It was my initial goal at the outset of this summer to write quite a bit more about what I’ve been up to in my totally awesome summer job. Unfortunately, I’ve been a lot busier than I anticipated, and now I’ve got to play a little bit of catch-up to share with you some of the fun things I’ve been doing.
Quick background: I’m working at Vanderbilt with a summer program for gifted high school students from across the country. There are 3 different sessions, intended for 3 different age groups (the last of which began on Monday, and will run for 3 weeks.) Students live on campus and take one class which meets for about 5-6 hours each day. One of my brilliant and talented friends from college designed the curricula for the mathematically themed courses, and has been teaching with the program for 5 years. I assisted him back in 2008 and he convinced me to come back again this year.
The 2nd Session class, Math and Music, is not only right up my alley (I love math! I love music!), but was also probably 2 of the best weeks of my year. I can’t write too much about all that we did without feeling like I’m totally ripping off my friend’s activities (he is, unfortunately, not currently a part of the math teacher blogosphere) but I wanted to share one that is quick, easy, and could be used even if you are not at all musically inclined.
(Fast forward through imaginary lesson that gets you and your students investigating the golden ratio. You could definitely do this in Geometry after talking about Geometric means and the golden ratio, or in Geometry after talking about rectangles and the golden rectangle, or in Algebra II after talking about the Fibonacci sequence. Whatever. Just get to the point where you’re talking about phi.)
So. It seems that there is something aesthetically pleasing about this ratio, to the point that we’re even calling it “the divine proportion.” It appears in the physical proportions of the body. It seems to have some inherent resonance with our senses, providing beauty and balance in works of art. Does this hold for music?
Let’s listen to a song.
Ben Folds – “Zak and Sara” from Rockin’ the Suburbs (2001)
OK, cool. So musicians/composers have a lot of different techniques for creating musical emphasis. Often in songs there will be a sort of musical “building up” to a point where the song reaches its climax – and you can hear it. In “Zak and Sara” it happens right after he sings “submariiiiiine” (in that nice falsetto voice) and before the drums kick back in to take us to the last verse. If I had to place a time on it, I’d say about 2:02-ish into the song.
The song is 3:17 long. What’s the ratio of the entire song length to the time it takes to get to the climax?
197 sec/122 sec = 1.615
Okay, nice. Pretty close to phi. (phi = 1.618) ((I don’t know how to make a wiggly equals on wordpress.))
At this point, we gave the kids some stopwatches and played 3 more songs that had ratios in the right ballpark. For the first, we had the students time the song up until the climax (pausing the song there) and then used the time and the golden ratio to predict how long the song would be. For the second, we told them how long the song was (before playing it) and asked them to predict when the climax would occur. The final song was just for fun and to give them another example – they could record whatever measurements they wanted or just listen.
The songs and measurements are below. Enjoy!
Sam the Sham and the Pharaohs – “Wooly Bully” from Wooly Bully (1965)
Climax: 1:25 (he literally screams.) Song duration: 2:17 137 sec/85 sec = 1.612
Steppenwolf – “Born to Be Wild” from Steppenwolf (1968)
Climax: 2:09 (drum riff) Song duration: 3:25 205 sec/129 sec = 1.589
Stevie Wonder – “Superstition” from Talking Book (1972)
Climax: 2:31 (scream) Song duration: 3:59 239 sec/151 sec = 1.583
Obviously, this doesn’t work for every song – songs with exceptionally long codas, lengthy solos, etc. are going to have ratios that are out of whack. But for your kind of average pop song it actually works quite well. Give it a try with a song from your music library.