Tortoiseshell Music Home of composer Nathan Curtis

For a while, I've been meaning to have a semi-regular feature on this blog, where I take a look at a particular piece of music and try to tell you what makes it important to me. I'll be tackling a wide range of music: my own compositions and works by other artists; pieces you might not have heard before and pieces that are probably familiar to you. I'll also try to keep up a schedule of at least one featured piece per week, ideally on Tuesdays.

Why Tuesday? Well, partly because of the piece I'm featuring to kick off this series: my composition Fanfare for Tuesday, for solo trombone. And why is it called Fanfare for Tuesday? Well, I started writing the piece on Tuesday, April 8th of this year, and finished it the next day. And I think Fanfare for Wednesday would be an inferior title; don't you agree? By now you're probably wondering what this fanfare sounds like. Well, look no further:

Fanfare for Tuesday

So what's special about this little fanfare? Well, for me, it represents an important step in exploring a more plastic conception of rhythm. To further explain what I mean, I want to first look at the way musicians learn about rhythm in the Western notated music tradition. Note the use of the word "notated" in the previous sentence: not only do I wish to differentiate notated musical traditions (a.k.a. "classical music") from primarily oral/aural musical traditions ("folk music"), but I specifically wish to draw attention to the way that the notation itself has come to shape our conception of rhythm.

At the lowest level of rhythmic organization¹, we have conceptual units which are typically called "beats" and "pulses" (for now, I am ignoring higher-level conceptual units like measures and phrases). Now, what do I mean by beats and pulses? Well, it's hard to explain a priori without falling into some sort of circular definition. Peter Westergaard gives an excellent explanation of rhythmic organization at all conceptual levels in chapters 7 and 8 of An Introduction to Tonal Theory, but I don't have the time or space to do him justice here. However, I can give some illustrative examples. In general, beats are what conductors are ostensibly² demarcating when they are marking time; they are the units which a marching band will be in step with; they are the points in time that you would typically accentuate with your movements when dancing along to music. (One caveat: musicians use "beat" to denote three different things: a durational unit spanning two points in time, a single point of time which is the onset of that durational unit, or the grid of durational units or points in time generated by the individual beats. I will not attempt to explicitly indicate which sense of "beat" I am using at any given time. You have been warned.) Pulses, in turn, are a more-or-less even subdivision of the beat; the pulse typically corresponds to the shortest note-values used in a given piece, and governs the length of most longer note-values as well. Confused? Well, it's time for our music lessons.

Of course, just as kindergarteners do not start learning math through axiomatic set theory, beginning music students do not start out with such a general concept of rhythm. What they first learn is that beats can be broken down into pulses through binary subdivisions. For example, a quarter-note beat can be divided in two, generating an underlying eighth-note pulse:

Ex. 1 (mp3)

Those eighth notes could be further subdivided in two to create a sixteenth-note pulse, now with four pulses for every quarter-note beat:

Ex. 2 (mp3)

Our notational system is designed around this idea of binary subdivision, so written note values and sounding beats and pulses seem to go hand in hand. But soon, we learn that beats and pulses do not have to be related strictly through binary subdivisions. Hearkening back to our first example, we could have that same quarter-note beat subdivided into three equal parts, which we notate as triplet eighth notes:

Ex. 3 (mp3)

Conversely, we could instead maintain the eighth-note pulse of the first example, and group them in threes to create a beat of dotted quarter notes:

Ex. 4 (mp3)

At this point, a sufficiently clever observer might ask, "What's the difference between these last two examples? Sure, the beats and pulses in the second mp3 are slower, but can't we control the tempo [i.e., the overall speed of the music, often measured in beats per minute] as well? If you increase the tempo of Example 4 by 50%, isn't the second half aurally indistinguishable from Example 3?" And, to a degree, they have a valid point: if you have a steady beat with a steady underlying ternary pulse, and can set the tempo as you desire, then the only difference the quarter-note/triplet-eighth-note notation and the dotted-quarter-note/eighth-note notation is a psychological effect on the performer, which may affect the performance but theoretically should not. However, this only applies when the beat and the subdivision of the beat are both steady, which brings us to our next level of rhythmic and notational complexity: the beat and/or pulse do not have to be steady. We can, for example, have a steady quarter-note beat which is alternately subdivided into two or three parts, for a pulse which shifts between eighth notes and triplet eighth notes:³

