The Rabbit Sequence

Casey Mongoven

 

 

I. About This Composition

 

This work is for two percussion players and two percussion instruments of indefinite pitch with crisp attacks and a fast decay;1 each player has one instrument, the instruments need not be of the same type. Cymbals, triangles and other instruments with a long decay and less crisp attacks are not to be used. The players should find a combination of instruments or an instrument that they find works best and sounds best for them.

 

The players should be situated in front of the audience a good distance from each other; they should be far enough from each other to create a significant spatial independence of each part. This spatial independence should, however, not be over-exaggerated. The players should consider the environment in which they are playing and adjust accordingly.

 

This work is based on a sequence known as the rabbit sequence (also known as the golden string, golden chain or infinite Fibonacci word). This is an infinite binary sequence that is related to the golden mean and the Fibonacci numbers. The sequence is a fractal. There are many ways of generating this sequence. For the purpose of performance, one way of generating this sequence will be concentrated upon in order to aid the performers in accurate execution. It is imperative that the performers understand the construction of the sequence in order to perform the work accurately and with flexibility.

 

 

II. Construction Of The Sequence

 

We will use 1s and 0s to represent the sequence (i.e. 1 0 1 1 0 1 0 1 ...); the 1s represent player one's part and the 0s player two's. From a performance standpoint, the best way to look at the construction of the sequence is as follows:

 

We start with a 1, this is called stage 0 (the initiator):

 

1

 

From here forth we replace each 1 in the sequence with 1 0, so that the next stage (stage 1) of the sequence is:

 

1 0

 

Now we have 1 0, a binary sequence. Moving on from here we replace each 1 with 1 0 as we did before, but now we replace each 0 in the sequence with 1 as well, so that the next stage of the sequence (called stage 2) is:

 

1 0 1

 

So far we started with 1, replaced that with 1 0, and then replaced 1 0 with 1 0 1. For each 1 we put 1 0 in its place, for each 0 we put a 1. Going onto stage 3 we get:

 

1 0 1 1 0

 

Moving further to stage 4 we get:

 

1 0 1 1 0 1 0 1

 

Here is a good visual representation of this way of constructing the sequence:

 

 

 

This shows the sequence up to stage 5, but the sequence can, of course, go on endlessly.

 

 

III. Performers' Task

 

The composite rhythm of the work is straight. By this is meant that the two voices combined create a constant rhythm comparable to straight sixteenth notes. Here is an example of the work at stage 5 in traditional notation – the sequence is written above the notes here to clearly show its relation to the music:2

 

File written by Adobe Photoshop¨ 5.0

 

It can be seen that player one plays the 1s in the sequence, while player two plays the 0s.

 

The tempo of the work is left up to the performers; it should, however, be at least 155 attacks per minute. The length of the composition is also left up to the performer – the composition will always contain a Fibonacci number of attacks; the performers can choose what stage of the sequence they want to perform and should indicate this to the audience (i.e. the title of the work in the program might read something like The Rabbit Sequence, Stage 7).3 The dynamic level should stay perfectly steady throughout and should be from mezzoforte to double-fortissimo (mffff). Both parts should be played at the same dynamic level.

 

If more than one stage is being performed on the same concert, then the different stages should be performed at the same tempo and with the same instruments. If the stages are to be performed consecutively, long pauses (at least 20 seconds) should be given in between the stages.

 

This work is to be performed with a metronome or click-track. Each performer should have an earphone in the ear opposite the other performer. The metronome will help the large-scale accuracy of the performance; it does not make the execution of this work easier for the performers. 

 

Each attack is to be thought of by the performers as exactly as long as 1 unit. It is not suppose to be thought of as a point, rather as a line. For instruments with decay times longer than each unit, where the sound is still decaying when the next attack occurs, the players should mute their instruments every time they have a rest.

 

 

IV. Exercises And Methods For The Performers

 

The following is an explanation of the methods and exercises I would like the performers to use in performing this work. Mastery of these methods will take hard work. Once these techniques are mastered, the performers will be able to perform this work with great ease and flexibility. Because of the somewhat complicated nature of these methods, I have relied heavily on traditional notation.

 

Here is a simple exercise that is fundamental in performing this composition:

 

File written by Adobe Photoshop¨ 5.0

 

Notice the rest in parenthesis at the beginning for player two. Although player two's part does in reality start with a rest, the player will find it easier not to think about it this way; player two should try to imagine his first note as the first note in the composition. Later, if the performers were to switch parts, the players would find it very easy if the composition is thought of in this way.

 

Note the stroke-patterns (L R L L R L R L ...); for some instruments this won't apply. The players should practice their parts both together and independently at various tempos.4 This exercise is crucial to performing this work, and needs to be completely internalized before moving on.

