#### A Math Forum Project

 ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary

Student Version

### Polyrhythms - posted May 21, 2001

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## Polyrhythms

When you listen to or play music, or better yet, dance to it, you are very aware of the rhythm. You might notice that every 3rd or 4th beat is played a little louder than the others, or that it is accented somehow. In some types of music, several different rhythms are performed at the same time. When that happens, it is called a polyrhythm. Polyrhythms are a very important feature of African drumming.

In this problem you will be investigating polyrhythms, and then have a chance to build your own so you can investigate the mathematics.

### To Do:

Play with the applet using the various rhythms. Experiment to find out what it means to play a 1:2 or a 1:3 rhythm. See how the different rhythms interact when you play them together. When a 1:2 rhythm and a 1:3 rhythm are played together, the polyrhythm is called a 2:3. The number of beats it takes a pattern to repeat itself is called the phrase length. Explore how long it takes different patterns to repeat.

### QUESTIONS

1. What is the ratio of the mystery polyrhythm? Explain how you know.

2. Create a polyrythm of your own using two of the rhythms in the applet. What is the ratio of the polyrhythm and its phrase length? Explain how you know.

3. A complicated polyrhythm has the ratio 2:3:4:5:6:7. What is its phrase length? How do you know?

The definition of the ratio and phrase length were not understood by all of the submitters. Some simply followed the pattern given in the instructions and when they had to develop their own polyrthym, they found that the pattern didn't work for all ratios. Some had trouble finding the ratio. Instead of thinking about the lowest common multiple (LCM) and a ratio as a fraction, they tried to use the pattern in the intro. This resulted in added numerators and the keeping of the highest denominator, and it didn't always work out.

The last question was the hardest for those who didn't understand the LCM and how it intertwined with the ratios. Some chose the phrase length of 1 or 7 either because they went with the highest denominator or the lowest numerator in the ratio as being equal to the phrase length. If there were no problems in the first two problems, then there was generally no problems with the third. The best solutions were able to use the logic that was needed in the third problem to be expressed throughout the entire problem (LCM, common denominators, fractions like ratios).

### Highlighted solutions:

 From: Mark J., age 13 School: Issaquah Middle School, Issaquah, WA

```1. What is the ratio of the mystery polyrhythm? Explain how you know.
The ratio of the mystery polyrhthm is 3:4.  When I built the same
song on the composite panel, the notes match with the two ratios at
1:3 and 1:4.  To find a ratio of two speeds, you take the smallest
number on the right side of the ratio and add the bigger number on
that side from the other number.  Therefore, 1:3 and 1:4 equal 3:4.

2. Create a polyrythm of your own using two of the rhythms in the
applet. What is the ratio of the polyrhythm and its phrase length?
Explain how you know.
My composite rythm has a ratio of 2:6.(One bar is 1:2, the other is
1:6).  The phrase legth of this rythm is 6.  This is because you
count how many note slots it takes before the rythm of sound repeats
itself.

3. A complicated polyrhythm has the ratio 2:3:4:5:6:7. What is its
phrase length? How do you know?  To find the phrase legth of this
rythm, you must fing the least common demonimater of all the
numbers.  Since 5 was one of the numbers, the number would have to
end in a 5 or 0.  Since 7 was the biggest number, I descided to check
by sets of 70's.  I went up till I got to 420, which all the numbers
go into.  So the phrase legth of that song is 420.
```

 From: Sharon L., age 13 School: Issaquah Middle School, Issaquah, WA

```1. What is the ratio of the mystery polyrhythm? Explain how you know.

3:4 The first pattern is 1:4 while the second pattern is 1:3. When
put together, they create the mystery polyrhythm, so the ratio of the
mystery polyrhythm is 3:4.

2. Create a polyrythm of your own using two of the rhythms in the
applet. What is the ratio of the polyrhythm and its phrase length?
Explain how you know.

The ratio of the polyrhythm is 2:4. Its phrase lengh is 4 beats. The
first pattern is 1:2 while the second pattern is 1:4, so when the
patterns are put together the ratio is 2:4. Since the LCM (least
common multiple) is 4, then the time it takes for the polyrhythm to
repeat is 4 beats.

3. A complicated polyrhythm has the ratio 2:3:4:5:6:7. What is its
phrase length? How do you know?

It's phrase lengh is 420 beats. All of the numbers least common
multiple is 420, so after 420 beats all the different patterns would
end and start all over again.
```

 From: Andy H., age 14 School: Issaquah Middle School, Issaquah, WA

```1. What is the ratio of the mystery polyrhythm? Explain how you know.

The ratio of the mystery poly rhythm is 3:4. I found this by
positioning the squares on the beat boxes of the composite polyrhythm
so they coincided with the beat box X's of the mystery. I them took
into account the number of beats for each polyrhythm and noticed it
took 12 beats in order for the beats to be simul taneous, or 3 sets
of the 4 original beats. 12 is also the LCM of the latter numbers in
the two combinations of two(1:3, 1:4), to which we could conclude
produced the phrase length in other cases..

2. Create a polyrythm of your own using two of the rhythms in the
applet. What is the ratio of the polyrhythm and its phrase length?
Explain how you know.

I made a polyrhythm with the beats 1:3 and 1:5. The result was a
polyrhythm that has a phrase length of 15 and a ratio of 3:5. the
ratio was the smaller of the latter of the two beat ratios in direct
relation to the other beat. the phrase length was the least common
multiple of the digits of that ratio

3. A complicated polyrhythm has the ratio 2:3:4:5:6:7. What is its
phrase length? How do you know?

The phrase length is 420. I found this by obtaining the least common
multiple of the six digits contained in the ratio.
```

### 4 students received credit this week.

Andy H., age 14 - Issaquah Middle School, Issaquah, WA
Mark J., age 13 - Issaquah Middle School, Issaquah, WA
Sharon L., age 13 - Issaquah Middle School, Issaquah, WA
Bradley M., age 12 - Issaquah Middle School, Issaquah, WA