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Chapter 3
Rhythm
The beat is the foundation of music. Even before birth, we hear the regular pulse of our
mothers’ heartbeats. We experience the beat as simultaneously linear (progressing through
time) and cyclic (repeating, as on a clock). Musical events—sounds and silences—occur
within a repeating framework called a meter. A rhythm is a pattern of note onsets that is
actually present in a piece. In traditional and popular music, repeated rhythmic patterns, or
time-lines, overlay the meter.
3.1
Measuring time
Watch the video “Tāla in Carnatic classical music.” In this video, the performer’s repeated
hand gestures indicate the tāla, or repeating rhythmic organization of time in the piece.
These same gestures accompany any piece using this tāla. Compare also the hand gestures
used in Western classical conducting (Figure 3.1).
The tactus is the basic pulse of a piece of music—it’s where you naturally tap your foot. 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.
Normally this event is something we can hear (a drum hit, or the beginning of a note) but
sometimes there is a silence—a rest. 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.
···
|
|
| |
|
|
|
|
|
|
|
|
···
Beats in the tactus are grouped into units called measures, or bars, just as seconds are
grouped into minutes. Musicians normally count beats by their position in the measure.
Here is an example of counting in four beats per measure.
···
|
1
|
2
|
3
|
4
|
1
|
2
|
3
|
4
|
1
|
2
|
3
| ···
4
Beats may also be subdivided. In Western classical and popular music, divisions into 2, 3, or
4 equal parts are common. This diagram shows a four-count measure with beats subdivided
15
CHAPTER 3. RHYTHM
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Figure 3.1: Curwen’s conducting diagrams.
into 2 or 4 parts. The “&” should be read as “and.” Practice clapping at a constant rate on
the beats marked “|” and speak each line of the drill along with your claps. Then practice
three-count measures, using the patterns 123, 1&2&3&, and 1e&a2e&a3e&a.
|
1
1
1
e
&
&
a
|
2
2
2
e
&
&
a
|
3
3
3
e
&
&
a
|
4
4
4
e
&
&
a
Figure 3.2: Duple and quadruple subdivisions of a four-count measure.
|
1
1
e
a
|
2
2
e
a
|
3
3
e
a
|
4
4
e
a
Figure 3.3: Triple subdivisions of a four-count measure.
In some pieces, there are intermediate levels of grouping between the measure and the
tactus and, in addition, higher-level groupings of measures. In Western music notation,
the time signature indicates the internal structure of the measure. For example, the time
signature 12/8 is normally played as four counts per measure, each subdivided into three
parts. H.W. Day’s “Tree of Time” (Figure 3.4) is a fanciful representation of time signatures.
Exercise 3.1. Do this exercise with a partner. Both of the patterns “123123” (that is, two three-count
measures) and “121212” (three two-count measures) occupy six pulses in the tactus. Each person
picks one of the patterns. Start by clapping six beats, then count the tactus using the two different
patterns simultaneously while keeping your claps synchronized. Continue counting until you are
comfortable working together. Now change the volume of your counting so that the “1” is loud and
the other numbers are almost silent. The pattern of “1s” you hear is called a two-against-three
CHAPTER 3. RHYTHM
17
Figure 3.4: A fanciful representation of meter in Western classical music.
pattern. It should sound like 1.111.1.111.1.111., etc., where the dots represent quiet beats.
The pattern repeats every six beats. Try the patterns 3-against-4, 2-against-4, and 6-against-4 in the
same way. If you start saying “1” together, how many beats, in each example, are needed before you
both say “1” at the same time again?
3.2
Rhythm and groove
Groove is a dif icult term to de ine. Most musicians agree that groove is essential to most
styles of popular music. It normally refers to characteristic repeating rhythm patterns
and accents that identify different styles of dance music, such as ska and reggae. Listen to
the audio, starting around 3:00, in which James Brown, “The Godfather of Soul,” discusses
grooves in his music in this 2005 interview with Terry Gross of NPR. He compares two
versions of the “I Got You” groove and describes how his groove changed with his 1965 song
“Papa’s Got A Brand New Bag.”
The irst count in every measure is called the downbeat and the last count is called the
upbeat, because it comes right before the downbeat of the next measure. The most basic
grooves in Western music use four-count measures. Counts one and three are called the
on-beats and counts two and four are the off-beats. “Backbeat” grooves, popular in blues and
R&B, emphasize the off-beats, while downbeat-oriented grooves have a primary accent on
the downbeat and a secondary accent on count three. In the NPR interview, James Brown
said that “Papa’s Got a Brand New Bag” was his irst song to give a strong emphasis to the
downbeat.
