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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 Sep 13, 2013 in Septimal Harmony, The Notes | 0 comments

The Blue Tritone

I have a favorite note. Don’t tell the others. It’s the septimal flat five, or septimal tritone. I call it 7b5 on the lattice.

There are many reasons why I love this note. One is that Jimi played it, and he’s my favorite musician of them all. Another is that this note is rarely discussed in music theory (try googling it and you will find a few references), which allows me to sort of plant a flag in it. But the biggest reason is that the 7b5 opens up a whole world of melodic and harmonic possibility, and unlocks the minor blues.

The ratio of the septimal flat five is 7/5. It’s a tritone, a note smack in the middle of the octave, between the 4 and the 5. Tritones are famously dissonant. There are three of them in the inner lattice — the 7b5, the #4+, with a ratio of 45/32, and the b5-, whose ratio is 64/45. The 7/5 blue tritone is the most consonant one, by which I mean it has the smallest numbers in its ratio.

Most traditional blues are built on major chords, the I, IV and V, with septimal, or blue, notes in the melody. The 7b3 is especially important — there are entire songs that hang out forever on this note. These blues are major in character — everything happens above the central spine of the lattice.

The 7b5 is different. It lives in the minor part of the lattice, below the central spine, which allows for a whole different set of chordal harmonies. The 7b5 is a blue note that works with songs in minor keys.

Here are a couple of striking examples. First, I invite you to listen to a bit of Dizzy Miss Lizzy, by The Beatles. This is a major blues, played with I, IV and V chords.

In that insistent riff, George Harrison is playing with four notes: the major third (3), the septimal minor third (7b3), the 2 and the 1. He bends the 2 and makes a 7b3, or a 3, or both.

George is exploring a delicious melodic zone that includes four major/blues melody notes in a tight group: the 2-, 2, 7b3, and 3, all in the span of two piano keys. As the I-IV-V progression rocks back and forth from left to right, between dominant and subdominant territory, the melody subtly shifts with it.

Listen again to the intro of the song. The riff repeats, but it’s not always tuned the same. The first two repeats are over a I chord. The riff is sharp, major-third-ish. On the third repeat, the chord changes to a IV, and I hear the tuning fall down into the pocket of the 7b3. It feels to me as though the IV chord allows George to lock into the 7b3, because that note is its seventh harmonic, a beautiful, consonant note. At that point the song goes blue.

Throughout the song, George goes back and forth between that major feeling (the 3) and that blue feeling (the 7b3), over all three chords. Ear candy.

Now listen to Jimi Hendrix exploring the same kind of space, but around the septimal flatted fifth (7b5). This is a minor blues. The chords are i, bVI and bVII.

There is an insistent riff in Voodoo Child (Slight Return) as well, and it’s a lot like the one in Dizzy Miss Lizzy. The pattern is the same, only moved down and to the right on the lattice.

George Harrison is bending the 2, to get the 7b3 and the 3 notes. Jimi Hendrix is bending the 4, to get the 7b5 and 5. It’s another compact, tasty melody zone. Hendrix explores it incredibly well on this song. He cooks up about a half dozen yummy tritone dishes in the space between 0:30 and 0:60.

If I go back and forth between the two songs, the distinction becomes clear. Dizzy Miss Lizzy is major, and the riff centers around the 7b3. Voodoo Child is minor, and the riffs center around the 7b5. Please do click back and forth between the videos.

Want to hear Eric Clapton and Steve Winwood explore the same territory? Here’s a ridiculously good version of Voodoo Chile (the long one from Electric Ladyland) from 2010.

Next: More Blue Tritones

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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 Jun 23, 2013 in Equal Temperament, Just Intonation, Septimal Harmony, The Lattice, The Notes |

The Septimal Minor Third

At the very end of the chorus, Be Love showcases one of my favorite notes, the 7b3.

Most of the notes of the inner lattice can be approximated on the piano, but not the septimal minor third. It’s in between the keys. Blues pianists can evoke it by trilling between 2 and b3, but only variable-pitch instruments can actually hit the note.

Here it is in context:

It’s fun to sing this part of the song, stretching out that septimal note and tasting its flavor.

