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Posted by on Mar 14, 2014 in Equal Temperament, Just Intonation, The Lattice | 3 comments

A Harmonic Journey: ET and JI Compared

The Harmonic Lattice can be viewed as a map of harmonic space. Music moves in harmonic space, just as it moves in melodic space (the world of scales and keyboards). The two spaces are very different from each other.

In melodic space, such as a piano keyboard, when two notes are close together, it means they are close in pitch.

In a harmonic space, such as the lattice, when two notes are close together, it means they are harmonically related.

“Harmonically related” means that one note can be converted into the other note by multiplying and dividing by small whole numbers. A note vibrating at 100 cycles per second is closely related to a note at 300 cycles per second. In melodic space, these two notes are far apart, but in harmonic terms they are right next door to each other — they harmonize.

In my video, Flying Dream, I animated the movement of one of my songs on the lattice. Now I’ve animated a composition of W. A. Mathieu’s.

Mathieu is the author of Harmonic Experience, an astonishing book that takes music back to its origins in resonance and pure harmony, and then uses the lattice concept to bring that harmonic understanding forward into the world of equal temperament. For me, the book opened the study of music like a flower.

The lattice, and my stop-motion animations, have given me a sort of musical oscilloscope. Instead of the music being some sort of black box, I can see inside it, get a visual image of what is going on harmonically. The new tool has made songwriting, improvising and arranging much easier.

I’ve animated Example 22.10 from the book. It’s intended to be an illustration of unambiguous harmony — the chord progression moves by short distances on the lattice, so it is clear to the eye and ear where you are. I think it’s a beautiful piece of music in its own right, a one-minute tour of a huge area of the lattice. It uses 28 different notes!

There are two versions of the video. The first one, in ET, has a soundtrack of Allaudin Mathieu playing the piece on his beautifully tuned piano. This is perfect equal temperament. It uses twelve notes to approximate the twenty-eight notes that the piece visits.

For the second one, in JI, I retuned the piano to the actual pitches of the lattice notes. Now, magically, the piano has all 28 notes. There is a whole new dimension to the music. In the JI version, I feel:

  • Slight vertigo when the music moves quickly
  • Satisfaction when a spread-out (tense) pattern collapses to a compact (resolved) one
  • A great sense of homecoming at the end
  • Stronger consonance and dissonance than in the ET version.

The four voices, from lowest to highest, are red, green, orange and yellow. It’s fun to follow one voice at a time.

This lattice is notated differently. It’s my usual system, but with letters instead of numbers. C is the tonic.


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Posted by on Oct 27, 2013 in Consonance, Just Intonation, The Lattice, Tonal Gravity | 2 comments

Tonal Gravity and the Major Scale

In my last post, I proposed a simple way to graph tonal gravity against the octave.

Overtonal notes, generated by multiplying, are restful, stable — they have positive polarity, pulling toward the center. Reciprocal notes, generated by division, are restless, unstable — they push. I call this negative polarity. Mixed-polarity notes have both, and I’ve chosen to simply add their overtonal and reciprocal components together to get the total polarity.

Here again is the graph of the 13 most central notes of the lattice.

Tonal Gravity 13-01

The stable notes are gravity wells, and the unstable ones are peaks. Melodies and harmonies dance in this gravity field. Higher points represent tension, lower ones resolution, and the lower they are, the more resolved and stable. The tonic major triad, most stable of all, occupies the lowest spots — 1, 3 and 5.

The polarity map of the major scale looks like this:

Tonal Gravity Major Scale-01The notes are all overtonal except the 4, which is strongly reciprocal, and the 6, which is mixed and slightly unstable.

Here’s a split screen video showing the major scale, against a tonic drone, on both the lattice and the octave. This is an example of how the lattice serves as a Rosetta Stone, a translator between harmonic and melodic space.

Can you hear the push/pull quality of the notes? Each note has its own feeling against the steady 1.



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Posted by on Oct 22, 2013 in Consonance, Just Intonation, The Lattice, Tonal Gravity | 0 comments

Putting Some Numbers on Tonal Gravity

I believe the sensation of tonal gravity is the most important driver of tension and resolution in tonal music, music that has a central key note.

The tonic is like a sun, creating a gravitational field around it. The lattice is a beautiful map of this gravitational field, in harmonic space.

Tonal gravity acts like real gravity, in that it’s strongest close to the center. The farther out you get, the weaker the effect.

