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Posted by on Jul 18, 2013 in Just Intonation, The Lattice, The Notes |

Mixolydian Mode

I’ve been quiet lately because I’ve been working on an animation of my song Real Girl. It’s a complicated one, a dance of harmonic tension and resolution. The bass and melody chase each other around the lattice like courting butterflies.

Meanwhile, there’s more to be extracted from the Be Love video.

A mode is a type of scale, characterized by the pattern of intervals between its notes. The note spacings stay the same no matter what key it’s in. When we say a song is in A major, we mean the tonic is A, and the mode is major. The major mode, also called Ionian, is the familiar Do-re-mi-fa-sol-la-ti-do.

Any combination of notes, covering an octave and organized in pitch order, can be a mode. There is a particular set of modes, often called church modes, that can be played on the white keys of the piano. The different modes start on different notes. The major mode goes from C to C; Aeolian, or minor mode, runs from A to A. Stick to the white keys, and the notes will be right for that mode.

Ionian and Aeolian are the commonest modes in modern Western music, but Dorian (D to D) and Mixolydian (G to G) are popular too.

I wrote the chorus of Be Love first. It’s in major mode. I wanted the verse to have a different feel, so I decided to make it Mixolydian — a favorite for several reasons. I love the name. And, as Mathieu points out in his book Harmonic Experience, it’s particularly easy to improvise over. It’s common in rock music from the seventies on — see the BTO clip in this post. My song Driving is mostly in Mixolydian mode.

The big reason I wanted the change was to make the chorus more of an anthem, by contrast. Mixolydian has a dark, beefy quality to me, and when the chorus comes around it sounds like the sun is coming out.

In equal temperament, there is only one difference between major and Mixolydian scales. The seventh degree is minor instead of major. Starting with G, and going up the white keys, you get G-A-B-C-D-E-F-G. The G major scale goes G-A-B-C-D-E-F#-G. Only the seventh is different.

In just intonation, the situation is a bit different. There are three b7s in the inner lattice:  b7, b7-, and 7b7. Which to choose?

Here is the major scale.


One possibility is to just drop the 7 to the b7:


I like the interpretation below. It uses the b7-, and also changes the 2 to a 2-.


This gives me an in-tune major flatted seventh chord, which I love. In the key of G that’s F major — lots of rock music uses this chord.

The notes of this scale (Western Mixolydian?) are in the same relationship to each other as the notes of the major scale, shifted one space to the left. The intervals are just as consonant as the major scale ones, only arranged in a different order.

It’s easy to write chord progressions in this mode. Major and minor triads form triangles on the lattice (major triads point up, and minor ones point down) and there are five in-tune triads, just like the major scale. Since the triangles are all connected, moving from one to another feels natural and is easy for the ear to follow.

Again I refer to Mathieu’s book. Have I lately? In Harmonic Experience he writes about “matchstick harmony.” Cool stuff.

Here is the scale, animated against a drone.



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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.

220px-Crayola-64It 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.



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Posted by on Feb 6, 2013 in Just Intonation, The Lattice, The Notes | 3 comments

The Untempered Chromatic Scale (Part 1)

The familiar 12-note scale is also called the chromatic scale. There are many ways to generate an untempered version of such a scale.

The oldest way seems to be Pythagorean tuning. When you multiply a frequency by 3, you get a new note, an octave plus a fifth above. Multiply by 3 again, and you get another new note, and so on. Dividing by 3 gives you another new note, the perfect fourth, dividing by 3 again delivers another new note, and so on. Do this enough times and you can generate a 12-tone scale.

This scale forms the central, horizontal spine of the lattice.

pythagorean lattice


The central six notes of the Pythagorean scale are highlighted. If you keep extending to the left and right you get all 12 notes, and then even more.

Trouble is, once you’re more than two or three steps away from the center, the notes are definitely dissonant. The ratios just get so big that the ear can no longer hear them as harmony. The Pythagorean major third, for example, is the next note to the right of the 6+, just off the central lattice. Its ratio is 3x3x3x3, or 81/1. The octave reduction trick allows you to divide by 2 until it’s in the same octave as the tonic, so the ratio becomes 81/64.

Harmony is built from small, whole number ratios, and these are not small numbers. Here is what a Pythagorean major third sounds like:

clarinet 81:64

Wow. Back when Pythagorean tuning was the norm, the third was considered a dissonant interval. I concur!

