Chord pizzas

The Groove Pizza uses geometry to help visualize rhythms. The MusEDLab is planning to create a similar tool for visualizing music theory by merging the aQWERTYon with the Scale Wheel. When you put the twelve pitch classes in a circle, you can connect the dots between different notes in a chord or scale to form shapes. My hypothesis is that seeing these shapes along with hearing the notes will help people learn music theory more easily. In this post, I’ll talk through some concept images.

First, let’s look at two different ways to represent the pitch classes on a circle. On the left is the chromatic circle, showing the notes in the order of pitch height (the way they are on a piano keyboard.) On the right is the circle of fifths. These two circles have an interesting relationship: the circle of fifths is the involute of the chromatic circle. Notice that C, D, E, G-flat, A-flat and B-flat are in the same places on both circles, while the other six notes trade places across the circle. Pretty cool!

The chromatic circle and the circle of fifths

The colors represent the harmonic function of each note relative to the root C. Purple notes are perfect (neither major nor minor.) Green notes are major or natural. Blue notes are minor or flatted. You could technically think of, say, B-flat as being the sharp sixth rather than the flat seventh, but that usage is rare in real life. G-flat is a special case–it’s equally likely to be the sharp fourth or flat fifth. I represented this ambiguity by making it blue-green. (We could make it blue if we knew it was flat fifth from Locrian mode, or green if it was the sharp fourth from Lydian mode.)

Once the Scale Wheel and aQWERTYon get combined, then whenever you play more than one note at a time, they will be connected on the circle. Here are some common chord progressions, and what their shapes can tell us about how they function. First, let’s look at the I-vi-ii-V jazz turnaround in C major.

Major scale chords

Seeing things on the circle really helps you understand the voice leading. You can see how the notes move very little from one chord to the next. To get from Cmaj7 to Am7, you just move the B to A while keeping the other three notes the same. To get from Am7 to Dm7, you move the G to F and the E to D while keeping the other two notes the same. To get from Dm7 to G7, you move the A to G and the C to B while keeping the other two notes the same. Finally, to get from G7 back to Cmaj7, you move the D to C and the F to E while keeping the other two notes the same. In general, any chord you can produce by moving the notes as little as possible from the current chord is likely to sound smooth and logical.

The pitch circle doesn’t represent musical “real life” perfectly–while pitch classes are circular, actual notes belong to specific octaves. That makes the voice leading harder to figure out, because you will need to introduce some jumps or additional chord voices to make it work. That said, thinking in terms of pitch class rather than pitch makes it easier to learn the concept; then you can work out the logistics of voice leading actual pitches from a place of understanding.

Next, let’s look at the Mixolydian mode I-bVII-IV-I turnaround that’s ubiquitous in rock, e.g. the “na na na” section in the Beatles’ “Hey Jude.”

Mixoydian mode chords

The circle of fifths view is more clear here. Getting from the Bb to the F is just a matter of rotating the little triangle clockwise by one slot. If you voice the C7 chord like a jazz musician and leave out the G, then the voice leading in this progression becomes exquisitely clear and simple.

Finally, here’s a more exotic-sounding progression from Phrygian dominant, the I-bvii you hear in Middle Eastern and Jewish music like “Hava Nagilah.”

Phrygian dominant mode chords

Seeing these chords on the circle of fifths is not very enlightening–while Western functional harmony keeps things close together on the circle of fifths, non-Western harmony jumps around a lot more. But on the chromatic circle, you can see exactly what’s happening: To get from C7 to Bb-7, B-flat stays the same, but all the other notes move one scale degree clockwise. To get from Bb-7 back to C7, B-flat stays the same while the other notes move one scale degree counterclockwise. This is very close to the way I conceptualize this progression in my head. It’s like the notes in Bb-7 are lifting or pulling away from their homes in C7, and when you release them, they snap back into place. You could also think of this progression as being iv-V7 in the key of F minor, in which case the Bb-7 is acting more like C7sus(b9 #5). Here the suspension metaphor makes even more sense.

Beyond the fact that it looks cool, seeing geometric representations of music gives you insight into why it works the way it does. The main insight you get from the circles is that perfect symmetry is boring. On the Groove Pizza, squares and equilateral triangles produce steady isochronous rhythms, like the four on the floor kick drum pattern. These rhythms are musical, but they’re boring, because they’re perfectly predictable. The more exciting rhythms come from shapes that don’t evenly fit the metrical grid. On a sixteen-step grid, pentagons produce clave patterns, while hexagons make habanera and tresillo.

