Math – NYU Music Experience Design Lab

The Groove Pizzeria

For his NYU music technology masters thesis, Tyler Bisson created a web app called Groove Pizzeria, a polyrhythmic/polymetric extension of the Groove Pizza. Click the image to try it for yourself.

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Groove Pizzeria

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Note that the Groove Pizzeria is still a prototype, and it doesn’t yet have the full feature set that the Groove Pizza does. As of this writing, there are no presets, no saving, no exporting of audio or MIDI, and no changing drum kits. You can record the Groove Pizzeria’s output using Audio Hijack, however.

Like the Groove Pizza, the Groove Pizzeria is based on the idea of the rhythm necklace, a circular representation of musical rhythm. The Groove Pizza is a set of three concentric rhythm necklaces, each of which controls one drum sound, e.g. kick, snare and hi-hat. The Groove Pizzeria gives you two sets of concentric rhythm necklaces, each of which can have its own time duration and subdivisions. This means that you can use the Groove Pizzeria to make polyrhythm and polymeter.

The words “polyrhythm” and “polymeter” are frequently used interchangeably, but they are not the same thing. Tyler’s thesis contains the clearest definition of the terms that I know of, which I paraphrase here.

Polyrhythm is two or more concurrent loops of equal duration. Each loop consists of a set of evenly-spaced subdivisions or rhythmic onsets. The loops contain different numbers of onsets, meaning that the subdivisions of each loop are not same length. Finally, the ratio of the number of onsets in each loop is not a whole number (otherwise one loop would just be an even subdivision of the other). When people talk about 4:3 or 5:2 polyrhythm, this is what they mean. In Western music, polyrhythms usually only occur for short time spans in the form of tuplets, but in West African drumming, polyrhythms are a core structural feature.
Polymeter is two or more concurrent loops of different duration. The onsets in each loop have the same duration, but each loop has a different number of onsets. This is much more common in Western music than polyrhythm. In Western music, you mostly see polymeter over short time spans in the form of hemiola or syncopation.

With these two definitions in mind, let’s take a look at the Groove Pizzeria interface. For each loop, you can control both the number of subdivisions (the number of onsets) in each loop and the length (duration) of each subdivision. The basic time unit in the Groove Pizzeria is one sixteenth note. Each of the “teeth” on the outer radius of each circle represents the duration of one sixteenth note. If you change the Time Units setting, you make the sixteenth notes shorter, and the radius of the circle gets smaller to preserve the cumulative distances between each tooth of the loop. The easiest way to understand the difference is just to draw some rhythm patterns on the grid, play with the sliders, and see what happens. Notice that the Groove Pizzeria visualizes the compound pattern formed by the two loops in the top left corner of the screen.

Here’s a 5:4 polyrhythm created by taking two loops that are the same length and dividing them into five and four steps respectively:

If you want a 5:4 polymeter rather than a polyrhythm, then you will need to adjust the number of time units in each loop as well. (The patterns aren’t perfectly symmetric so you can hear where they start and end.)

Here’s a less exotic sound, a 4:3 polymeter, also known as hemiola. On the left is a 4/4 hip-hop pattern. On the right, I made a 12-beat-long pattern that repeats four times in the same amount of time as it takes the hip-hip pattern to repeat three times.

Here’s a less familiar sound, an 11:5 polyrhythm. On the left, I made the closest thing to a hip-hop pattern that’s possible in 11/8 time, and on the right I made a simple quintuplet pattern. This will probably sound weird to you at first, but if you listen to it for a while, it will eventually start to make a wonky kind of sense.

How about some real-world examples? Genuine polyrhythm is unusual in popular music, but it’s not unheard of. James Blake uses a quintuplet hi-hat pattern in his song “Unluck.”

Here’s my Groove Pizzeria representation of this beat. On the left is the kick and snare playing a straight quarter note pattern in 4/4, and on the right is the hi-hat pattern (though it’s not playing back on a hi-hat sound.)

Hip-hop producers sometimes use polyrhythms to create specific varieties of swing. On drum machines, swing (sometimes called shuffle) shortens and lengthens each alternate beat. At zero swing, also known as 1:1 swing, the beats within each pair are the same length. At maximum swing, the first beat in each pair will be twice as long as the second beat in the pair. This is known as 2:1 swing, sometimes called “triplet swing” because it’s as if the first beat is two triplets long, while the second is one triplet long. In real life, you usually want your swing setting somewhere between these two extremes. (Click here for a more detailed explanation of swing.)

