(NOTE: see Ron Knott's Fibonacci Series page for lots of fascinating information on Fibonacci numbers.)
This article first appeared in 1/1, the Journal of the Just Intonation Network, Volume 6, Number 4, p.4 (1990).
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[1].
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 Fibonacci sequence has been used in music, for example as a rhythmic theme in some of the compositions of Bela Bartok. (I have also heard of Fibonacci ratios being used as the harmonic basis for just scales.) My first musical inspiration involving the Fibonacci sequence was a sort of fugue, where the theme is made of successive phrases whose lengths (in beats) keep increasing following the Fibonacci sequence. Each later voice enters in such a way that at some point two phrases of the later voice (say, length 3 and 5 beats) occur during the next phrase (length 8 beats) of the first voice. This idea eventually evolved into the piece "Rosier Sands," for the Partch Ensemble, which can be heard on Tellus 14: Just Intonation (formerly available from the Just Intonation Store). Figure 1 gives a condensed excerpt: each part has a central tone that begins each phrase, and some parts have more elaborate phrases than others, the phrases being subdivided using Fibonacci ratios.
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. Figure 2 illustrates this idea in piano notation (each of four layers playing the same melody in this example). 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. This is shown graphically in Figure 3.
The most obvious way to get Fibonacci gamelan layered rhythms is to have the highest layer play every beat, the next layer every two beats, the next every three, the next every five, etc. While this structure could be an interesting starting point, the result has very little of the reinforcement between layers that gives gamelan music such strength. The only way to get that sort of reinforcement with Fibonacci numbers is for each layer to have at least two different note lengths. So I was led to the idea of short and long notes (S and L for shorthand) at each layer, the highest layer playing one- and two-beat notes, the next two- and three-beat notes, and so on. Thus a long note (L) at one layer lasts for a long and short (L S or S L) of the next higher layer, and a short note (S) at one layer corresponds to a long (L) at the next layer.
Furthermore, I wanted the same pattern of long and short notes at every layer (for any number of layers), starting from the beginning. Some experimentation showed that this was only possible if the pattern begins L S L. The same-pattern decision requires that all the lower layers begin the same way, and the reinforcement decision says that higher layers must reinforce the lower layers, so certain later positions of the pattern are also determined by the two decisions. But choices remain, starting with whether the fourth note of the pattern should be short or long (giving L S L S L or L S L L S respectively, since the next note is then determined); this is shown graphically in Figure 4.
One could devise many different schemes (including randomness) for deciding all the remaining choices in this rhythmic structure. After some exploration, three different, simple schemes appeared to give the most aesthetic (to me) results. Since each choice corresponds to how a long note at one layer should be divided at the next higher layer (i.e., L S or S L ?), one scheme is always to put the long note first (L S). The musical effect is a familiar one, like every major downbeat being preceded by a pickup beat - at some layer. The second scheme is, given a choice, always to put the short note first (S L), the effect being essentially the reverse of the previous scheme. The third scheme relates the choice to the setting: you put the long note next to the more deeply reinforced of the two beats that delimit the duration being subdivided. Here the musical effect seems orderly yet surprising; this scheme most piqued my interest.
The rhythmic structures resulting from each of these schemes are shown in Figure 5. Certain features are common to all three: two short notes in succession (S S) do not occur, nor do three long (L L L). Most other schemes apparently don't have these features, but regardless of how the choices are made (in accordance with reinforcement and same-pattern), one never gets more than two shorts in a row or more than four longs in a row. Also, there are more long notes than short notes, in a ratio of approximately the Golden Mean (exactly, if the process were extended to an infinite pattern). Similarly, each layer plays the pattern more slowly than the next higher, by a factor of the Golden Mean. (Actually, the factor is a Fibonacci ratio based on the overall length of the pattern - which for a long pattern is an excellent approximation of G.)
The three schemes above can also be described in a different way, in terms of extending the beginning of the pattern. In the long-first scheme, given the first F(n) notes of the pattern (where F(n) is the nth Fibonacci number), if one takes the first F(n-1) notes and repeats them at the end, that correctly extends the pattern to F(n+1) notes. (Here "note" just means long or short duration.) In less precise terms, to extend, take the long first "half" and copy it onto the end. (So, starting with L S L, take the beginning L S and add it on to get L S L L S, then do the same again: add on the beginning L S L to get L S L L S L S L, etc.) For the short-first scheme, take the ending long "half" and repeat it to extend. (So L S L gains the ending S L to become L S L S L, then again gains the ending L S L to extend to L S L S L L S L, etc.) For the last scheme (long- deepest), take the beginning long "half" and reverse it onto the end. This gives a symmetric or nearly symmetric pattern at each step of extension, like a palindrome. For the long-first or long- deepest schemes, these extension rules apply not only to a single layer but to all layers at once.
