Harmonic Ordering in Quintet: A Use of Harmonics as Horizontal and Vertical Determinants
I'm not a theorist. All I can do is compose each piece as I must, and then try to figure out what I've done in hopes of making it easier to compose the next one. So what I say will undoubtedly seem terribly banausic, but that's what must come of asking a mere workman to speak. I would hope, of course, that what I've noticed in myself might also be possible of generalization into something of use to others. But that's for others to do - I can but speak of what I know from experience.
Each piece imposes new problems, of course, but there are necessarily some technical matters that carry over from one to another, being not compositional problems, but inherent modes of perception or thought. One such is a strong - and unwilled - tendency, while composing, to hear the notes in my imagination as if they were related by ideal harmonic ratios and creating (or at least strongly implying) resultant tones - both difference and summation. This seems to happen in the case of both harmonic and melodic juxtaposition. Indeed the first instance of it that I was forced to notice was melodic, and occurred while I was writing my first string quartet when a melodic succession of an E and an F seemed to demand that the next note be a quarter-sharp E in the octave above. This, I later decided, must have been an instance of a 16th harmonic reacting with a 17th to demand a 33rd.
Example 1: The microtone symbols I've used in my 72-note music since 1970
Some particularly perspicuous instances of that presented themselves in my recent Quintet for clarinet and strings, and I would like to discuss some of them. Before I do, I should show you my notation (like everyone else, I have invented my own) and the scale I use. The notation consists of combinations of the 8 signs given in Example 1 above. (I'm pleased with it: it seems to cause performers very little trouble - they pick it up very quickly.)
The scale (Example 2) is very like Prof. Karl Richter Herf's Ecmelic Scale and the so-called "harmonic scale" that Wendy Carlos seems to have recently begun to use. It can be transposed to begin on any of the 72 notes that my notation can describe, in a way analogous to that in which the diatonic scale can begin on any of the 12 chromatic notes. The ratios written above and below the notes give the relation I understand (and expect) each note to have with the Tonic. When I talk about harmonic relations between the notes, I shall speak of them always in terms of their harmonic number - the numerator of the ratio - since the frequency of the fundamental can always be factored out of the frequencies of any particular set of harmonics.
Example 2: The scale I've used since 1978, given in its C transposition.
me start with the first slow movement of the Quintet. When the piece had reached that point, it became clear that it would be in E
(Example 3) and that it would start with the pedal B and F on which the 1st
movement had ended (Example 4a, Tape Example 1). Those notes, in that spacing, would relate as
17 to 6. Immediately, the clarinet
wanted to enter on the 1/6-high A#, which is the 23rd harmonic, the summation
tone of the B and F (Example 4b, Tape Example 2). The clarinet line that then suggested itself
is given in Example 4c (Tape Example 3).
The harmonic numbers of its notes are: 23, 24, 21, 20, 22. This line made me hear the bass line in
Example 4d (Tape Example 4). But, as you
will have noticed, the B at the end of the bass phrase implies, with the
clarinet line, a resultant not of the 17th harmonic, but the 16th: which made the moment seem to demand an
Example 3: The scale in its transposition to E.
Example 4: The generation of the first material in Movement II: a. the pivot pitches. b. the resultant of those pitches (the clarinet's starting pitch. c. the clarinet's line. d. the resultant bass line
Tape Example 5 is a computer simulation of the whole complex. I think it's worth noticing in the tape example how the final clarinet note doesn't sound clearly like the 22nd harmonic until the F (which wants the clarinet to be on the 23rd) resolves to the E (which wants it to be on the 22nd).
There follow a couple of repetitions of the phrase, each time varied rhythmically in ways that cause the notes to anticipate or delay the moment when they would form the basic resultant structures.
Then follow 4 chords that take the piece to the key of B, and are so made that with the exception of the dissonant notes in the clarinet's pedal, any note is implied by some pair of the other notes of the chord, either as combination or difference tone. The clarinet holds its note from the first chord as a sort of decorated pedal, which is resolved to a resultant at the arrival of the second, varied, statement of the first section (Example 5).
