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Posted by on Aug 22, 2013 in Consonance, Recordings, The Lattice, Tonal Gravity | 0 comments

100 Girlfriends

There is a passage, in my song Real Girl, that clearly showcases both kinds of dissonance — the kind that comes from harmonic distance, and the kind that comes from reverse polarity.

This melodic passage occurs many times in the song, and it contains a rather dizzying series of tensions and resolutions. My friend Jody Mulgrew, who has an exquisite sense of pitch, experienced actual nausea the first time he heard the song. He told me, “I was wondering how to tell my friend Gary that I didn’t like his new song. Then, before the chorus, it started to sweeten up, and when the song was over I immediately hit the ‘replay’ button. I realized it was just tension and resolution.”

I think my friend was experiencing what I call tonal vertigo. His comment spurred some of my thinking on the nature of harmony, how it may be a byproduct of our orientation software. The “100 girlfriends” section is a roller coaster ride in the tonal gravity field. Here it is in its original form:

Now to slow it way down and take it apart.

The first dissonant melody move is to the 7. The interval is a major seventh, down a half step in pitch, and the harmonic distance is great enough (3×5=15) that the note is quite dissonant. But the bass, alternating between 1 and 5 as so many bass lines do, quickly moves to resolve the dissonance.

Note that there is still an unresolved, unfinished feeling. Even though everything you can hear is beautifully consonant, the ear still remembers that the real root of the chord is the 1. This memory is crucial to tonal music.

The next move creates a different kind of dissonance. This is the tension of reverse polarity.

First the melody moves to the 1. This note is right next to that 5 in the bass, and beautifully harmonious. But there is tension, because it’s a reciprocal note. The way to get from a 5 to a 1 is to divide by 3 — it’s one move to the left on the lattice.

Then it makes a crazy move, to the b6, that gives me vertigo. Not only is this note distant from the bass note (a factor of 15), but it’s the reciprocal version of the major seventh, its mirror twin, the minor second. You’re dividing by 15, rather than multiplying. Here’s the article that explains why this is such an important difference.

If this weren’t enough, the b6 is also a reciprocal of the root. Remember, even though the bass is the 5, the root is still the 1. The b6 is the mirror twin of the 3, an intensely reciprocal note. So the tension is very high.

And, in two moves, the melody has covered a lot of harmonic territory, all in the reciprocal, Southwest direction. No wonder Jody felt nausea! It’s an E-ticket ride.


Once again, the bass moves to save the day. The chord changes too — that 4 in the bass is the new root. The melody note magically becomes a minor third, not fully consonant, not fully resolved, but a lot better.

In the next post, the famous tritone! Then full resolution.

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Posted by on Aug 20, 2013 in Just Intonation, Recordings, The Lattice | 1 comment

Real Girl, Animated

Here is my third stop-motion animation of a full song.

Real Girl uses a custom nine-note scale. It occupies the Southeast quadrant of the lattice, the zone of the natural minor, with two added notes — the 7, which allows for a major V chord in the progression, and the 7b5, a blue note that is showcased often in the melody.

This scale contains a sharp dissonance, between the b6 and the 7.  I go back and forth between those two notes a lot, with a stop on the 1 in between to help ease the transition.

Watch how the melody and bass chase each other around. In the next few blog posts, I’ll slow this dance down, and show how the polarity flips create tension and resolution. When the melody is below and to the left of the bass, the energy is reciprocal, tense. Then one or the other moves so that the melody is above and to the right, the energy becomes overtonal, and the tension resolves.

Another fun thing to watch is the alternating bass. Roots and fifths are right next to each other on the lattice. The red lens swings like a pendulum throughout the verses.

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Posted by on Aug 6, 2013 in Consonance, The Lattice, Tonal Gravity | 0 comments


The following video compares the perfect fifth with the perfect fourth.

These notes are the next-door neighbors of the tonic. They are equally close to the center. They are both harmonious. Yet there is a great difference in their character.

The difference between these two intervals is polarity.

I learned this term from W.A. Mathieu, in his amazing book “Harmonic Experience.”