This cut-and-paste method of extending the rhythmic pattern naturally suggested that any melody following the rhythm could be extended the same way. This idea, using the long-deepest scheme, gave rise to the short guitar quintet that appears as the third section of my guitar suite, which appears on Rational Music for an Irrational World (also previously available from the Just Intonation Store). There a short eight-note melody is extended (after the ninth note) by reversing the first five notes, then reversing all eight notes, etc. This melody is played in five layers, based on either the root or the "fifth," interpreted within the current pentatonic scale, and the piece changes through five pentatonic scales. (The five pentatonics, relative to D, are 1/1 9/8 5/4 3/2 5/3, 3/2 15/8 21/20 6/5 21/16, 8/7 4/3 32/21 12/7 1/1, 3/2 15/8 33/32 39/32 21/16, and 1/1 9/8 5/4 3/2 7/4.) So the top five layers of the rhythmic structure correspond to the five guitar parts, and lower layers give the overall form of the piece.
I used the same construction in a piano piece called "Canon for Seven Hands" [see Fibonacci Suite page]. Of course, the piano needs to be retuned to a just scale, in C, specifically: 1/1 28/27 9/8 7/6 5/4 4/3 45/32 3/2 14/9 5/3 7/4 15/8. Figure 6 shows the beginning of the piece (all layers start together on a rest).
A 21-note melody is extended to a five-minute piece, again using the scheme of reversing the first long half onto the end to extend. The same melody is used in all seven layers, in the seven octaves of the piano (requiring three and a half performers). The melody uses all twelve tones, and is designed to combine variety with harmony. The structure determines that whenever a particular note of the 21-note melody is struck at some layer, the next higher layer always strikes one other particular note of the melody, and so on, up to the highest layer. Knowing this harmonic interconnection of the melody with itself, I could choose, to some extent, which tones would be struck simultaneously, according to their positions in the 21-note melody. All such chords occur on some branch of the inverted harmonic tree shown in Figure 7. For example, the chord at the beginning of the sixth measure corresponds to the whole leftmost branch. The lower tones in the slower layers are sustained while the upper melodies move on, giving harmonic combinations outside those in Figure 7.
The piece is tonally and melodically rather static, in that it never changes scale or tempo. Much of the textural variety comes from continually changing dynamics in various ways, such as softening the left-hand parts while strengthening the right-hand parts, or reducing the higher layers while bringing out the lower layers. This allows the same rather complex rhythmic- melodic-harmonic structure to be perceived in many different ways, illuminating different relationships. The timing for the various dynamic patterns is determined by deep layers of the rhythmic structure. In this way, the entire temporal structure of the piece, from the fastest time scale of the top melody down to the slowest time scale of dynamic changes, is unified by the same governing concept. There is a sort of continuum, through the layers, from melody to accompaniment to bass lines to structure, with no clear demarcations in between. A good analogy would be a crystal, where the way the molecules fit together on the smallest scale determines the overall shape of the crystal at its largest scale, with smaller crystals of the same type having essentially the same shape.
The Golden Mean occurs in the temporal proportions of the canon, both in the relative speeds between layers and approximately in the relative durations of long and short notes within a layer. While in the fastest melody the durations of long and short notes have the clearly perceptible ratio 2/1, by the seventh layer the ratio is 34/21 = 1.6190..., a good approximation of the Golden Mean. If this piece were realized by player piano, a la Conlon Nancarrow, then all the note durations could be based on the Golden Mean precisely, instead of on the Fibonacci ratios.
Before closing, I'd just like to mention a few surprising connections, without going into too much detail. The patterns of long and short generated by the three schemes above, if extended to infinite length, would give what mathematicians call a nonperiodic sequence, that is, the sequence is not simply repetition of any single short pattern. This, and the relation with the Golden Mean, bring to mind a two-dimensional analogue: Penrose tiles[2]. Penrose tiles are two simple two-dimensional diamond shapes (rhombi), one large and one small, that fit together to fill a plane, but only in nonperiodic ways (unlike square tiles, for example); here, too, the ratio of the number of large tiles to small tiles is the Golden Mean. The patterns formed by Penrose tiles have nearly five-way symmetry. In fact, their spacings in each of five directions give patterns of long and short spacing called Ammann bars, which look just like the Fibonacci gamelan patterns (...L S L S L L S L L S L S L L S...). Moreover, Robert Ammann, the discoverer of Ammann bars, describes these long-short patterns by the term "musical"!
Some recent experiments with certain metallic alloys, rapidly cooled from the molten state, show X-ray diffraction patterns with five-way symmetry, which is impossible for an ordinary (periodic) crystal. The evidence is mounting that these are "quasicrystals," that the atoms fit together in a way analogous to three-dimensional Penrose diamond shapes, in a nonperiodic way. Then the spacings between planes of atoms would again give "musical" patterns of long and short. In this way the Fibonacci-gamelan music contains the same patterns as in this new form of matter.
Furthermore, recent conversations I have had with space aliens have shown that their hyperspace drive relies on the same patterns... (No, really, all the previous relations are true.)
In short, these Fibonacci-gamelan rhythms are the result of combining the Fibonacci sequence with gamelan layers, requiring reinforcement and the same pattern at every layer, and choosing a simple scheme to fill the remaining gaps in the pattern. From these few ideas has serendipitously come a structure that I find rich and fascinating, and aesthetically satisfying.