Composing Harmonic Fantasy
by Hubert Howe
What makes electronic music pieces unique, and in fact “electronic,” is the nature of the sounds that they use. All electronic pieces explore some aspects of acoustic materials, and their treatment of these properties is quite different from what composers of instrumental music are able to do. In my electronic music, I have attempted to create sounds that have interesting and identifiable qualities that fit together under some general concept. With some exceptions, I have written pieces that have done this in three different ways: through structuring overtones of a sound individually, through filtering, and through exploring sounds with non-harmonic partials or non-12-tone tempered scales.
My process of working often involves imagining things in my head which I can’t always understand, and then going to the computer or to some piece of equipment and seeing what it sounds like. These are the infamous “test runs.” Sometimes, my idea turns out to be something quite common and not worth all that much in itself, but with refinement of the idea and additional tests, these ideas can turn out to be quite interesting.
I wrote Harmonic Fantasy because I received a commission from Winthrop University in Rock Hill, South Carolina, to compose a piece of electro-acoustic music for a residency that is planned there for November, 2004. Since I felt that their choice of me to do this was partly a result of other music I had written, it would be good to use some of the ideas I had worked with before, only extend them further.
Harmonic Fantasy is an “overtone” piece. I began thinking of something I had never tried: generating a tone in which different overtones had different rates of vibrato. This is quite complicated to implement, because it involves generating each overtone separately and then thinking up some way to control all those different vibrato rates. When I first did it, I was astonished at the richness of the sound that resulted. This seemed to be an idea worthy of further exploration.
To explain the problem a bit further, the amount of vibrato is set at two tenths of a semitone. If you think about it, as you must when you work with the computer, this amount represents a different value of Hz for each overtone. If the fundamental is middle C or 261.63 Hz, then a semitone is about 15.55 Hz, so 20% of this is 3.11 Hz. The first overtone is 523.26 Hz, so the amount of the semitone is 31.1 Hz, and 20% is 6.22 Hz. This process is used in the first section of the piece.
In determining the rate of vibrato, it should always be slow, and again I wanted to have something that would be different for each pitch, because out-of-sync qualities are usually more interesting than when everything is the same. I hit on the idea of using a subsonic pitch that is seven octaves below the fundamental. Thus, middle C has a vibrato rate of 2.04 Hz, while C# above it is 2.17 Hz.
Closely related to vibrato, in that it is also a pitch modification, is glissando. As the vibrato is different for each overtone, so did I also use a similar process with glissandos: each overtone of a sound introduced and delayed for a different amount, and then it begins a glissando to a different overtone of the next sound. I have used this process before, and I think of it as “dissolving” a sound. The tone emerges only after several overtones have been introduced, then it dissolves, only to re-coalesce into a new tone. This process is used in the second section of the piece.
These ideas of vibrato and glissando are combined with another idea which I have also used in several of my “overtone” pieces, and that is introducing each of the overtones of a sound not in an ascending or descending manner, but in an irregular pattern that emphasizes the harmony of the passage. As you know, overtones are individual tones that extend up from the fundamental in a harmonic series, and if they are played separately, we hear them as separate tones. Since Harmonic Fantasy uses 32 overtones for each tone (except when the overtone would exceed a value we could either hear or generate on the computer), this means that most overtones occupy the space five octaves above the fundamental.
I should also point out that almost all tones we hear in music have no more than 32 overtones, and many of them have far less. Overtones have been the source of historical tunings that led to the development of the scale, which only became equal tempered in the 17th century – and equal temperament is something that exists as an ideal only rarely achieved in actual music. (In fact, when it is achieved, it is usually through electronic instruments that are often criticized as having uninteresting sound qualities. As you go higher, the overtones get closer and closer together and become more “dissonant”.) When individual overtones are introduced separately, we can hear an actual “chord” that these create before they merge into the tone that we hear as the fundamental. The interplay between overtones and fundamentals is one of the most fascinating things about music in general, and it is a property that I have exploited to the maximum in my overtone pieces.
