EXAMPLE 8 Musical Sequences

What about the order in which the s and s occur, as we move from left to right in Figure 20.29? Is there any pattern to that?

From the limited part of the pattern that we can observe, we see the sequence as

You might think from the figure that the pattern continues repeating the group indefinitely. But such is not the case. The sequence of intervals between Ammann bars is nonperiodic: It cannot be produced by repeating any finite group of symbols. We can think of it as a one-dimensional analogue of a Penrose tiling. The notation is reminiscent of the melody pattern of songs: Many popular songs follow the pattern , with the first and the last sections having the same melody but the middle section being different. Consequently, a sequence of intervals between Ammann bars is known as a musical sequence.

There are some rules that musical sequences follow. Two s can never be next to each other, nor can we have three As in a row (see Exercises 54 and 55 on page 863). Just as any finite part of any Penrose tiling occurs infinitely often in any other Penrose tiling, any finite part of any musical sequence appears infinitely often in any other one. The order of the symbols is neither periodic nor random, but between the two—quasiperiodic.