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Base Phi Representations:

Base Phi Minimum Binary, Base Phi Maximum Binary

Casey Mongoven

August 13, 2006

 

the sequences: There exists a binary non-standard positional number system which uses the golden ratio as a base. In this system, powers of the golden ratio are used instead of the powers of ten used in the decimal system. Despite the base being irrational, every integer has a unique finite representation.

 

The golden ratio values are symbolized by Phi = 1.618033989…, and phi = .618033989…. The following diagram shows the values of the powers of Phi, and illustrates an interesting relation to the Fibonacci sequence:

 

Phi5

8 + 5phi

11.0901699…

Phi4

5 + 3phi

6.85410197…

Phi3

3 + 2phi

4.23606798…

Phi2

2 + 1phi

2.61803399…

Phi1

1 + 1phi

1.61803399…

Phi0

1 + 0phi

1.00000000…

Phi-1

0 + 1phi

0.61803399…

Phi-2

1 - 1phi

0.38196601…

Phi-3

-1 + 2phi

0.23606798…

Phi-4

2 - 3phi

0.14589803…

Phi-5

-3 + 5phi

0.09016994…

 

This relation to the Fibonacci sequence can be captured in the equation:

 

Phin = Fibn+1 + Fibn*phi

 

Here are the integers 1 through 11 in base Phi:

 

decimal

powers of Phi

base Phi

1

Phi0

1     

2

Phi1+Phi-2

10.01  

3

Phi2+Phi-2

100.01  

4

Phi2+Phi0+Phi-2

101.01  

5

Phi3+Phi-1+Phi-4

1000.1001

6

Phi3+Phi1+Phi-4

1010.0001

7

Phi4+Phi-4

10000.0001

8

Phi4+Phi0+Phi-4

10001.0001

9

Phi4+Phi1+Phi-2+Phi-4

10010.0101

10

Phi4+Phi2+Phi-2+Phi-4

10100.0101

11

Phi4+Phi2+Phi0+Phi-2+Phi-4

10101.0101

 

A radix point ‘.’ is used to separate the powers of Phi into those greater than or equal to 1 and those less than 1.

 

As a rule, the base Phi representations of integers shown above never contain the digit sequence “11” (or “1.1”). Put in other words, they never contain any consecutive powers of Phi (Phi4 and Phi3, for example). This system of representing integers and the piece based on it I will call Base Phi Minimum Binary.1

 

Because in this system 100 is equal to 011 (this follows from Phin = Phin-1 + Phin-2), it is possible to use a replacing-algorithm to eliminate every occurrence of the digit sequence “00” so that every integer also has an alternate unique finite representation which contains no instance of “00”. This system of representations and the piece based on it I will call Base Phi Maximum Binary.

 

Both sequences are shown below:

 

           Minimum                       Maximum

 

0             0.                            0.

1             1.                            1.

2            10.01                          1.11

3           100.01                         11.01

4           101.01                        101.01

5          1000.1001                      101.1111

6          1010.0001                      111.0111

7         10000.0001                     1010.1101

8         10001.0001                     1011.1101

9         10010.0101                     1101.1101

10        10100.0101                     1111.0101

11        10101.0101                    10101.0101

12       100000.101001                  10101.111111

13       100010.001001                  10111.011111

14       100100.001001                  11010.110111

15       100101.001001                  11011.110111

16       101000.100001                  11101.110111

17       101010.000001                  11111.010111

18      1000000.000001                 101010.101101

19      1000001.000001                 101011.101101

20      1000010.010001                 101101.101101

21      1000100.010001                 101110.111101

22      1000101.010001                 101111.111101

23      1001000.100101                 110101.111101

24      1001010.000101                 110111.011101

25      1010000.000101                 111010.110101

26      1010001.000101                 111011.110101

27      1010010.010101                 111101.110101

28      1010100.010101                 111111.010101

29      1010101.010101                1010101.010101

 

More terms (as many as 24,476) and other files for performing this work can be found at the following web addresses:2

 

http://caseymongoven.com/catalogue/B416.html

 

and http://caseymongoven.com/catalogue/B417.html

 

the pieces: This score is for two works (catalogue B416 and B417), and these pieces can be performed separately or on the same concert. The order of performance is not important. Freedom is given to the performers to determine compositional aspects of the works with certain restrictions to be explained.

 

In this work, each player is assigned a power of Phi. Each performer plays their note - the same note - every time their assigned power of Phi occurs in the integer being expressed. The following example of Base Phi Minimum Binary in traditional notation should help make this clear:

 

 

The above example is a short performance (only 11 integers) for 8 players.

 

There are 3 versions of this work: one for percussion instruments of indefinite pitch, one for percussion instruments of definite pitch, and one for pitched instruments (in which some percussion instruments are allowed as well). Each player plays the same type of instrument in every case. The three versions are explained separately below. First, the elements common in all versions will be explained.

 

number of integers played: Each member of the sequence is given equal duration. The duration of each member is up to the performers to decide, but should be at least 70 beats per minute (.857 seconds or faster). The players are to start with the representation for 1 and play through in increasing order (i.e. in decimal notation 1,2,3,4,5,6,7,…). A Lucas number of integers are to be played:

 

2,1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,…

 

One can imagine that very long performances are possible. If both works are played, the same number of integers should be played in both.