When you have 32 overtones to work with, all notes of the harmonic scale are available, at least in their “just” tunings. However, this is not as simple as it may seem, since several of the overtones, particularly the octaves, have several octave duplications, while others, particularly the minor second, minor third, tritone and major sixth have only one representative in the span. The following list shows which overtones are available for each of the notes in the 12-tone scale:
Interval Overtones unison, octave 1, 2, 4, 8, 16, 32 minor second 17 major second 9, 18 minor third 19 major third 5, 10, 20 tritone 21 minor sixth 11, 22 (or 23) major sixth 25 minor seventh 7, 14, 28 major seventh 15, 30 Left out 13, 23 (or 11 and 22), 26, 29 and 31
The following chart shows the intervals (in 8ve.pc form) above the fundamental for each of the first 32 overtones:
Partial Interval Partial Interval Partial Interval Partial Interval 1 .00 9 3.02 17 4.01 25 4.08 2 1.00 10 3.04 18 4.02 26 4.08/.09 3 1.07 11 3.05/.06 19 4.03 27 4.09 4 2.00 12 3.07 20 4.04 28 4.10 5 2.04 13 3.05/.06 21 4.05 29 4.10/.11 6 2.07 14 3.10 22 4.05/.06 30 4.11 7 2.10 15 3.11 23 4.06 31 4.11/5.00 8 3.00 16 4.00 24 4.07 32 5.00
Harmonic Fantasy is based on trichords, tetrachords, pentachords, and hexachords, so there are all kinds of harmonies ranging from a simple triad to a much more complex sounds. I established rules for introducing the overtones based on the following principles:
1. Choose whether the series will be ascending or descending.
2. Overtones that state the primary harmony are introduced, including both individual overtones and octave doublings, in the ascending or descending order chosen above.
3. If available, overtones that state a transposition of the harmony, or a subset of the harmony, are next introduced. First comes transpositions that have a tone in common with the stated harmony, then transpositions that start from a note not included in the basic set. If more than one transposition is used, then they are used in the order of whichever one has the most overtones available.
4. Finally, all the overtones not included by the above processes are stated, in the ascending or descending order chosen.
Since every chord is unique, there are many ways to approach this problem, and that is part of the challenge of working with concepts like this.
Let me give you some examples of how this works. These are all taken from actual notes in the piece. The first group of tones come from passages that state the tones with individualized partial vibrato.
Example 1 (15 seconds): The easiest and most familiar harmony to use, derived from the “chord of nature” itself, is the major triad. I use this in a descending series, as follows:
1. Notes of the chord itself are introduced. These are 32, 24, 20, 16, 12, 10, 8, 6, 5, 4, 2 and 1.
2. Next, I used partials 31, 26, 30 and 25. When combined with partials 5, 10 and 20 (the third), these form another major triad starting on E. The pair 31 and 26 and 25 and 30 are both close to being a major third and fifth from E, but next to one another, these are quite dissonant.
3. Next, a transposition of the chord to the seventh overtone (a major seventh) are introduced. These are partials 28, 22, 19, 14, 11, and 7.
4. Next, another transposition, to the ninth partial, is brought in. These are 27, 23, 18 and 9.
5. Finally, the remaining partials come in: 29, 21, 17, 15, 13.
Example 2 (7.5 seconds): The minor triad, horrendously dissonant with the “chord of nature”, has a very different structure. In this case, I used transpositions of the triad to different partials, but without all the octave duplications, so I could use several:
1. First, the transposition to the perfect fifth (partial 3): 18, 14, 12, 9, 7, 6, 3.
2. Next, transposition to the major third: 30, 24, 20, 15, 10.
3. Next, transposition to the minor third: 29, 23, 19.
4. Finally, the rest, with an incomplete attempt at transposition to the minor sixth: 32, 27, 16, 8, 26, 21, 17, 13, 5, 4, 2, 1.
