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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.

Next: Three Flavors of Seventh Chord

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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.

Next: Consonance and Dissonance

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Posted by on Jun 22, 2013 in Just Intonation, Recordings, The Lattice, The Notes, Tonal Gravity |

To the Far Northwest

The chorus of Be Love is solidly in major territory. The chords are I, V, IV, the classic backbone chords of the major scale.

The verse, however, is going to be back in the Northwest — I, bVII-, IV. I have to get back over there somehow, and I want to lead the ear strongly so that the move feels right.

Once again I’ll use a dominant seventh type chord. Here’s the shape again:

P1080225

The ear expects a I chord to come after this. Perhaps it’s one of those “nature abhors a vacuum” things. All the notes have tonal gravity that is pointing at the center, yet there’s nothing there. It’s as though the planets are orbiting the sun, and the sun is missing. The ear wants to put it there.

This “dominant seventh” shape is named after the dominant chord, the V7, where it shows up so often. It can be used elsewhere, and is the main tool used in classical music to change keys.

When I started to work this out, I already had a climax in mind for the chorus. In the last line, I wanted to leap all the way to the far Northwest, the ii- chord (lower case roman numerals mean it’s a minor chord), and have the melody sit still while the chords revolve around it. Here’s the ii-:

P1080430

 

It’s a long leap, but I can make it with one transition chord, the I7.

P1080421

Adding a seventh to the I chord sends a strong signal to the ear: “We’re going west!” Normally the next chord would be a IV.

This move is used all the time in music. It’s common for the bridge of a song to start on a IV chord — it’s a shift on the lattice that makes the bridge sound different from the rest of the music, much like my shift to major for the chorus of Be Love. Putting a I7 right before the bridge tells the ear to expect this shift.

I do something a little different, though. There is also a huge vacuum where the ii- ought to be — look how the two chord shapes fit into each other. Sure enough, going to the ii- after the I7 is satisfying and dramatic. Here’s how the sequence looks and sounds.

Notice how the melody leads the way again, by going to the 4. There’s another lovely passing chord, a stack of perfect fifths, right before the change.

P1080425

 

Next: The Septimal Minor Third

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Posted by on Mar 6, 2013 in Just Intonation, The Lattice, The Notes | 0 comments

Rosetta Stone

Almost all Western music, including my own, lives in the world of tonal harmony. This means:

  • There can be, and usually are, multiple notes playing at the same time.
  • There is a key center, or tonic, around which the notes are arranged. The tonic doesn’t always sound — it’s an intangible presence, the home from which you leave on your harmonic journey, and to which you will hopefully return.

The multiple notes can have different functions:

  • Roots are the fundamental notes of chords. A G chord has its root on G. Roots are local centers that move the ear around the lattice as they change.
  • Harmonies flesh out the chord. In a G major chord, the harmony notes are B and D. They stake out more lattice territory and add definition to the chord. Is it a G major, minor, seventh? The harmonies establish this.
  • Melodies dance in the harmonic field set up by the tonic, roots and harmonies. They have more freedom than the others. Melodies travel fast and light, and though they can sing the same notes as the others, they can also travel farther afield, further embellishing the chord, or leading the ear toward the next chord in the progression, or lingering on the last one after it has changed.

All this action is happening in two musical spaces at once.

Piano-keyboard

Melodic space is the world of scales. It’s organized in order of pitch. The piano keyboard is a perfect representation of melodic space.

full lattice all-01

Harmonic space is the world of ratios. Multiply a note by a small whole number ratio, and you have moved a small distance in harmonic space. Multiply by large numbers, and you have moved a large distance. The lattice is a map of harmonic space.

The two worlds are not the same. Often, they are opposites. The perfect fifth is a small move harmonically but it’s a mile in the melody — bass singers have to jump all over the place in pitch. Small melodic moves tend to be big harmonic ones. A chromatic half step, the distance between the 3 and b3, is only 70 cents, less than the distance between neighboring keys on the piano. But on the lattice, it’s a long haul — down a third, down another third, and up a fifth.

