In this article, I will try to explain the processes that I used to produce the sounds in my composition Timbre Study No. 7, written in 2008. I will also include examples the illustrate the sounds at different stages of their construction. The examples will be included with the article, along with a recording and the score of the work.
My electronic music belongs to that extremely rare breed of compositions that are synthesized, without relying on any prerecorded sounds or manual or automated processing techniques. Thus, it is probably of little interest to most people who presently compose what is known as electroacoustic music. I don’t advocate that others should attempt to emulate these processes; I don’t know if I would have done it myself if the kind of resources that are now available were around when I began this kind of work. Nevertheless, I offer this article as evidence that some worthwhile sounds can be created in the old-fashioned way, and there is also great pleasure in perfecting every aspect of the sounds in a composition according to similar and coherent processes.
Timbre Study No. 7 is one of my “overtone” compositions, in which every sound in the piece is produced by harmonic generators that create up to 32 harmonic or inharmonic partials. There are a few concepts that are applied to each sound, and all of the sounds have certain common characteristics. Basically, the unifying elements are clusters, overtone patterns, cross-fading of individual components, and in the middle sections, squashing the overtones into smaller intervals. The instruments that produce the tones have nicknames of unfold, shift, slide, squash, and seesaw. Since there are so many different things happening to each tone, the overall slow tempo is an essential aspect of the piece.
The terms “timbre study” usually signify that a particular type of sound or timbre is under consideration, and that is the case here. I have long been fascinated with the properties that harmonic partials can imbue in a musical context, and part of this composition focuses on those. But the piece also applies the ideas of overtone-shifting timbre changes to inharmonic partials. While “inharmonic” includes anything that is not harmonic, in this composition I am preserving the property that the overtones of a sound are always equally spaced. These sounds indeed have a “timbre” or tone color, but not the same kind of color that harmonic partials possess. The other major concept at work in this piece is the property of clusters, where each note in the score is interpreted as representing three, four or five notes that duplicate the harmony of its context. Thus, a passage based on a pentachord actually includes 25 separate pitches.
The composition is in seven sections, and the overall form can be described as follows: in the first section, overtones are unfolded into pentachords, with a movement that is both upwards and downwards. This is followed by a passage that plays trichords with shifting overtone patterns. The middle sections are concerned with tones in which 32 harmonic partials are squashed into the space originally occupied by the first seven harmonics, producing sounds with equally spaced overtones but no definite sense of pitch, or in some cases a sense of pitch that is in the subsonic range. The third section transitions into these sounds by beginning with the harmonic spectrum and then having the overtones slide individually and asynchronously to their inharmonic counterparts, and the sixth sections transitions back in a manner that seesaws up and back towards the harmonic spectrum. The middle sections consist entirely of the squashed tones, and the ending wallows in the rich, overtone-shifting sounds that inspired my interest in these processes in the first place.
While these processes are at the basis of the composition, another basic concept is that of clusters. Every note in the score is interpreted not just by the pitch that it designates but also by a cluster of notes that duplicate the harmony of the passage in which is occurs. The amplitudes of the notes in the cluster are cross-faded so that the first note is prominent over the first portion of the tone, the second over the next, and so on up to the number of notes in the cluster (three to five). This is one aspect of the piece that imbues the sense of the individual harmonies into the fabric of the composition, but there is more: almost all tones also have overtone patterns that also bring out the notes of the harmony at the beginning of the tone. Since 32 partials are used, every note of the chromatic scale is available in a quasi-just intonation beginning with the sixteenth partial. The pattern begins with those notes, followed by their octave duplications in the overtone series, and then followed by a transposition of the harmony before bringing in all the other overtones. This pattern is played twice over each tone.
Finally, the cluster structure is also embedded in the spatial locations of the tones. Since there are passages based on 3-, 4- and 5-note chords, the space between the two loudspeakers in stereo, or around the room in the quadraphonic and octaphonic versions of the piece, is divided into 3, 4 or 5 equal divisions. The first tone is located at the front left speaker, and the additional tones are moved to the next corresponding location. But within the overall spaces, each pitch is also assigned to a unique location, so there are a minimum of 36 to a maximum of 60 distinct locations for each sound.
