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Math and Music: Making Complex Meters Simple

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These are strange days: social isolation, toilet paper hoarding, entire families reworking classics from smash musicals into contemporary Covid-19 masterpieces… Since we’re all sitting at home with nothing but time on our hands and an entire internet full of distraction, this is the perfect opportunity for us to bust out some hyper-nerdy music theory tidbits for your knowledge and enjoyment.

And nothing gets our educational endorphins up quicker than an in-depth explanation of time signatures. So strap yourself in!

If you’ve been a music student for longer than five minutes, you probably know that there are two numbers to your stock-standard time signature. The top number (the numerator) tells you how many beats are in each bar, while the bottom number (the denominator) tells you what kind of note makes up the beat. The top number, in theory, can be any whole number, while the bottom number has to be a number from the pattern 1, 2, 4, 8, 16, 32, 64 and so on, because they correspond to a whole note, half note, quarter note, eighth note and on and on.

Theoretically, you could have a time signature thats like 47/64. But apart from being ridiculously stupid and complex for the sake of being complex and just stupidly unwieldy, it’s not really how time signatures are actually supposed to work. Let’s start with the basics.

(To learn all about time signatures, beats and rhythms, tonality, timbre, and everything else there is to know about musical notation, check out Soundfly’s online course, Introduction to the Composer’s Craft.)

This is 2/4. Say hello. It’s about as straightforward as a time signature can get. Two quarter note beats per bar. And importantly, the two beats can be evenly divided into two smaller beats each, two eighth notes to be precise, meaning there are four eighth note beats in the bar, grouped in twos. When we count the eighth notes, we count, “one-and-two-and, one-and-two-and.”

Because this time signature has two beats in it, we can say that this is a duple meter.

In the same way, 3/4 has three quarter note beats and 4/4 has four quarter note beats, each of which can be divided evenly into two eighth notes. We can call them a triple meter and a quadruple meter respectively. Probably the most common time signature ever is 4/4, so that should look and sound familiar.

So far so good? Great.

Ok so this is 6/8. It’s made up of six eighth note beats. Fantastic. Do we call it a sextuple meter? Don’t be silly, that would make sense, and this is music, where things make sense only 50% of the time and the other 50% is complete madness.

6/8 is a duple meter. Like 2/4. Why? Because when we count the eighth notes in a 6/8 bar, we don’t say “onetwothreefourfivesixonetwothreefourfivesix (gasp for air) onetwothreefourfivesix,” we count it “one-and-a-two-and-a, one-and-a-two-and-a.” We count two beats, divided evenly into three smaller beats. So six eighth notes in total, but they’re grouped into threes, creating two main beats. Therefore it’s a duple meter.

Building on this, we can say that 9/8 is a triple meter, and 12/8 is a quadruple meter. They have nine and twelve eighth beats in each bar, but those eighth note beats can be evenly grouped into three and four dotted quarter notes respectively. Like 3/4 and 4/4, 9/8 and 12/8 are triple and quadruple meters.

What distinguishes a meter like 6/8 from 2/4 is how the main beats are divided. One has beats that can be further subdivided evenly into two smaller beats, one has beats that can be further subdivided into three smaller beats.

Meters like 2/4, 3/4 and 4/4, with beats that can be divided into two, are called simple meters. Meters like 6/8, with beats that can be divided into three, are called compound meters.

This rule is generally true no matter what the denominator is in a time signature. 6/4 for example is still a triple compound meter, except the basic beat units are dotted half notes which can be subdivided further into quarter notes. 2/8 is still a simple duple meter, except the basic beat is an eighth note divided into two sixteenth notes.

(One important exception to this rule are meters like 3/8 or 3/16, which are often treated like a measure with a single compound beat in it. But don’t let that worry you for now).

But what about meters like 5/8 or 7/8 or stuff like that? They can’t be neatly grouped or subdivided. So how do you count those?

 

Let’s look at 5/8. One could count every single eighth note beat I guess, but we’ve already established that’s not really what we do. However… you can group the beats unevenly.

Specifically, you can group them in one group of two eighth notes making a quarter note beat, and one group of three eighth notes making a dotted quarter note beat. There are two beats, but they’re uneven. They’re irregular. And so that’s literally what we call 5/8: irregular duple.

We can continue this pattern and create triple irregular meters, like 7/8:

And 8/8 (which is an irregular meter. If you want to group eight eighth notes into four equal beats, you’ve just created 4/4):

And you can make quadruple irregular meters, like 10/8 and 11/8:

Irregular meters, though, have a neat quirk that neither simple or compound meters have. If I take 2/4, and swap the first and second quarter note beats around, there’s no difference to where the accented beats fall. It always ends up sounding like “one-and-two-and.”

But when I swap around beats in an irregular meter, it does change things. “One-and-two-and-a, one-and-two-and-a” is quite different to “one-and-a-two-and, one-and-a-two-and.” The same goes for triple irregular and quadruple irregular meters:

This is why I love using irregular meters. They really do help you to spice up your compositions without too much hard work. They have this weird little built-in glitch that will catch your listener’s ear, and they encourage you to think outside the box a little because your accented beats are uneven.

One neat compositional trick is to get your manuscript paper and, before you write a note, go across the page and write a bunch of spaced out barlines, and then write a weird mix of time signatures across the measures, like this:

Now, when you start coming up with some actual dots to put on the page, you have to make sure they make use of this weird succession of different note groupings, each with their own accented note. It’s a great way to get out of your own head and come up with something new.

Give it a red hot go. It’s not like you’re going anywhere any time soon…

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Tim Hansen
Tim Hansen

Tim Hansen is a composer, songwriter, music director, and storyteller inspired by a variety of gleefully dark sources. An enthusiastic and charismatic educator, Tim has created lessons for TED-Ed, is a composer in residence at Santa Sabina College, and is engaged with Musica Viva’s “Musician in the Classroom” program in Sydney. He is a co-founding director of W4 New Music collective in New York and has worked with ensembles including Synergy Percussion, the Song Company, Contemporaneous, Cadillac Moon, and TRANSIT. Tim holds a Masters in Music Theory and Composition from NYU.