Major and minor chords look like triangles on the lattice. They are closed loops, which I think contributes to their stable feeling. Up a major third, up a minor third, and down a fifth (due West on the lattice) brings you right back where you started. The intervals interlock, and they are all consonant, reinforcing the sense of harmonic rest.
Other combinations of notes are more open. Many chords look like straight lines on the lattice.
These chords often show up in my animations as fleeting transitional harmonies. In Flying Dream, for an instant, there’s even a stack of three major thirds:
To my ear, this stack of thirds has a distinctive sound, almost like cloth ripping.
A Suspended Second chord is a straight line, like an augmented chord, but on the horizontal axis. It’s a stack of two fifths. At the time the Sus2 and Sus4 chords were named, the 2 and 4 were usually “suspended,” or held over from the previous chord, as a tension to be resolved. Now they are often used as full chords.
The Police used these stacks of fifths a lot. Andy Summers’ guitar part for Every Breath You Take is full of Sus2 chords, alternating with major and minor thirds.
To me, Sus2 feels lightly stable, and wistful. All the intervals are overtonal, and quite consonant, but unlike the major and minor triads, the Sus2 doesn’t come full circle. If it keeps going, it will never return home, but climb on up the endless spiral of fifths.
It’s a great way to end a certain sort of song, finished but with a sense of longing.
The Suspended Fourth consists of a root, a perfect fifth, and a perfect fourth. The Sus4 looks the same on the lattice as the Sus2. The difference is the root.
This chord has a wonderful tension. The root establishes home. The 5 is overtonal, stable, with strong tonal gravity that attracts. The 4 is reciprocal, unstable, with a strong tonal gravity that repels.
Resolving to the 3 is satisfying indeed, as the unstable 4 slides from its unstable peak into the stable gravity well of the major third.
Sus4 chords are all over rock music. Pinball Wizard is a study, here it is by Townshend on acoustic guitar:
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.
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.
Reading this blog might give you the impression that I’m “against” equal temperament and “for” just intonation, or untempered music.
True, discovering untempered music has been like sailing to a new world. It’s delicious to have 20 or more notes to work with instead of 12, each with its own individual personality.
Equal temperament, however, is a fabulous invention. The lattice of fifths and thirds does not quite repeat. If you start with any note and go in any direction, you will soon encounter almost the same note again, but it will be off by a comma, a small interval, from the original note.
Equal temperament flattens out the lattice just a hair so it does repeat. Now there are only twelve notes to work with, and they imply the untempered ones in the ear. This innovation makes lots of things possible in music. Beethoven and Mozart could not exist without it.
It’s sometimes said that equal temperament and just intonation are incompatible with each other, because the notes will be out of tune. I say they can get along fine, you just have to show ’em who’s boss.
I submit for your consideration: Ray Charles.
Ray Charles’ piano is an equal tempered instrument. Ray Charles’ voice is most certainly not. He is singing the exact resonant notes, those blue notes, all tuned just like a gospel choir, which is what he grew up loving. Ray is boss. His voice establishes the tonality of the song. The backup singers, the horns and the standup bass all agree, this song is in the harmonic pocket, and it resonates.
That leaves the piano slightly out of tune, but who cares?
Notes that are slightly out of tune don’t necessarily sound bad — that’s the basis of the “chorus effect.” No two singers in a choir are exactly in tune with each other, and the resulting complexity is a huge part of the sound of the choir.
So if the tonality is established in the ear, maybe the equal tempered notes, which are only a bit off after all, will just enrich the sound a bit.
Listen to how “Hit the Road, Jack” starts off. First the piano intro. ET. Then the horns kick in, and they start to establish the soul of the tune. Then come the backup singers, that gospel choir. When Ray’s voice finally joins them, the pocket is waiting for him, and he proceeds to own it. The piano is now a background instrument.
I think that’s the secret. Put untempered instruments up front, and ET instruments more in the background. This asserts the untempered tonality in the ear.
Playing acoustic guitar and singing is a great playground for this. The acoustic guitar is, in its bones, an equally tempered instrument. Fretted instruments drove the adoption of ET in Europe, even before keyboards did. The voice is the archetypal untempered instrument. It can do anything.
If the guitar is boss, the song will be in equal temperament. If the voice is boss, you can establish any tonality you want (blues, Gypsy, whatever), and the guitar will tag along. You can retune it in the ear, just like Ray retunes his piano.
Here are some tricks for making friends with acoustic guitar (or any tempered instrument):
1) Sing solidly in tune, with the tonality coming from you, and not from the guitar. Don’t follow the guitar, lead it. The song is the melody, it is your voice, and you are accompanying that voice with guitar notes.
I like to think of the guitar as playing the grid lines on the map. The guitar notes are perfectly equally spaced, and are excellent reference points. The guitar tells me where I am. We completely agree on one note, the tonic. I use the tonic on the guitar as my true home base.
My voice is playing the actual territory.
2) Sing louder than the guitar.
This isn’t all that easy. The guitar is projecting outward, so it sounds louder to the audience than it does to me. The voice is right there in my head, so it sounds quieter to the audience than it does to me. If I sound balanced to myself, the audience will hear way more guitar than vocal. I hear this all the time at open mics.
I’ve found that in an acoustic setting, I have to sing twice as loud as my guitar (from my own point of view) for it to sound balanced out in front of me.
It gets easier with more JI instruments. In “Premature Nostalgia,” the fretless bass and backing vocals are all in strict just intonation. The guitar is truly a backing instrument, and the tonality of the song feels secure.
3) There is a third, more subtle thing you can do to bring the guitar closer to just intonation. The most clearly out-of-tune note on acoustic guitar is the major third. It’s already 14 cents sharp even when perfectly tuned, and the slightest unintentional string bend will take it into some really grating territory. Choose chord voicings that de-emphasize major thirds, and your guitar will sound a lot sweeter. I wrote an article illustrating this effect, here.
My new song video, Real Girl, contains many examples of consonance and dissonance, tension and resolution. In my last post, I extracted a phrase from the song and slowed it way down to illustrate how the bass and melody dance, creating and resolving tension in several different ways. Here is the last half of that analysis.
When we last left our heroes, they were on the 4 and b6, quite consonant relative to each other, but still unresolved because the ear remembers where the tonic is. Here is that clip:
Now the melody moves back to the 7. This interval, against the 4, is the dreaded tritone, the devil’s interval, and it’s dissonant indeed.
Then the bass moves up to the 1, lessening the dissonance, and the melody soon joins it, and all is consonant.
But there is still a sense of incompleteness, even though both the bass and melody are smack on the tonic, the most consonant interval of all. What’s up?
The answer is that the ear remembers that the root is still the 4, and we aren’t quite home yet. Getting there requires a cadence, or final resolution. Notice that in this next clip the bass note never moves, but the harmonies and the melody signal that the root has now moved to the 1 and we are home. The bass note has magically changed character.