There is a difference, though — unlike physical gravity, which only attracts, tonal gravity has two polarities — it can attract or repel. Briefly, if an interval is generated by multiplication, it will pull toward the tonic, as though to say, “You’re going the right way, you’re just not there yet.” If the interval is generated by division, the sensation is more of a push — it still points to the tonic, but now it’s saying “It’s over that way, go!”

Overtonal notes are stable, reciprocals are unstable. Reciprocal intervals create tension, overtonal ones create resolution.

The article on Polarity goes into detail, with examples.

The lattice can be divided into four quadrants, each with a characteristic tonal gravity. The northeast quadrant is entirely overtonal. This is the world of Major/Dominant: overtonal thirds, overtonal fifths.

The southwest quadrant is entirely reciprocal. Every interval is generated by division.

The northwest and southeast are zones of mixed polarity, overtonal on one axis and reciprocal on the other one. Here’s a map of the quadrants and the compass points:


Green is overtonal, stable, tonal gravity pulls.

Red is reciprocal, unstable, tonal gravity pushes.

Blue is mixed, gravity pulls on one axis and pushes on the other.

For a couple of years now, I’ve been chewing on how to represent the tonal gravity of the lattice mathematically. To describe what I experience, the equations should behave as follows:

  1. Overtonal intervals have one polarity, and reciprocals have the opposite polarity,
  2. Gravity gets weaker the farther one gets from the center, and
  3. Multiplying or dividing by 2 does not affect tonal gravity. This is to account for the octave phenomenon — going up or down an octave does not change a note’s position on the lattice.

Here’s my latest approach. I’m not presenting this as some kind of truth — but it nicely matches my own perceptions, and it leads to some interesting graphs. Any input you may have is welcome — feel free to comment, or email me from the Contact page. Here goes:

I will call the direction and magnitude of the tonal gravity field P, for Polarity.

Intervals are expressed as a ratio of two numbers, numerator and denominator, N/D. For example, a perfect fifth is 3/1, or N=3, D=1.

  1. For purely overtonal notes, of the form N/1: P = 2/N.
  2. For purely reciprocal notes, of the form 1/D: P = -2/D.
  3. For compound notes, with both overtonal and reciprocal components, add the overtonal and reciprocal gravities together: P = 2/N – 2/D.
  4. The ratio of the tonic, the 1, is 2/1.

I’d better explain that last one, because it would seem at first glance that the ratio of the tonic would be 1/1.

But what is the actual tonic? It has no specific pitch. It is not a ratio. It is an abstraction, the anvil upon which all notes are forged, the sound of one hand clapping. If a song is in the key of A, all of the A’s from subsonic to ultrasonic are actually octaves of the tonic, created by multiplying by two. It is impossible to say that any one of these A’s is “the” tonic — the tonic is “A-ness,” that thing which connects the numbers 110, 220, 440, 880, to infinity in both directions. I submit that the “1” of the lattice, which is a real pitch (or set of pitches, an octave apart, just like all the other notes) is in fact the octave, and its ratio is 2/1.

Here’s another drawing of the inner lattice. Instead of the note names, I’ve filled in the ratios, and the value of P.

Polarity central lattice-01

The green notes all have positive polarity, getting weaker as they get farther out. The red ones have negative polarity, also fading with distance. The blue ones have different polarities. Sometimes the overtonal part dominates, sometimes the reciprocal.

The b3, just southeast of the tonic, is a mixed-polarity note. Its ratio is 3/5, combining an overtonal fifth, P = .67, with a reciprocal third, P = -.40. If I just add the two gravities together, I get a positive net polarity of .27.

This makes sense. The minor third is considered to be a stable interval, though not as stable as the major third.

Both the major and minor triads consist entirely of stable intervals with positive P, which helps explain their special place in music.

So: now that I have some values for P, I can graph the tonal gravity of these 13 inner notes against the octave, in order of pitch.

Positive polarity is at the bottom, so that the feel is the same as real gravity. Unstable notes are up on mountain peaks, and when they resolve to stable ones they slide down into the gravity wells of the stable notes.

Tonal Gravity 13-01

There is that tasty melody zone I mentioned a few posts back. The whole region from 2- to 3 is stable.

The 4 is an isolated peak, and it’s easy to imagine a 4 sliding into the pocket of the 3, or the 5. This is what happens when a Sus4 chord resolves.

The 7 is lightly stable but hanging on by its fingernails — it’s called the leading tone, because it “wants” so badly to resolve to the 1. The tonal gravity of the 7 is usually thought of in terms of melodic pull — here’s a graphic demonstration that it has harmonic pull as well.

There is an unstable region from b6 through b7, with all the mirror twins of that stable melody zone. A melody will sound unstable, unresolved as long as it stays in that region.