The same concept applies to the pitch wheel. A square on the pitch wheel is a diminished seventh chord; an equilateral triangle is an augmented triad; and a hexagon is a whole tone scale. (Interestingly, this is true both on the chromatic circle and the circle of fifths.) These sounds are fine for occasional use or special effects, but they get tedious very quickly if you repeat them too much. By contrast, the harmonic devices we use most commonly, like major and minor triads and seventh chords, are uneven and asymmetrical. The same uneven seven-sided figure produces the major scale and its modes on the pitch wheel, and the “standard bell pattern” on the Groove Pizza. Food (ha) for thought.

Musical simples – Teenage Dream

I’m working with Soundfly on the next installment of Theory For Producers, our ultra-futuristic online music theory course. The first unit covered the black keys of the piano and the pentatonic scales. The next one will talk about the white keys  and the diatonic modes. We were gathering examples, and we needed to find a well-known pop song that uses Lydian mode. My usual go-to example for Lydian is “Possibly Maybe” by Björk. But the course already uses a Björk tune for different example, and the Soundfly guys quite reasonably wanted something a little more millennial-friendly anyway. We decided to use Katy Perry’s “Teenage Dream” instead.

A couple of years ago, Slate ran an analysis of this tune by Owen Pallett. It’s an okay explanation, but it doesn’t delve too deep. We thought we could do better.

Here’s my transcription of the chorus:

When you look at the melody, this would seem to be a straightforward use of the B-flat major scale. However, the chord changes tell a different story. The tune doesn’t ever use a B-flat major chord. Instead, it oscillates back and forth between E-flat and F. In this harmonic context, the melody doesn’t belong to the plain vanilla B-flat major scale at all, but rather the dreamy and modernist E-flat Lydian mode. The graphic below shows the difference.

Teenage Dream Eb Lydian circles

Both scales use the same seven pitches: B-flat, C, D, E-flat, F, G, and A. The only difference between the two is which note you consider to be “home base.” Let’s consider B-flat major first.

To make chords from a scale, you pick any note, and then go clockwise around the scale, skipping every other degree. The chords are named for the note you start on. If you start on the fourth note, E-flat, you get the IV chord (the other two notes are G and B-flat.) If you start on the fifth note, F, you get the V chord (the other two notes are A and C.) In a major key, IV and V are very important chords. They’re called the subdominant and dominant chords, respectively, and they both create a feeling of suspense. You can resolve the suspense by following either one with the I chord. The weird thing about “Teenage Dream” is that if you think about it as being in B-flat, then it never lands on the I chord at all. It just oscillates back and fourth between IV and V. The suspense never gets resolved.

If we think of “Teenage Dream” as being in E-flat Lydian, then the E-flat chord is I, which makes more sense. The function of the F chord in this context isn’t clearly defined by music theory, but it does sound good. Lydian is very similar to the major scale, with only one difference: while the fourth note of E-flat major is A-flat, the fourth note of E-flat Lydian is A natural. That raised fourth gives Lydian mode its otherworldly sound. The F chord gets its airborne quality from that raised fourth.

Click here to play over “Teenage Dream” using the aQWERTYon. The two chords can be played on the letters Z-A-Q and X-S-W. For comparison, try playing it with B-flat major. Read more about the aQWERTYon here.

“Teenage Dream” is not the only well-known song to use the Lydian I-II progression. Other high profile examples include “Dreams” by Fleetwood Mac and “Jane Says” by Jane’s Addiction. over the same chords. Try singing any of these songs over any of the others; they all fit seamlessly.

The chorus of “Teenage Dream” uses a striking rhythm on the phrases “you make me”, “teenage dream”, and “I can’t sleep”. The song is in 4/4 time, like nearly all contemporary pop tracks, but that chorus rhythm has a feeling of three about it. It’s no illusion. The words “you” and “make” in the first line are each three eighth notes long. It sounds like an attempt to divide the eight eighth notes into groups of three. This rhythm is called Tresillo, and it’s one of the building blocks of Afro-Cuban drumming.


Tresillo is the front half of son clave. It’s extraordinarily common in American vernacular music, especially in accompaniment patterns. You hear Tresillo in the bassline to “Hound Dog” and countless other fifties rock songs; in the generic acoustic guitar strumming pattern used by singer-songwriters everywhere; and in the kick and snare pattern characteristic of reggaetón. Tresillo is ubiquitous in jazz, and in the dance music of India and the Middle East.