 

Before it was mentioned that player one plays the 1s and player two the 0s in the sequence. The performers shouldn't allow this to confuse them in the discussion that follows. That view of the sequence and its relation to the work is best left aside for now. Each player will now use the sequence in its full form to aid them in performance – they will use both 1s and 0s.

 

The next step involves substituting the numbers in the rabbit sequence with "measures." Basically, player one will think of the 1s and 0s in the rabbit sequence as such:

 

File written by Adobe Photoshop¨ 5.0

 

So using player one's part as an example:

 

File written by Adobe Photoshop¨ 5.0

File written by Adobe Photoshop¨ 5.0

 

and so on, depending on the desired length of the composition.

 

Player two will think about the sequence as such:

 

File written by Adobe Photoshop¨ 5.0

 

 

therefore:

 

File written by Adobe Photoshop¨ 5.0

 

and so on...

 

Once the performers have gotten used to doing this, they should be able to play the work with just a sheet of numbers in front of them to guide them. Performers who have internalized the structure of the sequence to a higher degree might not even need this; they might indeed be able to visualize the sequence in their minds. A performer may want to expand the 1 and 0 as to contain more attacks. These numbers will always be adjacent Fibonacci numbers, with the 1 being the greater of the two. For example:

File written by Adobe Photoshop¨ 5.0

 

This will make longer performances and performances without a guide sheet easier. It is not incredibly important how many attacks the 1 or 0 are imagined as consisting of. This only serves as an aid to the performer. That is up to the performers and all will have different preferences; two performers playing this composition together might use different numbers from each other in performance.5 Once this method of performing the composition is understood, the performers will gain great flexibility and will be able to perform the composition at various stages on command.

 

 

 V. A Theoretical Performance

 

Let us suppose that two players have decided to play the composition at stage 9 with 390 attacks per minute on wood blocks at the dynamic level forte. After they have practiced the exercises given above and internalized the structure of the sequence, they decide to make a sheet of numbers to aid them in performance. Player one decides he wants to think of the 1 in the sequence as containing 5 attacks and the 0 as containing 3, player two chooses to do the same, so:

Player 1:

 

File written by Adobe Photoshop¨ 5.0

 

Player 2:

 

File written by Adobe Photoshop¨ 5.0

 

The performers internalize the patterns above. Since the performance is 89 attacks long at stage 9, the performers need to calculate how many numbers of the rabbit sequence they will need to substitute with the "measures" above to fit their total number of attacks. Player one plays 55 attacks, player two plays 34 (these numbers will always be adjacent Fibonacci numbers, with player one's being the higher number). Their sheet of numbers during performance looks like this:

 

Player 1:

 

1 0 1 1 0 1 0 1 1 0 1 1 0


Player 2:

 

1 0 1 1 0 1 0 1

 

In performance they simply read through the sequence replacing the 1s and 0s with the "measures" shown above, which they have practiced and internalized.

 

 

VI. Composition Lengths And Corresponding Stages

 

Here are the first 21 stages and their lengths in attacks:

 

Length in Attacks                 Stage

 

1                                              0

2                                              1

3                                              2                     

5                                              3

8                                              4

13                                           5

21                                           6

34                                           7

55                                           8

89                                           9

144                                         10

233                                         11

377                                         12                  

610                                         13

987                                         14

1597                                       15

2584                                       16

4181                                       17

6765                                       18

10946                                                19

17711                                                20

                                   

Here is a table to give performers an idea of the duration of the different stages at different tempos:6


Stage 3: 5                  Stage 4: 8                  Stage 5: 13               Stage 6: 21

 

bpm/length

 

155/1.9 sec               155/3.1 sec.              155/5.0 sec               155/8.1 sec.

195/1.5                      195/2.5                      195/4.0                      195/6.5

235/1.3                      235/2.0                      235/3.3                      235/5.4

275/1.1                      275/1.7                      275/2.8                      275/4.6

315/1.0                      315/1.5                      315/2.5                      315/4.0

355/.8                         355/1.4                      355/2.2                      355/3.5

395/.8                         395/1.2                      395/2.0                      395/3.2         

435/.7                         435/1.1                      435/1.8                      435/2.9

475/.6                         475/1.0                      475/1.6                      475/2.7

515/.6                         515/.9                         515/1.5                      515/2.4

555/.5                         555/.9                         555/1.4                      555/2.3

595/.5                         595/.8                         595/1.3                      595/2.1         

635/.5                         635/.8                         635/1.2                      635/2.0

675/.4                         675/.7                         675/1.2                      675/1.9         

715/.4                         715/.7                         715/1.1                      715/1.8

755/.4                         755/.6                         755/1.0                      755/1.7

795/.4                         795/.6                         795/1.0                      795/1.6




Stage 7: 34               Stage 8: 55              Stage 9: 89               Stage 10: 144

 

 

155/13.2                    155/21.3                    155/34.5                    155/55.7

195/10.5                    195/16.9                    195/27.4                    195/44.3

235/8.7                      235/14.0                    235/22.7                    235/36.8

