``` What is the condition that's required to make the sum of periodic waves periodic and what is the period? Here is a link to the desmos page from which the below screen shots were taken. https://www.desmos.com/calculator/eilanhax6u w=1680;h=1050;p=.66; document.write('This shows the green function and the black (sum of green, blue, black) function.

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'); document.write('This shows the blue function and the black (sum of green, blue, black) function.

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'); document.write('This shows the purple function and the black (sum of green, blue black) function.

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'); Here's an analogy that helps us understand this. Think of runners going around a track. (We're talking super human runners here.) The purple runner ("Speedy") goes around the track 13 times in .2 seconds. The time he completes his nth loop is .2/13*n. The blue runner ("Jogger") goes around the track 7 times in .2 seconds. The time he completes his nth loop is .2/7*n. The green runner ("Pokey") goes around the track 3 times in .2 seconds. The time he completes his nth loop is .2/3*n. time on the clock at the end of loop n for each runner n Speedy Jogger Pokey 1 .2/13*1 .2/7*1 .2/3*1 2 .2/13*2 .2/7*2 .2/3*2 3 .2/13*3 .2/7*3 .2/3*3 4 .2/13*4 .2/7*4 5 .2/13*5 .2/7*5 6 .2/13*6 .2/7*6 7 .2/13*7 .2/7*7 8 .2/13*8 9 .2/13*9 10 .2/13*10 11 .2/13*11 12 .2/13*12 13 .2/13*13 All three runners are together again at the start line after .2 seconds. If this analogy doesn't work for you, think of Earth, Jupiter and Saturn. What is the period between conjunctions? Here's a desmos that shows the three planets. https://www.desmos.com/calculator/p2ztpsjz6j Here's a desmos that shows the 20 year period between conjunctions. This occurs when all three planets are on a common line from the sun to the planets https://www.desmos.com/calculator/p2ztpsjz6j Finally, what does this have to do with music? The notes A5, C#5, and E5 when added are periodic. The frequencies of the notes on a piano are here ... https://en.m.wikipedia.org/wiki/Piano_key_frequencies Here's a desmos on this ... https://www.desmos.com/calculator/sokbrpa1rs ```