Polarity is the main driver of tension and release in tonal music. I think it’s much more important than harmonic distance, the other component of consonance.

Here’s how I think polarity works:

Almost all sounds are actually a bundle of waves, of different frequencies. The frequencies in these bundles tend to vibrate at multiples of each other — 2x, 3x, 4x, 5x some base frequency — a harmonic series.

Our ears are highly attuned to such relationships. They help us figure out which frequencies belong together, so we can analyze them and identify the source. If we hear two frequencies in lockstep, three cycles to one, it is very likely they come from the same object.

I think this is why we can hear harmony. It’s a byproduct of our built-in orientation software. The overtone series of a sound is a powerful source of information about the object that made it. The harmonic content tells us whether it’s a barking dog or a friend or rustling leaves. Something in us sorts out and analyzes many harmonic series at once, in real time, identifying sound sources, locating them in space, and even sensing their texture at a distance. It’s a phenomenal processor.

So the processor recognizes the 3-wiggles-to-1 dance of the perfect fourth. But something is wrong — the 4 does not belong in the overtone series of the 1. The overtone series is generated by multiplication, not division.

This is a strange input for the mighty processor.

  • The 4 has to belong to the 1, because the two are in step with each other at three beats for one. In nature, that’s a dead giveaway.
  • But the 4 can’t belong to the 1, because the ratio is the wrong way around — natural sounds do not contain 1/3 in their overtone series.
  • Maybe the 1 actually belongs to the 4. The 1 is three times the 4, so if the 4 were the base, all would make sense.
  • But every other clue, all the harmonics in the drone (and, importantly, the listener’s memory), are pointing to the 1, hollering “This is the basic frequency!”

What’s a supercomputer to do?

I feel this sensation as unrest or instability, a need for something to change. Either the 4 needs to resolve to an overtonal note such as the 5 or major 3, or the root needs to change. Moving the root to the 4 will resolve the dissonance, introducing a new reciprocal tension — now the root “wants” to go to the 1, because the ear remembers. There are videos of this here.

On the lattice, there are two basic polarities, overtonal and reciprocal. Some notes combine both energies.

Overtonal notes are created by multiplying the base frequency. They appear in the natural harmonic content of sounds. They sound stable, restful, resolved — more so the closer they are to the center of the lattice, where tonal gravity is stronger. Pure overtonal notes are in the Northeast quadrant of the lattice.

Reciprocal notes are created by dividing the base frequency. The mind recognizes them as being related to the base frequency, but they do not appear in its natural overtone series. They sound unstable, restless, tense — and, like the overtonal ones, the effect is stronger closer to the center.

Polarity is our sense of the tonal gravity field. It is how we orient ourselves in harmonic space.

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Posted by on Jun 21, 2013 in The Lattice, The Notes, Tonal Gravity |

The Power of the Seventh Chord

The V chord, the major chord based on the 5, is a powerful compositional tool. It points, very clearly and with a lot of tension, directly at the tonic. If you want to lead the ear to the I, the V chord is the top-of-the-line triad.

Why this is so is still a bit mysterious to me. It’s been discussed a lot. It seems to have both melodic and harmonic elements.

Melodies “like” to move short distances in pitch, and the move from the V to the I is elegant melodically. The 7, or major seventh, resolves up a half step to the 1. The major seventh is called a leading tone because of this very property. The 2 drops a whole step, also to the 1, and the 5 stays put.

In harmonic space, voices, especially roots, “like” to move short distances too. The shortest move of all is a fifth, and when the V goes to the I, the root moves down by a fifth. It seems natural that if the ear is anticipating the next chord, it will place its bet on the change that expends the least energy. All three notes could be seen as moving that same short distance, the easiest possible move.

I like to think of it in terms of tonal gravity. The tonic, the 1, is like a sun at the center of a solar system, and it exerts a gravitational pull. Moving away from it creates tension, collapsing into it creates resolution. Just as with gravity, the closer in you are, the stronger the force. The V is right next to the I, harmonically, so the tension is very strong.