Before discussing the individual instruments, it is important to mention that all notes are played with a strong crescendo and diminuendo, starting and ending at one-fourth the amplitude and reaching the maximum at the mid-point.
This is the simplest instrument in the group. Overtones are introduced one at a time in eight groups of four, beginning with partials 29-32, then 25-28, 21-24, 17-20, 13-16, 9-12, 5-8 and 1-4. Each overtone has an envelope that makes it prominent at its individual time in the overall pattern, except that there is a “pause” between each group of four. The first four are heard over the first sixteenth of the duration, then there is a pause of a sixteenth of the duration, then the next four partials enter, and so forth.
Let’s listen to one example of such a tone, the first heard in the piece [example 1]. This example is just of a single tone. Now let me play the actual tone used in the piece, which is a cluster of five tones stating the harmony 01457(0) [example 2].
These sounds are similar to many I have used in past compositions, and they are incredibly rich and interesting. The shifting, which I also call overtone patterns, is accomplished by processing each harmonic partial by a “shift” function, an envelope that reaches full amplitude at a particular point in the cycle of the duration and then recedes down to one-fifth the level at three-quarters of the duration, and then rises to full amplitude again. The function is entered at 32 evenly-spaced intervals in its cycle, so that the same shape can be used for all harmonics. The function is sampled cyclically, so that it can occur any number of times over the course of the note’s duration, but in this composition, the number of cycles is always two.
In order to understand the “pitch content” of the overtone series, you need to compute the closest 12-tone pitch class for each overtone. Thus, the fundamental, second, fourth, eighth, sixteenth and thirty-second partials are all octaves (all powers of two). The third, sixth, twelfth and twenty-fourth partials are all perfect fifths, or interval 7. Overtones 5, 10 and 20 are major thirds. 7, 14 and 28 are major sevenths. Starting with the sixteenth partial it is possible to obtain all the notes of the chromatic scale, although some are out of tune. The overtones that are the most out-of-tune are 11 and 22, which are a very flat tritone, interval 6 (while 23 is sharp), and 13 and 26, which are pretty much equidistant between a major and minor sixth. 27 is closer to the major sixth, but if you need the minor sixth, you have to choose between 25, which is quite flat, and 26, which is sharp. Overtones 15 and 30 are closer to the major seventh than these are to those intervals.
Except for the opening and the middle sections that use “squashed” overtones, all tones in the composition use these shifting overtone patterns. The important quality is the order in which the overtones are introduced, which are heard distinctly. I have created individual sequences for each of the many harmonies that are used in the piece, but the sections where these are most distinctly heard are all based on trichords. The sequences were all created according to the following principle: first, introduce the overtones that state the harmony, starting from the sixteenth partial and moving up, then the eighth to the fifteenth, then the fourth to the seventh, then two, three, and one (if these are present). Second, introduce one or more transpositions of the harmony with any of the overtones not yet used. Finally, bring in all unused overtones, again from the 17 through 31 (remember that 16 and 32 are always used in the first group), then 9 through 15, and so on until all have been stated.
Let me give you some examples of this process. For instance, the trichord 034 is represented by the following sequence: 16, 19, 20, 32, 8, 10, 4, 5, 2, 1 (0340 and octave doublings of 0 and 4), 24, 28, 30, 12, 14, 6, 7, 3 (0347), 22, 27, 31, 11 (0346), and 25, 26, 29, 17, 18, 21, 23, 9, 13, 15 (the rest). Let me play this sound [example 3]. Here is another example, the trichord 035, which is used on the last notes of the piece. First we have 16, 19, 21, 32, 8, 4, 2, 1 (0350), 20, 22, 27, 10, 11, 5 (0354), 26, 28, 17, 13, 14, 7 (0358), and 24, 25, 29, 30, 31, 18, 23, 9, 12, 15, 6, 3 (the residue). Here is the sound [example 4]. While it may be difficult to hear the trichords as “melodies,” the slow and interesting change of timbre is certainly the most prominent aspect of these sounds.
