Chapter 2: Rhythm

1.1: Basics

• The tactus is the basic pulse of a piece of music.
• In music with a regular rhythm, each pulse, or beat has the same length.
• When tapping along with the tactus, we normally expect some musical “event” to happen at every beat we tap. This synchronization of our expectations with the beat is called entrainment.
• The tempo, or speed, of the tactus is measured in beats per minute. Tempos of about 120 beats per minute are typical for pop songs, while a comfortable walking tempo is around 100 beats per minute.
• Pulses in the tactus are usually grouped into larger units, called measures. The number of beats per measure usually stays the same throughout a piece.
• The tactus may also be subdivided into faster pulses. Subdivisions of 2, 3, or 4, pulses are common. The tatum is the smallest subdivision of the tactus. It’s named for Art Tatum, who was famous for playing fast.

Finding the tactus, the measure, and their subdivisions

The most common grouping of the tactus is four beats per measure. Subdivisions of the tactus are normally two, three, or four tatums.

Compare Sister Rosetta Tharpe’s “That’s All” with Otis Redding’s “Respect” (made popular by Aretha Franklin). Both have four-beat measures. In “That’s All,” beats are subdivided into three tatums, while in “Respect,” they are divided into four tatums (i.e. subdivided into two twice).

1.2 Rhythm patterns

Simple clapping or drumming rhythms may be represented by drum tab, with x for a clap and - for a rest. Both symbols take one beat (that is, one pulse in the tactus).

Measures are indicated by bars, like so:

|xx--|xx--|xx--|

This means that there is a four-beat measure, and you repeat the pattern clap-clap-rest-rest three times. You could also write this pattern as |xx--| ×3.

The downbeat is the first beat in the measure. It’s usually clapped louder than the other beats.

The formula for the number of binary codes

The length of a binary code is the number of symbols in that particular code. For example, x--x-x- has length 7. How many binary codes of a given length are possible? Here’s a chart, using the symbols 0 and 1:

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Formula: There are 2ⁿ codes that have length n.

Two algorithms for writing binary codes

Algorithm 1. To write the codes of length n+1, write the codes of length n twice, then add 0’s to the beginning of the first group of codes and 1’s to the second group of codes.

Example: writing the codes of length 3

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Algorithm 2. Use the 2ⁿ formula to find the number of codes you need. Build a list of the codes by repeating 0, 1, 0, 1 in the right column, 0, 0, 1, 1, 0, 0, 1, 1, in the next column, 4 0’s, 4 1’s in the next column, 8 0’s, 8 1’s, and so on until all columns are complete.�Example: writing the codes of length 3

Example: Drum patterns and binary codes

Every drum pattern may be written as a binary code using the symbols x (strike) and - (rest). Each x or - takes the same amount of time, which I’ll call a “beat.” An example of an 8-beat pattern is x-x-x---.

Question: How many 8-beat drum patterns are there?

Answer: there are 2⁸ = 256 drum patterns with 8 beats.

Question: List the drum patterns with 4 beats.

Answer: There are 2⁴ = 16 total patterns. Use Algorithm #2 on the previous slide to list them, using x instead of 0 and - instead of 1.

xxxx -xxx�xxx- -xx-�xx-x -x-x�xx-- -x--�x-xx --xx�x-x- --x-�x--x ---x�x--- ----�

The “John Cage” rhythm!

Thinking about music like a mathematician

Suppose you want to fill the space of 4 beats with some combination of

• Notes that last one beat (write as “1” or 𝅘𝅥 )
• Notes that last two beats (write as “2” or 𝅗𝅥 )

For example, 112 represents two 1-beat notes followed by a 2-beat note, for a total of 4 beats. You could also write the pattern as 𝅘𝅥 𝅘𝅥 𝅗𝅥

Question: How many different ways can you fill 4 beats? � Answer: five ways. They are 1111, 112, 121, 211, 22.� In music notation, that’s 𝅘𝅥 𝅘𝅥 𝅘𝅥 𝅘𝅥 𝅘𝅥 𝅘𝅥 𝅗𝅥 𝅘𝅥 𝅗𝅥 𝅘𝅥 𝅗𝅥 𝅘𝅥 𝅘𝅥 𝅗𝅥 𝅗𝅥�

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Clapping Music

Here is the basic pattern for Steve Reich’s �“Clapping Music” (1972): xxx-xx-x-xx-�The first player repeats this 12-beat pattern until the end of the piece. The second player claps 12 different patterns, clapping each pattern 12 times.

xxx-xx-x-xx- -x-xx-xxx-xx�xx-xx-x-xx-x x-xx-xxx-xx-�x-xx-x-xx-xx -xx-xxx-xx-x�-xx-x-xx-xxx xx-xxx-xx-x-�xx-x-xx-xxx- x-xxx-xx-x-x�x-x-xx-xxx-x -xxx-xx-x-xx� xxx-xx-x-xx-��

Player 2 claps the original pattern 4 more times at the end.

The resultant rhythm

The resultant rhythm is the pattern of claps (or strikes) and rests that are heard when two people clap or play together. For example:

Player 1 x--x--x-�Player 2 -xxx---x�Resultant xxxx--xx

The resultant has an x whenever either or both of the players clap and a rest if they both rest. Of course, the music is slightly louder when both players clap.

Notice that the resultant rhythm changes every time the second player claps a different pattern.

A mathematical response to Clapping Music

As a mathematician, I asked several questions about this pattern

• How many 12-beat clapping patterns are possible?

