I keep working my way closer to recording in strict just intonation. Here’s one I did today, of my song Breakup Songs.
The acoustic guitar is in equal temperament, and the bass and vocals are untempered. I love singing harmony when the fretless bass is playing lattice notes. Sometimes I feel like I’m sliding along a groove in the tonal gravity field.
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.