Matherhythm - Follow-up Material
Contrastheatre
E-mail: [email protected]
Contents
1 Division
1
2 Exercises
3
3 Highest Common Factors
3
4 Exercises
4
5 Euclid’s Algorithm
4
6 Exercises
5
7 Music
5
8 Exercises
6
9 Matherhythm
7
10 Exercises
1
10
Division
We will start by reviewing the concept of division. Division consists of forming groups. In a division
you are given two terms, the dividend and the divisor. Thus, dividing the dividend by the divisor
is to see how many groups of the size of the divisor I can form with the dividend. Division is a way
of grouping objects. For example, if I have 12 as dividend and 3 as divisor, the division 12 ÷ 3 is
equivalent to find how many groups of size 3 I can form with 12 objects. In this case, the answer
is 4. The number of groups is called the quotient. Below you have an illustration of division as
formation of groups.
111111111111
→
111k111k111k111
Figure 1: Exact division as formation of groups.
Sometimes, after forming all the groups we see that there are elements left over. Those elements
are called the remainder. When a division has no remainder is called even division or exact
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division. Otherwise, it is called division with a remainder. In the previous example, 12 ÷ 3 is
an exact division. We can write
12 ÷ 3 =⇒ 12 = 3 × 4.
If we want to divide 17 by 5 we find it is a division with remainder.
5
3) 17
− 12
2
Here 2 is the remainder and 5 is the quotient. In this case we can write
17 ÷ 5 =⇒ 17 = 5 × 3 + 2.
Performed as formation of groups this division we have the following:
2
11111111111111111 →
1
1
1
1
1
1
1
1
1
1
1 1 1
1
1
1
1
Figure 2: Division with remainder as formation of groups.
Groups have been formed by rows.
Let us go a little bit more abstract. Let a be the dividend, b the divisor, q the quotient, and
r the remainder of a division. The relation among those numbers through division is given by the
following relation
a=b·q+r
(1)
There is a point to make here. When dividing two numbers, we form as many groups as we
can. That implies that the remainder is always less than the divisor. In terms of our equation, that
means 0 ≤ r < b.
2
Exercises
1. Classify the following divisions as exact or with remainder: 1 ÷ 1, 3 ÷ 2, 121 ÷ 11, 0 ÷ 7,
1089 ÷ 33, 1405782039 ÷ 2, 1405782039 ÷ 3.
2. Perform the division 20 ÷ 4 by forming groups.
3. Perform the division 21 ÷ 4 by forming groups. Specify the dividend, divisor, quotient and
remainder in terms of the groups formed.
4. We are given 19 objects. We want to divide 19 by 3. After performing the division we obtain
5 groups of 3 elements each plus a remainder of 4. Is this division correct?
3
Highest Common Factors
The highest common factor is a concept easy to understand. Given two numbers they have a set
of factors, also called divisors. The highest common factor (hcf from now on) is just the highest
common factor found in both numbers. The hcf always exists because 1 is a factor of every number.
Here we have an example of the hcf. Take numbers 45 and 75. The factors of those numbers
are:
• Factors of 45: {1, 3, 5, 9, 15, 45}.
• Factors of 75: {1, 3, 5, 15, 25, 75}.
We easily see that the set of common factors is {1, 3, 5, 15}, the highest being 15. Therefore, the
hcf(45, 75) = 15.
This method of calculating the hcf consists of enumerating all factors of both numbers. This is
a good idea to practice division because you have to do many divisions to find those factors. If the
numbers are low, that is all right. When they are high, it becomes cumbersome.
3
Here there is a question to make: How do we find the lists of factors of a number? First, we
have to find its prime factors. By the way, what is a prime factor? A prime number, I may
recall you, is a number only divisible by 1 and itself. Examples of prime numbers are 3, 5, 7, or
749027409284729381. Every number is the product of a series of prime factors. Normally, you find
the prime factors of a number by trial and error. You check if your number is divisible by 3; if not,
you carry on with 5, the next prime number, and so on. When you a division is exact, you obtain a
smaller number, and you continue trying with more, bigger prime numbers. For the number of the
our example, their decomposition in prime factors is
45 = 32 · 5
75 = 3 · 52
In view of that decomposition it is easy to realize that the hcd is formed by the common factors
raised to the least exponent. In our case the common factors are 3 and 5, and the least exponents are
1 in all cases. Therefore, hcd(45, 75) = 3 · 5 = 15. This idea relates the hcd with the decomposition
in prime factors. However, it is not a good method to calculate the hcd. Again, when the numbers
are high, finding the prime factorization is very hard to obtain.
