Computer Variations is my first acknowledged computer piece, which I wrote during the same year that I began teaching at Queens College. The piece is a traditional set of variations, beginning with a theme that is transformed, and sometimes literally repeated, in each variation. Following the theme, there are seven variations.
At the time that I wrote it, the computer synthesis techniques available were quite limited, and the processes that I used in each variation are remarkably simple, although their implementation is very accurate and unsparing. All of the tones are generated by simple oscillators, the only timbre variation created by mixing waveforms with different harmonic partials. The theme is a three-voice tune lasting only 34 seconds. The ensuing variations make use of different envelope shapes, timbre changes, reverberation, and amplitude and frequency modulation to produce sharply defined sounds. All sounds are located in various places between the two speakers, sometimes traveling from one to the other. Reverberation is sometimes used to create the sensation of traveling into the distance, but I don’t think this process worked very effectively; nevertheless, the reverberated sounds are clearly differentiated from the others. The fifth variation consists of four-note chords that fade in and make asynchronous glissandos to the note in the next chord. The sixth uses four instruments that each have very different envelopes, ranging from half a sine wave to shapes that are mostly decay to mostly rise, with all notes also having other distinguishing qualities, such as amplitude and frequency modulation, and very exacting rhythms. The last variation is similar to the fifth without the glissandos: each note simply fades in and sustains for a different duration, so that the effect is a changing mosaic of chords.
In resynthesizing this work from the old computer outputs I found in my basement in the summer of 2004, I can recall the day-to-day problems that plagued my work when this was done. Three different computer synthesis languages were used, ranging from Music 4B, a similar program I wrote for the IBM 7040, and Music7, which I wrote for the XDS Sigma-7 computer at Queens College. While all these procedures could be translated into csound, I had a hard time remembering what all the different statements meant. The only storage medium that large data files could be saved on were magnetic tapes, and I had to travel to both Princeton University and Bell Telephone Laboratories to convert these tapes to sound. I then had to splice the magnetic tape segments of no more than two minutes each into the final result, which is 9 and a half minutes long. What a difference 37 years makes!
Macro Structure was written in 1971 but never adequately realized, since I had envisioned it for a four-channel playback system. At that time, it was synthesized and performed in stereo.
The basic idea of a "macro structure" is that a single line in a computer program specifies an entire series of operations, usually with different data that are expanded in the operations. In this composition, there are three basic ways in which this idea is realized: (1) Each "note" specifies a cluster of pitches transposed to begin from the indicated pitch. Different sections change from trichords to pentachords and to hexachords. (2) Each cluster has an envelope which allows each of the components and the totality to have prominence over a portion of the note. (3) The developing process continues through the entire piece, in a palindromic structure. The pitch of each component begins steadily, then starts vibrato. As the piece continues, both the amount and the speed increase, then decrease after reaching the climax in the middle.
Newer computers have allowed for synthesis in four discrete channels, and the work has been resynthesized. My original conception has now be realized, although it can only be heard in a concert with four channel playback facilities. In order to fully appreciate it, you will also need to sit as near to the middle of the playback venue as possible.
Freeze uses sounds created by filtering a pulse-like wave with variable filters that slowly sweep back and forth between two harmonics, constantly producing a change of timbre. Tones are often modified by other processes: amplitude modulation, travelling back and forth between the two speakers, and reverberation. The speed of variation of each property slowly changes over the course of the tones. The piece is in four sections, building to a climax at the end of the third section. In the first three sections, the speed and dynamics of individual tones increase by a factor of one-third over the course of the duration, which exactly parallels the amount of change over the entire section. In the fourth section, these properties decrease, so that the overall structure is a large crescendo or increase followed by a smaller diminuendo or decrease. The speeds of variation are different within each octave, with higher octaves faster than lower ones. There is a constant ratio of 1 to 5 between the speeds in each octave. This produces a distinct character for the tones within each octave. In the middle two sections, which are so similar that they appear more to be one large section, sequences are introduced, and the articulative properties are applied not to the individual tones within the sequence but to the whole, so that timbres change, for example, at a different speed from the attacking of new tones in the sequence, and both speeds change. The title "Freeze" is intended to reflect the redundancy of materials on which the piece is based, as if these are "suspended" or frozen while what changes is our perspective of them.
Timbre Study No. 3 is the third movement of Three Studies in Timbre, composed between 1970 and 1973.
Timbre is the overtone structure of a sound. Each of my studies in timbre is based on some different conception of musical timbre, particularly conceptions that can be explored more successfully in computer music synthesis than with other methods of sound generation.
Timbre Study No. 3 is based entirely on the use of harmonic partials that fade in and out in different ways over the course of each tone. Only the first twelve partials are used. One group of instruments continually attacks tones at a speed which is deliberately out of synchronization with the tempo. Another group gives tones a strong attack followed by an exponential decay, while a third group produces a crescendo followed by a diminuendo. The last and most important group plays long tones in which individual overtones cyclically fade in and out, producing a melodic interaction with the other music. Variations in timbre are coordinated to location changes and amplitude variations.
Timbre Study No. 3 is recorded on Opus One No. 47.
Canons is a four-movement piece written in 1974, shortly after I encountered frequency modulation synthesis. In this manner, simple timbral variations can be produced easily, and they are used extensively in the piece.
FM synthesis had been invented a few years earlier by John Chowning at Stanford University. Chowning’s outstanding contributions to computer music are among the most important in history. Long before Yamaha had adapted it to their first venture in electronic music synthesizers and even before his article on the subject was published in Computer Music Journal , an earlier version of his article was circulating among members of the computer music community. This was where I learned it, and this work was the piece in which I carried out all my initial experiments. Because the work was so long, I was unable to record or present the full work, but the fourth movement was recorded on Opus One No. 47.
In Canons 4, there are two independent sub-parts within each part. In addition, there are two modes of articulation for tones in each sub-part, providing the basic contrast of sounds heard in the piece: short tones with percussive attacks which fade away quickly, and long tones which increase and decrease both in amplitude and in beating. The overall harmonic effect is one of continuous change through complex chords in which each tone fades in and out at a different time and at a different speed. Rhythmically, the piece starts at a moderate tempo and progresses to a constantly slower unfoldment of materials, increasing the density while maintaining the same overall rate of change.
