
ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary 
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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
 What is the ratio of the mystery polyrhythm? Explain how you know.
 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.
 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).

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.

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.

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.
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