Throughout my school days I never particularly enjoyed studying for tests and taking tests, but I usually did really well on them. And I took hundreds of them.
Although I was an A-B student from grade school through graduate school, my proudest grade was a C. As a freshman when I looked at the curriculum for Bachelor of Business Administration, I saw calculus during my senior year. For three years calculus loomed on the horizon. Reading that catalog it looked like and felt like a formidable obstacle to obtaining my degree. But I had three years to worry about that.
My senior year arrived. I had passed nearly all my courses with flying colors, but now I was staring calculus right in the face. My college had a math lab where tutors were available from 8am to 8pm to help the students. I lived in that lab. I spent countless hours with assignments and exam preparation. When the final exam arrived, I was barely hanging onto a C. The final wasn't easy. I had studied for days and I gave it my best shot, but I just didn't know. There were functions on that final I had never seen. I didn't leave the room with a warm, fuzzy feeling. Just before Christmas I booted up my semester grades. Amongst the As and Bs I saw a C!! A solid C!! I was ecstatic. It was one of the best Christmas gifts ever. Calculus was in my rear view mirror. Four months later, at the age of 57, I walked the stage to receive my degree. Cum Laude.
But my proudest test grade was at another college for a different degree. On my way to a Bachelor of Music Education in 1976, I had to pass several different music theory courses. Each one was harder than the previous class. Music theory is basically the function of notes and chords in tonal Western music. When you're listening to tonal "Classical" music, if you freeze the music at any point, the notes make musical sense both horizontally and vertically. When you look at the vertical chord created by the horizontal melodies and harmonies, there is a mathematical name for each chord. The names include both words and numbers. Not just any words and numbers, but the right words and numbers. These numbers are specific to that chord. Within the music, however, there are certain "non-harmonic tones" that don't belong to a certain chord. They belong to the composition, just not to the chord.
One of my favorite types of music is the fugue. The fugue is highly complex and profoundly beautiful. The composition involves the same melody entering part by part as the music evolves. Layer upon layer of a single melody are woven into a magical musical masterpiece. Because of the nature and the structure of a fugue, it creates a myriad of non-harmonic tones. Just like there is a name for each chord, there is a name for each of these tones--passing tones, leading tones, neighboring tones, anticipations, suspensions, etc. You know which is which by its proximity to the chord, the way it approaches and leaves the chord.
In this particular advanced theory class the exam was to name each of those tones. The professor simply copied a page of a Bach fugue and circled the non-harmonic tones. All we had to do was name them. Most of the class failed the exam. Several passed with a C. A few made a B. And one or two students made an A. I made 100! I named 100% of the tones correctly. That was forty years ago and I'm still pumped about it. It was a very proud academic moment for me.
Last evening I was listening to a fugue from J.S. Bach's Mass in B Minor. I was enjoying the music, but I was thinking about that test. I'm sure those students who failed that exam passed the course, earned their degree and have led rewarding and productive music careers. But I can't imagine that any of them enjoy listening to a fugue any more than I. The whole is certainly beautiful, but the sum of its parts is quite remarkable as well.
Calculus is the mathematical study of change. And that's about all I remember about it.
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