Improvisation on the Overtone Series is based on the idea of employing the overtones of fundamental frequencies in a manner analogous to the use of independent pitches. When beginning the work, I made two assumptions pertaining to the entire piece: (1) Overtones should be audible as separate tones, yet still contribute to the "color" of the fundamental frequency, which would last over a longer duration. (2) The order in which the overtones occur would be determined by the "harmony" of the passage in which the fundamental occurs. Only harmonic partials are used.
Within this framework, I designed two kinds of computer "instruments". The first "brings out" the partials one at a time by means of an amplitude control function. The shape of this function is a rise over the first 3/16ths of a cycle, a sustain for the same duration and a decay for the rest of the cycle. The "cycle" is the duration over which all partials enter and leave the tone in this fashion. Each tone in the piece usually goes through several such cycles over the course of its complete duration.
The second instrument attacks each partial (i.e. a sine tone) separately and sustains it for 3/8ths of the duration of a cycle; after this duration, the tone dies away completely, so that there is never the impression of a "floating background" as with the first instrument. In this case the impression of the fundamental does not emerge until a few tones have sounded.
Each of these instruments normally employs 16 partials, stretching from the fundamental up over a range of five octaves. In the last section of the piece, there is a passage where only partials 8 through 16 are used, on an instrument otherwise identical to the first type described above. The fundamental emerges clearly, but the timbre is more nasal in character.
All "harmonies" (simultaneous sounds) are derived from 3-note and 4-note chords. Sometimes more complex chords are formed by combining these, but in that case the overtone pattern is a combination of the two chords forming it. The overtone patterns I selected were chosen by placing overtones that create the "harmony" in question at the beginning of the cycle, so that they are heard as a separate "chord" but still blend in with the fundamental as part of the timbre of the sound. Following these tones, which always occur in descending order, the other overtones occur in a manner that tends to be dispersed over the entire five-octave range of the first 16 partials.
The piece is in three sections, which might be heard as independent "movements" of the entire work, although I intend for all sections to be heard together. The piece was synthesized in 1977 using the XDS Sigma-7 and IBM 370/168 computers at Queens College and the City University of New York.
Improvisation on the Overtone Series is recorded on Opus One No. 53.
The basic idea for Luminescence is revealed by the opening sound, an F above middle C, which is modified by a filter that moves from a point above the second harmonic to about the sixth harmonic and back with a speed that starts slowly (at zero), increases up to a moderate rate of speed, and then decreases to the original point. This creates a "shimmering" effect that suggested the title to me.
Each tone in the composition undergoes similar spectral changes, produced by moving a filter from one point in the spectrum to another, and back, in a varying periodic motion. The resulting effects sound different depending on where in the overtone spectrum the change occurs. There is always an interaction between individual overtones, which can be heard as separate tones, and both other overtones and the fundamental frequencies of other pitches in the same musical context. The rates at which the spectral changes occur are always in the subsonic range, and they are several octaves (nine or ten) below the fundamental frequencies of the notes on which they occur. The frequency continuum is thus structured on multiple levels: subsonically, in the rates of change of filters; in the mid-range, in the fundamental frequencies that constitute the composition's pitch structure; and in the high range, in the individual overtones that are emphasized in a periodic manner.
The piece is in six sections, each one evolving into the next. The opening states four all-interval tetrachords in four octaves that give the basic chords of the piece, and these are embedded in much of the remaining sections. The chords in this passage are a transposition of the voices.
The second section begins with a series of rapidly-modulated pentachords that are each stated in five descending octaves. These are followed by a passage that states the same chords in a contrapuntal manner, and it leads to a climax in section 3, a fast passage full of repeated notes, downward glissandos and crescendo-diminuendos.
The following section, 4, is a slow, contemplative passage that returns to the filtering, and the chords, of section 1. All the notes are stated in a single octave, but the filtering occupies four overlapping areas in the three octaves above the fundamentals. There are slow changes, but the speeds of the changes are different within each octave. In the second half of section 4, the chords and voices are interchanged. Section 5 is a slow climax of this material. Frequencies that had delimited the boundaries of the filters in the previous section are now introduced as pitches, and the notes are spread over a five-octave range. The filtering continues as before, only this time extending above the range of the highest notes on the piano. This leads to section 6, which continues the same filtering patterns but returns to the original tetrachords that have been used throughout. The notes in the lowest octave, brought into focus by the last note, is filtered from the fundamental up to five octaves above and back.
My Piece for Five-Octave Keyboard was written in 1985 and 1986, at a time when I was beginning to be enthralled by the possibilities of live performance on the new MIDI keyboards, particularly the Yamaha DX-7. These instruments were the first that offered the opportunity to play music with the same range of expression as acoustic instruments. One of their main limitations was that the keyboards were limited to five octaves or 60 keys, unlike the 88 keys on the normal grand piano. Therefore, this became the main limitation of the piece. The timbre that I wanted to use didn't really matter that much to me, as long as the sounds had decaying characteristics similar to a piano. In fact, I rather liked the possibility that the tones could be sustained a little longer than on the piano, but only marginally. Now, however, I feel that the piece is best played either on a normal grand piano, or on an instrument with weighted keys and the same sort of feel to the pianist as a piano. The DX-7 is usually placed on a flimsy stand in performance, and it or the pedals can easily become knocked around in a performance as physical and virtuosic as this piece demands, so I am delighted when I can hear the work on a grand piano.
