Jan 15, 2024
Our third day on campus began with a lecture by James Myer of the City University of New York. I have been looking forward to his talk since last summer when he spoke to our students and impressed me with ability to communicate very effectively with our students. James fancies himself a mathemusician—a term I had never heard before I met James. As many of you may know, mathematical ability and musical ability seem to go together (although you would never believe that if you heard me try to play the piano!). You can read about James’ background at his website, James Myer.
James began by asking our participants, “Who plays a musical instrument?” and was impressed to see a sea of hands. Subsequently, James asked, “Who believes mathematics and music are related?” One of our students shared her thoughts about the correspondence between subdivisions, e.g. 1/2, 1/3, etc. of a string, e.g. on a double bass, and the frequency the string produces when closed at those subdivisions. James demonstrated this phenomenon via harmonics on the double bass.
James kicked off the actual talk by investigating the claim that musical notation is simply a graph of frequency over time via a beautiful animation of the first movement (“Vivace”) of Bach’s Double Violin Concerto BWV 1043. Then, we were all astounded by an animation displaying Bach’s “Crab Canon” played forward, backward, forward & backward simultaneously, and finally, forward & backward simultaneously on a Möbius strip, a miraculous surface with only one side. The students and I were amazed at how wonderful this piece of music sounded no matter which direction it was played.
Thoroughly convinced that musical notation is simply a graph of frequency over time, we proceeded to explore the claim that harmony is frequency in ratio. We heard how various frequency ratios sound together, and learned some ratios are more enjoyable than others. Generally, we tend to enjoy a ratio (sufficiently close to) a “small” rational number, i.e. the numerator & denominator are each small. However, there is sometimes a context in which “bigger” ratios sound pleasant, e.g. in jazz harmony. James demonstrated this idea while playing chords on his electric bass.
James hinted at a mystical connection between voice leading (the art of choosing the next note(s)) and geodesic motion on a manifold, i.e. traveling along a geometric object in the most efficient way possible.
What followed was an explanation of the background underlying the award-winning “mathemusical” app/game, named JirafaGraph, that James designed, brought to life by James’ friend, Kyle McGrath. By numbering the notes C = 0, C#/Db = 1, … B = 11, we afford ourselves a mathematical perspective on chromaticism. What’s fascinating is that this perspective suggests 0 = 12, and so while performing arithmetic operations, e.g. 7 + 6, we must remember 12 = 0: 7 + 6 = 13 = 12 + 1 = 0 + 1 = 1 (mod 12). Equipped with the requisite background, James demonstrated how to play the game, and then allowed the audience to download the game and play themselves for a little while. Everyone seemed to enjoy!
Along the way, two T-shirts and one poster were awarded to three students who provided the correct answers to various questions James posed during his talk.
James wrapped up his talk with the following advice: “Be a mathemusician!” James also provided us with a ink to notes on today’s talk as described on his website: “Here are slides from a talk relating mathematics & music (featuring an award-winning app/game I designed, and which Kyle McGrath brought to further fruition) for the 2024 Math League International Summer Tournament at The College of New Jersey: James Myer's Music Talk.”
After a short break, we held our first of two individual rounds. In these rounds, students are given seven minutes to solve difficult questions working on their own. There are seven questions on each round and each question is worth 10 points. The fun for those of us watching the students comes when we collect their answers, announce the correct answer, and hear them cheer if they get the question correct. The questions are quite challenging, but our participants continued to impress us with their ability to solve many of them.
After a break for lunch, we returned to Decker for Professor Teddy Einstein’s lecture. Teddy Einstein is a visiting professor of mathematics at Swarthmore College. Of course, the first question I asked him when I introduced him to our students was the obvious one: “Are you related to Teddy Roosevelt?” Turns out, Teddy is NOT related to the 26th President of the United States AND Teddy Einstein had never been asked this question before. All right, to be honest I purposely avoided the obvious question about his last name that many of our students asked immediately after Teddy replied to my question. Professor Einstein’s answer to the students’ questions about his last name was, “I never comment on this question in public.” He may not be related to Albert Einstein, but he definitely wowed us with his talk on permutations, an important topic in abstract algebra. He first explained what a permutation of numbers is—an arrangement of a group of numbers in some order. For example, if we begin with the numbers 1, 2, 3 in that order and then arrange them as 3,1,2, that’s one permutation of them. Teddy then asked our students how many different permutations there were of 1,2 3. Our students easily figured out that there are 6 permutations of these three numbers. (Can you find them all?) He next gave each student a paper cutout of a square with the vertices labeled 1, 2, 3, and 4. He then proceeded to discuss ways of arranging the order of the numbers at the four vertices by either rotating the square 90 degrees, or flipping it on any of its lines of symmetry. Eventually, he was able to prove that there are exactly 8 possible permutations of the numbered vertices when reading the numbers from the top left vertex to the top right vertex to the bottom right vertex to the bottom left vertex. Professor Einstein completed his talk by asking the students the following question: “Can you think of how we might take symmetries of other shapes to get permutations of 1, 2, 3, . . .?”
After this talk, we completed the second part of our individual rounds. Once again we challenged our participants with seven difficult math problems that I doubt I would have been able to solve when I was in high school. After this round was completed, we took a break and had time for some recreational activities before dinner.
Tonight was the second night of our talent show. Adam Raichel was back by popular demand to host our big show. (I heard that one student requested him!) To be completely honest, Adam is far more than a talent show host. I have known Adam since he was 11-years old and an 8th grade student. I soon learned what a brilliant person Adam is. Three years after I first met Adam, he graduated from high school and began his undergraduate studies at Princeton. Four years after this, he began his studies at Harvard Law School after receiving a perfect score on the Law School admission Test. Despite this background, Adam still comes to The College of New Jersey every summer to be a valuable part of our program. Once again, he was able to organize our talent show and amuse us at the same time. Our performers tonight included the usual assortment of singers and pianists and Rubik Cube experts, but also included violinists, comedians (yes, Adam isn’t the only one who makes us laugh), a talk show host, a Chinese calligraphy demonstration, and two card trick performers. What an evening!
点击这里查看 Day 3 照片(部分)