 

number of players: The number of integers to be played determines the number of players required. If for example, 11 integers are to be played in the minimum representation, this will require 8 players because 10101.0101 has 9 digits and the second to last digit is never used in the minimum representation (this can be seen in the example in traditional notation above; Phi-3 has only rests). Note, however, that for maximum representations a player is needed to represent every digit.

 

spacing on stage: In performance, regardless of the number of players needed, there should be a person standing in for the power of Phi not used in the minimum representations - with instrument if possible.

 

There are 3 options for spacing in this work. In each spacing, the players line up in a straight line from stage left to stage right. The bigger the space where playing and the more reverberation, the larger the space between the performers should be. Space is always measured from the center of the body. The positions should be marked on stage beforehand as in one of the following options:

 

1. Even spacing is used. There should be at least 1.4 meters, performer to performer. For example:

 

Φ4----Φ3----Φ2----Φ1----Φ0----Φ-1----Φ-2----Φ-3----Φ-4

 

2. Even spacing with a radix point is used. There should be at least 1.4 meters between each performer and the space representing the radix point should be at least 2 times the distance between each performer and can be much greater. It is optional to use an object to represent the radix point; if used, this should be placed directly in between performers Φ0 and Φ-1. For example:

 

Φ4----Φ3----Φ2----Φ1----Φ0----‘.’----Φ-1----Φ-2----Φ-3----Φ-4

 

3. Spacing based on the golden ratio is used. In this spacing, the distance between the players increases from right to left based on the golden ratio. The space between the two 2 right-most players cannot be less than 75 centimeters. If x represents the space between the players at the right-most end, then the distance for between the players is figured as such:

    

         Phi3*x            Phi2*x      Phi*x    x

 

Φ2--------------------Φ1------------Φ0-------Φ-1----Φ-2

 

In the 3rd spacing option, the smaller space must be between the smaller powers of Phi as shown. This option requires more space to perform than the others.

 

Any of these spacings given as examples may be inverted from left to right; the lowest value can be on the left or right side.

 

synchronization: Some electronic means of synchronization must be used in performance for the sake of large scale precision. This can be a conductor who is connected to a click track, a click track for the performers, or a blinking light. If a conductor is used, he should be placed in a position that does not block the sound field of the performers.

 

articulation, dynamic, manner of playing: The group should agree on a dynamic level which will stay the same for the duration of the work. Each player plays their note exactly the same each time: exactly as loud and in the exact same manner as the other players. The posture of the musicians should be as uniform as possible. The attack should be crisp. For instruments with decay times longer than the duration of one member of the sequence, the note should be dampened each time it is played. Each note should last an absolute maximum of 1 member of the sequence (i.e. in the example above in traditional notation, at maximum as long as a quarter note).

 

rehearsal, performance suggestions: After the performers have agreed on how to perform the work, preparations must be made by each player. Each player should practice their part without the group beforehand. Although it is up to the performers, I would recommend not using standard notation in performing this work. In faster versions it might be necessary for some performers to write out their part in some standard notation. But it is best to perform the work with a list of numbers in front of oneself, with the part to be performed in bold print:

 

1             1.                            1.

2            10.01                          1.11

3           100.01                         11.01

4           101.01                        101.01

5          1000.1001                      101.1111

6          1010.0001                      111.0111

7         10000.0001                     1010.1101

8         10001.0001                     1011.1101

9         10010.0101                     1101.1101

10        10100.0101                     1111.0101

11        10101.0101                    10101.0101

12       100000.101001                  10101.111111

13       100010.001001                  10111.011111

14       100100.001001                  11010.110111

15       100101.001001                  11011.110111

16       101000.100001                  11101.110111

17       101010.000001                  11111.010111

18      1000000.000001                 101010.101101

 

The 0s are, of course rests, and 1s attacks. Such notation for this work is available at the web addresses given above with up to 24,476 members.3

 

The difficult task of performing long versions of this work can be made easier by recognizing the regularity and patterns in the digits. Numbers indicating the length of the runs of 1s and 0s might be hand-written into the margin of the score. The players at the ends must be prepared to play long runs of notes without losing count. It can be very helpful to practice this work at slower tempos. Also of great importance is to check one’s counting by listening to the players directly on one’s left or right side, because their parts are most visible in the score.

 

Players who find the lack of barlines (and standard notation) in the work confusing might draw in lines into the score as such:

 

1             1.                            1.

2            10.01                          1.11

3           100.01                         11.01

4           101.01                        101.01       .

5          1000.1001                      101.1111

6          1010.0001                      111.0111

7         10000.0001                     1010.1101

8         10001.0001                     1011.1101     .

9         10010.0101                     1101.1101

10        10100.0101                     1111.0101

11        10101.0101                    10101.0101

12       100000.101001                  10101.111111   .

13       100010.001001                  10111.011111

14       100100.001001                  11010.110111

15       100101.001001                  11011.110111

16       101000.100001                  11101.110111   .

17       101010.000001                  11111.010111

18      1000000.000001                 101010.101101

 

The score might also be turned on its side for a player who finds it confusing to read music vertically.

 

The conductor or director overseeing the group in rehearsal should keep a printed list of the members of the sequence as such at hand: