Base Phi Representations:
Base Phi Minimum Binary, Base Phi Maximum Binary
Casey Mongoven
August 13, 2006
the sequences: There exists a binary nonstandard 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:
Phi^{5} 
8 + 5phi 
11.0901699… 
Phi^{4} 
5 + 3phi 
6.85410197… 
Phi^{3} 
3 + 2phi 
4.23606798… 
Phi^{2} 
2 + 1phi 
2.61803399… 
Phi^{1} 
1 + 1phi 
1.61803399… 
Phi^{0} 
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:
Phi^{n} = Fib_{n+1} + Fib_{n}*phi
Here are the
integers 1 through 11 in base Phi:
decimal 
powers of Phi 
base Phi 
1 
Phi^{0} 
1 
2 
Phi^{1}+Phi^{2} 
10.01 
3 
Phi^{2}+Phi^{2} 
100.01 
4 
Phi^{2}+Phi^{0}+Phi^{2} 
101.01 
5 
Phi^{3}+Phi^{1}+Phi^{4} 
1000.1001 
6 
Phi^{3}+Phi^{1}+Phi^{4} 
1010.0001 
7 
Phi^{4}+Phi^{4} 
10000.0001 
8 
Phi^{4}+Phi^{0}+Phi^{4} 
10001.0001 
9 
Phi^{4}+Phi^{1}+Phi^{2}+Phi^{4} 
10010.0101 
10 
Phi^{4}+Phi^{2}+Phi^{2}+Phi^{4} 
10100.0101 
11 
Phi^{4}+Phi^{2}+Phi^{0}+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
(Phi^{4} and Phi^{3}, 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 Phi^{n}^{ }= Phi^{n1} + Phi^{n2}), it is possible to use a replacingalgorithm 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:
Φ^{4}Φ^{3}Φ^{2}Φ^{1}Φ^{0}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 Φ^{0} and Φ^{1}.
For example:
Φ^{4}Φ^{3}Φ^{2}Φ^{1}Φ^{0}'.'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 rightmost players cannot be less than 75 centimeters. If x
represents the space between the players at the rightmost end, then the
distance for between the players is figured as such:
x Phi^{2}*x Phi*x x
Φ^{2}Φ^{1}Φ^{0}Φ^{1}Φ^{2}
In the 3^{rd} 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 handwritten 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 stageleft, then the traditional notation should be inverted on the
page).
The director can help in
that he periodically gives the beat number using a handsignal 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 socalled gray area of pitchdefiniteness, such as
tomtoms, 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 tomtom). 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 + phi^{n}^{}
^{ }
Here is an example for
violins in the 10^{th} temperament (1 + phi^{10}), and one for
cellos in the 11^{th} (right side). These examples use only open
strings:
player 
string IV tuned
to 
Phi^{6} 
185.00 Hz 
Phi^{5} 
186.50 Hz 
Phi^{4} 
188.02 Hz 
Phi^{3} 
189.55 Hz 
Phi^{2} 
191.09 Hz 
Phi^{1} 
192.64 Hz 
Phi^{0} 
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 
Phi^{6} 
207.65 Hz 
Phi^{5} 
208.69 Hz 
Phi^{4} 
209.74 Hz 
Phi^{3} 
210.80 Hz 
Phi^{2} 
211.86 Hz 
Phi^{1} 
212.92 Hz 
Phi^{0} 
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 + phi^{n}.^{}
^{ }
1 + phi^{1}

833 cents 
1 + phi^{2} 
560 cents 
1 + phi^{3} 
367 cents 
1 + phi^{4} 
236 cents 
1 + phi^{5} 
149 cents 
1 + phi^{6} 
94 cents 
1 + phi^{7} 
59 cents 
1 + phi^{8} 
36 cents 
1 + phi^{9} 
23 cents 
1 + phi^{10} 
14 cents 
1 + phi^{11} 
9 cents 
1 + phi^{12} 
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 