My Music, for Non-Musicians

I have been away from the conservatory and intense musical environments for over two and a half years now. I have played my music for many people who do not consider themselves musicians; in fact, the majority of the people I have played my music for during this time have been what I would consider non-musicians. The reactions to my music were totally varied, probably much as they would be at a conservatory.

But the music I have written since my years at the conservatory is significantly different from that which I write now. And truthfully, I do not yet know how this music will be received, although all the people I have kept in contact with from New England Conservatory enjoy it.

None of the non-musicians who I have personally played my music for have had a negative reaction.1 Many of their reactions have surprised me, in that I truly felt like they understood my aesthetic. Others admitted it was very hard for them to understand, and wanted some further explanation.

This essay is written for non-musicians, although it will also be of interest to musicians. I have simplified the explanation as much as possible, without watering down the content significantly. The text still might not be completely easy to understand. The ideas behind my music cannot be made simple enough to make them a piece of cake to understand. I will try my best.

A Basic Explanation of My Music

In the vast majority of works that I have written in the last year and a half (works B1-B58), I have used mathematical sequences. These sequences are fractal; they have self-similar patterns, or put more simply, they contain “copies” of themselves within themselves. For example, the works Fibonacci Sequences nos. 1-5, Vertical Para-Fibonacci Sequences nos. 1-8, Beatty Sequences nos. 1-6 and others. Works are always named after the sequence they are based on. The works Golden Ratio nos. 1-8 are based on a mathematical proportion – the golden ratio. The Internet contains a large amount of information about the mathematical properties of these sequences and the golden ratio for those who are interested in learning more about them.2

The idea in my works is to create a musical model of these sequences in the purest, most musically interesting and sonically pleasing way possible. It is important to realize that music is not a mathematical sequence. This may sound like an obvious statement, but I have heard many misleading remarks from people saying things like “music is mathematics.” Currently, in the search engine Google, I see 2,500 hits under “music is math.” To me music is no more mathematical than anything else. Mathematics can be used as a tool to analyze many things – such as the laws of our solar system, the stock market, or the proportions of a church – but math is math and music is music. You could not, for example, listen to 2 + 2 = 4. When I compose, I use mathematical properties to create musical ones. There is no formula for composing this kind of music; I have to make all decisions myself, although I use a mathematical sequence or a ratio in choosing the pitches.

The tunings that I use in my works are my own invention; I do not use the notes of the traditional keyboard. My music has no octaves. Two notes an octave apart are related to each other in a 2:1 frequency ratio. On the keyboard, notes an octave apart have the same name – for example the lowest note on the piano is called A and so is the note 12 pitches higher. I use the notes “in between” the notes on the keyboard.

A Musical Example with Explanations

In my work Fibonacci Sequence no. 6, I used the first 8 members of the Fibonacci sequence. The first known mention of the Fibonacci sequence in the Western world was in the year 1202 by a mathematician named Leonardo Pisano “Fibonacci.” He posed an algebraic problem having to do with rabbits breeding in ideal conditions, the answer to which was the Fibonacci sequence.

In the Fibonacci sequence, the sum of two consecutive numbers equals the next; you add the last two numbers to get the next. The sequence starts:

0, 1, 1, 2, 3, 5, 8, 13, ...

It can be seen that 0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5 and so on. The sequence is a fractal and is infinite. It is so abundant with interesting properties that an entire mathematical journal, The Fibonacci Quarterly, is devoted to this sequence.3

Here is a slightly modifed score to Fibonacci Sequence no. 6,4 in which I used the first 8 numbers of the Fibonacci sequence. In red are simple explanations of the notation. The gray squares represent the notes. The reader should listen to this work while looking at the score, in order to hear the relation between the notation and the music. One thing that might come as a shock to some listeners is the extreme brevity of the work, which lasts 1.688 seconds.

WAV (.9 MB), MP3 (.03 MB)

1. This column shows the pitch of each note in the graph expressed in Hertz, or, cycles per second. 993 Hertz means that the sound wave cycles 993 times each second.

2. In most of my works, the position of the sound for each note is derived from the sequence used. I use 30 degree speaker placement. Imagine a round clock where the listener sits in the center facing 12 o’clock; the speakers would be at 11 o’clock and 1 o’clock. So -28 degrees means that it is almost fully panned to the left, 28 degrees indicates that the sound comes from the right. 0 degrees is, of course, in the middle.