Ex. 5 (mp3)

Or we can have a steady eighth-note pulse which is alternately grouped in twos and threes, for a beat alternating between quarter notes and dotted quarter notes:

Ex. 6 (mp3)

These last two examples are in fact quite different; there is no way that we can uniformly speed up one of the examples to exactly match the other. And we've only begun to scratch the surface of what is possible when we allow the beat and/or pulse to change. For example, we could allow both the beat and the pulse to shift at different points in time:

Ex. 7 (mp3)

Or they can both be shifting at the same time:

Ex. 8 (mp3)

Or they can get even weirder:

Ex. 9 (mp3)

And we've still only considered binary and ternary subdivisions of the beat. We can throw in higher subdivisions, like a quinternary subdivision of quarter notes into quintuplet sixteenth notes:

Ex. 10 (mp3)

However, when we start with a constant pulse and create groupings of more than three pulses at once, we usually perceive the beat as some intermediate (and possibly irregular) grouping of 2s and 3s, so we haven't opened up many new possibilities in that direction.

Now, I have been talking about triplets and quintuplets, and showing examples of them in action, but what do we mean by "tuplets" in general? A tuplet is usually thought of as a (non-binary) grouping of notes which occupies the same amount of time as one of the basic note values. What's a triplet eighth note? Well, 3 of them together are exactly as long as a quarter note. And what's a septuplet thirty-second note? 7 of them together are as long as an eight note. For many musicians, even those who are familiar with a good bit of the contemporary repertoire, this is the full extent of their trained understanding of tuplets, and taken together with the ideas introduced in the preceeding paragraphs, may be a decent summary of their full conception of notated rhythm. But to get to Tuesday, we need to go a couple of steps further.

So far, we've been looking at tuplets as subdivisions of basic binary note values. But do we always have to start with basic binary note values? What if we want to have, say, five tuplet eighth notes which occupy the space of a dotted quarter note? (note: a single dot adds 50% to the length of a note) Well, there are at least two widely used ways of notating this. One way is to explicitly specify, as part of the tuplet marking, (which ordinarily consists of a number, indicating the number of durational units grouped together, and an optional bracket demarcating the particular notes which fall into that grouping) the note value which is equal to the length of the whole tuplet together. So, for five tuplet eighth notes in the space of a dotted quarter note:

The other option is to think about this tuplet in terms of its constituent units, which are notionally eighth notes. A dotted quarter note is equal to three eighth notes, so we have five tuplet eighth notes in the space of three eighth notes. Five in the space of three. We can express this as a ratio:

In either notational convention, we may choose to omit the extra details (the note value in the first convention, the denominator of the ratio in the second example) if the tuplet is a division of a basic note value like a quarter note or half note, but if the duration of your tuplet groupings is constantly changing, you're better off overnotating in this case. I personally prefer to use ratios when applicable, but that's mostly just an aesthetic preference. Either way, when you throw in the idea that tuplets don't have to add up to basic binary note values, you reach the limit of rhythmic understanding for many experienced performers of contemporary music, and also you have the tools for understanding nearly everything I have written prior to Fanfare for Tuesday.⁴ But we're not done yet.

All this time, we've been treating tuplets as groupings of notes. Recall:

What's a triplet eighth note? Well, 3 of them together are exactly as long as a quarter note.

In my answer, I assiduously avoided talking about an individual triplet eighth note. Of course, from the definition I gave, it is immediately clear that a single triplet eighth note is one-third of the length of a quarter note. But the way we are trained to think about rhythm, that's not how we're supposed to think of it. A triplet eighth note is supposed to be part of, well, a triplet. They're like quarks; you don't encounter triplet eighth notes in isolation, only "bound" in a group of three triplet eighth notes.⁵ And there are reasonable justifications for thinking about tuplets as groupings: for example, the way most people perceive time, the easiest way to conceive of a timespan which is one-third as long as a given time-span is to divide that longer time-span into three equal pieces.

