Warming up with math

In our last episode, I recounted some experiences thinking about the use of the Golden Ratio and the Fibonacci Sequence in musical structures. I focused in that post on the question of finding a mathematically "golden moment" in a work like Beethoven's Symphony No. 5, but for the teaching task that inspired all of this, I never even got around to Beethoven. (My students will get to hear me talk about the Golden Ratio in the Beethoven later in the semester.)

On the special "Golden Ratio" day in question, I was more focused on providing some direct musical encounters with the ratio that students could feel for themselves. We actually first experimented with seating the students in Fibonacci rows as follows:

X

X

X    X

X    X    X

X    X    X    X    X

X    X    X    X    X    X    X    X

Then we tried some counting-off games to help them feel how each row was the sum of the two rows in front. 

But I had also started playing around with composing some music that used these relationships, specifically: 1, 1, 2, 3, 5, 8, 13, etc. It wasn't too long before I'd come up with the following:

It actually had begun with just numbers as lyrics (1, 1, 1 2, 1 2 3, 1 2 3 4 5, 1 2 3 4 5 6 7 8), but I soon realized it might make an interesting vocal warm-up, so there you go. Of course, in real life, I probably would just notate the first three bars as one 3/8 bar, but these meters and the beaming patterns help to reinforce the additive process that's going on. Here's what it might sound like sung by creepy robots (but I have found it to be a fun and useful choral warm-up with humans):

And here's what it looks like notated fully. (If the embedded score doesn't work well, go here.)

One of the surprisingly rewarding things about doing this was realizing that the last three measures ascend to the 3rd, the 5th, and the octave, and of course, a major triad is constructed from root, 3rd, and 5th. That is pretty cool, although if you include the 2/8 bar, that one ascends to the 2nd scale degree which doesn't fit so nicely into the triad. 

Notice that, unlike the subtler principle of putting a structurally important musical moment 55/89 (61.8%) of the way into a piece, this use of the Fibonacci Sequence is right there on the surface. For better or for worse, the performer and listener can directly feel the asymmetry caused by the additive process. When a composer, intentionally or not, puts a surprisingly subdued oboe cadenza 254 seconds into a fiery 411-second movement (see end of previous post), it's highly unlikely that anyone could experience that golden relationship so consciously, though we may at some level feel its rightness. Of course, some would say that is part of the magic.

I have one more composition that came out of this whole experimenterience...and I'll reveal that soon.