The piece also exemplifies my concern with generating the same or similar musical structures from different elements, and having sections where each of those elements appears separately and in combination with the others. It also reflects my use of the property of cycle-of-fifths equivalence and its inversion; indeed, there are entire sections of the piece that are related by this property. The work is in the form of a palindrome, where the middle section could actually be heard as a slower movement. It opens with a grand statement on pentachords, which span the entire range of the keyboard. This is followed by a long passage marked "as fast as possible" which uses tetrachords in combinations of only three octaves at a time, but which sometimes has near-impossible stretches for the performer. This gradually dissipates into a long thinning passage that contracts to one octave and comes to rest on D and F# (also the last pitches of the piece). Then begins a slower, jazzy passage based on trichords that use only three octaves at a time. The climax of this occurs in the middle, where the trichords are continued, but the pentachords of the opening section return in all octaves. The midpoint begins the second half of the palindrome, although it is heard simply as a continuation of the same texture. It continues to the slower, trichordal passage, which leads directly into the thin single-octave passage that ended the second section, only now this expands into the return of the fast and complicated section six, the cycle-of-fifths equivalent of section two. The piece ends with a return of the pentachords similar to the beginning of the piece in a full, five-octave statement.
For those interested in such technical matters, cycle-of-fifths equivalence and its inversion, also known as the multiplicative operations M7 and M5, is similar to inversion (M11), in that these are the only operations in the twelve-tone scale that allow chords with a different pitch content to be produced (and only on some chords). Whereas in inversions, minor seconds become major sevenths, etc., in cycle-of-fifths equivalence minor seconds exchange with perfect fifths. The materials that the piece is based on include trichords, tetrachords and pentachords, and the particular combinations all generate the collections 0134 and 0358, which are themselves related by cycle-of-fifths equivalence.
Piece for DX-7 II Synthesizer Ensemble was written in 1988, after I had just acquired a Yamaha DX-7 synthesizer. I became intrigued by the sounds it could produce, not just the imitations of musical instruments, but particularly the interesting and unusual "bell-like" sounds it could create by using simple fractional ratios. This work is a reflection of my obsession with one of those sounds, called "BellWahh A" by the manufacturer. The attack of the sound is like a sharp bell, but as that component decays, another vocal-like component enters and builds slowly. I found that this sound worked particularly well with the type of harmony that this piece uses, in which complex chords are built up slowly. Each tone has components in at least two octaves, and one of the sounds near the end, a low D at 18.32 Hz, is the lowest sound I have ever seen used in a piece of music.
The work has now been synthesized entirely by the computer.
Procession is essentially a slow march. I envisioned describing a slow-moving procession moving past a fixed location with an unrelenting motion toward a foreboding destination. The work is in three parts, with the faster midle section at twice the tempo of the slower outer sections.
The piece is scored for three synthesizers in live performance. Two of them play a sustaining timbre, with a rich sound containing many harmonics. The original sound I used was a mix of stringlike and brasslike timbres, but any rich sustaining sound will do. The other synthesizer plays a sustaining tubular bell sound, rather like a tuned chime, but extending over a much greater range than conventional chimes.
The piece was written in Tuscaloosa, Alabama in 1989, when I was a visiting professor at the University of Alabama.
Timbre Study No. 5 was composed and synthesized entirely during my stay at the Gubbio 1991 festival in Gubbio, Italy. In the piece, the overtones of each tone are generated and controlled individually in order to create complex timbre changes. For each tone, a series of overtones is stated at the beginning of the sound that reflects the harmony of the surrounding passage. Tones are stated in basically three ways: (1) a cyclic pattern that states the overtone series at least twice over the course of the duration, (2) a complex envelope that states the series once with a changing timbre, and (3) a pattern that states each overtone individually, as a separate tone. Most sections use the first 16 or 24 partials to create the complete sound. Near the end, there is a passage that uses only high overtones, with no energy at the fundamental frequency and all the overtones concentrated in the same frequency area, as the fundamental frequencies reach into lower and lower octaves, until the entire series is introduced once again. Throughout the piece, there is a fascinating interplay and tension between the overtones and the fundamental frequencies or pitches produced by the series.
Symphony No. 2 was composed in 1992. It is a five-movement work with a palindromic relationship between the movements as well as the sections within the movements. The first and last movements are fast, the second and fourth slow, and the middle movement, marked "Scherzando", is moderately fast. While a palindrome is, in some sense, easy to follow, it is a challenging and interesting concept to work with, since it requires that passages that are initially heard as the opening statement of a musical idea later be heard as a conclusion or summing up, or for some other function, and vice versa. For this reason, there are some sections that are grouped together, so that the palindrome applies, for example, to a group of three sections rather than to each section individually.
The piece employs my own method of composing based upon ideas that depart markedly from the other two major methods used in the twentieth century, namely tonality and 12-tone serialism. In 12-tone music, a complex and frequently jarring musical surface is often underlaid by a simplistic background structure. Composers have resorted to such ridiculous constructions as simultaneous multiple dynamic levels and rhythms that are as impossible to play as they are to hear. Early 12-tone music employed a conscious avoidance of tonal references (such as triads or seventh chords), and to this day composers go to great lengths to avoid octaves. Since the total chromatic is always stated, differentiations are made mainly by the ordering of elements, and the entire system is governed by a logic of permutations rather than combinations that generate new elements.
In tonal music, compositions are based on simple harmonic structures (major and minor triads), and dissonances always resolve into consonances. While the dissonances are often the most interesting and important aspects of tonal compositions, for the most part tonality has not evolved to the idea of basing compositions on more complex structures than triads, and some composers who have attempted to do so have created logical inconsistencies that create impossible dilemmas.
My compositions are based on small collections of notes (3, 4, 5 or 6) related by only their intervallic structure. These are combined into groups called arrays, which possess various structural, common tone, and ordering properties that allow events to be structured in several dimensions at once. While the music may be very complex at times, the basic elements (mainly trichords, tetrachords and pentachords) are easily to perceive and to understand. This method also forces a consideration of pitch duplications not present in 12-tone music, since groups of notes combined with other groups generate new notes sometimes duplicating tones already present. On the surface of the music, this process results explicitly in pitch repetitions and octave duplications that parallel the larger structures.
The goal is a musical texture in which each note is simultaneously related to every other event in its context in several different ways. The four relationships that make up all the arrays used in the piece are the multiplicative operations of identity (M1), inversion (M11), cycle-of-fifths equivalence (M7) and inversion of cycle-of-fifths equivalence (M5), which are the only single-interval cycles within the equal_tempered scale that are capable of generating the total chromatic. Passages in the piece are based on sets of trichords, tetrachords, pentachords and hexachords related by these properties. In particular, long passages in corresponding sections are related by cycle-of-fifths equivalence.