2¹² - 1 (not allowing ------------)

• Of those patterns, are any particularly “bad” choices?

Yes. For example, xxxxxxxxxxxx, x-x-x-x-x-x-, and x---x---x--- are bad because the composition ends too early. In general, any pattern with repeats is bad for this reason.

• Does the Clapping Music pattern have any mathematical properties that make it “good,” or is Reich’s decision purely subjective?

If variation in the resultant rhythm is “good,” a “good” pattern would have approximately the same number of hits and rests.

• Which patterns are as equally “good” as Reich’s patterns?

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Baby Clapping Music

As a class, we recreated “Clapping Music” using four-beat patterns. We tried x-x- and xxx-. The second composition doesn’t end until the second person has clapped four different patterns, while the first pattern has only two different variations, x-x- and -x-x, so the composition is very short.

As groups, I’d like you to decide on a 6-beat rhythm pattern. Write down the six-beat patterns formed by the same process as the patterns clapped by the second person in “Clapping Music.” Then, choose a “band name” and write your pattern on the board with your group name. Write all the variations that are clapped by the second player. For extra credit, perform the composition.

Are some groups’ patterns the same? Similar? Which ones seem to be “better” or at least most like Reich’s Clapping Music?

Circular notation

If a pattern is repeated, it makes sense to write it on a circle, with each blob representing a clap. Here are the first four patterns in Clapping Music:

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Write your group’s drum pattern on a circle and put it on the board. Perform your pattern and discuss the types of resultant rhythms you hear.

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Conclusions

• Although there are 2⁶ = 64 possible rhythm patterns with 6 beats (including the “empty” rhythm ------, some patterns produce more interesting compositions than others.
• For most rhythms, there are 6 different variations of the pattern that are clapped by the second players.
• If a pattern consists of repeated smaller patterns, then the piece ends early.
• In particular, a pattern may have 1, 2, 3, or 6 variations. Those are the only numbers that divide 6.
• There are 2³ patterns with two repeats, 2² patterns with three repeats, and 2¹ patterns with six repeats. However, the patterns ------ and xxxxxx have each been counted 3 times, so the number of rhythms with repeats are 2³+2²+2¹-2-2 =10 patterns with repeats, and 64-10 = 54 “good” patterns with no repeats!
• Each of the “good” patterns has 6 variations. If we count a pattern and its variations as essentially the same, there are 54/6 groups of good patterns.

EXERCISE. A pattern is reducible if it is made of repeated shorter patterns and primitive otherwise (this is what I called “good”). For example, x--x-- and xxxxx are reducible and x-xxx- is not. Find the number of reducible and primitive patterns of up to 7 beats. Include the empty rhythm in your count of reducible rhythms.

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We’ll use the Online Encyclopedia of Integer Sequences to investigate our data.

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How many 12-beat reducible patterns are there?

EXERCISE. A pattern is reducible if it is made of repeated shorter patterns and primitive otherwise (this is what I called “good”). For example, x--x-- and xxxxx are reducible and x-xxx- is not. Find the number of reducible and primitive patterns of up to 7 beats. Include the empty rhythm in your count of reducible rhythms.

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Use the Online Encyclopedia of Integer Sequences to investigate the sequence�

2, 2, 6, 12, 30, 54, 126 ...

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How many 12-beat primitive patterns are there?

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What do mathematicians call this type of pattern?

4020

A binary sequence with primitive period n

Equivalent patterns

We’ll say that two patterns are equivalent if one is a variation of the other that is produced by repeatedly shifting the first beat of the other to the end (that is, “Clapping Music style”). For example, xx-xxx and x-xxxx are equivalent.

• True or false: if two patterns are equivalent, they have the same number of beats.
• True or false: if two patterns are equivalent, they have the same number of hits (x) and the same number of rests (-).
• True or false: x----- and --x--- are equivalent
• True or false: x--x-x and x-xx-- are equivalent
• True or false: x-xx-x and x-xx-- are equivalent
• True or false: --xx-x and x-xx-- are equivalent
• True or false: if two patterns have the same number of hits and the same number of rests, then they are equivalent.

Classes of equivalent patterns

There are 14 “classes” (collections) of equivalent patterns with 6 beats.

• Which class did you select when we did this exercise last week?
• Which class do you like best? �List all the equivalent patterns in the class you selected.
• Of all the equivalent patterns in this class, which do you think would be the best starting pattern, and why?
• What is the significance of the yellow shading?

1.3 Duration

Western musical notation specifies how long a note or rest lasts. In a four-beat measure, a whole note occupies four beats, etc.

Whole note

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Half notes

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Quarter notes

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Eighth notes

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Sixteenth notes

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Fibonacci numbers

The Fibonacci numbers (called Hemacandra numbers in India) are

1, 2, 3, 5, 8, 13, 21, 34, 55, …

where each number is the sum of the two previous numbers. The number of ways to fill any given number of beats with notes of length 1 and 2 is a Fibonacci number. Precisely, there are�1 way to fill a duration of 1 beat�2 ways to fill a duration of 2 beats�3 ways to fill a duration of 3 beats�5 ways to fill a duration of 4 beats�8 ways to fill a duration of 5 beats, and so on…..�This fact was discovered by the Indian scholar Hemacandra c.1150 AD.

The domino-square tiling problem

Theorem: Suppose n is a counting number. The number of ways you can tile a nx1 rectangle with 2x1 dominoes and 1x1 squares is a Fibonacci number for every value of n.

This picture suggests why...