3. The release of a note is the amount of time at the very end of the note where the sound “gets quieter.” Usually when a note is played, if you look at the sound on a more detailed level, the sound does not stay the same volume for its entire duration. In this work, during the last .081 seconds of each note the amplitude falls to zero. Here is a diagram of a sound wave showing the slope of the release:

One of the best questions I have been asked by someone about my music came from composer Troy Wayne. The question was simple: what do you accomplish in so little time? My answer was: I accomplish no more or no less than I need to. I plan for these works to be this short. I find such works packed full of excitement; they do not need to be any longer than they are. I would recommend the reader look next at the works Vertical Para-Fibonacci Sequence no. 6, Lucas Sequence no. 1, and Complementary Beatty Sequences no. 1.

I realized this work with a program called Csound, written by Barry Vercoe. Csound is a programming language for designing sound files. Thus far I have written two works for acoustic instruments in this style; Horizontal Para-Fibonacci Sequence no. 8 and Fibonacci Sequence no. 7. Both of these works have yet to be performed.

Visual and Natural Fractals

The Mandelbrot set is probably the most famous fractal that exists. It has also been called by some mathematicians the most complicated mathematical object yet discovered. Such fractals seem immediately accessible to people today, even to those totally uninterested in mathematics. Here is an image of the Mandelbrot set:

Nature is abundant with self-similar patterns, as Benoit Mandelbrot points out in his book The Fractal Geometry of Nature. Some patterns in the Mandelbrot set look strikingly similar to naturally existing features of the earth. The pictures of the Martian surface coming back from NASA and the ESA now confirm that the surface of Mars is, like that of the earth, abundant with self-similarity.

Sonic fractals are a much younger phenomenon than visual ones, and have not yet gained the same kind of public recognition. It is doubtful that they ever will have the same impact on popular culture that visual ones have had. The reasons for this are numerous. The primary reason is that it is a much more difficult task to consciously hear self-similarity than to see it. I believe that the majority of humans are primarily visual learners and perfect the sense of sight to a more sophisticated degree than hearing. People blind from birth do not have this problem.

Mood and Emotion

Many non-musicians seem very in tune with the emotional side of music. One of the most common questions I have been asked about my music was: can you hear the mood of the composer, as you can in Beethoven or Brahms? Can you hear whether the composer was sad, happy, angry or frustrated? I do not know how easy it is to hear the mood of any composer through his music, but I know my mental state has a lot to do with how I compose a particular piece on a particular day. Sometimes when I have had a really bad day I compose much more agressively; louder, faster and more dissonant music. When I am in a better mood the music I compose generally has a more stable sound. But hearing these differences in my music is clearly more difficult than hearing them in Brahms or Beethoven, as their style is based on a much more familiar tradition with many popular connotations.

Influences and Inspirations

More than anyone else, my friends and teachers have inspired me to compose how I do. The list would be long if I included every name, but I will list a few: Joseph Johnson, Martin Near, Stephen Sikorski, Bob Kasenchak, Daniel Mutlu, Troy Wayne, Kevin Mendoza, Marcin Bela and of course, my teacher Alan Fletcher. Their encouragement and support has helped me a great deal.

Other composers have influenced me much less in style than in dedication, conviction and passion: Johann Sebastian Bach, Arnold Schoenberg, Anton Webern, Bela Bartók, Ludwig van Beethoven, and John Cage to name a few; this list could truly go on and on. All these composers possessed a driving passion to do what they did. They all brought their styles to the highest degree of perfection.

In Closing

I wish all people that hear my music an enjoyable listening experience. I do not compose my music for specialists, I compose it for everyone. It has been my experience and the experience of many others that music in general becomes more enjoyable with time and familiarity. I urge listeners to open their minds and give it time.


1. I generally give a very brief explanation of the golden ratio and Fibonacci numbers before playing my music. I have probably personally played my music for about 20 to 30 non-musicians in this time. I have had a “negative reaction” from only one musician in the last two and a half years.

2. Links to information from the ODP: Fibonacci numbers, golden ratio.

3. The Fibonacci Association's Official Website.

4. Only the format of the score was modified; the content is the same. The score to this work can be downloaded in the catalogue. Link to catalogue entry for this work: B53.

February 29, 2004

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