, 22 cents sharp.

The other comma I’ve discussed, the Great Diesis, shows how to extend the lattice north and south. This one shifts the pitch by 41 cents. It’s the shift that results when you go up or down by three major thirds. Equal temperament flattens this comma out too, but the adjustment is more extreme. Every major third in ET has to be sharp by about 14 cents in order for three of them to add up to an octave, a noticeable difference in pitch.

The pairs created by the Great Diesis have different note names. The b6 in the lattice above is at 814 cents, and the #5 is at 773 cents — 41 cents flat. In the key of C, these notes are Ab and G#, and they are played with the same black key on the piano, between G and A. Until I started studying just intonation and the lattice, I had no idea why one would want to think of these as different notes. It’s not an old-fashioned or obsolete distinction. It’s very useful, when writing or arranging, to know where you are on the lattice, and it’s just as useful in ET as it is in JI.

Now I can add two more blocks to the north and south.

And here’s the whole thing. The colors are arbitrary.

The chromatic scale, a block of 12 notes, has tiled the plane. The note names get pretty crazy — triple flats indeed! But they exactly describe the pitch of every note, in just intonation. Start with the major scale, 1-2-3-4-5-6-7. Every sharp (#) adds 70 cents to the original note, and each flat (b) subtracts 70 cents. Each + adds a Didymic comma, 22 cents, and every – subtracts 22 cents.

Posted by on Feb 17, 2013 in The Lattice, The Notes | 0 comments

## Extending the lattice

As I’ve analyzed my songs on the lattice, and written new music using it as a tool, I have found that I have a certain palette of notes in my mind, a territory of the lattice that I can hear and think with. The notes in this portion are distinct individuals for me. Each one has its own personality, a distinct mix of attraction, repulsion, beauty and function. I’ve described and given examples of many of them.

When I wrote Flying Dream in 1981, I was consciously trying to write a song that used all twelve notes of the chromatic scale. The first part of making the Flying Dream animation was to reverse-engineer my own song, figuring out with my new tool (the lattice) what I had been instinctively hearing at the time.

It turned out that I had been hearing about 18 notes in the song, including blue notes, and notes up in the northern part of the lattice. That made sense. I remember, as a kid, being disappointed to find out that there were only 12 total notes in music to work with. It seemed to limit the possibilities, like being stuck with the 8-color Crayola box.

The music I love to listen to, and make, has the big 64-color box with built-in sharpener. What’s up with Mick Jagger’s “Oooooh,” at the beginning of Gimme Shelter, or that guitar lick in Dizzy Miss Lizzy? These notes can’t be found on the piano, unless you have a pitch bend wheel. Check out this clip of Ray Charles bending notes in 2000 — now there’s a use of technology! The mystery of those notes, and others like them, has stuck with me, and now I feel like I’m getting to know them as friends.

Posted by on Oct 28, 2012 in Background, Just Intonation | 0 comments

## Untempered Music

For almost two years now, I’ve been exploring the nature of music almost full-time. I threw out everything I knew, started with the most basic thing I could think of, the number 1, the origin of the musical universe, and worked my way from there.

My explorations quickly led to the underpinnings of musical harmony, the natural notes that can be expressed as ratios of small whole numbers. These are the notes people generally played and sang, before twelve-tone equal temperament (12ET) came along. 12ET is a clever tuning system, a collection of 12 notes that are slightly retuned from the natural ones, mathematically fudged so that you can play fixed-pitch instruments in any key, and change keys without retuning.

Before 2011, I’d never seriously questioned those 12 notes. They are “The Notes,” after all. They’re the ones on the piano, and they’re what your guitar tuner tunes to. They’re right, right? Well, not really. In the past two years, I have met and made friends with a whole color palette of new, untempered notes, no two alike, each with its own function and personality. In the process, I’ve discovered new ways of thinking about and visualizing music that have greatly increased my enjoyment of it. I have even made friends with equal temperament again, after a long journey away. This website will tell about the journey. Welcome!

Gary Garrett