Fibonacci-Type Sequence
Casey Mongoven
April 8, 2005
Instructions
1. This work is to be played on a percussion instrument of indefinite
pitch; anything is acceptable.
2. The player chooses two relatively prime positive integers x
and y (1 and 1 for example).
3. The player chooses a unit of duration d, which does not change
throughout the piece; 1 second for example.
4. The player chooses a dynamic level, which will not change throughout
the whole piece; double fortissimo for example.
5. The player constructs a Fibonacci-type sequence {x, y, x
+ y, x + 2y, 2x + 3y, 3x + 5y,
5x + 8y, 8x + 13y, ...}; for example {1, 1, 2, 3,
5, 8, 13, 21, 34, 55, 89, 144, ...}. In a Fibonacci-type sequence, each term is
the sum of the two letzte: fn =
fn -1 + fn -2. The player may choose how many terms
of the sequence are to be performed.
6. The player imagines a number line on which the numbers of the
Fibonacci-type sequence described above are marked omitting the first member
and plays one note on each of the points in the Fibonacci-type sequence. Each
note should be exactly as long as each other; for instruments with a decay time
longer than one unit of duration, the note should be muted so that it lasts
exactly as long as d. Instruments with a very short decay time need not
worry about this.
7. The player must use some electronic device for synchronization (a
stopwatch or click track, for example).
8. The piece can go on as long as desired. It can be given personal
performances as well as public or private. The proportion of the adjacent
durations between the attacks get closer and closer to the golden ratio the
further into the sequence one plays.
An Example Performance
It just so happens that I am in the middle of performing this work as I
write this. I intend for this performance to go on for about 21 hours. I chose
to start with 1 and 1, and to use 1 second as a durational value for
convenience. My sequence is {1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,
610, 987, 1597, 2587, 4181, 6765, 10946, 17711, 28657, 46368, 75025} – notice I omit the first
term; notice that, due to the additive properties of Fibonacci sequences, the
resulting distances between notes do indeed form the original {1, 1, 2, 3, 5,
8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2587, 4181, 6765, 10946,
17711, 28657}. I am
playing it on a Remo practice pad at a dynamic level
of forte. I made some calculations before I started and am using a
stopwatch to aid in performance. Here is a table showing the calculations I made,
starting the performance on the 10 second mark of my stopwatch:
|
# |
h |
m |
s |
|
1 |
0 |
0 |
10 |
|
2 |
0 |
0 |
11 |
|
3 |
0 |
0 |
12 |
|
5 |
0 |
0 |
14 |
|
8 |
0 |
0 |
17 |
|
13 |
0 |
0 |
22 |
|
21 |
0 |
0 |
30 |
|
34 |
0 |
0 |
43 |
|
55 |
0 |
1 |
4 |
|
89 |
0 |
1 |
38 |
|
144 |
0 |
2 |
33 |
|
233 |
0 |
4 |
2 |
|
# |
h |
m |
s |
|
377 |
0 |
6 |
26 |
|
610 |
0 |
10 |
19 |
|
987 |
0 |
16 |
36 |
|
1597 |
0 |
26 |
46 |
|
2584 |
0 |
43 |
13 |
|
4181 |
1 |
9 |
50 |
|
6765 |
1 |
52 |
54 |
|
10946 |
3 |
2 |
35 |
|
17711 |
4 |
55 |
20 |
|
28657 |
7 |
57 |
46 |
|
46368 |
12 |
52 |
57 |
|
75025 |
20 |
50 |
34 |
At each of these points in time I play a note.
In traditional notation the beginning of the piece would look like this:
Although I chose to use the classic Fibonacci sequence for my first
performance, any Fibonacci-type sequence is possible; adjacent terms of all
Fibonacci sequences begin to converge to the golden ratio very quickly. The
table below shows this convergence for the classic Fibonacci sequence. The
numbers in the left column are divided by the numbers in the middle column,
resulting in the decimal shown in the right column:
|
1 |
1 |
1.0000000000 |
|
1 |
2 |
0.5000000000 |
|
2 |
3 |
0.6666666667 |
|
3 |
5 |
0.6000000000 |
|
5 |
8 |
0.6250000000 |
|
8 |
13 |
0.6153846154 |
|
13 |
21 |
0.6190476190 |
|
21 |
34 |
0.6176470588 |
|
34 |
55 |
0.6181818182 |
|
55 |
89 |
0.6179775281 |
|
89 |
144 |
0.6180555556 |
|
144 |
233 |
0.6180257511 |
|
233 |
377 |
0.6180371353 |
|
377 |
610 |
0.6180327869 |
|
610 |
987 |
0.6180344478 |
|
987 |
1597 |
0.6180338134 |
|
1597 |
2584 |
0.6180340557 |
|
2584 |
4181 |
0.6180339632 |
|
4181 |
6765 |
0.6180339985 |
|
6765 |
10946 |
0.6180339850 |
|
10946 |
17711 |
0.6180339902 |
|
17711 |
28657 |
0.6180339882 |
The golden ratio to 30 decimal places reads:
0.618033988749894848204586834366
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