A Guide To My Notation


Brief History


When I started writing the work Beatty Sequence no. 1a (catalogue entry B1), I turned to 5 previous pieces as models for notation (Golden Strings I-V, A26-30). I had decided to start using graphic notation at the end of my time at the conservatory, although all the music I wrote at the conservatory used traditional notation. From April to December 2002 I wrote 21 works using a new notation, they were all hand-written.


In February 2003, while living in Berlin, I decided to start notating my music on the computer. In this process I changed several things about the appearance of my notation. One year later I re-notated works B1-B3 and B5-B22.


Over the last 2 years my notation has evolved continually. Its present state is a culmination of trial and error, input from my friend Ron Knott, and notational challenges arising from the nature of certain works. I have always considered the graphic representation of music to be an important and indispensable tool for understanding and finding a way into works that are unfamiliar.


Basic Features


A score contains two main sections: the graph and the key. The graph stands alone, as not to clutter its appearance. The key gives the parameters of the composition.


Time is represented on the horizontal axis in the graph, as in traditional notation. In my works using sequences, squares or rectangles are used to represent frequencies and other compositional parameters. The duration of one square is given near the top of the page. In this typical example (catalogue B32), the vertical placement of the square in the graph tells you the frequency of the note (given in Hertz), its pan value, the wave form and the loudness (dynamic); the actual values used are found in the key. Other things that can be found in the key section include the temperament used, the attack and release values, the length of the work, dedications and any other important information pertaining to the structure of the composition.


The above example is based on a sequence; some of my works are based solely on a mathematical ratio, as in the following example (cat. B36).


In this example, the two pitches used are represented by long rectangles. The length of the entire composition is given at the top. The key is integrated into the graph.

Arrows in my notation represent linear changes from one value to the other. In the case of waves, the wave literally morphs into the other wave.


The symbol Φ (big Phi) in my notation represents the value

(1 + √5)/2 = 1.618033989…. This is an irrational number.

φ = 1/Φ = Φ – 1 = .618033989….

φ (little phi) is used here to indicate the location of the climax of the piece at triple-forte

φ(19 sec.) = 11.74264… sec.

φ is also often used to indicate the temperament of the work, as in the first example.



In addition to their vertical placement on the graph, higher pitches are always represented by darker shades.


In this example (B25), glissandi are represented by a line graph. As in the above example, the key is integrated into the graph.


Any time the value of a parameter changes during a composition, that parameter is given in italics (in this example Hz, wave, and dynamic).



Colors are occasionally used to represent parameters. In the example on the left (B83), colors are used to represent pan values. All green squares have a pan value of 30°.


Notice that the pan value is not given in italics here, although it changes during the composition. This is an exception to the rule, as the pan parameter is used to represent two sequences. One sequence is played on the left speaker, and one on the right.


I have also used colors to

represent different wave forms. This can be found in the work Fibonacci Rabbit Sequence no. 4 (B92).


As a rule I only use colors when I feel that they are necessary for clear presentation.



Sequences are rendered, in most cases, in one to one correspondence. If the sequence begins {1, 1, 2, 2, 2, 3, 2, 3, 4, 3, 4, 3, 3, …}, for example, repeated notes are usually blended together.

About The Parameters


Before I discuss the parameters I will digress shortly to terminology. Above I used the terms frequency, pan value, wave form, and loudness. These are to be distinguished from the terms pitch, spatial location, timbre, and amplitude. Frequency is the number of cycles per second a wave makes, pitch is the psychoacoustic perception of this – the perceived highness or lowness. Amplitude is, in the case of electro-acoustic music, the strength of the signal. Loudness is the psychoacoustic perception of amplitude. Wave form describes simply the shapes of the waves, and timbre (tone color) is the psychoacoustic perception of wave form.


I write my music to be listened to with speakers at 30 degrees left and right. The distance to the speakers varies from speaker to speaker, but is typically 50 to 70 inches (130 to 180 cm). With larger speakers the distance can be increased. I experiment with the placement in the room before I mark the ground with a red X in the middle of the speakers. The back of the listeners head should be at the red dot. The pan values given are a reflection of the amplitude of the signal between the left and the right speaker. Spatial location is, in most pieces, simulated through stereo effect.


I use a scale of standard dynamic markings from ppppp to fffff, a total of 12 dynamic values. I use this for practical purposes; in reality I use practically infinite dynamic levels. The dynamic level given in a score is the closest standard dynamic marking to the dynamic level used. ppppp represents the absolute quietest sounds we can hear (0 decibels for example), fffff represents for me the loudest sounds I am willing to use in my music (somewhere around 100 dBA according to my own tests), I have not yet used this level. I used ffff  (about 92 dBA) in my work Fibonacci Rabbit Sequence no. 2 (cat. B72).


It is important that the stereo volume is set properly as not to distort dynamic curves. It has also proven dangerous for me to experiment with changing the volume levels in the course listening to several works. I blew out an electrostatic speaker in 2002 listening to Golden Ratio no. 1 (cat. B9). For the two reasons above, the volume setting on the stereo should never change when listening to different pieces. It needs to be set at the beginning, before listening. I have this level marked on my own equipment, all my pieces were written at the same amplitude level. Setting the level can be tricky for those without a sound level meter. For this reason I have provided a setting file on my DVDs. This setting file, which is made from the same files I use when composing to judge loudness, includes an extremely quiet tone ppppp and a very loud one at about ffff (my second highest dynamic level). The listener must find a good balance on their system. ffff should be extremely loud.


The four basic wave forms charaterize the timbre in my music. They are used purely, and sometimes as mixtures of each other, in which they are added together at different strengths, morphing into one another.


Wave cycles are measured on a 360° scale. 180° indicates that that the wave starts “in the middle” of the cycle.


I represent the ascending type sawtooth wave as –saw.


My temperaments are based on irrational numbers, which means working with infinite decimals. The frequencies used in my works are generally rounded off to 6 decimal places for practicality. In most cases I use higher resolution in the files I use to produce the digital audio. In some cases the exact frequency can be given (see Golden Ratio no. 11, B87 or Reciprocal Fibonacci Constant no. 1, B97).


Instrumental Compositions


I have written a number of works for acoustic instruments in this style, and the notation used is practically the same. Performance notes are included with these works. See Horizontal Para-Fibonacci Sequence no. 8 (cat. B43) for example.


Closing Words


I have sought to make my notation logical, and rather than explain how I notated this music in even greater detail, I would like to invite the listener to discover this for themselves by listening to and becoming familiar with the music. Perhaps the easiest and most advantageous way to become familiar with a notation is simply to listen to the music while watching the score.


September 22, 2004


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