Symphony No. 2 is based upon a particular group of arrays in which the set of notes A-Bb-B-E occur in special ways. As a result, these notes should always be followed as though they were guiding the rest of the music, which in fact they do. This process is somewhat analogous to the tonal sense of key, since all the harmonies have something to do with these notes, but not of course in the tonal sense where these are "more important" than the others and serve as the resolutions of other tones.
In listening to a composition like this, the listener should make an effort to avoid hearing events as in other music, as if the piece consisted of a tonal piece with wrong notes. Central to this process is discarding the idea of the resolution of notes in a dissonant chord into a consonance. Although many events in the piece are reminiscent of tonality -- and indeed, I have often striven to use elements like triads -- the traditional concepts of tonal resolutions and the sense of key are not present. Dissonances (indeed, any structures) are simply stated, and the intervallic relationship between the constituent notes should be perceived as the essence of the total structure. Rather than looking for resolutions, the listener should direct his or her attention to looking for similarities between the present sounds and others in the immediate and longer-range context, keeping in mind that, in inversions, minor seconds are exchanged with major sevenths, and in cycle-of-fifths equivalence, with perfect fifths.
The piece is scored for chamber orchestra consisting of one flute, oboe, clarinet, bassoon, french horn, trumpet, and piano, and strings in sections (at least two on each part). In particular, the cello and string bass parts are different, and the string bass part can be played completely on cello if a string bass is not available.
Meditation is my first completely microtonal composition, based entirely on 19-tone equal temperament. 19-tone temperament has been praised by theorists for many years because of its extremely well-tuned intervals, particularly major thirds and perfect fourths and fifths; these intervals are much more “in tune” with pure intonation than 12-tone equal temperament. On the other hand, major and minor seconds are more “out of tune” than in 12-tone temperament, and the basic 19-tone “minor second” is 63 cents, or about two-thirds of a semitone, producing some strange sounds. One thing that particularly fascinates me with 19-tone tuning is the fact that, for most pitch sets, new structures are produced by eighteen different multiplicative operations. These operations consist of expanding the intervals in a set by a given number of semitones, and they provide the most comprehensive method for incorporating pitch relationships in the system.
With the aid of a computer, I investigated the complete vocabulary of 19-tone pitch sets and developed a method of constructing arrays based on these relationships. These were the preliminaries that were completed before composed the piece itself.
Meditation, as the title implies, is a slow, contemplative work that begins from a single tone, combines it with other tones, builds to larger and faster materials, and ultimately returns to a single tone as in the beginning. The basic sound is a vocal-like tone produced by three-carrier FM synthesis so that two formants are emphasized. Throughout much of the piece, the sound undergoes a crescendo and diminuendo with a corresponding timbre change that parallels the basic structure of the piece. There is no amplitude or frequency modulation in the synthesis of the sounds; all the beating that is present is a natural result of the intonation of the tones.
There are five sections in the piece in a palindromic relationship and a 2:1 tempo change between each, increasing at first and then decreasing. In the beginning, tones start from the middle octave (the first note is middle C) and expand outward into other octaves. In the second section, where certain highlighted tones travel between the loudspeakers, the basic “theme” of the piece is stated. In the middle section, tones are attacked with a more "bell-like" envelope, and the exact midpoint is a climax. After that point, material returns in a compressed form, and the piece parallels the opening sections, returning to a single octave and single tone as in the beginning.
The piece was composed in 1993 and synthesized completely with the csound program.
Since no sound can be produced "spontaneously" by a computer except after considerable thought and programming, I must begin by explaining that I use the term "improvisation" to describe a piece that is spontaneously conceived, even though its execution takes a much longer time. These works may veer off in unexpected directions without necessarily returning to earlier themes. In Improvisation No. 2 the same two musical passages presented twice comprise the entire work: there are four sections, and the outer two and middle two present the identical music in different ways.
The unifying idea behind the piece is the overtone structure of chords -- the overtones of all tones in the chord considered together rather than as the sum of the four or five separate tones. This idea is also combined with glissando to create the impression of a constantly shifting sense of pitch. The piece begins with overtones unfolding so that only after several seconds can the listener actually perceive the chord, and as soon as this happens, the pitch begins to change and fade out. The next section introduces the overtones attacked suddenly in a bell-like fashion, with the pitch again moving as soon as it is established. The third section presents all the overtones together in a shifting pattern, but with each tone making a slow glissando just after the pitch is established. A climax occurs as all tones coalesce to a single pitch, which then dissipates. The final passage presents a single-tone "drone" against the high overtones of the other notes of the chord moving in a separate rhythm. At the end, only the "residue" of the chord remains as it dies away.
The opening passage of my Quintet (8 measures) states a succession of chords (the four all-interval tetrachords) from which the entire composition is derived. Each subsequent passage in the work relates directly to these chords, in the same order in which they are stated here. While there are two movements, the work is a single unit with a palindromic structure. The beginning of the second movement is the exact midpoint, and the succeeding passages relate, in reverse order, to the first movement.
The basic tempo of the piece (first stated at [3]) is 60 beats per minute, four beats of which occupy four seconds. Divided into sixteenth notes, there are sixteen ticks in a measure. Other tempos relate to this by dividing the same four seconds into different subdivisions: 12 yields a tempo of 45 beats per minute (60 times 12 divided by 16), and 20 yields 75 (60 times 20 divided by 16). The piece begins at the slower tempo, and the main body of the first movement is at the faster tempo; the median tempo is used in transitions.
The opening passage ([1] and [2]) expands the original chords into several octaves with increasing complexity and rhythmic interaction between the instruments. At [3], a transition occurs at a faster tempo that states the basic materials for the main body of the first movement. At [4], the tempo increases again, and there is a lengthy passage expanding and developing this material. At [5], a softer and less intense interlude affords the opportunity for solos on each of the instruments. This leads to a climax at [6], which is the loudest and most complicated section of the piece. When this has finished, the tempo slows, and a soft conclusion ([7]) functions as a transition to the second movement.