Writing and arranging a song is sort of like designing (rather than solving) a crossword puzzle. There are two intersecting, independent universes, Up and Down. To design the puzzle, you work back and forth between the two, massaging them until they don’t conflict, and each one makes sense on its own.

All of the notes live in both harmonic and melodic space. They may have a foot in one more than the other — the roots tend to move small distances on the lattice, the melodies usually move small distances in pitch, and the harmonies tend to bridge the two, moving melodically while staking out the form of the music on the lattice. But every note moves in both spaces, all the time.

Rosetta_stone_(photo)A great advantage of the lattice is that it serves as a sort of Rosetta Stone, a bridge or translator between the two worlds.

The Rosetta Stone was carved in 196 BC and rediscovered in 1799. It immediately became famous because it repeats the same text three times, in three different languages. It was the key that allowed scholars to decipher Egyptian hieroglyphs.

The lattice bridges the two musical spaces by means of the patterns it presents to the eye.

When two or more notes are plotted on the lattice, they will form a particular visual pattern. Any time you see this pattern, no matter where on the lattice it is, the relationship between the notes of the pattern will be exactly the same, in both harmonic and melodic space.

3-01For example, this pattern shows an interval of a major third. The ratio of the frequencies of these two notes is 5/4 (or 5/2, or 5/1 — twos don’t count, they just shift the note by an octave). Any time you see two notes in this formation, no matter where they are, you know they have the following relationship to each other:

  • Harmonic space: When the notes are sounded simultaneously, they will have the characteristic sound of a pure major third.
  • Melodic space: When you move from one note to the other, you are traveling a distance of 386 cents, or about four semitones on the piano.

Getting familiar with these patterns, and learning to recognize them wherever they are, has made it easier for me to think in harmonic and melodic space at the same time, which makes writing and arranging music much easier.

Next: Tonal Gravity

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Posted by on Nov 17, 2012 in The Lattice | 0 comments

The Tonic Major Chord

The tonic is the center of the lattice. A drone note on the tonic establishes the center of that particular musical universe.

Adding a major third and a perfect fifth (5/4 and 3/2) further reinforces the center and starts to carve out some territory on the map.

This is the tonic major chord:

In my view, the tonic major helps the ear grab onto the center, by adding two notes that point directly at it. The ear has more information to work with.

The mind has amazing real time mathematical ability. Maybe a more accurate way to say this is that the mind has an amazing ability to quickly analyze and predict physical phenomena. The physical phenomena can be described by math. I don’t think the mind is working with arithmetic calculations at blinding speed, like a computer. It’s more of a massively parallel, holistic analog processor, that achieves a similar result.

Willie Mays used to catch fly balls with his back to the plate. Here’s a famous one:

Mays watches the ball start its flight, calculates the parabola it will follow (fine tuned by the conditions that day), and sets out at top speed for the spot, 400+ feet deep in center field, where he knows it’s going to land. He doesn’t (can’t!) look at the ball until it’s almost upon him. Marvelous.

So the ear hears a note, another one at 3x the frequency (remember octaves don’t count, 3/2 works like 3/1 in this regard), and another one at 5x. All three notes are direct signposts, pointing exactly at the tonic. Here we are, says the mind.

This may be why the equal-tempered major third gives me that slight queasy feeling. The tonic is the tonic, all right, but that equal-tempered third doesn’t point right at it! It’s close enough that the ear correctly identifies it, but it’s actually pointing at a note about 1% sharp of the tonic, and something sounds subtly off, like day-old sushi.

Here it is again: pure third, ET third, pure third. The middle note, the ET third, has a ratio of about 5.04/4.

JI3 vs ET3

Is it slight tonal vertigo? Where is home?

Next: Compound Notes

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