But why does it have to be that way? Once we get used to playing triplets and other tuplets, it becomes easier to conceptualize a single triplet eighth note in isolation. Why not just say that a triplet eighth note is one-third as long as a quarter note? Or, since a quarter note is twice as long as an eighth note, say that a triplet eight note is two-thirds as long as an eighth note? In fact, we can generalize: a single triplet of any note value is two-thirds as long as the original note value. For the more mathematically-inclined of you, we might even think of the "triplet" as an operator, which multiplies the length of a note by 2/3. And in general, the "tuplet" operator multiplies the length of a given note by some specific ratio. This is one of the reasons why I prefer to use ratios when writing complicated tuplets: in a "5:3" tuplet, each note is 3/5 as long as its normal value, and so on.

And now that we have a way to think about individual tuplets, why don't we treat them as individual notes? Why should triplet eighth notes always have to come in threes? What if I only want two triplet eighth notes, followed by some binary note values or even a different tuplet? Or, to borrow some terminology from computer programming, why can't we treat tuplets as first-class objects? Well, for the most part, it's just been a convention to treat tuplets as second-class objects, only found in groupings of a particular size. As I said, our system of notation is centered around binary subdivisions. And in tying music instruction so closely to notation, it usually gets taken for granted that your tuplets are going to fall into groupings which are temporally equal to an integer multiple of some binary subdivision of the basic note values. Sometimes, it is even perceived as a part of the musical syntax: a grouping of only two triplet eighth notes is considered an error, like a sentence fragment or an unclosed HTML tag.⁷

But in automatically labelling such notations as ill-formed, we are completely ignoring the concepts of beat and pulse which the notation is supposed to support. Let's think about what a grouping of two triplet eighth notes means, from the perspective of beats and pulses. We might start with a steady beat that is subdivided into two equal parts (quarter-note beat, eighth-note pulse). Then, while keeping the beat steady, we shift to a subdivision of three pulses per beat (quarter-note beat, triplet-eighth-note pulse). Then, while keeping this new pulse steady, we shift the beat to a grouping of two pulses (triplet-eighth-note pulse, triplet-quarter-note beat). If all of the previous examples of shifting the beat and the pulse around have been clear, then this should be pretty clear as well. And, if you're sufficiently comfortable switching between various note/beat groupings in your head, then going from quarter-note/eighth-note to triplet-quarter-note/triplet-eighth-note and back should be the most natural thing in the world. At least, it's natural to me, and I know I'm not the only one. Switching gears between different beat/pulse groupings in simple-but-not-necessarily-binary ratios is absolutely essential to performing a wide swath of music in the Minimalist family tree, for example, all the way from Philip Glass to Michael Gordon.

Now, we do have to take some care in our treatment of tuplets as first-class objects. Henry Cowell was among the first to explore this possibility, writing at length about the idea in New Musical Resources. He developed an alternate notation for individual tuplets which allowed them to be used with nearly complete freedom: a triplet eighth note might be followed by a quintuplet sixteenth note, and then two septuplet eighth notes. But few if any musicians could be expected to accurately perform such a passage without extensive study or mechanical assistance, and as a result, much of the music that further explores these rhythmic possibilities has been intended strictly for mechanized performance, like Conlon Nancarrow’s player piano studies and Kyle Gann’s Disklavier studies.

But I'm a human, and I tend to write music to be performed by humans like me, so I can't go to the outer limits of Cowell's insights. Instead, I need to keep the underlying beats and pulses in mind when I write. Isolated tuplets with constantly changing denominators are unfeasible for performance, but groupings of tuplets – complete or incomplete – with denominators that vary among only a handful of small numbers are much easier to handle. After all, they're just different groupings of pulses that we already recognize. And this is what goes on in Fanfare for Tuesday. At the beginning, I work with a steady quarter-note beat, which is variously subdivided into two, three, four, or five pulses. Once these subdivisions have had a chance to sink in, I start playing around with the groupings. At first I only play around with different groupings of the binary subdivisions -- the eighth notes and sixteenth notes. But then I introduce groupings of two or four triplet eighth notes. I don't go so far as to incorporate incomplete groupings of quintuplets as well, though I reserve the right to add a few of those in a later revision. Or to use them in a separate piece.