The second movement begins with a slow, lyrical passage ([8]) based on trichords. This is the purest and simplest passage in the entire work. It is followed by another passage ([9]) only slightly more complex. There follows an extended development ([10] and [11]) that alternates loud and soft passages and which begin to retrace the steps that led from the beginning of the piece. At [12] and [13], the tempo changes to that of the end of the first movement, followed by an even faster passage ([14] and [15]) that corresponds to the middle of the first movement. At [16], the tempo slows to that of the first transition at [3], the material from there being stated in compressed form. The ending retraces the material from the opening passages all the way back to the beginning, gradually diminishing and contracting until just a single chord is left.
The basic material of the piece consists of the four chords stated at the beginning, which are related by certain basic structural operations, and these are the only chords that exist with various specific properties. These chords are fragmented (the opening passage states them as a single note followed by a trichord) and combined with other notes to form new chords that usually possess similar unique qualities. In some later passages, successions of four different chords related by the same operations which state or present each of the original chords in some unique way are used, thus producing lengthy passages of considerable complexity all derived from a single chord.
Improvisation No. 3 is another of my pieces based on the sounds of natural overtones in changing patterns and unusual ways. In addition to using the overtones of chords as a single unit and cyclic variations of the overtones of a single tone, in this work I have created a type of "bell" sound by "squashing" the first sixteen overtones into the space occupied by the first fifteen, producing inharmonic relationships among the components. Only the first and fifteenth overtones are "in tune".
The piece begins with a big crashing bell sound, which presents all the harmonic content of the piece simultaneously, with both components and chords decaying at different rates. Then a slow, low passage emerges which gradually builds as it moves into higher octaves. This is followed by a passage combining middle- and low-octave glissandos with much higher overtones of the same chords descending downward over a six-octave span. The climax of the piece presents the bell sounds against constantly-shifting chords. The conclusion combines all of these materials into one mosaic spanning eight octaves: high chordal overtones drifting downward in patterns, middle-register bells, and a low constantly-moving "drone" with overtones extending up five octaves into the bell sounds.
My Nonet is in two movements, approximately 15 and 12 minutes respectively. The opening of the first movement, three phrases based on groups of 3, 4 and 5 notes, states the materials on which the entire piece is based. The first movement is a kind of sonata form based on two contrasting ideas. The first, presented in 5/4, begins immediately after the slow introduction. Several contrasting passages build to a climax and lead to the second idea, presented more slowly and softly in 4/4. After a slow transition, a lengthy development ensues, in which these two ideas are combined in different ways, along with some digressions. The development continuously intensifies up to the recapitulation, which presents a cycle-of-fifths transformation of the original idea, while the developmental processes (loud interruptions occurring within softer passages) continues. These processes extend through the return of the second idea, leading to a slow and rather lengthy coda. The ending summarizes all the materials presented in the movement.<.p>
The second movement is a very slow series of excursions that are spun out of the opening duet between the oboe and violin. The opening intensifies very gradually as new elements are added in each successive phrase. Finally a faster, louder passage begins, interrupted by silent punctuation marks. This passage wanders through many different areas while leading to quieter, more intricate harmonies. These "wanderings" are nevertheless connected by a common thread. After a brief return to the opening texture, a melancholy recapitulation of the opening occurs. As in the first movement, this is a cycle-of-fifths transformation. This leads to a slower transformation of the second section, which moves through a similar series of excursions. The conclusion recalls the opening of the first movement, stated so that it recalls the entire piece.
I have always been fascinated with the process whereby the overtones of a sound, which are in fact separate tones, fuse into a single pitch, with the fundamental frequency being the pitch perceived and the overtones being perceived as the timbre or tone color of the sound. All of my Timbre Studies deal with this idea in one way or another.
This composition is basically a study in filtering and glissando. It originated from the fantasy of listening to the upper overtones of a sound and filtering them sharply to focus on a specific pitch area. These areas are then structured as pitches themselves, with a similar type of content to the lower notes that provide the overtone material. This fantasy is realized explicitly in the middle section, where one hears a low passage that proceeds on its course and a higher, faster passage that creates a counterpoint to the slower passage in the filters. As for the outer sections of the piece, an analogy is created between fixed pitches as relating to fixed filters and variable filters to glissandos. Contrast the variable filter (which creates a sort of "wa-wa" or "yeow" effect) against the fixed pitches at the end of the first section with the glissandos against the fixed filters in the beginning of the last section. Overall, the piece is a large crescendo-diminuendo, beginning and ending on a single note, growing and shrinking along the same lines. The frequency content that is structured in the piece exceeds that of the piano keyboard, from the lowest note to higher than the highest note.
Improvisation No. 4 is the fourth of a series of pieces that deal with the overtone structures of sounds, as well as vibrato and glissando, in unique ways. In this work, each tone in the upper octaves consist of eight partials, in the middle 16 partials, and the lowest 24 partials. The overtones are always generated individually and in ascending or descending order. The overall shape of the piece is a palindrome, starting at the very slow tempo of 10 beats per minute and increasing (usually doubling) the tempo of each successive section until the middle, which reaches the tempo of 120 beats per minute, and then reversing the process.
In the opening section, the overtones are stated in a manner that "unfolds" the tones: the highest overtones enter individually one at a time, and the sense of the pitch of the tone does not enter until several of them are sounding together. They decay in a similar way. After an initial delay, the pitch of the tone begins a very small vibrato, the speed of which is harmonically related to the fundamental: it is eight octaves below (in a subsonic range).
The second (and penultimate) section develop what I describe as a "dissolution" of the tone. The overtones enter as a group, and then individually begin a glissando to the corresponding overtone of the next note. There are four lines, each of which trace a separate path, and some of the glissandos are very long and cover a small interval while others are short and cover much wider intervals, although each line exists within a separate octave.
The third and fourth sections use an instrument that produces all overtones together but with increasingly longer attack and decay times, producing a "swelling" effect that simultaneously makes a crescendo and diminuendo. Once the pitch is established, a small vibrato like that of the unfolding sound enters. In the fourth section, the overtones of the sound make a slight glissando up and down rather than the vibrato. As the section proceeds, the amount of glissando increases.