Unlike the basic grouping of pulses in the tactus into measures, a groove can be highly
complex, with several musicians playing interlocking rhythms, each with different accent
patterns. A hip-hop beat is an example of what I’m calling a groove. We can study both the
pattern that each instrument plays and the combination of all these patterns. In addition,
musicians sometimes intentionally play a little ahead or behind the tactus—this kind of
variation is called microtiming or swing. Although microtiming is an important feature of
dance music grooves, it is dif icult to write down in music notation. Musicians normally
CHAPTER 3. RHYTHM
18
learn it by ear. Just to keep things simple, we won’t be studying microtiming in this class, but
it is an important part of groove in a lot of music.
Exercise 3.2. Practice both on-beat and off-beat clapping with different kinds of dance music in
four counts per measure. Find a piece that clearly emphasizes the on-beats and another piece that
clearly emphasizes the off-beats. Do you ind it easier to clap to “Papa’s Got a Brand New Bag” on the
on-beats, or on the off-beats?
3.3
Notation
Drum tablature, or drum tab, is a common way to notate rhythm. Sequences of drum hits can
be written with x’s, indicating hits, and .’s (periods), or rests, indicating that the drum is
not sounded. Each symbol occupies the same amount of time, a pulse in either the tactus or
some regular subdivision of the tactus. For example, the notation x..x..x. means “hit rest
rest hit rest rest hit rest.” This is a common pattern in music with four counts to a measure,
subdivided once. Practice clapping the pattern while you speak the names of the counts:
Count: 1 & 2 & 3 & 4 &
Clap: x . . x . . x .
Exercise 3.3. Write the pattern for a four-count measure, with hits (a) on the downbeat only (b) on
the upbeat only (c) on the on-beats, and (d) on the off-beats. Each x or . occupies one count. How
would your patterns change if each hit or rest takes half a count?
Exercise 3.4. How many patterns of hits and rests are possible in an eight-beat measure? Of those,
how many have three hits? (Don’t try to do this from scratch—use the results of the last chapter!)
Steve Reich’s “Clapping Music.” We can use drum tab to visualize Steve Reich’s “Clapping
Music.” Player A claps the same rhythm over and over until the piece ends.
Player A:
x x x . x x . x . x x .
Player B progresses through a number of patterns, playing each one eight times. The irst
pattern is the same as Player A’s, then the pattern is repeatedly shifted by one beat.
(1)
(2)
(3)
(4)
x
x
x
.
x
x
.
x
x
.
x
x
.
x
x
.
x
x
.
x
x
.
x
.
.
x
.
x
x
.
x
x
.
x
x
.
x
x
.
x
x
.
x
x
.
x
x
x
When two drum patterns are played simultaneously, a third rhythm is heard, called the
resultant rhythm. It is the rhythm that has rests whenever both of the original drum patterns
had a rest and drum hits otherwise.
Drum tab is convenient for inding resultant rhythms. Any beat that has an x in either A
or B (or both) has an x in the resultant rhythm. Here are patterns A and B in the previous
example, plus the resultant rhythm. The seventh beat is the only rest in the resultant rhythm.
Player A:
x x x . x x . x . x x .
CHAPTER 3. RHYTHM
Player B:
resultant:
19
x . x x . x . x x . x x
x x x x x x . x x x x x
Led Zeppelin’s “Kashmir” Here is a simpli ied version of the Kashmir pattern. The drums
hit on the numbered counts.
Count:
Guitar:
1 & 2 & 3 & 4 &|1 & 2 & 3 & 4 &|1 & 2 & 3 & 4 &|1
x x . x x . x x|. x x . x x . x|x . x x . x x .|x
Exercise 3.5. Explain why the sequence repeats after three measures.
Circular notation.
Because grooves repeat, it makes sense to visualize patterns on a circle, just as we visualize
hours on a clock. The downbeat goes at the top and the other beats proceed clockwise. Dots
show when the drum is hit.
Here are the irst four patterns in “Clapping Music” that are clapped by player B.
1
3.4
2
3
4
The beat class circle and modulus
In music, time is of the essence. How should we measure it? Rather than using minutes or
seconds, we’ll use a unit of time that is relevant to the piece itself. Since the tactus is often
subdivided, using the tactus as a “ruler” to measure time would mean using a lot of fractions.
To avoid this, let’s measure time using the smallest level of subdivision, called the tatum.