The V chord isn’t the last word, however. It’s possible to crank it up, by adding another tense note.


The 4 and the 5 are the closest notes to the 1, in harmonic space. These two notes have the strongest tonal gravity of all. Their effect is different — 5 is the strongest overtonal note, and 4 is the strongest reciprocal note. Both point straight at the tonic.

Melodically, the 4 is two half steps below the 5. This makes it a flatted or minor seventh, added to the V chord. So the final chord is called a V7.

Of all the notes we could add to the V chord, the 4 creates the most tension, and it’s pointed directly at the tonic. I say this is the source of the power of the dominant 7th chord.

In Be Love, I add even more tension before I’m through. The melody dances around, and right before the final resolution, it lands on the 6.


I’ve added yet another tense note to the mix. It’s not as strong as the 4, but it jacks up the gravity another notch. The root is on 5, so the 6 is two half steps up from it melodically. This makes it a ninth chord — start with the basic major triad, and add a seventh and a ninth.

Now I’m set up as strongly as possible for a return to the tonic, and sure enough when the drop happens it lands with authority. I’m in major land now, and the chorus will feel entirely different from the verse.

Here’s the whole effect:



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Posted by on Mar 11, 2013 in Consonance, Just Intonation, The Lattice, Tonal Gravity | 1 comment

Tonal Gravity

I believe that the great driving force in tonal music, that creates the drama and story of the music itself (independently of any lyrics), is the longing for home.

Home is the tonic. If a song is in the key of A, all the A’s in their various octaves will sound like home.

Although there are many exceptions, most music begins on the tonic, to show the ear what key the piece is in, and ends on the tonic, to bring the listener home again. In between, the music wanders, out and back again, creating tension and resolution.

One of the beauties of the lattice is that it shows a clear graphical display of this tension.

It’s as though the tonic creates a sort of gravitational field around itself. It acts a lot like real gravity. The chords and notes move in this gravitational field, like planets and moons around a sun. The gravitational field follows a few basic rules:

  1. Movement away from the center creates tension; movement toward the center gives a sense of resolution.
  2. Notes that are overtonal from the center, generated by multiplying, located to the right and up, will feel more resolved. Notes that are reciprocal, generated by dividing, to the left and down, will feel unresolved.
  3. The closer you are to the center in your journey, the stronger the sensations of tension and resolution are. The field is stronger closer in, just like real gravity.
  4. The closer together two notes are, the more consonant, or harmonious, they will be when sounded together. The farther apart they are, the more dissonant they will be, the more they will clash.

Roots generate local gravitational fields. I think of them as Jupiter to the tonic’s Sun. When the root is on the 5, for example, it shifts the gravity field to the east on the lattice, and the 2 and 7 become harmonious, consonant notes, rather than dissonant ones. The tonic still has great influence, so the entire chord feels unresolved — a 5 chord pulls very strongly toward the 1 chord, a property that is heavily relied upon in Western music. As long as the 5 is the root, though, the 2 and 7 will be consonant harmonies, because they are close to the 5 on the lattice.

Here is a movie to show how that works. The music starts with a tonic chord. Then, one at a time, the 2 and 7 are introduced. These notes are dissonant, and create a sense of tension against the tonic.

Then the root moves to the 5, and the character of the 2 and 7 changes. Now they form a major chord based on the 5, a harmonious configuration. They have become moons of Jupiter. Hear how the dissonance goes away? But there is still plenty of tension, as now there are three notes venturing away from the center, pulling the ear back toward home.

Then the root moves back to the 1, and the 2 and 7 collapse back in toward the center. There is a sense of arrival.

This movie illustrates another observation: consonance / dissonance and tension / resolution are not the same thing. They both relate to distance on the lattice, but they do not necessarily track together. When the root moves to the 5, the dissonance goes away, but there is a new tension, a drive to resolve toward the center. The ear remembers where home is, and longs for it.

These principles can be consciously used to create desired effects when writing and arranging. Resolution and consonance give the music beauty, and tension and dissonance give it teeth.




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