I have always been interested in tones containing harmonic partials, particularly when I am able to control their frequencies and amplitudes independently. When a tone is generated by carefully introducing each partial higher than the fundamental independently, as fascinating thing happens. First, you hear the individual partials as tones in themselves, but after several of them are sounding, a new tone emerges – the fundamental – and then, instead of hearing independent tones, we hear the collection of them as the timbre of the sound. My composition Emergence plays with this concept as one of its central ideas.
Another property which I have also used in past compositions is that of playing the partials of a tone in a specific order, in order to group certain partials together so that they produce an independent sonority to contrast with the other tones in a particular musical context. Every harmonic partial can be thought of as an interval above the fundamental. These are all pure intervals, and they are out of tune with respect to our 12-tone equally-tempered scale. Some of them differ by very small amounts, which we might not notice if they are presented successively. When they are presented simultaneously, we are more likely to notice pitch discrepancies, which we might hear as being out of tune. For example, the difference between the eighth and ninth partials and the ninth and tenth partials are both major seconds, but one is flat and the other is sharp. These ratios of frequencies were originally used to create scales, and it was only after the discovery of logarithms that the mathematics of equal temperament could be worked out.
In order to think of almost all of the partials as “notes” in order to create “harmonies” from them, I assigned each partial to the closest note of the scale that they were next to. While the low intervals up to the eighth partial produce all the intervals of the major triad (or, rather, the dominant seventh chord), it is only above the sixteenth that you begin to obtain useful minor seconds and thirds and other notes of the scale. According to the scheme I devised, all harmonies are created by starting with the sixteenth partial. The sixteenth is regarded as the “zeroth” interval, with 17 the half step, 18 the whole step, 19 the minor third, 20 the major third, 21 the perfect fourth, 23 the tritone, and 24 the fifth. The complete scale is shown in table 1.
Partial Interval Partial Interval Partial Interval Partial Interval 1 0 9 .02 17 .01 25 <.08 2 0 10 .04 18 .02 26 >.08 3 .07 11 <.06 19 .03 27 .09 4 0 12 .07 20 .04 28 .10 5 .04 13 >.08 21 .05 29 >.10 6 .07 14 .10 22 <.06 30 .11 7 .10 15 .11 23 >.06 31 >.11 8 0 16 0 24 .07 32 0 Table 1: Partials as intervals above the fundamentals (in 8ve.pc form).
To explain this a bit further, while it is clear that all intervals except the octave are out of tune with the equally tempered scale, I accept the perfect fifth, major third, major seventh, and major second uncritically. The eleventh partial is more out of tune with the tritone than these are with respect to their intervals, but if we need a to accept tritone among the partials, a choice must be made. The other interval where a difficult choice is necessary is the minor sixth, but I took the 26th since it has an octave duplication.
The complete process I devised to state a harmony from the partials involves stating the harmony upwards from the sixteenth partial to the 32nd, which is an octave duplication of the sixteenth, and then the lower intervals, upwards by octaves; then a transposition of the harmony, if possible; and finally the residue from the lowest to the highest. This last step is important, because stating the remaining partials in this manner makes them less noticeable than the ones involved in creating the harmony, but they contribute to the formation of the fundamental.
There are three different instruments which I have used in the piece, all of which use harmonic partial sequences. The first one, which opens the work, attacks the partials individually. The second one plays a cyclic series of the partial sequence, and the third is a complex envelope.
Here is an example of this process with the harmony 0357, which is the first sound heard in the composition. The sequence starts with 16, 18, 21, 24, 32, 8, 9, 10, 4, 6, 2, 3, 1, which are the basic intervals of the 0357 chord. The next partials are 22, 28, 30, 17, 11, 14, 17, 7, which create a transposition of the harmony a tritone above the fundamental. The residue is partials 5, 9, 10, 13, 20, 23, 25, 26, 29 and 31.
Before playing an example of this sound, I need to explain one further important aspect of the instrument, which is the amount of overlap between the partials as they are presented. If there is no overlap between the partials, there will be no perception of the fundamental; we will just hear a series of separate tones. In fact, since the partials are so spread around and so few contiguous partials are stated, we don’t get much of the fundamental until at least six or eight partials are overlapped. The overlapping is a parameter of the instrument, so that I can control the amount of this perception.
Finally, there is one more important aspect of the instrument: the number of cycles of the overtone series that is played on a given note. I wanted this to be completely flexible, where I could have any number of cycles as well as fractions of a cycle. On a basic level, this will be perceived as the speed of the sequence of tones, and in the context of the music there are different speeds going on at different times.
Example 1 plays the first tone heard in the piece, which is an E-flat in the second octave below middle C with the 0357 harmony and an overlap of eight partials [example 1].
While some sense of the fundamental emerges after the fundamental itself is stated at the thirteenth partial, it isn’t very persistent, and we mostly hear a series of disparate tones, although the overlapping of the partials does create a sense of “cluster” that is more prevalent at the end of the tone, when the partials are more close together. Example 2 presents the same tone, but with the overlapping increasing to 14 partials [example 2]. From the time that the first series unfolds at the thirteenth stated partial, while we still hear new partials entering, the fundamental is more prevalent, and we can begin to hear the newly entering tones as part of a change in harmony as much as the entrance of a new tone. This is exactly the kind of borderline perception that the piece abounds in.