(In music notation, the tatum might be a pulse of eighth or sixteenth notes.) I will call the
duration of a pulse in the tatum a beat. Each pulse in the tactus lasts a whole number of
beats.
The fact that a beat is actually a duration of time—not a speci ic point in time—means that
we ought to measure musical time starting at zero, just like a ruler starts at zero, so that
“beat 0” refers to the beginning of the irst measure and “beat a” refers to the time a beats
later.
Any pattern of hits corresponds to a pattern of time intervals (durations) that
measure the time from the beginning of one hit to the next. For example, the pattern
x..x..x.x..x..x. is described as “3+3+2+3+3+2” and shown in Figure 3.5. The hits in
the pattern fall on beats 0, 3, 6, 8, 11, and 14. The multiple levels of counting in Figure 3.5
CHAPTER 3. RHYTHM
20
give some idea of the complexity of musical time. The tatum is, here, twice as fast as the
tactus.
beats
count
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
1 & 2 & 3 & 4 & 1 & 2 & 3 & 4 &
Figure 3.5: Two repeats of the pattern x..x..x.
When we translate between the tactus and the measure, certain points in time are associated
with one another. Just as a clock represents the organization of time within a day, the beat
class circle represents beats in a musical measure. Figure 3.6 shows a repeated rhythm
pattern, called a timeline, on a beat class circle with eight beats. The blobs indicate which
time points are hits.
6
7
5
0
4
1
3
2
Figure 3.6: The timeline x..x..x. represented on the beat class circle.
Exercise 3.6. Write the rhythms in Figure 3.7 in drum tab and circular notation.
Rhythms of one- and two-beat notes
merengue bell part (Dominican Rep.)
cumbia bell part (Columbia)
mambo bell part (Cuba)
bintin bell pattern (Ghana)
also bembe shango (Afro-Cuban)
Rhythms of two- and three-beat notes
lesnoto (Bulgaria)
bomba (Puerto Rico)
guajira (Spain)
12-beat clave (Cuba)
Figure 3.7: Rhythms from around the world
De inition 1 (Beat class). The beat class (bc) of beat a is its position in the measure,
calculated from the beginning of the measure.
Let beat a be the time a beats after the beginning of the irst measure of a piece. A measure
with n beats has n beat classes,
0, 1, 2, . . . , n − 1
Every downbeat starts at bc 0. Notice that beat class and “count” is not the same thing: the
downbeats are counted “1” but start at bc 0. The situation is similar to the relationship
between “the 1900s” and “the twentieth century.”
CHAPTER 3. RHYTHM
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Suppose you start at the irst downbeat of a piece. After 54 beats, what is the beat class?
Since moving backward 12 beats has no effect on the position in the measure, we start
by subtracting as many multiples of 12 as we can, then looking at what’s left. Since
54 − 12 − 12 − 12 − 12 = 6, beat 54 is in beat class 6. It occurs 6 beats after the beginning of
a measure. Of course, we can do this more ef iciently using remainders: 6 is the remainder
when 54 is divided by 12.
In general, suppose there are n beats in a measure. Beat a falls in beat class R, where R is
the remainder when a is divided by n. Mathematicians call this remainder the modulus.
De inition 2 (Modulus). If a is an integer and n is a positive integer, then the modulus of a
relative to n is the remainder when a is divided by n. If R is the remainder, we write
a mod n = R
The beat class formula.
If there are n beats per measure, beat a falls in beat class a mod n.
Two beats have the same beat class if and only if they are in the same position in the
measure, which is the same thing as being separated by a whole number of measures.
Therefore, beat a and beat b belong to the same beat class if and only if the distance between
them, a − b, is divisible by n. (In general, an integer m is divisible by an integer n if m/n is an
integer.) Mathematicians call this relationship modular congruence:
De inition 3 (Modular congruence). Two numbers a and b are congruent modulo a positive
integer n if their difference, a − b, is divisible by n. We write a ≡ b (mod n).
The triple equals sign ≡ is read as “is congruent to.” Note that “their difference is
divisible by n” means that (a − b)/n is an integer. For example, 23 ≡ 51 (mod 7) because
(51 − 23)/7 = 28/7 = 4, which is an integer. It is true that a ≡ b (mod n) if and only if
a mod n = b mod n.
Theorem 3 (The beat class theorem). Suppose there are n beats in a measure. Beats a and b
lie in the same beat class if and only if a ≡ b (mod n).
For example, suppose there are seven beats to a measure. Then
• Beat class 0 corresponds to the downbeats, which occur on beats 0, 7, 14, 21 ….
• Beat 100 falls on beat class 2 because 100 ÷ 7 = 14R2.
• Beat 28 falls on beat class 0 because 28 is divisible by 7 (28 ÷ 7 = 4R0).
• Beats 100 and 849 belong to the same beat class because they are separated by
849 − 100 = 749 beats, which equals exactly 107 measures. Equivalently, use modular
congruence and the beat class theorem: 100 ≡ 849 (mod 7) because (100 − 849)/7 is
an integer. Another way to solve the problem is to use modulus. 100 is in bc 2 (see
above) and, since 849 ÷ 7 = 121R2, 849 is also in bc 2.
Exercise 3.7. Evaluate (a) 144 mod 13 and (b) 169 mod 13. Determine whether the equations are
true:
(c) 25 ≡ 61 (mod 6)
(d) −4 ≡ 4 (mod 3).
CHAPTER 3. RHYTHM
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Exercise 3.8. Suppose there are six beats to the measure. Which beats are downbeats? To which
beat class does beat 50 belong? Beat 90? Are beats 36 and 63 in the same beat class?
Exercise 3.9. Suppose there are four beats per measure and you clap on every third beat, starting
on beat 0. Find the sequence of beat classes on which your claps fall. Practice counting in four and
clapping every third beat.
Exercise 3.10. In Kashmir, there are eight beats per measure and the guitar pattern repeats xx.
every three beats. Find the beat classes of the hits in the guitar part of Kashmir.
Applications of modulus in music. As we have seen, there are pieces where a pattern
repeats some number of times resulting in a duration that equals a whole number of
measures. An example is the Kashmir pattern, which has three beats, where there are
eight beats per measure. Eight repeats ill a whole number of measures. This is expressed
mathematically as 3 · 8 ≡ 0 (mod 8).
In general, if p is the length of a pattern, repeating the pattern r times takes p beats, and this
equals a whole number of n-beat measures if
rp mod n = 0
In Carnatic music, a tihai often ends a piece or a solo. A tihai consists of a rhythmic pattern
that starts and ends on a hit and repeats three times, with some ixed number of beats
between the repeats. A tihai starts on bc 0 and ends on bc 1, so its total duration equals
some number of whole measures plus one beat. Here is an example where there are eight
beats per measure: xxxxx.xxxxx.xxxxx. In this case, the pattern is xxxxx and the gap
is one beat, so the total number of beats is seventeen—that is, one more than a multiple of
eight. The following also is a tihai with eight beats per measure:
Count:
Play:
1 & 2 & 3 & 4 &|1 & 2 & 3 & 4 &|1 & 2 & 3 & 4 &|1
x x x . x . . .|. . x x x . x .|. . . . x x x .|x
If n is the number of beats in a measure, p is the length of the pattern, and g is the length of
the gap, then the tihai’s length is (p + g + p + g + p) and satis ies the tihai equation
(3p + 2g) mod n = 1
Finding values of p and g that work for a given n is not straightforward. For example, if p is
an even number and n = 8, then (3p + 2g) is even, and therefore (3p + 2g mod 8) is even, so
the tihai equation (3p + 2g) mod n = 1 cannot be solved.
Exercise 3.11. Explain why
np mod n = 0
for any integer values of p and n (n > 0). What musical fact does this re lect?
Exercise 3.12. Verify that, in a six-beat measure, a 3-beat pattern and a 5-beat gap works for a tihai
ending. Write a tihai of this type in drum tab. Explain why it is impossible to use a 4-beat pattern
when there are six beats per measure. Is it possible to have a 3-beat gap in a 6-beat measure?
CHAPTER 3. RHYTHM
3.5
23
Solutions to Exercises
3.1. The 2-against-3 pattern repeats after 6 beats. The 3-against-4 pattern is 1..11.1.11.. (12
beats), the 2-against-4 pattern is 1.1. (4 beats), and the 6-against-4 pattern is 1...1.1.1...
(12 beats). In general, the number of beats in the pattern is the least common multiple (lcm) of the
number of beats in each count.
3.3. (a) x... (b) ...x (c) x.x. (d) .x.x If each hit or rest takes half a count, the patterns are
(a) x....... (b) ......x. (c) x...x... (d) ..x...x.
3.4. Hits and rests are binary patterns, and the number of binary patterns of length n is 2n . Therefore,
there are 28 = 256 patterns. The eighth row of Pascal’s triangle tells us that there are 56 eight-beat
patterns that have three hits and ive rests.
