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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:

     

     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:

 

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 orientation on the page should be the same as the performers on stage (i.e. if the smallest power of Phi is stage-left, then the traditional notation should be inverted on the page).

 

The director can help in that he periodically gives the beat number using a hand-signal in performance, or saying the number out loud in rehearsal. The director should make the performers aware of the most important structural points in the work. At the same time he should stress that the players should not play any differently at these points. The most important structural points in these works occur around Lucas numbers (2,1,3,4,7,11,18,…).

 

for percussion instruments of indefinite pitch

 

All instruments used must be of the same type and dimensions. The instruments must blend well together and sound very similar to one another; no instrument should stick out in the group. The instruments must be capable of producing a crisp attack with no perceivable pitch. Instruments such as maracas, ratchets and cymbals are not to be used. Instruments which are in the so-called gray area of pitch-definiteness, such as tom-toms, are acceptable if a technique can be found which suppresses the perceived pitch (a rimshot, for example, can be effective in covering the perceived pitch of a tom-tom). Performances on unconventional instruments are encouraged.

 

for percussion instruments of definite pitch

 

All instruments used must be of the same type and dimensions. The players must agree on a single pitch for the group to use; each performer plays the same pitch. This pitch should be at least 18.5 Hertz. The pitch may contain complex overtones as long as one and the same pitch is predominant from all instruments. The instruments must blend well together and sound very similar to one another; no instrument should stick out in the group. The instruments must be capable of producing a crisp attack. Examples of acceptable instruments include crystal glasses or glass bottles (struck with a mallet), handbells, steel drums, crotales, and timpani. Performances on unconventional instruments are encouraged.

 

for pitched instruments

 

All instruments used must be of the same type and dimensions. The instruments must be able to be tuned – and played - to a high degree of intonational precision (within 2 cents). Unacceptable instruments include all standard orchestral brass and woodwind instruments. The instrument must be able to produce a sharp attack. Most plucked and bowed instruments are acceptable. Percussion instruments such as crystal glasses may be used, timpani are not acceptable.

 

The group decides on a temperament and tuning – this should suit the instruments. The temperament must be based on the following equation, in which n is an integer 1 through 12:

 

1 + phin

 

Here is an example for violins in the 10th temperament (1 + phi10), and one for cellos in the 11th (right side). These examples use only open strings:

 


player

string IV tuned to

Phi6

185.00 Hz

Phi5

186.50 Hz

Phi4

188.02 Hz

Phi3

189.55 Hz

Phi2

191.09 Hz

Phi1

192.64 Hz

Phi0

194.21 Hz

Phi-1

195.79 Hz

Phi-2

197.38 Hz

Phi-3

198.99 Hz

Phi-4

200.60 Hz

Phi-5

202.24 Hz

Phi-6

203.88 Hz

 

player

string I tuned to

Phi6

207.65 Hz

Phi5

208.69 Hz

Phi4

209.74 Hz

Phi3

210.80 Hz

Phi2

211.86 Hz

Phi1

212.92 Hz

Phi0

213.99 Hz

Phi-1

215.07 Hz

Phi-2

216.15 Hz

Phi-3

217.23 Hz

Phi-4

218.32 Hz

Phi-5

219.42 Hz

Phi-6

220.52 Hz


The lowest to highest pitches shown in the diagrams above can be inverted as well, so that the smaller powers of Phi get the lower pitch.

 

This table shows the approximate value in cents of the temperaments which can be used. This table is only for convenience and should not be used to calculate the tuning; this should be done with a calculator using 1 + phin.

 

1 + phi1

833 cents

1 + phi2

560 cents

1 + phi3

367 cents

1 + phi4

236 cents

1 + phi5

149 cents

1 + phi6

94 cents

1 + phi7

59 cents

1 + phi8

36 cents

1 + phi9

23 cents

1 + phi10

14 cents

1 + phi11

9 cents

1 + phi12

5 cents

 

Only instruments which can be tuned to a very high degree of accuracy (guitars or cellos, for example) should use the temperaments with the smallest intervals. Instruments should be tuned offstage using the same articulation used in the performance.

 

If stringed instruments are used, the players should all play on the same string. Exceptions to this rule are the first 3 temperaments, which can use different strings; here the players should choose the strings carefully in a way to create the greatest uniformity in sound. The players should either 1) all use open strings or 2) use no open strings. The temperaments 10 through 12 can be used effectively tuning the same open string slightly apart on the different instruments, as shown in the examples. This has the advantage of great intonational accuracy.

 

Dedication

 

This work is dedicated to English mathematician Ron Knott, who introduced me to these sequences.

 

Notes

 

1. The titles on programs should read as one of the following:

 

Base Phi Representations:

Base Phi Minimum Binary, Base Phi Maximum Binary

 

or Base Phi Representations:

Base Phi Maximum Binary, Base Phi Minimum Binary

 

or Base Phi Representations: Base Phi Minimum Binary

 

or Base Phi Representations: Base Phi Maximum Binary

 

2. More information on Fibonacci numbers, the golden ratio, and these sequences can be found on Ron Knott's page:

 

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phigits.html

 

3. To give one an idea of how long different versions of this work would be, here is a small table showing various possibilities. The values given are rounded to the nearest second:

 

 

number of integers used

 

 

 

 

 

 

 

 

 

bpm

76

123

199

322

521

843

1364

2207

3571

5778

9349

15127

24476

70

65 sec.

105

171

276

447

723

1169

1892

3061

4953

8013

12966

20979

90

51

82

133

215

347

562

909

1471

2381

3852

6233

10085

16317

110

41

67

109

176

284

460

744

1204

1948

3152

5099

8251

13351

130

35

57

92

149

240

389

630

1019

1648

2667

4315

6982

11297

150

30

49

80

129

208

337

546

883

1428

2311

3740

6051

9790

170

27

43

70

114

184

298

481

779

1260

2039

3300

5339

8639

190

24

39

63

102

165

266

431

697

1128

1825

2952

4777

7729

210

22

35

57

92

149

241

390

631

1020

1651

2671

4322

6993

230

20

32

52

84

136

220

356

576

932

1507

2439

3946

6385

250

18

30

48

77

125

202

327

530

857

1387

2244

3630

5874

270

17

27

44

72

116

187

303

490

794

1284

2078

3362

5439

290

16

25

41

67

108

174

282

457

739

1195

1934

3130

5064

310

15

24

39

62

101

163

264

427

691

1118

1809

2928

4737

330

14

22

36

59

95

153

248

401

649

1051

1700

2750

4450

350

13

21

34

55

89

145

234

378

612

991

1603

2593

4196

370

12

20

32

52

84

137

221

358

579

937

1516

2453

3969

390

12

19

31

50

80

130

210

340

549

889

1438

2327

3766

410

11

18

29

47

76

123

200

323

523

846

1368

2214

3582

430

11

17

28

45

73

118

190

308

498

806

1305

2111

3415

450

10

16

27

43

69

112

182

294

476

770

1247

2017

3263

470

10

16

25

41

67

108

174

282

456

738

1193

1931

3125

490

9

15

24

39

64

103

167

270

437

708

1145

1852

2997