4
Exercises
1. Calculate the hcf of the following pair of numbers by finding their prime factors: 33 and 11;
144 and 60; 121 and 275; 1089 and 924.
2. With the ideas presented above, how would you calculate the hcf of three numbers? Use your
ideas to calculate the hcf(35, 49, 77).
5
Euclid’s Algorithm
How can we calculate the hcd in a better way? Euclid, a Greek mathematician who lived 400 years
B.C., invented a way to do it. Euclides wrote down the equation of the division:
a = b · q + r.
Every common factor of a and b -Euclid observed- must be a common factor of the remainder
r. Let d such a common factor. Then:
a
b
r
= ·q+ .
d
d
d
If numbers ad and db are natural numbers, so will dr be. In particular, if d is the hcd of both
numbers, that property is still true. Therefore, hcd(a, b) =hcd(b, r). However, we should not stop
here. We can carry on applying this idea until the remainder is 0. At that moment, the last quotient
obtained is the hcd.
Let’s how this algorithm1 works. Take a = 75 and b = 45, as before. We start off by dividing
both numbers:
75 = 45 · 1 + 30
1
Algorithm is a mathematical term for method of calculation.
4
Here the remainder is 30. Is the remainder equal to 0? No. In that case, we continue doing
divisions:
45 = 30 · 1 + 15
Again , is the remainder null? No. We proceed to the next division.
30 = 15 · 2 + 0
Is the remainder null now? Yes. We are done. The hcd is the quotient of this division, namely,
15.
Again, this process can be thought through formation of groups. Let’s illustrate this by calculating the hcf of 16 and 12. We start by writing 12 ones followed by 4 zeros. This is equivalent to
performing the first successive division 16 by 12.
111111111111k0000 →
→
1 1 1 1k1 1 1 1
0000
1111
1 1 1 1k1 1 1 1 1 1 1 1
0000
→
1
0
1
1
1
0
1
1
1
0
1
1
→
1
0
1
1
Figure 3: Calculating the hcf through Euclid’s algorithm and formation of groups.
Since the groups were formed by putting elements one below another, the hcf is the number of
columns in the final box, that is, 4.
6
Exercises
1. Calculate the hcf of the following pair of numbers by using Euclid’s algorithm: 33 and 11; 144
and 60; 121 and 275; 1089 and 924. Compare how long it took you to the method of finding
the primer factors. Which one is the fastest?
2. Calculate the hcf of 24 and 18 by applying Euclid’s algorithm and formation of groups.
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Music
In music everything arises from a fixed amount of time. We call it time span. It can be represented
by a rectangle.
Figure 4: The timespan.
That time span can be divided in equal parts. Those parts are called pulses.
5
Figure 5: Pulses in music.
Each little box in the figure is a pulse. Pulses have the same length in time.
A rhythm is formed by placing notes on certain pulses.
Figure 6: The rhythm.
The empty boxes are rests. They are not played, but have to be counted.
Here we are going to study a particular type of rhythms: timelines. A timeline is a rhythm
played throughout the whole piece. It serves as temporal and structural reference. In the Figure
below there are several timelines. Notice that those timelines have different length in terms of
number of pulses.
Figure 7: Timelines of different lengths.
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Exercises
1. Play the timelines given in Figure 7. Try different speeds, but start always from a slow speed.
2. Compose your own timelines and play them.
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9
Matherhythm
Let’s now play with maths and music together. Let’s fixed a 12-pulse timespan. If I want to play
12 notes, then I have to divide 12 by 1:
12
= 12,
1
and resulting rhythm would be
x x x x x x x x x x x x
Here an x denotes a note. This is very easy since it is clear that every note must go to one pulse.
If I want to play 4 notes, then I have to divide 12 by 3:
12
= 4,
3
and resulting rhythm would be
x . . x . . x . . x . .
Here an . denotes a rest. If we want to obtain this rhythm through division, all we have to do is
divide 12 by 3 with ones (notes) and zeros (rests) as follows.
111100000000 →
1 1 1 1k 0 0 0 0
0000
→
1111
0000
0000
Figure 8: Making rhythms through division.
Again, since the groups were formed vertically, we produce the rhythm by reading the groups by
columns. The resulting rhythm is [x . . x . . x . . . x . x . . .]
If I want to play 6 notes, then I have to divide 12 by 2:
12
= 6,
2
and resulting rhythm would be
x . x . x . x . x . x .
If I want to play 8 notes, then I have to divide 12 by...
12
= 8,
X
Oops! It turns out that 8 is greater than 6. No divisor of 12 can be greater than 6.
I can’t do it... Then, how?
We’ll use the so-called evenness principle:
Evenness Principle: Distribute the notes among the pulses as evenly as possible.
7
Let’s how this principle works. If we have this rhythm:
x x x x x x x x . . . .
it is clear that its notes are not evenly distributed. However, if we move a few notes here and
there, then this new rhythm
. x x . x x . x x . x x
does have its notes evenly distributed. How did we do it? By calculating the hcf with Euclid’s
algorithm and formation of groups.
111111110000 →
1 1 1 1k 1 1 1 1
0000
→
1111
1111
0000
→
101101101101
Figure 9: Making rhythms through the hcf and formation of groups (12 pulses and 8 notes).
The resulting rhythm here is [x . x x . x x . x x . x ], which is not the rhythm we obtained
above. This interesting fact has to do with rotation of rhythms. The evenness of a rhythm
depends on the distances between its notes. Therefore, the rotation of an even rhythm is still
even. In our case, both rhythms are the same up to a rotation.
Since the hcf of 12 and 8 is 4, our rhythm is formed by a small rhythmic cell, [. x x],
repeated 4 times.
Finally, notice that when the number of notes is a divisor of the number of pulses, applying
the evenness principle is the same as performing division. In that sense the evenness principle
is a generalization of division.
If we want to play 7 notes... we already know we have to apply the evenness principle
again. The resulting rhythm would be
x . x . x x . x . x . x
Let’s see how we obtained this rhythm.
111111100000 →
1
0
1
1
0
11
00
1
1
0
→
1111111
00000
→
11111
00000
11
→
101101011010
Figure 10: Making rhythms through the hcf and formation of groups (12 pulses and 7 notes).
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This rhythm, as already mentioned, is a rotation of [x . x . x x . x . x . x]. In particular, it
is a rotation by 3 positions to the right.
Rhythms produced by using the evenness principle are called Euclidean rhythms.
Finally, here we have the main four timelines we obtained by considering the divisors of
12 and applying the evenness principle.
7-note
4-note
8-note
6-note
rhythm:
rhythm:
rhythm:
rhythm:
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
Figure 11: The four timelines.
Although these rhythms admit a mathematical explanation, they actually exist. They
come from a rich rhythmic tradition, the Ghana tradition. In particular these rhythms are
the timelines for a piece of music called gamamla. They are played on a double bell called
gankogui.
Figure 12: Gankogui, an African bell.
The gankogui produces two sounds, a low-pitched sound and high-pitched one. The notes
of the timelines are distributed among the two bells. Written in musical notation, the final
score for the gamamla is given below; notes below the line mean low bell, whereas notes
above the line means high bell.
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Figure 13: Gamamla.
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Exercises
1. Explore the rotations of the 6-note and 8-note timelines.
2. Explore musically the rotations of the 7-note timeline [x . x . x x . x . x . x]. Which
are the most interesting ones? Do you have an explanation?
3. Compose a piece of music formed by 4 timelines following the ideas in the text. Choose
16 as your number of pulses. Play the piece. Make the necessary adjustments in terms
of rotations of rhythms.
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