The Well-Tempered February - Commentary #18
Fugue in E Major, BWV 854
Color Theory Part 2
In 1931, a Belgian primary school teacher who also happened to be a violinist wondered why his young students were so enthusiastic about making music, and at the same time so uninterested in playing mathematical games. It was a good question, so Georges Cuisenaire decided to model mathematical learning in the same way as musical learning, using very clear visual symbols for each number between 1 and 10, in the same way that the musical staff uses unambiguous positions on a staff to represent the 7 notes A, B, C, D, E, F, and G. He sawed 10 rods into 10 separate lengths, from 1 cm to 10 cm. He colored them to make them distinctive. And so he developed the Cuisenaire rod:
We used to play with these rods ourselves, at a place called The Pickley Wizard, a preschool in Oxford based on Montessori lines in the 1960s. (Montessori had also developed colored rods to teach mathematics in her first primary schools, but in the 60s Cuisenaire rods were an official fad.) My contemporary relative Kate, who has taught mathematics to children in England all her life, remembers doing the same thing. What made these rods so satisfying is that you could use color to do math. A yellow plus a yellow was an orange (5+5=10). If you needed one more to get from brown to teal, you just added a white (8+1=9). Two reds was a purple (2+2=4, but also 2x2=4). Put a purple below a dark green, and you only had space for a red (6-4=2). What is more, if you went for broke and put a teal with a yellow (9+5) it was exactly the same length as an orange and a purple (10+4). And so on.
We could play with these for hours. Nick Gray, now an ethnomusicologist and expert on the gamelan, would sit with me at age 4 and construct 3-D Cuisenaire cubes, working out area and volume without being told that's what we were doing. Becca and Deborah Jenkins, sisters and children of the presiding genius of the crèche, later both became musicians. I can't speak for the others but I bet half of them are musicians and the other half are mathematicians. Because we spoke the language of representation. We understood, without having anyone to explain it, what a symbol was. And that meant that we could read the world.
What does this have to do with Bach, or with color theory? Everything. I teach small children the piano, and I make sure that in every lesson they are saturated with color. They choose a pencil from a multi-colored set (oddly, most of them choose the turquoise one). They play with farm animals or doll's houses or marble runs, but mostly they play with color. And then they play music. Music, if I had to simplify (and I do), is color + rhythm. That's basically the Prelude in C. And the one in C minor. And the one in E. When you go to the faculty lounge at the Tampere Department of Communication Sciences, you are greeted with this:
Lego, of course, besides being Danish, is color + mathematics. That means, I think, if you solve for color in both equations, that Music + Mathematics = Lego + Rhythm.
Mu = c + r L = c + Ma
c = Mu - r c = L - Ma
Mu - r = L - Ma
Mu + Ma = L + r
And why is this one teal? Because the piece is a system of three-part rods. It begins with a quarter rest and an eighth rest (3/8ths of the bar). The short theme then announces itself with two notes: an eighth and then a quarter (1/8+2/8=3/8). That's three beats of the four-beat bar (3/8+3/8=3/4). This theme is then distributed among three voices (3/8x3/8=9/8). And teal is 9...
University of Tampere, Finland
Cuisenaire rods (1 cm x 1-10 cm)