"Fibonacci Gamelan Rhythms"
What possible connection could there be between the traditional gamelan music of Indonesia and the twelfth-century Italian mathematician Leonardo Fibonacci? The two would seem unrelated, perhaps unrelatable. But in attempting to combine ideas from both of these sources, I discovered some fascinating rhythmic patterns, which are also related to a new form of matter called quasicrystals.
("Hold on there! Isn't this journal supposed to be about Just Intonation? What's this article doing here?" you may be wondering. My main excuse is that I've used these rhythms in compositions in Just Intonation. Besides, the concept of ideal proportion, as in the frequencies of acoustically pure musical intervals, can be extended to the much lower frequencies of rhythmic structure.)
The Fibonacci sequence (or series) of numbers begins:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
Each number is the sum of the previous two numbers, and the sequence is endless. These numbers, and especially ratios of two successive Fibonacci numbers, show up in a wide variety of situations, including nature. For example, the petals of a Monterey pine cone are arranged in spirals crossing in both directions: eight spirals in one direction and thirteen in the other. Similar patterns arise in the seeds of sunflowers and in other plants whose leaves grow in a spiral around a central stem; each successive leaf may be on the opposite side (1/2 way around) or may be 2/3 of the way around, or 3/5, etc. Nature seems to be biased in some ways in favor of the Fibonacci numbers.
A later idea was to use Fibonacci numbers in a layered rhythmic structure like that of traditional Indonesian music. Many years before, a friend had taken me to a performance of Balinese music, and I was entranced by the beauty of the instruments of the gamelan (Indonesian ensemble of gongs, metallophones, etc.), by the fluid melodies, the harmonious scales, and the layers of sounds. In simplified terms, each layer consists of instruments of a certain pitch range, playing at a certain tempo. The highest-pitched instruments play fastest, the next highest play half as fast (i.e., every other beat), the next layer half again as fast, all the way down to the low gongs that strike only once every 16 or 32 beats. Every layer strictly reinforces all higher layers in this simplified view - there is no syncopation (unlike real Indonesian music). Whenever a low note strikes, higher notes at all layers also strike.
The ratios of successive Fibonacci numbers form another sequence:
1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ...
or in decimal form:
1.0, 2.0, 1.5, 1.66..., 1.60, 1.625, ...
which gets closer and closer (over and under) to a certain irrational number called the Golden Mean (I'll use "G" for short):
G = (1 + sqrt(5))/2 = 1.6180339887...
If you take a rectangle of length G and height 1, and cut off a big square of length 1, the small rectangle left over has the same proportions as the original rectangle (i.e., G - 1 = 1/G). The ancient Greeks considered this number to be the perfect proportion; the Parthenon was designed with the proportion (width to height) of the Golden Mean. Many people since, up to the present, have attributed aesthetic qualities to the Golden Mean.
The rest here.