In late Medieval times, the 5/4 tuning became more popular, and major thirds began to be regarded as a consonant interval. Here’s the 5/4, for comparison:

clarinet 5:4

Ah. Using the prime number 5 allows a more consonant chromatic scale, with much smaller ratios. Here it is on the lattice:

12-tone lattice


All of these notes have been covered in previous posts. Each one has a unique personality. Some are more consonant, some more dissonant, but all twelve have small enough numbers in their ratios to be perceived as harmony by the ear.

Here are their ratios, and pitches in cents:

1 — 1/1 — 0

b2 — 16/15 — 112

2 — 9/8 — 204

b3 — 6/5 — 316

3 — 5/4 — 386

4 — 4/3 — 498

#4+ — 45/32 — 590

5 — 3/2 — 702

b6 — 8/5 — 814

6 — 5/3 — 884

b7 — 9/5 — 1018

7 — 15/8 — 1088

Here is the scale, compared with the 12 notes of equal temperament.

untempered chromatic


When I first drew this scale, on graph paper, I was startled by how narrow the major-minor half steps are, compared with the equal-tempered versions. Take a look at the b3 (minor third) and the 3 (major third). They are only 70 cents apart, not the 100 cents of the piano scale. Same with the b6-6 and b7-7. It’s such a small move in pitch, such a large move harmonically!

The Mozart movies I made a month ago or so illustrate the major-minor pair quite well. The blandification of equal temperament has obscured a beautiful detail of harmonic music. So much is gained with ET, and so much is lost or obfuscated.

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Posted by on Jan 16, 2013 in Just Intonation, The Lattice | 0 comments

Melodic Space, Harmonic Space

Throughout my musical education, I’ve been taught that music happens in a linear space. This is the space so beautifully laid out on the piano keyboard.


Music teaching is organized around scales. In most Western music, the full scale consists of twelve notes, equally spaced. Other scales, such as the seven-note minor and major scales, are subsets of this full, “chromatic” scale. Due to octave magic, a mysterious and crucial aspect of our inner perception, when we get to the thirteenth note, we have multiplied the original note by two, and the sequence starts over again.

So, fortunately for musical analysis, melodic space can be described in one octave. It takes about ten of these octaves to cover the range of human hearing.

On the piano keyboard, melodies look the way they sound. When the pitch goes up, you move up the scale, and when the pitch goes down, you move down the scale. Short distances (the shortest is from one key to the next, a half step), feel short. Long distances (more than about three half steps) feel long. This is a good and useful space for visualizing melody.

Harmony, not so much.

Musical nomenclature, as I’ve pointed out before, has grown like an old city over the years. As music theory changes, bits and pieces of the old terminology are appropriated and redefined by new thinkers. The result is a cobbled-together mass that has a lot of weird contradictions and misleading names.

I think one of the most regrettable bits of confusion comes from the word interval.

The distance between two notes on the keyboard is called an interval. When my melody moves by an interval of a minor third, it has covered a distance of three half steps. When I move by a major third, I’ve covered four half steps. The major interval is bigger than the minor one — that’s why it’s called “major.” No problem! The move feels bigger when you sing it.

The problem comes when you start to think about harmony — two or more notes sounding simultaneously. The word “interval,” with the same connotation of pitch difference, is also used to describe the distance between harmony notes. Yet in the world of harmony, the interval, or pitch distances don’t make any intuitive sense at all.

For example, two notes a fifth apart (seven half steps) sound wonderful when played together. C and G are two such notes. They are closely related to each other, harmonically. So are C and F, which are a fourth apart (five half steps). These are the best consonances there are, except for unisons and octaves.

So what about the note in between them, an interval of six half steps?

Yep, none other than the dreaded tritone, the devil’s interval, definitely a dissonant note.

If the linear scale were the best way to think about harmony, wouldn’t the tritone be between the fourth and fifth in consonance? Why would three notes in a row, next-door neighbors on the scale, be so drastically different from each other harmonically? The scale gives no clue. You just have to remember.

Perhaps there is a more intuitive way to visualize harmony, one that puts harmonically related notes closer to each other, and puts the notes that are harmonically farther apart … farther apart?

I think there is indeed a harmonic space as distinct from a melodic space. This space can be illustrated on the lattice. It’s not a good model for melody — scales do a much better job. But it’s a great model for visualizing harmony — what you see corresponds intuitively to what you hear.

The interplay between these two spaces creates the beautiful dance that is harmonized music.

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Posted by on Nov 13, 2012 in The Notes | 0 comments

The Major Third

Multiplying a note by 2 creates an octave, and multiplying it by 3 creates a perfect fifth.

Multiplying by 5 gives yet another new note, the pure major third.5-1

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.

Here are the bugle notes:bugle scale

Two sidebars before I go on.

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?


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