This is the gravitational field in which the music moves, a sort of tonal skate park.

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Posted by on Sep 24, 2013 in Consonance, Equal Temperament, Just Intonation, Septimal Harmony, The Lattice, The Notes | 7 comments

Three Flavors of Seventh Chord

Chords and other collections of notes have consistent, recognizable shapes on the lattice. A major chord is a triangle sitting on its base, a minor chord is a triangle on its point. Yesterday’s post has videos showing these chords.

In the songs I know and write, the next most common chords after major and minor triads are seventh chords.

By convention, a “seventh chord” means a triad, with a minor seventh added. If the added seventh is a 7, or major seventh, it’s called a “major seventh” chord.

A minor seventh is an interval of ten half steps, or two shy of an octave. There are three different minor sevenths in the inner lattice, and each one makes chords with a different sound and function — that is, if you are playing in just intonation, or untempered. In equal temperament, the minor sevenths all sound the same, but there is still profit in knowing that they are different, because they function differently in chord progressions.

The three notes are:

  • The b7, at 1018 cents. The ratio is 9/5.
  • The b7-, or dominant-type seventh, at 996 cents. The ratio, octave reduced so it lands in the same octave as the tonic, is 16/9.
  • The 7b7, at 969 cents. This is 7/4, the harmonic, or barbershop seventh, a consonant note that appears in the actual harmonic series of the tonic.

Here are some movies in just intonation, so you can hear the differences.


First, the b7, added to a minor chord.

A pretty sound, I like it! In equal temperament, this note is at 1000 cents, 18 cents flat of the b7, a clearly audible difference. Here’s the same movie in ET:

Both the b3 and b7 are decidedly flat. The b3 especially sounds different, a lot more dissonant and “beating.”

I wrote a post a while ago, exploring this minor seventh and how it sounds in an untempered chord progression. It’s here.


The next minor seventh is enormously important. This is the dominant-type seventh, b7-, 996 cents. It is fortunate that it is so close to the equal tempered note, 1000 cents, because that means its effect is barely diminished in ET — and it is a really important note in classical music.

The reason it’s called a dominant-type seventh is because it most often shows up with the dominant, or V chord. The note two steps south of the 5 is the 4 — and when you add a 4 to a V chord you get this:


The 4 is a powerful note in this context. It has strong tonal gravity, with reverse polarity. I’ve written several posts about this — here’s one  about polarity, here’s one about the use of dominant seventh chords.

Here’s how the chord sounds when it’s built on the 1, in just intonation.

There is strong dissonance when that seventh comes in, and it’s dissonance with a purpose — the chord “wants” badly to resolve somewhere. In this case, it wants to resolve to the 4, the empty space in the middle of the chord. The 1, 3 and 5 are all in the harmonic series of the 4 — that is, they all appear in its “chord of nature,” the overtones that accompany a natural sound. So these notes sort of point to the 4. They point to the 1 even more strongly, though, until that b7- comes into the picture.

When you add the new note, the b7-, something new happens. This note points hard to the 4, and in a different way. It’s as though it says, “home is over there, go!”

Here’s a more detailed discussion.

The entire note collection “wants” to collapse to its center, like a gravitational collapse. The b7- helps to locate that center on the 4.

This effect is often used to move the ear to a IV chord. For example, if you want to start the bridge of a song on the IV, it helps to hit a I7 first. If you’re playing a song in G, and want to go to a C chord, a quick G7 will make the change seem more inevitable. Here’s that move in slo-mo.

The pull of the dominant-seventh-type chord is so strong that it is the sharpest tool in the kit for changing keys, or modulating. Classical composers use it for this constantly.


The last of the three is a beauty. This is the 7b7, the quintessential note of barbershop harmony, the harmonic seventh, 7/4. The b7- is highly dissonant, the b7 rather neutral, and the 7b7 highly consonant. It sounds (and looks) like this:

This is a resolved chord. In fact, if the consonance and stability of an interval are determined by the smallness of the numbers in its ratio, these are the four most consonant notes of all — 1/1, 3/1, 5/1 and 7/1.

Here is another opportunity to compare just intonation with equal temperament. The harmonic seventh and the dominant seventh sound exactly the same in ET. I believe that a good composer knows, consciously or not, which one is meant.

A good example is the “… and many more” ending so commonly added to Happy Birthday. It is clearly not a dominant type — it’s intended to mean the end of the song, even to put a stronger period on it than the major triad by itself. It’s a quote, or a parody of blues harmony. Play it on the piano and it will be tuned exactly like a I7 chord, but the ear can tell, by context, that there is no move expected, to the IV or anywhere, because it’s heard that little melody a thousand times, and it belongs at the end of a song.