The middle of the piece (actually comprising three sections in length and taking two minutes) introduce a very low note, G in the lowest octave of the piano, that, over the course of the complete passage, makes a downward glissando of an octave, to G off the end of the piano. The entire music played here makes a similar downward glissando of the same interval, which counteracts the motion of the three sections themselves, which move up an octave over the passage.
After the mid-point of the piece, the instruments and tempos of the first sections return in reverse order, except that the material is compressed and occupies fewer measures. The climax of the piece occurs in the next to last section, after which the final section unfolds its tones as a sort of coda.
The piece was composed in 1998 and synthesized entirely by the csound music synthesis program.
Cacophony is a work based on instruments that generate overtones individually and in different ways. Tones in the octave of middle C and below use 32 partials, spanning a space of six octaves above the fundamental. Tones in the octave above middle C use 16 partials, and those two octaves above use 8 partials. The music is constructed so that sound usually fills the entire spectrum from the lowest note on the piano (some notes go below that) to above the highest note. The title appropriately reflects the depth and breath of the sounds in the piece.
One group of instruments unfolds the harmonic partials of the tone in an ascending order, another group does so in a descending order, and a third group does so in an uneven manner, so that the opening of the sound states the "harmony" of the sound of the context in which it appears. Partials 1, 2, 4, 5, 6, 8 and their higher octaves state a major triad, for example; but for more complicated chords, the partials above 16 can state any harmony, no matter how complex. Since these are harmonic partials, their statement is in pure intonation, even though the music is in equal temperament. (Several partials are unusable in this way because they are too much out of tune with notes of the tempered scale.) Beyond this idea, there are different types of dynamics, processes, and envelopes applied to the sounds. One instrument repeats the tones where each pitch has a unique speed. Another instrument plays shifting cycles of the overtone pattern, and a different one states the overtones in a complex envelope. The "shimmering" sounds heard in the beginning and ending sections unfold the overtones of a chord in the low registers using overtones several octaves above the fundamentals.
The music is based on a series of interlocking array structures including a controlling array and two others, so that the entire series states the total chromatic and each has certain unique properties. This entire structure is only presented in the last section; in the earlier passages, each successive section includes a restatement of one of the elements in the preceding section, but played by a different instrument, so that its identity is not always immediately apparent. The restating idea gives a sense of continuity and progression. There are seven total sections. The tempo begins at 15 beats per minute doubles in each section up to the mid-point; afterwards, it is halved to return to the opening tempo at the end. Many of the passages include two statements in different time signatures.
One of my concerns in computer music is the development of interesting, original sounds. The sounds in this piece were conceived by imagining what would happen if the octave were replaced by something different. The ratio between the frequencies that are an octave apart is 1:2. In this work, the 1:2 octave ratio is replaced by the value of the square root of 2 and the square root of 3; the next number in this series would be the square root of 4, or 2, which is the usual value. (This idea was suggested to me by John Chowning’s piece Stria, in which he employs an analogous sound derived from the golden mean of 1:1.618.)
Instead of the usual harmonic series, the overtones in this work exist in the ratios of the square roots of 2 and 3. For the square root of 2, the series consists of the octaves and the square root of 2 values in between: 1, 1.414, 2, 2.828, 4, 5.657, 8, 11.314, 16, 22.627, 32, 45.255 and 64. For the square root of 3, the partials are in the ratio 1, 1.732, 3, 5.196, 9, 15.588, 27, 46.765, and 81. (Not all sounds use the entire series; when the upper partials are above the range of human hearing, they are not generated.) In addition to using these harmonic series, the work also derives 12-tone equal-tempered scales from these values. For the square root of 2, the scale is quarter tones, but the square root of 3 produces a more unusual series where 12 steps fit into about a major tenth and the step is about 5/6th of a half step. Within a usual range of frequencies less than the span of keys on the piano, there are 12 octaves in the quarter-tone sections and 8 in the square root of 3 sections.
In addition to these sounds, the piece also employs frequency-modulated sounds using the ratios of 1:1.414 and 1:1.732. These sounds can be described as “bell-like” and are employed with a “crescendo-diminuendo” spectral envelope that changes the timbre, whereas the square root tones unfold the partials in an upward direction.
The piece is in four continuous sections with a palindromic structure. The first half uses the square root of 2 and the second the square root of 3. It begins slowly with the tempo accelerating to 10 times the original tempo, then holding that value through the middle and decelerating after the middle. In the mid-point of the piece, where the transition between the square root of 2 and 3 occurs, there are two passages that are identical except one is based on the square root of 2 and the other on the square root of 3.
Mosaic employs filtering of the overtones of the sounds that occur. There are several different ways in which the computer "instruments" that perform the music do this, but all use similar principles and methods. The piece begins with tones that "unfold" the overtones in an upwardly ascending manner. Tones in different octaves have more area in which to operate, so they naturally ascend to higher overtones. The piece begins (and ends) with a single tone, after which more complex sonorities unfold. We hear each of the overtones in the series enter as the emphasis rises, and they interact with the other tones in the context. The second and, in some ways, most important instrument employs three resonances that oscillate above and below their mid-points, thus producing a kind of diphthong effect. The speed of these oscillations increases and decreases over the tone's duration. This instrument is used throughout the middle section, although it is combined with others. The third instrument employs the same three resonances as the second, but instead of oscillating they descend at different speeds to the fundamental, thus "dissolving" the sound into the fundamental. The fourth instrument, used only in parts of the middle sections which play "chorale" passages, simply unfold the overtones in a manner similar to a brass instrument, which introduces gradually higher overtones as the tone increases in amplitude.
The piece is based on a series of interlocking arrays that produce a cycle that exhausts all of the possible combinations in which a particular collection of tones (the octachord that excludes 02350) can be generated. The arrays are all based on either trichords and pentachords, and the interlocking manner in which the trichords are imbedded in the pentachords is what suggested the title to me. In each passage, vestiges of the previous passage appear, sometimes in the same and sometimes in different rhythms.
