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Posted by on Jun 19, 2013 in Just Intonation, The Lattice, The Notes |

Leading the Ear

Be Love was written and arranged on the lattice. I consciously used the lattice as a tool to make the music do what I wanted it to do.

Working with this song has taught me a lot about leading the ear.

Different parts of the lattice have different sounds. The upper right, the northeast, is major scale territory. Music in this zone sounds major, you know, that uplifting, stable, “happy” majorness. The northwest region, up and to the left, has a darker, dramatic sound, not like minor, but with its own flavor. It shows up a lot in rock. A great example is BTO’s Taking Care of Business. The progression is I, bVII-, IV, I. (I use numbers for notes and roman numerals for chords.)

I wanted the song to start in the northwest for the verse, and then move eastward for the chorus, and then go back again, and I wanted to choose notes that would lead the ear on the journey.

Here’s the beginning. The chords plant a flag in the Northwest.

The music stays there for a while, and then it starts to move. The chord progression changes, and the guitar melody reaches out to the east and starts to rope in more territory.

Finally, right before the chorus, the V chord takes the song firmly into dominant territory.

Notice how the melody leads the way into the far east. When the melody goes to the 2, in advance of the chord progression, it sets up tension. The tension is resolved when the root moves up to the 5 and creates a more consonant interval.

One of the pleasures of these lattice movies is watching the fleeting, exotic harmonies that are formed as the melody dances around the basic chords. This chord is a type of sixth chord.


When 4 is the root, 2 is its sixth degree. I call the interval between 4 and 2 a Pythagorean sixth, because it is generated entirely by multiples of 3 — a characteristic of Pythagorean tuning. The ratio, octave reduced, is 27/16. It sounds different than the 5/3 sixth, and is tuned sharper — 906 cents instead of 884.

The Pythagorean sixth chord leads the ear to the east. The tension of the 2 in the melody is resolved by moving all the music up to meet it.

Now there’s a new tension, against the tonic, which is in the back of the listener’s mind all the time. I will want to resolve this tension by collapsing to the center, but first I want to increase it as much as possible. I want to dive into the chorus from a great height.


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Posted by on Feb 5, 2013 in Just Intonation | 0 comments

The Major Scale in Cents

The simplest untempered tuning of the major scale is:

1 — 0 cents

2 — 204

3 — 386

4 — 498

5 — 702

6 — 884

7 — 1088

Here’s how that tuning compares with the equal tempered scale:



The black numbers show the pitches of 12-tone equal temperament. They are equally spaced, like inches on a ruler.

The red numbers show the tuning of the untempered major scale. They are spaced in the way they naturally turn out when you generate them with small whole number ratios. As is so often the case with the natural world, they don’t line up too well with the nice human grid lines we love.

The way I see it, when you play in equal temperament, you’re playing the grid lines on the map.

When you sing or play the untempered notes, you are visiting the actual territory.

I’ve read that it’s not possible to combine the two, but I disagree. It’s a matter of showing the ET instrument who’s the boss. My favorite example is Ray Charles. Here’s a video from 1976. He’s playing the piano, laying down those grid lines, and the rest of the band is too, but when he sings, his voice owns the sound, and the sound becomes him. A great, dominant singer will infuse the whole combo with that soul.

Another great example is Ella Fitzgerald. Want some goosebumps? Check this video out.



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Posted by on Feb 1, 2013 in Just Intonation, The Lattice, The Notes | 0 comments

The Untempered Major Scale

In harmonic space, the clearest name for each note is its ratio — 5/3, 3/2, 4/3, etc. Precise and unambiguous. But the ratios don’t give a very good idea of how to go about playing or singing those notes.

There are a few instruments, such as the left hand keyboard on the accordion, that are organized for harmonic thinking.


The rows are arranged in fifths, and the first two rows are a major third apart, just like the lattice.

For most music making, though, we need to know the pitch of the note. Instruments and voices tend to live in the world of melodic space — scales and pitches.

Here’s what the simplest, most harmonically consonant major scale looks like on the lattice:

One can convert the ratios to pitches using the formula for cents. The ET values are in parentheses.

1 = 1/1 = 0 cents (0)

2 = 9/8 = 204 (200)

3 = 5/4 = 386 (400)

4 = 4/3 = 498 (500)

5 = 3/2 = 702 (700)

6 = 5/3 = 884 (900)

7 = 15/8 = 1088 (1100)

Note that the 1, 2, 4 and 5 are very close to their ET equivalents. Most ears would be unable to tell the difference.

The third, sixth and seventh, however, are all noticeably flat. Or perhaps I should say their ET namesakes are noticeably sharp.

I think this explains a lot about rock music, which depends heavily on power chords (roots and fifths with no thirds) and 1-4-5 chord progressions. The notes of a 1-4-5 power chord progression are 1, 4, 5 and 2!