Example 3 (7.5 seconds): This example shows the tetrachord 0247, also a descending series:
1. First, transposition to the fundamental: 32, 24, 20, 18, 16, 12, 10, 9, 8, 6, 5, 4, 2, 1.
2. Second, transposition to the major third, assuming that note from the first group: 30, 26, 23, 15, 13.
3. Third, transposition to the minor second, including the fifth (above the second) from the second group: 22, 19, 17, 11.
4. Finally, the rest: 31, 29, 28, 27, 25, 21, 14, 7.
Example 4 (7.5 seconds): The next example is of the hexachord 012346, another descending series:
1. First, the transposition to the fundamental: partials 32, 23, 20, 19, 18, 17, 16, 10, 9, 8, 5, 4, 2, 1.
2. Next, transposition up a fifth, which includes a common tone with the first collection (1, represented by partial 17): 28, 27, 25, 24, 14, 12, 7, 6, 3.
3. Finally, the rest: 31, 30, 29, 26, 22, 21, 15, 13, 11.
The next group of tones come from passages that state these overtone series in connection with glissandos. These are all associated with ascending series. The first one, based on the trichord 013, shows the way in which the overtones “dissolve” and the coalesce into another tone.
Example 5 (37.5 seconds): The series is based on the trichord 013:
1. First, the primary harmony is stated in partials 1, 2, 4, 8, 16, 17, 19, and 32.
2. Next, the transposition up a fifth: partials 3, 6, 7, 14, 24, 26 and 28.
3. Third, the transposition up a fourth: partials 21, 23 and 25.
4. Next, the subset 02 transposed up a second: partials 5, 10, 20, 9 and 18.
5. Fifth, the remaining partials: 15, 30, 27, 29, 31.
Example 6 (30 seconds): The next shows the pentachord 02457 in the same manner:
1. First, the original collection: partials 1, 2,3, 4, 6, 8, 9, 10, 11, 12, 16, 18, 20, 22, 24, 32.
2. Second, a transposition of the subset 025 to the tritone: partials 15, 23, 25 and 30.
3. Third, a transposition of the subset 027 to a minor third: 7, 14, 19, 21 and 28.
4. Finally, the remaining partials: 13 17 26 27 29 31.
Example 7 (15 seconds): The last example shows the trichord 014 in a descending series on an instrument that combines both the glissando and individual partial vibrato:
1. First, the notes comprising the harmony are stated: partials 32, 26, 24, 16, 13, 12, 8, 6, 4, 3, 2, and 1.
2. Second, transposition up a fourth: partials 29, 23 and 21.
3. Next, a related interval, the perfect fifth, transposed up a major sixth: partials 27, 20, 10 and 5.
4. Next, a related form, the major triad, transposed up a tritone: partials 22, 17, 14, 11, and 7.
5. Finally, the remaining partials: 31, 30, 28, 25, 19, 18, 15, and 9.
The following table shows each of the partial series used in the piece, with the vertical separator “|” indicating breaks in overtone groups:
1. Trichords, 014-group (descending)
| Chord | Overtone Series |
|---|---|
| 014 | 32 26 24 16 13 12 8 6 4 3 2 1 | 29 23 21 | 27 20 10 5 | 22 17 14 11 7 | 31 30 28 25 19 18 15 9 |
| 034 | 32 30 24 16 15 12 8 6 4 3 2 1 | 29 27 21 | 28 26 22 14 13 11 7 | 20 19 10 5 | 18 17 9 | 31 25 23 |
| 037 | 18 14 12 9 7 6 3 | 30 24 20 15 10 | 29 23 19 | 32 27 16 8 | 26 21 17 13 |5 4 2 1 | 31 28 25 22 11 |