“Teenage Dream” alternates the Tresillo with a funky syncopated rhythm pattern that skips the first beat of the measure. When you listen to the line “feel like I’m livin’ a”, there’s a hole right before the word “feel”. That hole is the downbeat, which is the usual place to start a phrase. When you avoid the obvious beat, you surprise the listener, which grabs their attention. The drums underneath this melody hammer relentlessly away on the strong beats, so it’s easy to parse out the rhythmic sophistication. Katy Perry songs have a lot of empty calories, but they taste as good as they do for a reason.

Ultralight Beam

The first song on Kanye West’s Life Of Pablo album, and my favorite so far, is the beautiful, gospel-saturated “Ultralight Beam.” See Kanye and company perform it live on SNL.

Ultralight Beam

The song uses only four chords, but they’re an interesting four: C minor, E-flat major, A-flat major, and G7. To find out why they sound so good together, let’s do a little music theory.

“Ultralight Beam” is in the key of C minor, and three of the four chords come from the C natural minor scale, shown below. Click the image to play the scale in the aQWERTYon (requires Chrome).

Ultralight Beam C natural minor

To make a chord, start on any scale degree, then skip two degrees clockwise, and then skip another two, and so on. To make C minor, you start on C, then jump to E-flat, and then to G. To make E-flat major, you start on E-flat, then jump to G, and then to B-flat. And to make A-flat major, you start on A-flat, then jump to C, and then to E-flat. Simple enough so far.

The C natural minor scale shares its seven notes with the E-flat major scale:

Ultralight Beam Eb major circles

All we’ve really done here is rotate the circle three slots counterclockwise. All the relationships stay the same, and you can form the same chords in the same way. The two scales are so closely related that if noodle around on C natural minor long enough, it starts just sounding like E-flat major. Try it!

The last of the four chords in “Ultralight Beam” is G7, and to make it, we need a note that isn’t in C natural minor (or E-flat major): the leading tone, B natural. If you take C natural minor and replace B-flat with B natural, you get a new scale: C harmonic minor.

Ultralight Beam C harmonic minor

If you make a chord starting on G from C natural minor, you get G minor (G, B-flat, D). The chord sounds fine, and you could use it with the other three above without offending anyone. But if you make the same chord using C harmonic minor, you get G major (G, B, D). This is a much more dramatic and exciting sound. If you add one more chord degree, you get G7 (G, B, D, F), known as the dominant chord in C minor. In the diagram below, the G7 chord is in blue, and C minor is in green.

Ultralight Beam C harmonic minor with V7 chord

Feel how much more intensely that B natural pulls to C than B-flat did? That’s what gives the song its drama, and what puts it unambivalently in C minor rather than E-flat major.

“Ultralight Beam” has a nice chord progression, but that isn’t its most distinctive feature. The thing that jumps out most immediately is the unusual beat. Nearly all hip-hop is in 4/4 time, where each measure is subdivided into four beats, and each of those four beats is subdivided into four sixteenth notes. “Ultralight Beam” uses 12/8 time, which was prevalent in the first half of the twentieth century, but is a rarity now. Each measure still has four beats in it, but these beats are subdivided into three beats rather than four.

four-four vs twelve-eight

The track states this rhythm very obliquely. The drum track is comprised almost entirely of silence. The vocals and other instruments skip lightly around the beat. Chance The Rapper’s verse in particular pulls against the meter in all kinds of complex ways.

The song’s structure is unusual too, a wide departure from the standard “verse-hook-verse-hook”.

Ultralight Beam song structure

The intro is six bars long, two bars of ambient voices, four bars over the chord progression. The song proper begins with just the first half of the chorus (known in hip-hop circles as the hook.) Kanye has an eight bar verse, followed by the first full chorus. Kelly Price gets the next eight bar verse. So far, so typical. But then, where you expect the next chorus, The-Dream gets his four-bar verse, followed by Chance The Rapper’s ecstatic sixteen-bar verse. Next is what feels like the last chorus, but that’s followed by Kirk Franklin’s four bar verse, and then a four-bar outtro with just the choir singing haunting single words. It’s strange, but it works. Say what you want about Kanye as a public figure, but as a musician, he is in complete control of his craft.