Example 5: Traditional chords, Movement II, mm. 121-122
Without the clarinet and its pedal, they sound as in Tape Example 6, with it, as in Tape Example 7 includes it. I find it interesting that the clarinet note in the 4th chord, being dissonant (that is, not a resultant of the other notes), seems too loud, unless programmed a bit softer than in those chords where it's consonant (that is, a participant in the resultant procedure).
Some more development of this material leads to a second large section, where the process occurs in a somewhat looser way, enriched with non-essential notes (Example 6). Tape example 8 is a, lamentably metronomic, simulation of it.
Now, I would be hard put to tell you just what I was thinking while working on this. Obviously, I was not applying some mechanical process: it would be apparent to anyone doing a casual analysis of the movement that not all the notes arise from so simple a process of generation. I certainly followed instinct when what it suggested seemed stronger and more meaningful than what my intellectual mind could invent. Usually, in fact, that instinct won't let me do otherwise: my verbal, analytic, mind is always powerless to overcome the intuitive (not without doing damage to the piece, it would seem). Like the American Congress, its essential role is to advise and consent.
Example 6: Second material, Movement II, mm.148-144
An example of this occurs in the section in F# (Example 7). I've written the harmonic numbers beside the notes of the accompanying voices. You'll notice that certain chords don't work out as resultant-chords: the second half of the third beat of the second measure, for example, or the first chord of the third measure. For that matter, the last chord of the second measure would work as resultants only if the bass were an octave higher. What this means, I can't be sure of. I often find it happening that way. Perhaps the ear, so long as it has a clear and simple set of relationships to hear, isn't terribly fussy about which octaves the notes are sounded in unless the transposition causes conflicts with other notes, equally cogently related.
Example 7: Movement II, mm. 149-152
And I've no idea of the harmonic generation of the viola line. As with the first chord of the 3rd measure, that's the way it goes, I may have had, when composing it, a conscious explanation of why it goes that way, but if I did, it has now vanished from my mind. Tape Example 9 gives the chords of that section. Tape Example 10 includes the viola line.
I don't remember this matter of resultants being in my mind while I was working on the 3rd movement. The consistent use of parallel 2nds,however, must no doubt have the effect of suggesting the tonic that would be the resultant of them all if they were dwelt on longer.
In the fourth movement, I found it useful to test the chords of the chorale-like sections by a concept of resultants that may or may not have any justification. But it did help the piece arrive at a series of well-ordered chords that had the repose combined with a sense of direction that seemed to be necessary. The first set is shown in Example 8a. The note names in the boxes indicate the tonic of the scale from which the notes of the chord are drawn. I'll speak of them after I discuss the chords themselves.
Example 8: First chordal section, Movement IV, mm. 248-252. a. the actual chords played. b. the implied resultant chords.
I'm not knowledgeable enough in psychoacoustics and perception to speak with categorical assurance about this. But it did seem to help if the adjacent notes of these complex chords related in such a way that, in isolation, they might produce difference tones that would, in the aggregate, form clear and simple chords.
It seemed further desirable that those implied resultant chords should relate smoothly and directionally, if the actual ones were to do so, as in Example 8b. The actual chords are presented in Tape Example 11. The chords implied by the difference tones of adjacent pairs are heard in Tape Example 12. I must wonder whether the coherence of the passage isn't somehow a function of our subliminal awareness of what is to be heard in Tape Example 13.
Now, about that sequence of keys. As I said, each chord of the sequence we just heard is made from the notes of the scale named in the box. The sequence is the fundamental one that has permeated the whole piece.
The piece is not in any way a chaconne, but that pattern of keys, the first 6 odd-numbered harmonics of the paramount key of G, transposed as necessary to begin on any of themselves, or - as happens in at least one case - using the harmonics of one key to imitate those of another, or - in another section - beginning on a more distant harmonic (the 15th). Often in permuted order, or abbreviated, in some places varied and developed in fairly obvious ways, this sequence governs the whole key structure of the piece, the large-scale movement of the paramount keys of large sections and whole movements, and that of subordinate sections within the large ones.