As Eddie Van Halen said in his terrific Guitar Magazine interview, “Really, the best songs are still based on I-IV-V, which is so pleasing to the ear. Billy Gibbons [of ZZ Top] calls me now and then, and he always asks, ‘Eddie, have you found that fourth chord yet?’ [Laughs].”

Of course the I-IV-V is inherently satisfying, it’s that great rocking chair between reciprocal and overtonal territory, in the most consonant part of the lattice, as close to the tonic as you can get (harmonically).

But I think the special appeal for rock music is that, in ET (and guitar is essentially an equal-tempered instrument), the 1, 4 and 5, and also the fifth of the 5 (the 2) are all in tune. Whatever one might think of ZZ Top’s simplicity (and some do scoff), it’s undeniable that they are fiercely in tune, and harmonically their music strikes a deep chord in the psyche, pun intended.

If the bass and rhythm guitar stick to those roots and fifths, the voices and lead guitar can play all the other notes, because they can be bent and wiggled until they sound right.

This works just as well in traditional country music. Let the bass player nail down those roots and fifths, and the voices (and the fiddle) can sing in-tune harmonies as sweet as you please.



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Posted by on Nov 25, 2012 in The Lattice | 0 comments

The Major Scale

The notes 1, 2, 3, 4, 5, 6 and 7, clustered at the center of the lattice, constitute a major scale. This tuning uses the smallest ratios (the ones with the lowest numbers) available for each position in the scale. It goes back at least to Ptolemy in the 100’s AD.

I find it visually beautiful. It’s like a cat’s cradle.

Here it is again, with a drone on the tonic, to show how the notes resonate with the drone. Each one has its own flavor, its own harmonic character.

Notice how the melody never moves from a note to the note next door. It always moves two grid segments. This is a first look at the difference between harmonic space and melodic space.

Melodies “like” to move up and down on a linear scale. They want to go to a nearby note when they move — that is, near by in pitch. We hear, and sing, small movements in pitch better than we hear leaps.

Harmonies “like” to go to nearby notes too, but harmonic space is different than linear, melodic space. The 1 and the 5 are harmonic neighbors. In fact, they are as close together as notes can be, harmonically, without being the same note — a single factor of three. But they are far apart melodically — the 5 is almost at the midpoint of the scale.

1 and 2 are melodic neighbors, It’s easy to for the voice to move from one to the other. But they are far apart harmonically — two factors of three. A small move in pitch can produce a large harmonic jump.

Arranging a melody and chord progression involves interweaving the notes so they work in both spaces. The melody will tend to move up and down by small melodic steps, close together on the scale. The chords will tend to move by small harmonic steps, close together on the lattice.

It’s a bit like designing a crossword puzzle, working “up” against “down” until it all fits. The lattice is a wonderful tool for visualizing this dance.


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Posted by on Nov 22, 2012 in The Lattice, The Notes | 0 comments


Musical nomenclature has been cobbled together over the centuries like a medieval city. Different systems leave their imprint in convention, later developments try to be compatible with accepted names, and the whole thing ends up confusing and contradictory.

Take enharmonic equivalents, for example. G# and Ab are the same note on the piano, the black key between G and A. So why do you sometimes call that note by one name, and sometimes by the other? The answer actually leads to some deep realizations about music, and it comes back to just intonation. In untempered or just music, G# and Ab are not the same note, and which one you choose becomes important. It’s important in ET too — the music establishes a context, and the ear figures out which note it’s supposed to be. But if you grew up with ET, and have no idea that there used to be two different notes there, the names can be confusing. How do you imply one note or the other? Which one is right in a given situation? Why bother? It’s a huge part of writing chord progressions that make sense, but ET by itself isn’t going to tell you what to do. You have to dig deeper for that.

I’ve slowly evolved a personal system I’m very happy with. It’s based on the lattice.

The great advantage of this approach is that it’s entirely unambiguous. Every note on the infinite lattice has a unique name, and that name tells you exactly what its pitch is, and where it is on the map.

The seven notes I’ve covered so far form the core of the system. I’ve dropped all the word names and just use numbers:

1 — the tonic, 1/1

2 — the major second, 9/8

3 — the major third, 5/4

4 — the perfect fourth, 4/3

5 — the perfect fifth, 3/2

6 — the major sixth, 5/3

7 — the major seventh, 15/8

The rest of the notes are named by adding accidentals to modify the pitches. I’ll quantify these later, and explain how they work, but approximately they are:

b — flat by about 2/3 of an equal-tempered semitone

# — sharp by about 2/3 of a semitone

— flat by about 1/5 of a semitone

+ — sharp by 1/5 semitone

7 — flat by 1/2 semitone.

The basic notes occupy the center of the lattice. These seven notes form the major scale.


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