275/7.4                      275/12.0                    275/19.4                    275/31.4

315/6.5                      315/10.5                    315/17.0                    315/27.4

355/5.7                      355/9.3                      355/15.0                    355/24.3

395/5.2                      395/8.4                      395/13.5                    395/21.9       

435/4.7                      435/7.6                      435/12.3                    435/19.9

475/4.3                      475/6.9                      475/11.2                    475/18.2

515/4.0                      515/6.4                      515/10.4                    515/16.8

555/3.7                      555/5.9                      555/9.6                      555/15.6

595/3.4                      595/5.5                      595/9.0                      595/14.5

635/3.2                      635/5.2                      635/8.4                      635/13.6

675/3.0                      675/4.9                      675/7.9                      675/12.8

715/2.9                      715/4.6                      715/7.5                      715/12.1

755/2.7                      755/4.4                      755/7.1                      755/11.4

795/2.6                      795/4.2                      795/6.7                      795/10.9




Stage 11: 233           Stage 12: 377           Stage 13: 610           Stage 14: 987

 

 

155/90.2                    155/145.9                  155/236.1                  155/382.1

195/71.7                    195/116.0                  195/187.7                  195/303.7

235/59.5                    235/96.3                    235/155.7                  235/252.0

275/50.8                    275/82.3                    275/133.1                  275/215.3

315/44.4                    315/71.8                    315/116.2                  315/188.0

355/39.4                    355/63.7                    355/103.1                  355/166.8

395/35.4                    395/57.3                    395/92.7                    395/149.9

435/32.1                    435/52.0                    435/84.1                    435/136.1

475/29.4                    475/47.6                    475/77.1                    475/124.7

515/27.1                    515/43.9                    515/71.1                    515/115.0

555/25.2                    555/40.8                    555/65.9                    555/106.7

595/23.5                    595/38.0                    595/61.5                    595/99.5       

635/22.0                    635/35.6                    635/57.6                    635/93.3

675/20.7                    675/33.5                    675/54.2                    675/87.7

715/19.6                    715/31.6                    715/51.2                    715/82.8

755/18.5                    755/30.0                    755/48.5                    755/78.4

795/17.6                    795/28.5                    795/46.0                    795/74.5

 

 

 

Stage 15: 1597        Stage 16: 2584        Stage 17: 4181       Stage 18: 6765

 

 

155/618.2                  155/1000.3               155/1618.5               155/2618.7

195/491.4                  195/795.1                  195/1286.5               195/2081.5

235/407.7                  235/659.7                  235/1067.5               235/1727.2

275/348.4                  275/563.8                  275/912.2                  275/1476.0

315/304.2                  315/492.2                  315/796.4                  315/1288.6

355/269.9                  355/436.7                  355/706.6                  355/1143.4

395/242.6                  395/392.5                  395/635.1                  395/1027.6  

435/220.3                  435/356.4                  435/576.7                  435/933.1

475/201.7                  475/326.4                  475/528.1                  475/854.5

515/186.1                  515/301.0                  515/487.1                  515/788.2

555/172.6                  555/179.4                  555/452.0                  555/731.4

595/161.0                  595/260.6                  595/421.6                  595/682.2

635/150.9                  635/244.2                  635/395.1                  635/639.2

675/142.0                  675/229.7                  675/371.6                  675/601.3

715/134.0                  715/216.8                  715/350.9                  715/567.7

755/126.9                  755/205.4                  755/332.3                  755/537.6

795/120.5                  795/195.0                  795/315.5                  795/510.6

 

 

 

First 9 stages of the rabbit sequence:

 

 

1

 

10

 

101

 

10110

 

10110101

 

1011010110110

 

101101011011010110101

 

1011010110110101101011011010110110

 

1011010110110101101011011010110110101101011011010110101

 

 

 

Notes

 

 

1. For example temple blocks, castanets, claves, woodblocks or even clapping can be used. Unconventional instruments can be used as well, but they must fit the criteria of having indefinite pitch, a crisp attack and fast decay.

 

2. The examples in traditional notation in these instructions are only a tool to help the performers understand how this work is constructed and performed. Once this has been understood, any ideas of traditional notation should be completely abandoned in the performers' minds.

 

3. The performers shouldn't shy away from extremely short or long versions of this work. Short performances can be extremely dramatic as well as long ones. Longer performances are without a doubt more challenging for the performers.

 

4. The performers may want to trade parts as an exercise.

 

5. It is, of course, more difficult the more attacks you have representing each number. If the 1 is imagined as consisting of 21 attacks, and the 0 of 13, the difficulty rises significantly (this of course depends on the length of the composition as well).

 

6. The performers should not be influenced by these tempos or stages in choosing their own. That would be unfortunate. The tempos here are only meant to provide the performers with a general idea of the lengths of different stages of the composition at different tempos.

 

 

May 10, 2003