3.5. The drum pattern repeats every measure (eight beats) and the guitar pattern repeats every three
beats. The entire pattern repeats after lcm(8, 3) = 24 beats, which equal three measures.
3.6. In general, a duration of one is written “x”; two is written “x.”, and three is written “x..”.
merengue: xxx.xxx.xxx.x.x.; cumbia: xx.xx.x.xx.x.x.x; mambo: x.x.xxxx.xxxx.xx;
bintin: x.x.xx.x.x.x; lesnoto: x..x.x.; bomba: x..x..x.; guajira: x..x..x.x.x. (this is
the rhythm of “America” in West Side Story); clave: x.x..x.x.x..
3.7. (a) 144 mod 13 = 1 because 144 ÷ 13 = 11R1 and (b) 169 mod 13 = 0 because 169 ÷ 13 = 13
(remainder is 0). Determine whether the equations are true:
(a) 25 ≡ 61 (mod 6) because (25 − 61)/6 = 6, which is an integer.
(b) −4 ̸≡ 4 (mod 3) because (−4 − 4)/3 = −8/3 is not an integer.
3.8. The downbeats are beats 0, 6, 12, 18, etc. Beat 50 is in beat class 2 because 50 ÷ 6 = 8R2.
Beat 90 is in beat class 0 because 90 ÷ 6 = 15R0.
3.9. You clap on beat classes
0 mod 4 = 0
3 mod 4 = 3
(3 + 3) mod 4 = 2
(2 + 3) mod 4 = 1
(1 + 3) mod 4 = 0, etc.
So you clap on beats 0, 3, 2, 1, 0, 3, 2, 1, …
3.10. The sequence of beat classes is 0, 1, 3, 4, 6, 7, 1, 2, 4, 5, 7, 0, 2, 3, 5, 6.
3.11. Since np mod n is the remainder when np is divided by n, and p is an integer, np mod n = 0.
Musically, a pattern of length p that is repeated n times, where n is the length of a measure, results in
a duration equal to a whole number of measures.
3.12. In this case,
(3p + 2g) mod n = (3 · 3 + 2 · 5) mod 6 = 19 mod 6 = 1
The possible patterns are x.x.....x.x.....x.x and xxx.....xxx.....xxx. Since
(3 · 4 + 2g) mod 6 = (12 + 2g) mod 6 = 2g mod 6 is even, it cannot equal 1. Since
(3p + 2 · 3) mod 6 = 3p mod 6, which equals either 0 or 3, a 3-beat gap is not possible.
CHAPTER 3. RHYTHM
24
Homework for Chapter 3
Directions: Write your homework neatly on the front side of the paper. Please do not write
on the back. Staple your papers together and cut off any messy edges.
3.1. Draw four beat class circles and notate the patterns
x...
.x..
..x.
...x
on the circles. How are the pictures related?
3.2. Suppose a piece of music has 12 beats per measure. Assume this is true for problems (a)-(h).
(a) Beat 0 is the irst downbeat. List four other beats that are downbeats.
(b) List four beats that belong to beat class 5.
(c) List four beats that have the same beat class as beat 270.
(d) To which beat class does beat 100 belong?
(e) To which beat class does beat 120 belong?
(f) Do beats 45 and 165 belong to the same beat class? Show work.
(g) Do beats 45 and 265 belong to the same beat class? Show work.
(h) Suppose A and B are beats and A − B = 36. Do beats A and B belong to the same beat class?
How do you know?
3.3. Determine whether the following are true or false. Show work.
(a) T/F: 357 ≡ 379 (mod 11)
(b) T/F: 30 ≡ 44 (mod 4)
3.4. Suppose N is a positive whole number and 0 ≡ 8 (mod N ). What are the possible values of N ?
(Hint: there are four possible values.)
3.5. List four numbers that are congruent to 0 (mod 2) and four numbers that are congruent to 1
(mod 2). What is the common name for the numbers that are congruent to 0 (mod 2)? The numbers
that are congruent to 1 (mod 2)?
3.6. Suppose A is a positive integer. Explain why A mod 10 equals the last digit of A.
3.7. Suppose there are eight beats per measure and you clap every sixth beat, starting with beat 0.
List the beat classes that you clap on, in the order you clap them. Do you clap on all the possible beat
classes?
3.8. Suppose there are seven beats per measure. Use the tihai equation to show that a 3-beat pattern
and a 3-beat gap work for a tihai ending. Find a gap length that works for a 4-beat pattern in a 7-beat
measure. Check your work by writing the patterns in drum tab. Extra credit: ind another gap length
that works for a 4-beat pattern in a 7-beat measure.