One way to get a swing ratio in between 1:1 and 2:1 is to use a quintuplet grid. If you think of the first three quintuplets in each group as being one “beat” in a pair and the last two as being the “beat” in the pair, you get the equivalent of 5:3 swing. Slynk explains how to set this up in Ableton:

Here’s a neo soul groove I made using pentuplet swing:

For an even narrower swing ratio, you can use septuplet swing. It’s the same idea, except now you’re grouping together the first four septuplets into one “beat” in the pair, and the last three septuplets into the other “beat”. This gives you a 4:3 swing ratio. This is pretty close to no swing at all, but it’s noticeably “off,” in a way that gives you a nice J Dilla “drunken drummer” feel. Slynk explains again:

Here’s a neo soul groove I made using septuplet swing:

Beyond complex rhythms, the Groove Pizzeria can teach another useful musical concept called event fusion. When a rhythm gets fast enough, you stop hearing individual beats and start to hear a continuous thrum. The transition happens at around twenty beats per second. If you play the rhythm even faster, the thrum becomes a steady pitch, and the higher the tempo, the faster the pitch. Here’s how you can experiment with event fusion yourself. First, put a clap on every sixteenth note. Next, reduce the number of time units to a small number (5 is fine) and set the tempo to 300 bpm. Now reduce the number of steps. Listen for the point when the claps fuse into a single tone. You can control the pitch of this tone by changing the number of steps.

If you think of more interesting music learning or creation applications for the Groove Pizzeria, please let me know. Happy drumming!

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.

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.”

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.”

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.

Seeing classic beats with the Groove Pizza

We created the Groove Pizza to make it easier to both see and hear rhythms. The next step is to create learning experiences around it. In this post, I’ll use the Pizza to explain the structure of some quintessential funk and hip-hop beats. You can click each one in the Groove Pizza, where you can customize or alter it as you see fit. I’ve also included Noteflight transcriptions of the beats.

The Backbeat Cross

View in Noteflight

This simple pattern is the basis of just about all rock and roll: kicks on beats one and three (north and south), and snares on beats two and four (east and west.) It’s boring, but it’s a solid foundation that you can build more musical-sounding grooves on top of.

The Big Beat

View in Noteflight

This Billy Squier classic is Number nine on WhoSampled’s list of Top Ten Most Sampled Breakbeats. There are only two embellishments to the backbeat cross: the snare drum hit to the east is anticipated by a kick a sixteenth note (one slice) earlier, and the kick drum to the south is anticipated by a kick an eighth note (two slices) earlier. It isn’t much, but together with some light swing, it’s enough to make for a compelling rhythm. The groove is interestingly close to being symmetrical on the right side of the circle, and there’s an antisymmetry with the kick-free left side. That balance between symmetry and asymmetry is what makes for satisfying music.

Planet Funk (eighth notes)

View in Noteflight

This pattern reminds me of Saturn viewed edge-on. The hi-hats are the planet itself, the snares are the rings, and the lone kick drum at the top is a moon. To make the simplest funk beats, all you need to do is add more moons into the kick drum orbit.

It’s A New Day

View in Noteflight

The Skull Snaps song isn’t too well known, but the break that kicks it off is number five on the WhoSampled list. The Planet Funk template has some extra kick drums embellishing particular beats. The kick on the downbeat (the topmost slice) has a kick anticipating it a sixteenth note (one slice) earlier, and another following it an eighth note (two slices) later. The snare drum hit to the west is anticipated by two more kicks. All that activity is balanced by the southeast half of the pizza, which is totally kick-free. Like “The Big Beat,” “It’s A New Day” is close to being symmetrical, with just enough variation to keep it interesting.

When The Levee Breaks

View in Noteflight

This Led Zeppelin classic embodies the a wesome majesty of rock. Rhythmically, though, it has more in common with funk. The crucial difference is beat three, the southernmost point on the pizza. In rock, you usually have a kick there. In funk, you usually don’t. The Levee break has a kick a sixteenth note before beat three, which is quite a surprise. Try moving that kick a slice later, and you’ll hear the groove lose its tension and interest. Like “It’s A New Day,” the Levee break sets up the second snare hit with two kicks. There’s another interesting wrinkle, too, a kick that immediately follows the first one. The result is another symmetrically asymmetrical drum pattern.

Planet Funk (sixteenth notes)

View in Noteflight

If you put a hi-hat on every slice of the pizza, you get a busier version of the basic funk groove. With twice as many hi-hats, you can slow the tempo down and still have an energetic feel.

So Fresh, So Clean

View in Noteflight

This OutKast banger has a fascinating drum machine pattern. The snare and hi-hat stick to the Planet Funk pattern above, but against all this predictable symmetry, the kick drum is all over the place. To understand what’s going on here, you need to know something about the concept of strong and weak beats. Strong beats are where you expect drum hits to fall, and weak beats are where you don’t expect them. The more times you have to divide the circle in half to get to a given beat, the weaker it is. The weakest beats are the even-numbered pizza slices. In the first bar, pictured above, every single even-numbered slice has a kick on it. This is, to put it mildly, not typical. Usually the base of your beat is stable and predictable, and the higher-pitched ornaments are more unpredictable. That’s what makes “So Fresh, So Clean” so cool.

Nas Is Like

View in Noteflight

While this track is best known for its samples, and deservedly so, the underlying drum machine rhythm is pretty remarkable too. Like the OutKast song above, the snares and hi-hats are mostly stable, with most of the variation in the kick. I won’t verbally analyze all four bars of the pattern, but if you play with it, you’ll see the idea of balanced symmetry and asymmetry at work.

Amen Break

View in Noteflight

The Amen break is the most complex rhythm here, and it’s a post unto itself to really explain the whole thing. The important thing is to compare the simplicity of the hi-hatsadditional sound, an open hi-hat in the last bar. Displacement!