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 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:
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!”
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.
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:
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.
I often have the pleasure of singing at one of Steve Key’s Songwriters At Play showcases. These are held several times a week in San Luis Obispo area. There’s a lot of talent in the county, and Steve books many good traveling acts. These tend to be excellent shows.
Brian Jeffrey is one of those local talents. Last month Steve had a showcase at Shell Cafe in Pismo Beach, Brian was the featured act and I played four songs. I didn’t know it but Brian videoed one of my songs, and he just put it up on YouTube. I like it a lot!
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.
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.
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.