The piece was composed in the Fall of 2000 and synthesized by the csound program.
LongGong is based on the sound of the same name that is one of the presets on the Yamaha DX7-II synthesizer, which I have enjoyed since I first heard it. It consists of three sine-wave carriers and one modulator that controls two carriers. The frequencies (ratios) of the three sine waves are 1.0, 1.42 and 2.14, and the two modulators create c:m ratios of 1.21:1.75 and 0.76:1.75. These processes create a series of which the first 22 elements are the partials .54, .76, .99, 1.21, 2.29, 2.51, 2.74, 2.96, 4.04, ..., 9.96.
All qualities of the composition, including both pitch and time elements, are created from the elements of this sound. The piece begins with a single tone that states the basic LongGong sound, followed by an echo that transposes the frequencies to each of the elements to the series. The time values are the values of the series transposed to seconds. Following this, the entire remainder of the composition is a fractal based on this sound. The elements of the series are transposed to minutes, and a complete series of tones that plays the entire series transposed to each of the elements of the series, in a slowly increasing series of time values, changing from 9.96 minutes to 9.96 seconds. Thus, the piece grows in intensity and frequency towards the ending. There is no chromatic scale, nor are there any traditional rhythmic values. It is my personal homage to this sound.
The overall shape of my Chamber Concerto is a sonata form, with an introduction, two main thematic groups, one presented in 5 and the other in 4, a development, recapitulation, and coda. The introduction consists of two brief phrases, the first based on trichords and presented in the string quartet, and the second based on pentachords and presented in all instruments except the piano. This passage is restated several times in the piece at structurally important points. The opening four measures are completely integrated into the following four, establishing a connecting principle that is used throughout the piece.
The first theme begins at 2, stating the main idea, which is a series of pentachords connected by trichords. A brief transition connects to the second theme at 4. This leads to a brief digression on the winds, followed by another in woodwinds and strings. A concluding passage at 7 leads to a restatement of the introduction, which begins the development. The development starts with an introduction, followed by three fortissimo passages that combine materials from the first theme group and a similarly long passage, starting at 15, that does the same with the second theme group. This leads to a climax at 18, after which the recapitulation begins at 19. The recapitulation reflects the process of combining materials begun in the development, alternating forte and piano passages. This leads to a climax at 25 and the coda at 26, which uses the same materials as the introduction and ends on the combined chords from the beginning.
Throughout the piece there are tempo and time signature changes that maintain strict proportions of 3 to 4 to 5. When the time signature is 3/4, the tempo is 48; when it is 4/4, the tempo is 60, and when 5/4 it is 75. This maintains the principle of dividing a measure of four seconds into 3, 4 and 5 beats, which reflects the overall structure of passages based on trichords, tetrachords, and pentachords.
The piece is based on a series of pentachordal arrays that exclude the collection 0235 (0), or the notes C-D-E-flat-F, a group of trichordal arrays that are imbedded in these arrays, and another group of tetrachordal arrays that use the same interconnecting arrays. These arrays are the only ones that possess these properties.
Both of my compositions entitled "Cacophony" deal with sounds that cover a vast stretch of frequencies, from below the lowest to above the highest notes on the piano. In this work, based on filtering, there are two different structures of pitches and rhythms. At the low end are the source tones, which provide a continuously evolving musical passage of between three and five octaves, and at the high end are the filters, which operate on the overtones of the lower sounds and provide their own evolving musical passage, often with different rhythms and pitches from the source tones. For much of the composition, there are independent pitches and rhythms structured over an eight-octave range.
There are several kinds of filters that are used in the piece, all of which fall into two basic categories: fixed and variable. The fixed filters resonate specific areas of the frequency continuum (specified and distinguished as pitches), but the only variable is the amplitude of the filter. Many passages alternate between four different fixed filters, fading in and out between them. Another type of fixed filter is the complex envelope, in which a number (four in this piece) of resonances is attacked within each note each with a different envelope. The variable filters change in frequency over time, producing effects sounding like the diphthongs "wa-wa" and "yeow". In this case the important variables are the time span of the change and the amount of frequency covered, which ranges from less than an octave to three octaves. Sometimes the variable filters are keyed to move through specific overtones of the fundamental frequency, and at other times they articulate a separate structure from the source tones in their own rhythms.
The piece is in twelve sections with the overall shape of a palindrome. The outer sections are the slowest, each lasting over three minutes, and the tempo doubles five times as the piece progresses from the beginning to the middle and is cut in half five times from the middle to the end (the fourth and fifth, and correspondingly the eighth and ninth sections are in the same tempo). The music proceeds by inclusion, whereby parts of the preceding passage are incorporated into the next one, and so on. The music in the two halves of the piece are related by cycle-of-fifths equivalence.
The piece was synthesized in January and February of 2002 using the csound program.
Iridescence was initially composed in November 2002 in response to my friend Dinu Ghezzo's request for a piece to be included on a concert he was planning, and revised in April 2003 . It is based on sounds that employ both fixed and variable filters that produce many different kinds of colorful "shimmering" effects, which suggested the title to me.
The piece is based on a succession of two sets of arrays (excluding the sets 0135(5) and 0356(4)) that include common elements and complement each other, so that successions and combinations of them produce unique properties. It begins slowly in a single octave and expands outward in stages, ultimately building to a climax that spans a 7-octave range (the entire range of the piano keyboard). After this, a series of shorter passages emerge that make the transition from the first to the second group of pitch materials, and the piece progresses in a quasi-palindromic but compressed fashion to the ending. Many sections of the piece repeat literally some of the music from the preceding section, integrating that material into a new context. Whenever this occurs, the repeated material is reverberated.
The tempo of the piece, while starting slowly, accelerates to four times the original tempo in the middle and ends in a faster tempo that it began.
All tones in the piece employ filters that are in the range of three octaves above the fundamental (except for tones in the highest octave of the piano, where they are up to two octaves above). These resonances articulate a different set of pitches from the fundamental, but a series that is related to the overall passage. One group of instruments employs variable filtering (this is part of the "iridescent" quality of the tones), while another uses fixed filters and variable amplitude modulation. Detuning among components produces a kind of chorus effect, and there is also a very slight variable vibrato. All speeds of these variable qualities occur at subsonic frequencies in ranges of eight to ten octaves below the fundamental.