Example 3 gives another example of the use of this instrument, this time in a different context. There are five cycles of the partial sequence presented over the course of the duration, so the speed of the attacking partials is greater. The overlapping factor is 12 [example 3]. Of course, this occurs in a context where many notes are entering and leaving, so it is hard to judge the effect of that context from this example alone.
The second instrument in the piece plays the partial sequence in order, but in this case there is always the sense of the fundamental frequency and the overtones creating the timbre of the sound, because they are all always present. The overtones are brought out from the background by means of a bell-curve-like amplitude function, which in turn brings out each of the partials in the order of the series. The timbres all sound unique. Here is an example of the first use of this instrument in the piece [example 4]. The pitch is C three octaves below middle C, and the harmony is 014, so the partial sequence is 16, 17, 20, 32, 8,10, 4, 5, 2, 1; followed by 20, 24, 27, 10, 12, 5, 6, 3 (t(3)); and the residue is 7, 9, 11, 13, 14, 15, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31. This instrument is also capable of playing multiple cycles of the partial sequence, but on this note it happens only once. Notice also one other property: this note moves from the right speaker to the left and back to where it began over the course of its duration. The movement follows the cycle of the overtones, so wherever the note begins it moves to the right (unless it begins at the extreme right, as this note did) and left and back to where it began.
To contrast that note with a different partial sequence, let me play another note which occurs a bit later in this passage. The pitch is F# in the third octave below middle C, and the harmony is 047, so the overtone sequence is 16, 20, 24, 32, 8, 10, 12, 4, 5, 6, 2, 3, 1; followed by 18, 22, 26, 9, 11, 13 (T(2)); and the residue is 7, 14, 15, 17, 19, 21, 23, 25, 27, 28, 30, 31 [example 5]. I think you will agree that the timbres are very different.
One of the things that this instrument can do is to play any number of partials (up to 32), and there is one passage in the piece that uses only partials above 16. Example 6 plays a B above middle C. The harmony is 0146, and the overtone sequence is 16, 17, 20, 22, 32, 8, 10, 11, 21, 24, 28, 30, 12, 14, 15, a total of 15 partials [example 6]. Here is another example of the same overtone sequence, but the pitch is C-sharp two octaves below middle C [example 7]. The pitch is more perceptible in the second example, but the shifting timbre is what captures most of the listener’s attention.
Complex envelopes present the same overtone sequences, but with just one cycle, and each envelope is simply an exponential attack and decay reaching prominence at a particular portion of the duration. There is never a question of the pitch, but the timbre change is quite evident. These are useful more for short tones. Example 8 is the first such tone that occurs in the piece: the pitch is a B two octaves and a semitone below middle C, and this duration is quite long, 15 seconds [example 8]. Example 9 gives a more usual example: G an octave and a fourth below middle C, with a duration of 2.75 seconds [example 9]. At the very end of the piece, a short passage returns as a kind of echo of the previous music. Example 10 demonstrates one of these notes, middle C [example 10].
Now let me describe the composition itself. It opens with the first note I played above, the E-flat that is part of a 0357 harmony, and this is the only note for the first 7.5 seconds. Then the other notes of this chord enter, one at a time – G, F and C. After these have been stated for a brief time (and they continue), other notes forming a new harmony are introduced, and the section continues for a minute and 121 seconds. As each new note enters, the speed of the partials attached increases slightly, so that they are going almost twice as fast at the end of the section. The increase in speed means that the tones are undergoing more than one cycle of the partial sequence. Also, as the section proceeds, the amount of overlap of the partials increases from 8 to 14 partials, so that by the latter part of the section, the fundamentals are much more prominent than they were at the beginning.
The second section (all sections overlap the previous one) introduces the second instrument I described above, which plays fading partial sequences where there is always the sense of a fundamental, and the listener’s attention is focused on the changing timbres. All these notes are in the third octave below middle C, the lowest complete octave on the piano, so the overtones are in the range where we can hear them quite precisely. The third section introduces the complex envelope instrument, and the music expands to cover a five-octave range, with both long and short notes contrasting with each other.
The fourth section, which is two minutes long, the longest section to maintain the same texture in the piece, uses just the instrument that attacks partials independently, and it is a fascinating interplay between the individual overtones and the fundamentals that emerge out of them. The next part, which is also quite long, uses fading partial sequences, but only partials from the sixteenth and above that are the initial part of the sequence in which the “harmony” is stated. This passage begins with fundamentals in the middle C octave, but the partials are in the fifth octave above middle C, which includes the highest note on the piano. As it continues, the music drops down one octave and then another, and when it is ready to drop another octave, the section 6, the climax, occurs. This passage includes both the individually-attacking instrument and the fading partial series, with the latter in the lowest octave and the individual partials in the two octaves above. The final section is a quiet echo of the preceding music, including notes that occurred kin that passage in the same order, but in different octaves on the complex envelope instrument. The last note dies out as a solo, as the piece began.