| 047 | 32 24 20 16 12 10 8 6 5 4 2 1 | 31 26 30 25 | 28 22 19 14 11 7 | 27 23 18 9 | 29 21 17 15 13 |
2. Trichords, 013-group (ascending)
| Chord | Overtone Series |
|---|---|
| 013 | 1 2 4 8 16 17 19 32 | 3 6 7 12 13 14 24 26 28 | 21 23 25 | 5 10 20 9 18 | 11 22 | 15 30 27 29 31 |
| 023 | 1 2 4 8 9 16 18 19 | 3 6 7 12 13 14 24 26 28 | 21 23 25 | 5 10 20 | 11 22 15 30 | 17 23 27 29 31 |
| 025 | 1 2 4 8 9 11 16 18 22 32 | 3 6 12 13 15 24 26 30 | 23 25 29 | 5 10 20 | 7 14 28 | 17 19 21 27 31 |
| 035 | 1 2 4 8 16 19 21 32 | 3 6 9 11 12 18 22 24 | 5 10 17 20 23 | 7 14 28 | 13 26 | 15 30 26 27 29 31 |
3. Tetrachords, 0124-group (descending)
| Chord | Overtone Series |
|---|---|
| 0124 | 32 20 18 17 16 10 9 8 5 4 2 1 | 30 27 25 24 15 12 6 3 | 31 28 26 14 13 7 | 29 23 22 21 19 11 |
| 0247 | 32 24 20 18 16 12 10 9 8 6 5 4 3 2 1 | 30 26 23 15 13 22 19 17 11 | 31 29 28 27 25 21 14 7 |
| 0234 | 32 20 19 18 16 10 9 8 5 4 2 1 | 30 28 26 24 15 14 13 12 7 6 3 | 31 29 27 25 23 22 21 17 11 |
| 0357 | 32 24 21 19 16 12 8 6 4 3 2 1 | 30 22 20 18 15 11 10 9 5 | 31 29 28 27 26 25 23 17 14 13 7 |
4. Pentachords, 01235-group (ascending)
| Chord | Overtone Series |
|---|---|
| 01235 | 1 2 4 8 9 16 17 18 19 21 32 | 3 6 11 12 15 22 24 25 30 | 5 7 10 13 14 20 23 26 27 28 29 31 |
| 02457 | 1 2 3 4 5 6 8 9 10 11 12 16 18 20 22 24 32 | 15 23 25 30 | 7 14 19 21 28 | 13 17 26 27 29 31 |
| 02345 | 1 2 4 5 8 9 10 11 16 18 19 20 22 32 | 13 14 15 23 26 27 28 30 | 3 6 7 12 17 21 24 25 29 31 |
| 02357 | 1 2 3 4 6 8 9 12 16 18 19 22 24 32 | 7 14 21 23 25 28 | 5 10 11 13 15 17 20 26 27 29 30 31 |
Hexachords, 012346-group (descending)
| Chord | Overtone Series |
|---|---|
| 012346 | 32 23 20 19 18 17 16 10 9 8 5 4 2 1 | 28 27 25 24 14 12 7 6 3 | 31 30 29 26 22 21 15 13 11 |
| 024679 | 32 23 20 19 18 17 16 10 9 8 5 4 2 1 | 28 27 25 24 14 12 7 6 3 | 31 30 29 26 22 21 15 13 11 |
| 023456 | 32 23 21 20 19 18 16 19 9 8 5 4 2 1 | 30 28 27 24 17 15 14 12 7 6 3 | 31 29 26 25 22 13 11 |
| 023579 | 32 26 24 21 19 18 16 13 12 9 8 6 4 3 2 1 | 30 25 22 20 17 15 11 10 5 | 31 29 28 27 23 14 7 |
All instruments used in this work employ these methods of introducing separate harmonics individually in these disparate orders., and they employ either vibrato or glissando. The climax occurs in the fifth section, which has an instrument that combines both vibrato and glissando; but instead of making the glissando up or down to the next pitch, these notes simply go up by a minor third and then back down. This is the section based on hexachords, so the texture is thicker anyway, and many of these events occur together, as chords.
The piece is in six sections, beginning with a thin trichordal texture and building by accretion to more complicated harmonies and textures. Each successive harmony is formed by adding one note to the chord from the preceding section, until a hexachordal texture is reached. The piece grows dynamically in a manner like 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 sketched while I was visiting Singapore to attend the International Computer Music Conference in 2003, but I could not produce it until I returned home. It, as almost all of electronic music, was synthesized using the csound program.
