1A. The scales produced from the Division Operation were only ascending in pitch. In fact, producing a melody with the Division Operation and the Fibonacci numbers is impossible (why?). The cyclic nature of the Modulo Operation will create an upward and downward melodic motion, and to demonstrate this the first step is to get a list of the first forty Fibonacci numbers. (Students can generate these numbers, but Music Algorithms will also produce the list. Although the site has a thorough help menu, you can also click on the following link to get annotated instructions on what to do for the current situation.)
1B. Ask how many keys are on the piano (since it is likely that some in-class will know there are 88 keys). This will be referred to as the full range of the piano: Keys 1-88.
1C. The problem is how to wrap the forty Fibonacci numbers around the full range of the piano; Initially, think of the Fibonacci numbers Mod 88. However, the first number (0) would then be at Key 0, since (0) Mod 88 = 0. We could call Key 0 a silent note, but we could just as easily add 1 so that the first number (0) ends up at Key 1. So, we really want (0) Mod 88 + 1 = 1. Now we have the first number (0) at Key 1. Similarly, (1) Mod 88 + 1 = 1 + 1 = 2, and(21) Mod 88 + 1 = 21 + 1 = 22. By the time we get to the twelfth Fibonacci number (89), things start to get more interesting: Where would (89) go on the piano range using the Modulo Operation? What about the thirteenth number (144), and so on?
1D. At this point, either by teacher-guided whole-class discussion or via small group work, the method of wrapping the Fibonacci numbers onto the full range of the piano should be made clear. Click for a detailed explanation of how the Modulo Operation would work in the current example.
1E. Once it is clear to students where the first forty Fibonacci numbers would go and why (that is, how the Modulo Operation works), the teacher or a student can go to the Music Algorithm website and use the applet to play the notes. You can also click on the following link to get annotated instructions on how to use the applet to play the notes.