Harmonic Fantasy is based upon very rich sounds, all consisting of 32 harmonic partials, which extend five octaves above the fundamental (except on high tones, where the partials exceed the limits of human hearing). The harmonics are introduced one at a time in an irregular series that emphasizes the harmony of the context in which the tone appears at the beginning of the series, followed by a transposition of the series, and finally by the remaining partials. Following the introduction of the individual partials, the tones undergo either vibrato or glissando in precisely controlled ways. Vibrato is applied to the partials in an individual, out-of-sync fashion at a subsonic speed that is seven octaves below the fundamental (thus, middle C would be about 2 Hz). The partials of glissandos are also delayed by a distinct amount and move individually to the corresponding partial in a new tone. This creates the effect of the sound dissembling before your ears, only to re-coalesce into a new tone. In the second section of the piece, these tones create a three-part melodic context, but in the later sections where these are used, the tones move up a minor third and back to the original tone over the context of the tone's duration.
The piece is in six sections, beginning with a thin texture of trichords and building by accretion to more complicated harmonies and textures. Each new harmony is formed by adding one tone to the chord from the previous section, until a hexachordal texture is reached. The piece grows dynamically in a manner similar to Ravel's Bolero, reaching a huge climax in the fifth section. The concluding sixth section extrapolates three-note chords from this passage into a new structure and concludes softly.
Harmonic Fantasy was commissioned by Winthrop University in Rock Hill, South Carolina. It was sketched while I visited Singapore in October, 2003 but could not be produced until I returned home.
Chamber Concerto No. 2 is similar to my first Chamber Concerto in that it is in two movements, the first fast and the second slow and both being based the same materials. The woodwind quartet and string quintet are treated mostly as separate groups, with the piano complementing each of them.
The first movement begins with a full passage, which is dissembled into several interlocking smaller sections, leading to a large climax. That passage is again dissembled into smaller sections, which in turn lead to some excursions away from where it began but then back into the same milieu, leading to an even bigger climax.
The second movement is in four large sections, each based on a similar and unique group of hexachordal arrays. These are made up of several smaller units, each of which is presented in a separate section and combined to form the whole. The orchestration reflects this organization, with passages for strings alternating with passages for woodwinds and the piano present in each. When combined, the whole ensemble plays. The ending is a large climax summarizing the entire process.
The first movement was sketched while I was visiting Singapore in the Fall of 2003, but I couldn’t write a note of it until returning home. The second movement was written in April of 2004.
In composing this work, I sought to create a new kind of sound containing clusters of non-harmonic partials, all compressed into a small span, and to combine that with the idea of a musical fractal. I composed a musical passage which develops according to its own logic, divided into twelve sections. Different sections span three, four and five octaves. I then created a series of computer instruments in which each note plays a complete passage consisting of all the notes in its own section, as sine tones (inharmonic partials), squeezed into the interval of a perfect fifth. Each of the tones enters in the same rhythm as the notes in the complete passage, the only differences being that the sequence is transposed to the level of the pitch in the score, and all of the partials take place within the time-span of the duration of the note. Different sections consist of 12, 28, 30, 48, 56, 64 and 68 components.
The work was composed in 2005 and synthesized with the csound program.
Inharmonic Fantasy is based on the idea of squeezing the harmonic partials of a tone into a smaller interval, where the tone loses its sensation of pitch but still retains a distinct identity. The piece begins with a single long tone, with harmonic partials fading in and out in a pattern; but before long, they “dissolve” through glissandos into a sound that has no real pitch. The partials are always equally spaced, and the fading process always keeps going. As the piece continues, partials are compressed into smaller and smaller intervals, ultimately squeezing the five octave range of the first 32 harmonic partials into the space of an octave and a fifth. In the middle section of the piece, tones are attacked like gongs, and while they do not make glissandos, successive tones are further compressed. Glissandos return at the end, where the piece closes out with harmonic partials, as it began. The piece was synthesized with the csound program.
In computer programming, a macro is a series of codes that can be defined in a prototype and then called in just one line. In Macro Structure 2, each note brings forth a series of notes that duplicate the harmony of its context. The piece progresses from sections based on trichords through tetrachords, pentachords and hexachords by adding one or more notes to the preceding chords, and then reverses the process. Each measure occupies the same duration of ten seconds, but successive sections subdivide the basic duration into a greater number of beats, so the tempo appears to increase and then decrease. Each tone consists of harmonic partials that enter at progressive delays and are sustained for half the duration, then drop out in the same order. In the stereo version, each tone begins in a specific location and then moves to one loudspeaker and then to the other, finally returning to the same location where it began. Notes at the edge (located entirely in one speaker) are completely dry (unreverberated), but as they move to the center, the reverberation increases, partly creating the impression of the note receding into the distance (only partly because, if the sound did move in physical space, it would get softer). In the quadraphonic and octaphonic versions of the piece, the center front location is the only dry spot, and reverberation increases as the sound moves to the back.
Inharmonic Fantasy No. 2 is based entirely on sounds containing inharmonic partials, that is, overtones that do not so much create a timbre for the sound as they create a kind of cluster above the fundamental. The overtones have the same relationship, entering in the same rhythm and in the same pitch relationship, as all of the notes in the passage in which they occur. Thus, the sounds themselves are a kind of fractal. There are two ways in which they are presented: continuously, as a kind of complex envelope, and attacked as separate tones. Underlying everything is a very slow vibrato that expands from zero to a perfect fourth in the middle of the piece, making just one cycle over each entire section.
The piece opens with a soft and slow passage that unfolds the basic idea, and which underlies most of the piece, except for the middle section. This is followed by a faster and denser passage, and then a less dense but even faster passage that introduces the attacking tones. There is a short pause in the middle of this section, after which a climax occurs that uses both instruments together. After this reaches its apex, the underlying tones similar to the beginning are left, and the piece ends quietly.
The work was composed in 2007 and synthesized with the csound program.