Bifurcate Me Baby [selection] (1995)

The implementation of this piece was accomplished with a chaotic function (the logistic difference equation):

X = P*X*(1-X)

In this equation, X varies from 0.0 to 1.0 and P varies from 0.0 to 4.0.

With each iteration of the function, the previous output value for X is fed back to compute the next value. As P increases, the behavior of the function increases in complexity. When P is between 0.0 and 1.0, the functions "dies" (all output gravitates toward zero). Between 1.0 and 3.0, the output values for X converge to a single curve that ascends in value with the value of P. At 3.0 the function bifurcates resulting a in limit cycle of two X values that oscillate. As P continues to increase, the interval between the two values increases until, at 3.5, a second bifurcation takes place to produce a "four cycle." This behavior continues as P increases to about 3.6 where the cycles become so long and complex that they are difficult to follow. This is the first chaotic region. In spite of the complexity, every value of P (even to differences in the tenth decimal place) has a corresponding pattern that is clearly recognized as an analog to motivic variation.

In this piece I used a function that produced and arch form by interpolating P from 2.0 to 4.0 and back again over the duration of the piece. It is two and a half minutes long. I used the golden section to determine the proportion between the rising and falling gestures. The rise occupied the longer period.

The output of the function was used to generate the pitches, rhythms, dynamics and articulation in a piece for Yamaha Disklavier.

Colony (1994) [selection] (1995)

In this work I use a technique called "granular" synthesis. A large number of small sounds are assembled in sound masses in a manner that parallels the work of pointillist painters who created images from small colored dots. The frequencies, amplitudes, timbres, and distribution of the grains cause larger sound structures to emerge.

This work continues an interest in granular synthesis that dates from 1974 and was revived in 1993 with the composition of "The Voyage of the Golah Iota." In 1974, I was inspired by the writings of Iannis Xenakis and by a conversion I had in Bowling Green, Ohio with one of his students, Bruce Rogers. Bruce played me fragments of works in progress that exhibited sounds unlike anything I had heard before. During the summer of 1974, I composed a series of studies I called "Particulations" using a FORTRAN program of my own design. One of these pieces was included in a concert at the first International Conference on Computer Music at Michigan State University.
Five Summer Scenes (1985-86)

This set of short pieces was composed at the National Music Camp in Interlochen, Michigan and at Melbourne University in Australia. In changing hemispheres I enjoyed a series of summers and springs uninterrupted by snow and ice. The five movements are intended to celebrate summer moods of relaxation, playfulness, and reflection in the sun.

1. The voice in the bell
   
2. Closing time at the glass menagerie
   
3. Road unsealed next 17 km
   
4. A concert on the sand
   
5. An unexpected call
   

The works were composed using "algorithms" of my own invention. I designed a Pascal program to control the dynamic aspects of the middle and large dimensions. Gesture, movement toward climax, contrast, variation, and many other composition techniques were defined in exhaustive detail and cast into computer code to carry out artistic methods and procedures. The small dimensions - intervals of pitch and duration - were organized using serial techniques. Like all of my algorithmic composition programs, these underwent a sustained period of evolution and refinement that occupied more than a year from concept to first acceptable results.

The system of hardware used to realize these pieces included an Apple Lisa 2/10 microcomputer, an IDAL RS232-to-MIDI interface, and digital synthesizers manufactured in Japan by Yamaha. The synthesizers (DX7 and TX8/16) constitute an orchestra of 144 separate parts organized in nine ensembles of 16 parts each. With my instructions, the computer "composed" sequences of notes and "conductor" functions that were translated into MIDI codes and transmitted to the interface. The interface acted as time keeper by holding each musical event until its action time arrived then passing it on to the synthesizers.

The instruments were created by means of timbral interpolation. The personality of each instrument is determined by 155 numeric parameters. For each movement I designed a small palette of 2-5 tone colors. The primary colors were expanded into a characteristic spectrum of 32 timbres for each movement. The secondary instruments were made by interpolation between corresponding parameters in pairs of primary instruments. Adjacent instruments are similar and a continuous scale of tone colors results. In "The voice in the bell," for example, instrument 1 is a synthetic human voice and instrument 32 is a tubular bell.

Fractal Mountains (1988-89) [selection]

With his discovery of fractals, Benoit Mandelbrot introduced a new class of mathematical models and a new branch of mathematical science.   He may have stimulated a new branch of music as well.   A growing number of composers have shown interest in the musical application of fractals.   "Fractal Mountains" represents my own first attempt to integrate fractal techniques into interactive performance and composition.

A significant feature of fractals is their self-similarity.   Consider the coastline of an island.   On a map we see seemingly random curves that have been shaped by the elements of nature.   If we look closer we see that, with magnification, smaller curves are similar in shape to the larger curves.   Continuing the process of magnification to ever smaller snapshots reveals a unity that obtains to the very grains of sand on the island's beaches.   The ramifications of such a process for the organization of musical form and structure seem almost overpowering.

Most of the practical literature on fractals is in the field of computer graphics with focus on the generation of realistic images for still pictures and animation. One of the simplest fractal models is the two dimensional outline of mountain ranges.   Initially, we draw lines to represent the major peaks and valleys.   These lines are subdivided by a recursive process to produce the next level of detail.   When we continue subdivision through generations of ever shortening lines, an image emerges that reminds us of the patterns found in natural landscapes.   Figure 1 illustrates the several layers of subdivision used in "Fractal Mountains" to arrive at pleasing melodic contours.

Musical sequences can be generated by mapping the meeting points of each pair of lines onto time and pitch.   Each vertex represents the attack point of a note.   A second fractal function is used to translate dynamics into MIDI key velocity.   The durations of notes are set by another fractal function that shapes texture by controlling the number of notes that are sounding at once.   To return to our island analogy, we are determining whether the beach is covered with rocks, pebbles, or fine sand.

Figure 1. Subdivision of lines to simulate mountain horizon.

The piece is performed on a Yamaha TX816 and a pair of EMU Proteus/1 synthesizers controlled by a Macintosh computer and a MIDI wind controller.   The micro tonal pitch system in "Fractal Mountains" is achieved by dividing the octave into 96 equally tempered steps.   MIDI channels 1-8 in the TX816 and Proteii are tuned in intervals of 12.5 cents by freezing the pitch bend wheel for each module at the beginning of the piece.

The numbers on the vertical axis of the graph are scaled to a range of six octaves and then rounded to the nearest eighth of a semitone.   The whole number part determines what MIDI key will be pressed and the fractional part determines which of the eight channels will receive the note.  

The harmonic shape in "Fractal Mountains" comes from the tendency of the fractal model to be attracted to particular numbers and thus to particular pitches and MIDI channels.   The pull of these "strange attractors" is one of the most interesting feature of fractals.   In music they produce a sense of tension and release that is appealing physically as well as esthetically.   Phrases begin with most notes in a single channel.   As the phrase progresses, the notes fan out among the eight channels and the richness of the micro tonal palette is revealed.   Near the end of each phrase the notes move toward a different channel and cadence in relative consonance.   The center of gravity is constantly changing.

Like the tuning system, the timbres are designed with a rich inner life.   Voices vibrate and beat against other voices.   Each tone color consists of a sustained sound that rises slowly from the beginnings of notes.   At higher key velocities the onset of notes are punctuated by bell sounds.   The timbres were constructed by interpolation between archetypes that were designed empirically.   The resulting orchestra creates a "scale" of timbres with subtle variation from one end of a limited spectrum to the other.

The major features of this tonal landscape are controlled by notes played on a MIDI wind controller and by the time delay between those notes.   The single notes from the soloist are passed through the Macintosh and translated into a multi-voice texture.   Imagine that we are touching peaks and valleys on a blank canvas and that the mountains appear automatically.

More specifically, each note from the soloist is recorded by arrival time, pitch, and key velocity.   Adjacent notes provide the endpoints of lines that represent the slopes of the mountain.   The fractal algorithm constructs an accompaniment gesture that traces the space between a pair of points.   A three-dimensional interpolation takes place in time, pitch, and loudness.

Some special rules of interaction between the soloist and accompaniment algorithm were found useful.   Short time periods (< 400 milliseconds) between notes in the solo part overburdened the accompaniment algorithm and synthesizers and produced objectionably thick textures.   Long time intervals (> 20 seconds) caused correspondingly long accompaniment gestures during which the soloist lost control over the evolution of the piece.   These extreme time periods are ignored by the fractal algorithm.

In one version of "Fractal Mountains" I superimposed two ranges of mountains as illustrated in Figure 2.   The ranges become lines in a monumental contrapuntal structure.

Figure 2.   Mountains in counterpoint

The computer program for   "Fractal Mountains" was written in C.   These programs communicate with MOXC, a system for connecting the actions of the soloist to reactions by the accompanying synthesizers. MOXC is part of Roger Dannenberg's "MIDI Toolkit," public domain software available from the Center for Art and Technology at Carnegie Mellon University.

MOXC consists of a parser, an interpreter, and a scheduler.   The parser receives signals from the MIDI Horn and informs the interpreter that musical events have occurred.   In the interpreter, the composer writes programs that decide what these events mean and what reaction to trigger in the accompaniment.   The reactions may be immediate or delayed for a period by passing them to the scheduler.   The scheduler keeps track of pending events and executes each event when its delay period expires.

Goss (1993) [selection]

The composition method for the solo violin part in "Goss" is similar to that of "Summer Song." A different function was used, the Gosper Curve shown in Figure 9. In a private communication, Douglass McKenna points out that he is the author of this function. I am happy to correctly identify it as the McKenna E Curve. I tracked down my original attribution to a piece of software for exploring L-systems by Paul Bourke from New Zealand. Bourke identifies this function as the "Quadratic Gosper Curve.

Figure 1. The McKenna E Curve

To reduce symmetry, Figure 9 was compressed toward the upper right corner as shown in Figure 2.

Figure 2. Compressed toward upper right corner

Next the image was twisted, warped and stretched by changing the turning angle to 107 degrees to give the image in Figure 3.

Figure 3. Twisted, warped and stretched

Finally, Figure 3 was unraveled into a continuous line (Figure 4) to form the pitch and time contour for the solo part.

Figure 4. Raveled into pitch time contour

The atonal accompaniment material in "Goss" is generated with an algorithm borrowed from my earlier pieces, "Fractal Mountains" and "Mountain Song." The violinist plays the solo part on a Zeta violin that is outfitted with a MIDI translator. A Macintosh computer "listens" to the notes from the solo part and generates the accompaniment automatically. The close harmonization of the solo part is also automatic using notes from the scale. The soloist exercises control over the harmony and accompaniment with two foot pedals and with dynamic balance that is tied to bow pressure.

The curve in Figure 4 was stretched to a duration of five minutes. The vertical axis was mapped onto the scale shown in Figure 5.

Figure 5. Scale for Goss
This scale is a projection of the interval series 2 2 3 over two and a half octaves. The scale is different in each octave although there is a hexatonic quality that was inspired by listening to
traditional Chinese flute music in Taiwan during the summer of 1991.

When the pitch mapping was complete, I created a MIDI file and loaded it into Finale 2.6. I used floating quantization to limit rhythmic subdivision to eighth and sixteenth notes and eighth note triplets. This limit on rhythmic complexity along with the conservative pitch range reflects my intent that this piece be accessible to young players.
Goss was commissioned and performed by Mark Litwin.Jabber (2006-2007) [selection]

Jabber is a rendering of Lewis Carroll’s classic poem, “Jabberwocky” that I committed to memory as a child. The sole audio source is my voice speaking the poem.  The text is completely intelligible in parts of the piece.  It is also subjected to granulation and shifting in time and pitch. I limited myself to the resources of a laptop computer that included a digital camera as well as the usual built-in microphone.

This work exists in several forms.  The present version is audio only in stereo to meet the constraints of the CD medium. A concert version is presented in surround sound with interactive processing of sound and image.  The sole image is my disembodied face captured with the computer camera and transformed with digital video processes that are akin to what is done to the audio.  Lighting is controlled using colors reflected on my face from the computer screen.  The third version is a standalone computer application that captures and analyzes a performer’s reading and then transforms the sound algorithmically in real time.  In this version, the choice of text is left to the performer.Phrase Structure Seven (1981) [selection]

As the title suggests, this is one of a series of pieces.  It was composed during my second of three residencies at the Computer Music Project at the University of Melbourne in Australia.  The original form of these pieces was quadraphonic.

The main goal of these works was to make synthetic instruments convey the dynamism and musicality of human performers of traditional instruments.  The piece consists of phrases created with algorithms that I call “event types.”  Each event type is characterized by a specific texture, melodic contour, rhythmic articulation, dynamic shape and pitch structure.  Each phrase evolves and eventually joins with and gives way to the next. The joints can sharp and articulated with silence or smooth and covered.  Often, a new phrase will be signaled by the introduction of new instruments chosen from a large synthetic orchestra that I created for the series.

Pitch patterns are controlled by Markov chains that govern sequences of intervals within each voice and within the ensemble as a whole.  This results in a clear and consistent style of harmonic and polyphonic motion that is influenced by my long standing interest in the pre-serial works of Anton Webern.Refractions (1991) [selection]

This piece explores the concept of "hyperinstruments."   A hyperinstrument possesses a wide range of pitch, timbre, and expressive qualities.   In this medium, the definition of a musical instrument expands.   Musicians interact with a computer to produce sonic effects of orchestral proportions and to control the shape of the composition.   Each member of the ensemble functions as a conductor as well as a performer.   Unlike performances with prerecorded tape, the "accompanist" is "aware" of interpretive nuances and, at the signal of the human musician, reacts and contributes to the discourse.   The Prism Saxophone Quartet commissioned the first version of this work in 1989 and their name suggested the title as well as much of the composition process.    As a prism transforms light, the computer and synthesizers illuminate and transform the music played by the performers.  

Although the piece was written for a quartet of Yamaha WX7 MIDI wind controllers, the revised version of 1991 can be performed by any number of musicians up to sixteen.   This version is for solo MIDI Horn.   This version was released shortly on Opus One Records.

Please note that "Refractions" is a structured improvisation.   There is no score for the solo version.   The structure of the piece is embodied in a computer program that responds to the music played by the soloist.Shovel It (1995) [selection]

This piece came out of a series of conversations I had with several colleagues about the efficacy of using a computer to compose.  One colleague contended that any composer who uses computers and synthesizers is a slave to those media and is inevitably under the control of the machines.  Further he stated that computer composers must accept what the machine gave them.

As a composer who has used computers for more than 30 years, I could not disagree more vigorously.  In fact, electronic media place fewer restrictions on a composer than writing for full orchestra.  There is a body of technique to be learned and skills to develop.  Once that is done, the mature composer is completely in control.

“Shovel It” was written in reaction to pieces composed by several of my colleagues.  Ironically, their lack of experience with the medium forced them into exactly the limitations that they ascribed to all composers who use computers.

Three Studies in Frequency Modulation (1974)

1. Inharmonics
   
2. Percussion Music
   
3. Polyphon
   

This set of short pieces was among the first I composed after taking up my position in the TIMARA Department in Oberlin.  They serve as an exploration of a then new and revolutionary synthesis technique based on frequency modulation (FM).  The theory of FM was known for many years and formed the basis of transmission the super-audio signals of FM radio. John Chowning of Stanford University discovered the potential of FM for sound synthesis in the audio domain. In particular, FM allowed the synthesis of timbres that possessed the dynamic inner life of acoustic instruments.  In addition, FM greatly reduced the computing time for complex synthetic instruments.

In this work, I explore three subsets of FM.  In Polyphon, I take the simplest case where a single sine oscillator modulates another.  The frequencies of these oscillators are maintained in whole number ratios so that all timbres are harmonic.  In Inharmonics, the ratios are non-integral resulting in timbres that are clangorous.   In both of these movements, I combine envelopes that control timbre complexity and articulation.  Different combinations of envelopes are perceived as different instruments.  In the original four-channel version, the instruments were isolated to a single position on the surrounding sonic circle.  Combinations of envelopes are kept long enough to give the impression that “players” have entered the space to play phrases that interact thematically with the other members of the ensemble.  In Percussion Music, I replace the sine modulator with a white noise generator.  This results in an ensemble where timbres range from thunderous rumbles to bells that are dainty and sweet.

To the Edge (1993) [selection]

This piece was composed with a mathematical model from the new field of "chaos."   Chaos theory [ James Gleick.   Chaos: Making a New Science , Viking Press, New York, 1987] was developed to describe continuous phenomena that appear to be random but actually contain a substantial amount of complex order.  

"To the Edge" is based on a simple equation:   x = p*x*(1-x) .   However, the behavior of this equation is decidedly not simple.   In this equation, x ranges from 0.0 to 1.0 and p ranges from 0.0 to 4.0.   After each computation of the equation the resulting x is "fed back" to be used in the next computation.   The number series' that emerge vary in complexity according to the value given to p .   In general, complexity of increases with p but there are "islands" of simplicity.   For me, the most interesting range for p is 3.5 to 4.0 shown in the graph below:

Figure 1

If we take a value for p near the left margin we encounter four lines.   Mapped into music these values of x represent a set of four pitches that will be repeated without variation as long as p remains fixed.   As p increases, we move to the right.   First we encounter a split where the four notes become eight.   Further on, the graph becomes dense and the musical patterns increase in complexity.   The vertical white bands represent areas of relative simplicity (three notes, six notes).   The dense regions appear random but they contain many small identifiable patterns that become motives when they are expressed musically.   The form of this piece is created by moving in a single gesture from left to right, proceeding "to the edge."