If, as my experience would seem to suggest, we can hear pairs of melodic pitches suggest their combination tones, I should think we must be equally able to hear them suggest difference tones. As we seem automatically to derive a putative fundamental from a vertical combination of appropriate harmonic tones, so, I should think, must we do for a melodic succession of such harmonic tones. Thus, it would seem to me, a section of music, made up of harmonically related pitches ( and it's my experience that the ear will try to find some such relation, however far-fetched and nonsensical that may have to be, between any set of pitches presented to it), must imply a single fundamental which can then, in an analytical context, stand surrogate for the whole complex that implies it.
I should think that a succession of such fundamentals, if harmonically related, must themselves imply a single superior fundamental no matter how many layers of subordination one may need to contrive. (This is a way the Schenkerian intuition could be salvaged for music using harmonic combinations passing the 5-limit of the triad for its structural materials.)
In the light of this, let's look at the first movement (Example 9). The example is merely a chart of the key sequence of the movement, giving the measure number each key change occurs in. The first ten measures are governed by a parent tonic of G: The keys appearing in mm. 1, 4, 5, 6, 7, and 8, have as their fundamentals harmonics 1, 3, 5, 7, 9, and 11 of G. In measure 11, the same sequence is repeated, this time in relation to 1/4-low C#, the 11th harmonic of G. In m. 21, it repeats again, rising above D, the 3rd harmonic of G. In m. 27, yet again - above 1/12th-low B, the 5th harmonic of G. At m. 34, it arrives at A and sticks there for a while, even though it suggests the keys of the sequence as it goes along. In mm. 54-55, it runs quickly through the sequence, this time based on 1/6th-low F, the seventh harmonic of G, and arrives at 1/4-low C# ( the 11th harmonic) for a development section.
Here, for some reason, it chose only to imitate (with certain variations) the sequence, not to present it exactly - the keynotes are the nearest (granted that variation) to harmonics of 1/4-low C# that can be derived from the G scale. I suppose this happens because my instincts wanted the freshness of new keys while emphasizing relations that would imply the overall tonic.
A similar thing happens in mm. 95-108, where the most distantly related parent key is reached: the 1/12th-low F#, the 15th harmonic of G - not a member of the governing sequence at all, but still in the G family.
Here again, the tonics are not quite those of the F# family. The C# is but it can also be heard as the 45th of G (quite distant, I admit), but the 1/6th-low C must be more reasonably related to G, as its 21st, than to F#, as its 11th.
Example 9: Key sequence of Movement I. The numbers above the notes are those of the measure in which the key appears.
At m. 109, - the nearest thing to a recapitulation that this movement has (the real one occurs in the third movement) - takes us briskly through the most fundamental series (that of G) and then we go back to the overall tonic (which is, of course, G). I think the move from 1/4-low C# to G at the end serves very well to get us home, but keeps us from settling down there. So what we have is a series of smaller blocks, each suggesting a generating tonic (Example 10) and those tonics are themselves drawn from the harmonic series of one fundamental, G. This seems to me the way the movement ties itself together, supporting the span of its development with a harmonic device of some strength and cogency.
Example 10: The sequence of governing tonics in Movement I.
The numbers indicate the measure in which the set of modulations governed by that tonic begins.
Whether such a thing as I discussed in connection with the second movement is the governing process in such a tenuous situation, I've no idea. But the keys of my pieces do, in general, keep choosing themselves from among the harmonics of the main key (for the larger divisions) and the harmonics of those (for the smaller divisions).
It's obviously not cut and dried. As this quick glance at the first movement shows, the main key seems to have some sort of over-riding power to prevent too great excursions into the distantly related. There also seems to be some analogue of parallel keys and major/minor modes in operation, though that is only lightly apparent here - in the 1/4-low C# section of the first movement, for instance - and I've chosen not to discuss it.
I must apologize for giving you little more than a quick tour of the piece. I do hope that it has no been wasted: that perhaps it made sufficient sense that some one would find the idea interesting enough to take it and work it into a proper analytic and prescriptive theory. (Unless, of course, someone has already done so, I, in my ignorance, have been urging a well-known concept.)
Tape example 14 is the first performance of the piece, given by the Boston new music group, Dinosaur Annex, on March 20, 1988.