But the signal is so much clearer when the tuning sends the message too! The 7b7 is at 969 cents, a third of a semitone flatter than the piano key.

By the way, I think this is why a common definition of “blue note” is “sung flatter than usual.” I believe the blue notes are the world of multiples of seven, and these just happen to be flatter than the closest notes in the worlds of 3 and 5, the basic lattice.

Here is a video of the 7b7 chord that starts with the harmonic seventh, goes to the equal-tempered seventh, and back to the 7b7.

Quite a difference. ET works because it implies the JI note, and the ear figures out what it’s supposed to be hearing. But the visceral impact is lessened a lot — in this case, IMO, completely.



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Posted by on Sep 23, 2013 in Just Intonation, The Lattice | 0 comments

Chords on the Lattice

A chord is a collection of three or more notes sounded at the same time. Arpeggios, in which the notes are sounded one after the other, are considered chords too. Two notes sounded at once are generally called an interval rather than a chord.

Chords make patterns on the lattice. A given kind of chord will look the same no matter where it is.

The most common chords are the major and minor triads (a triad is a three-note chord that is a stack of major and/or minor thirds). Here is what a major triad looks and sounds like on the lattice:

The major triad is an upright triangle. It even looks stable. It’s made of three interlocking intervals — in this case, from 1 to 3 (a major third), from 3 to 5 (a minor third), and from 1 to 5 (a perfect fifth).

Anything that looks like this on the lattice is a perfectly-in-tune major chord.

A minor triad is an upside-down triangle. Minor triads look like this:

Major and minor triads interlock to form the hexagonal lattice of fifths and thirds. This generates another lattice, a lattice of chords. W.A. Mathieu goes into great detail in Harmonic Experience, extending the chord lattice a long ways out and showing how music wanders on it. Here is an illustration based on my own lattice:

Chord Lattice

I use roman numerals for chord names, because the relationships between chords stay the same no matter what key I’m in. For example, the progression C-F-G is exactly the same as the progression G-C-D, at a different pitch. Both are I-IV-V progressions. This convention uses capital letters for major chords, and lower case for minors. I add a little twist by adding + and – to show commas; this allows a unique name for every chord on the infinite lattice.

It’s illuminating to track a chord progression on this lattice. The famous “Heart and Soul” progression, I-vi-IV-V, is what Mathieu calls “Matchstick Harmony.” The lines move like the matches in those matchstick puzzles. Progressions that move by these small harmonic distances are intuitive and easy to follow. The last move, from IV to V, is also easy for the ear, making this chord progression as natural as breathing. Start playing it on the piano and you will instantly have a crowd. In the key of C, it goes C-Am-F-G.

The chord lattice adds another dimension to lattice thinking. Watch the Flying Dream video for a good example. The progression travels far afield, exploring many of these major and minor triangles before finally coming home.

Other chords make other shapes that also repeat all over the lattice. For example, there are at least three different kinds of minor seventh chord. Here’s an article distinguishing them.

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Posted by on Sep 15, 2013 in Just Intonation, Septimal Harmony, The Lattice, The Notes | 0 comments

More Blue Tritones

I want to show you some lattice movies of how I’ve used blue tritones in my own music.

Real Girl has several examples. The clearest is a guitar lick in the chorus:

That 7b5 is tasty over the bVI chord. For an instant, it makes a “barbershop seventh,” the 7th harmonic of the root.

Here is a vocal example from the same song:

The melody visits the blue tritone on the way up, and again on the way down. I especially like it on the word “like,” the blues flavor of the septimal note comes through loud and clear without it being strictly blues at all. For me, this fusion of septimal notes to the European collection is the great contribution American music has made to the world. I wrote an early article on this, with some examples, here.

These bits of melody that visit the 7b5 are very similar to the ones that incorporate the 7b3. The septimal flatted third is the melody note of major blues tonality. It functions as the seventh harmonic of the IV chord, just as the 7b5 is the 7th harmonic of the bVI chord. Here’s an example from Flying Dream:

Hear the similarity? Try going back and forth between this video and the guitar lick in the first video.

One of the beauties of the lattice is that the patterns repeat everywhere. If you move a pattern to a different part of the lattice, the new notes will have the same relationship to each other, but the musical context will change and it will convey a different feeling. This is a splendid compositional tool, and helps me greatly in understanding harmony.

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