The passage illustrates the harmonic function of the note. It’s the septimal flatted seventh of the 4, also called the harmonic seventh or barbershop seventh. This is a beautifully consonant note, a great addition to a major chord. It’s generated by multiplying by seven. I use it here as a harmonic seventh over the IV.

Relative to the 1, the 7b3 is a compound note. To get there, you divide by 3, and then multiply by 7. The ratio is 7/6, octave reduced. The pitch is 267 cents, between the 2 and the b3. Here it is on the scale. The colored notes are in just intonation, the black ones are in equal temperament.

Scale with 7b3

Over a I chord, the 7b3 sounds bluesy, restless, gutsy — it’s the insistent melody note in Taking Care of Business. It’s at the heart of the guitar riff in Dizzy Miss Lizzy (George often bends it up to the major third), it’s Jagger’s haunting first “ooooh” of Gimme Shelter. Gimme Shelter

The Stones’ music is a feast of 7b3’s. So is Led Zeppelin’s. These septimal notes are found everywhere the blues has left its impression.

There’s an old question: why do the minor melody notes of the blues sound good over major chords? The web is full of discussions as to why this is so.

I think it’s because the blue minor third is not the b3, but the 7b3. The regular minor third is a reciprocal third, and harmonically it doesn’t fit with major chords — it’s in a different part of the lattice.

But the 7b3 is an overtonal 7th, built on the 4, generated by multiplication. The major notes are made by multiplying by 5. Times 5 and times 7 go together very well. The harmonic seventh chord is a thing of beauty.

There’s an implication for blues guitar. You can’t play this note in the classic minor pentatonic blues box. You can play a b3 (bend it a little to tune it up), or a 3 (bend it harder), but not a 7b3, it’s flat of the b3, and you can’t bend down.

minor pentatonic

You can play a 7b3 by grabbing the 2, one fret below, and gently bending up to it. The following box works great for septimal notes. They’re all laid out under the ring finger. Bend them by less than a half step.

major pentatonic

This box makes it easy to play the classic bit of melody, 7b3 – 2 – 1. All three songs I linked to earlier have this melody in their bones — BTO, the Stones, the Beatles. It’s everywhere. Here it is in Be Love.

Next: Mixolydian Mode

 

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

The Compass Points

There are two basic directions on the lattice: multiplication and division.

If I start with a note, and then multiply it by 3, or 5, or 7, I will get a harmony note with overtonal energy. Such a note is in the natural overtone series of the original note.

Overtonal energy is stable, restful, it belongs where it is and wouldn’t mind staying there.

If I divide by 3, 5 or 7, I get a completely different kind of note. I call this division energy “reciprocal,” after W.A. Mathieu’s suggestion in his amazing book Harmonic Experience.

Reciprocal energy is restless, unstable. The note wants to move, or for the music to come to it, until it is overtonal.

On the lattice of fifths and thirds, there are two axes, fifths and thirds, and two directions, overtonal and reciprocal.

This makes four total directions one can move on this lattice. Each direction has own characteristic flavor, or energy. I use the following names for these energies, mostly after Mathieu.

  • Dominant = East = Overtonal fifths
  • Subdominant = West = Reciprocal fifths
  • Major = North = Overtonal thirds
  • Minor = South = Reciprocal thirds

Compass Points

Every interval has its own unique recipe of moves in these four directions. The perfect fifth has pure dominant energy, the major third pure major. The minor third, b3 on the lattice, is a compound note — dominant and minor.

It’s interesting to look at the minor third (b3) from the viewpoint of tonal gravity. On the horizontal axis, dominant/subdominant, the b3 is overtonal, stable, restful. On the vertical axis, major/minor, the note is reciprocal, unstable, restless.

Tonal gravity is stronger the closer you are to the center. To make a minor third, you multiply by 3 (an overtonal jump of a fifth), and divide by 5 (a reciprocal jump of a third). I know, 3 generates fifths and 5 generates thirds, a confusing coincidence.

Fifths are closer to the center, harmonically, than thirds are, so the overtonal energy is stronger than the reciprocal.

This makes the minor third a stable note, although less stable than the major third. Songs can end on a tonic minor chord and they will still sound finished.

Next: Leading the Ear

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