In mapping the graph in Figure 1 onto music, each point could become a note.   Obviously, the density of points is too great for a solo marimba piece so the graph was replotted with fewer points as shown in Figure 2.

Figure 2.

Even with this thinning of the points the graph remained unsuitable for transformation into music.   If all of the points are presented as they are computed, a uniform pulse would occur and the result would be without rhythmic character.   Further thinning is necessary.

Another function was created to be supperimposed on the graph of Figure 2.   This function (Figure 3) was made with a small number of points computed from the x = p*x*(1-x) equation.   These points were used to create the boundries of a mask that further thinned the points shown in Figure 2.   The points in the shaded area were discarded and the points that fell within the boundries of the mask remained for musical rendering.

Figure 3.

The chaos function was implemented in MAX, an object-oriented programming system with an interactive graphical interface. [ Puckette, M. Zicarelli, D., MAX: An Interactive Graphic Programming Environment, Opcode Systems,   Menlo Park, CA (1992).]  Notes generated with a MAX patch were recorded in to a MIDI file and transcribed into standard notation using Finale.

“To the Edge” was commissioned and performed by Deborah Moore.

Voyage of the Golah Iota (1993) [selection]

When I work on a piece I often construct a metaphor that will guide me through its composition.   I use an electronic dictionary, thesaurus, and encyclopedia to sprout the seed of an initial idea.   In this case, I began with the words "grain" and "particle."   The thesaurus led me to "iota" and the dictionary told me it was a noun meaning "a very small amount, a bit."   The dictionary also reminded me that "iota" is "the 9th letter of the Greek alphabet" and that it is of Phoenician origin.   A search of the encyclopedia under "Phoenicia" brought me to a picture of a galley ( golah   in Phoenician).

Further reading informed me that there is speculation that the Phoenicians used one of these ships to cross the Atlantic and visit the mouth of the Amazon many centuries before the birth of Christ.   If the speculation is correct, the Phoenicians would have predated Columbus' voyage to the New World by more than a thousand years. 

My metaphor was now in hand.   The shape of my piece would mirror a voyage from the Mediterranean to South America on a mythical ship named the "Golah Iota."   This tiny galley would experience, through my sounds, a journey from the calm waters off North Africa through heavy seas and winds to the tranquil shores of what is now Brazil.

The implementation of this model was accomplished with a chaotic function (the logistic difference equation):

X = P*X*(1-X)

In this equation, X varies from 0.0 to 1.0 and P varies from 0.0 to 4.0.   Here is the graph of the function:

With each iteration of the function, the previous output value for X is fed back to compute the next value.   As P increases, the behavior of the function increases in complexity.   When P is between 0.0 and 1.0, the functions "dies" (all output gravitates toward zero). Between 1.0 and 3.0, the output values for X converge to a single curve that ascends in value with the value of P.   At 3.0 the function bifurcates resulting a in limit cycle of two X values that oscillate.   As P continues to increase, the interval between the two values increases until, at 3.5, a second bifurcation takes place to produce a "four cycle." This behavior continues as P increases to about 3.6 where the cycles become so long and complex that they are difficult to follow.   This is the chaotic region.   In spite of the complexity, every value of P (even to differences in the tenth decimal place) has a characteristic structure that is clearly recognized as an analog to motivic variation.

In "The Voyage of the Golah Iota," I used a function that produced and arch form by interpolating P from 1.0 and 4.0 and back again over a period of time  I used the golden section to determine the proportion between the rising and falling gestures.   The rise occupied the longer period. 

In "Golah," X values (the grains) were produced with a density of 10 per second when P was at 1.0 and 1000 per second when P reached 4.0.   The values of X where scaled and mapped onto a seven octave, micro tonal pitch range (approximately 192 tones per octave).   Sound was synthesized in 16 MIDI channels with two Yamaha TX816 synthesizers overlaid with two Proteus/1 modules.   The texture was then "smeared" with an Alesis MIDIVERB II.   The controlling program was implemented with MAX.