Again, these sounds are not precisely those used in the composition, because of the clusters: these are just components of the cluster. The notes of the cluster are stated simultaneously but with an envelope that emphasizes the first note over the first third of the duration, the second over the next third, and the third over the last; and these components are further clarified by spatializations. Here is an example of the first tone above from a actually sound used in the piece (F# 5.06) [example 5], and here is a sample of the second, the last note in the piece (E 5.04) [example 6]. I think you can hear that the combination of the overtone patterns and the clusters imbue these sounds with the color of the harmony.
“Slides” are used to effect the transition from harmonic to inharmonic spectra. Each tone begins as a harmonic spectrum and sustains it for a quarter of the overall duration. All these sounds also have the amplitude-shifting overtone patterns as the “shifting” sounds from the previous section. Then each partial begins a glissando to the corresponding partial of a sound that reduces the overall distance between the original 32 partials to a smaller number, which gradually shrinks from 31 at the beginning of the section to 7 at the end. The fundamental begins its glissando first, and then each successive partial delays its entrance by an additional 1/32 of the quarter of the duration, until by the time the note is half over, all partials are making the glissando. The duration of each of the glissandos is also a quarter of the overall duration, so that by the last quarter of the tone, all partials are resting on the new frequencies. During the entire process, the overtone-shifting amplitude patterns continue to bring out the partial sequence that began, although by now it is on an inharmonic spectrum. Here is an example of the first sound, in which the overall span changes from 32 to just 31 partials, not much of a difference, and thus you hardly hear much of a change. The pitch is A 6.09 [example 7]. Here is one of the last ones, where the span changes from 32 to 15 (E-flat 6.03) [example 8]. To emphasize the clusters that are actually used in the piece, here is the first example as a cluster. In this case the harmony is 013, and the notes are introduced in the order 1, 0 and 3, so the first note heard is a B-flat 6.10 [example 9]. Corresponding, here is the last note, where the harmony is 035 and the notes are introduced in the order 0, 5, 3 [example 10]. Notice that the shifting overtone patterns are clear on both the harmonic and inharmonic spectra.
The middle parts of the composition are based on “squashed” sounds, in which all the 32 harmonic partials are squeezed into the space occupied by the first seven partials. Rather than a recurring shift function, each harmonic has an envelope that makes it prominent over a portion of the duration corresponding to the order specified for the overtones. The fundamental is the same as the pitch of the corresponding harmonic series, but the compression means that the fundamental is not heard as a pitch but rather simply as the lowest tone in the collection. The tempo of section 4 is double that of the opening and two-thirds as fast as section 5, so these tones are much shorter. Here is an example of a single tone from section 4, A-flat 7.08 [example 11], and here is one from section 5, G 8.07 [example 12]. Section 4 is based on tetrachords and 5 on pentachords, so here are the same examples of these tones in their respective clusters, 0146 for the A-flat [example 13] and 02456 for the G [example 14].
The “seesaw” sound is one which begins from a set of squashed partials, then expands slowly up to a wider interval, and then returns to the original. The section moves from a squash factor of 32 into the space of 7 partials and gradually expands to 32 by the beginning of section 7, but since notes do not occur on the first beat, the first tone begins from the frequency 7.67 times the fundamental, expanding to 9.67. Here is an example of this tone, F# 9.06 [example 15]. Here is the same example with its harmony in context, 0369 [example 16].
The score is published together with this article. The approximate starting time of each measure in seconds is printed beneath each measure. If you follow it, please keep in mind that only the root of each cluster is indicated, so that each note is interpreted as a three- to five-note cluster. The recording is also included, so that you can hear what the whole thing sounds like. The principal version of the piece, however, is in eight channels, so that the spatial separation will become even more evident and dramatic in that presentation.