Put more simply, Fanfare for Tuesday explores some of the rhythmic possibilities that open up when you allow your beats and pulses to shift between multiple disparate yet logical and palpable groupings. While the listener may not be able to precisely identify the triplets and quintuplets when listening in the moment, I think the gist of these relationships is recognizable, and the details could be teased out through repeated close listenings: "Well, I can tell that there are “short” notes and “shorter” notes, and they seem to line up with the bigger notes somehow...oh, three of the “short” notes are as long as one of the “big” notes, which is also as long as five of the “shorter” notes. And over here there are more of the “short” notes, but they're not in a group of three...there are actually four notes there, and then we go back to the “shorter” notes..." Even without being aware of the precise numbers involved, I think the experience is like a sort of bizarre bicycle ride: you start off pedaling at a certain pace in a certain gear, and then you start changing gears and/or pace. Sometimes, when you change from a lower gear to a higher gear, you slow down your pace of pedaling by a similar factor so that the actual speed of the bike stays constant. Other times when you change gears, you keep pedaling at the same pace, so the speed of the bike suddenly increases by that same factor. And once you've gotten used to the different gears and paces and speed, you start mixing them up even more freely. Only you're not really in control of all these changes; I'm the one who's inflicting them on you.

Now, Fanfare for Tuesday is not the first time I've played around with shifting beats and pulses: in pieces like Trinkle Dance and Recombinant, I wind up shifting gears quite a bit. But whenever I shifted gears, I hewed to the binary-centric paradigm imposed by the notation. If I went from a quarter-note beat to a triplet-quarter-note-beat and back again, the number of triple-quarter-note beats would always be a multiple of three. But when I opened the door to allow incomplete tuplets in, I did more than elevate those tuplets to the status of first-class notational objects: I could now treat all imaginable beat/pulse groupings as first-class musical objects. In so doing I wound up transcending a barrier that I hadn't even been aware of in previous compositions, because I was letting the notation guide my ideas rather than the other way around.

Of course, this is all (comparatively) easy for me to say in retrospect. I did not specifically set out to subvert my dominant rhythmic paradigm when I started writing Fanfare for Tuesday, neither did I arrive at the idea of using incomplete tuplets sheerly through my own inspiration. I just started off writing an innocuous little fanfare, inspired by the penetrating sound of the muted trombone. And when I started out, everything fell neatly into a quarter-note grid. But at some point, I the music I imagined involved two triplet eighth notes followed by a larger note, and rather than simply completing the triplet with a third eighth note tied to the longer note, I decided to see leave the triplet incomplete. My decision to do so, furthermore, was informed by a couple of then-recent posts by Kyle Gann about incomplete tuplets and their implications, and enabled by Darcy James Argue's post explaining how to make the notation work in Finale.⁸

In many ways, Fanfare for Tuesday wound up being an ideal vehicle for trying out some of these new ideas about rhythm. With the sort of fanfare I was writing, I could stick to fairly simple melodic ideas – lots of repeated pitches, or alternation between two or three pitches – which made it easier to put my focus on rhythmic invention, much like the Michael Gordon excerpts that Kyle Gann cites. And writing for solo trombone freed me from worrying about the relationship of multiple parts, and also allowed me to write without regard for time signature. One of the corollaries of the assumption that tuplets must always occur in complete groups is that the time signature – which tells you how long each measure is, and the organization of the beats and pulses to some degree – should always have a denominator which is a power of 2: 4/4, 3/2, 6/8. Incomplete tuplets can break this rule: Kyle Gann's I'itoi Variations have time signatures like 2/3, 5/6, and 7/12. And many musicians will balk at that. Quite often, if you take a professional musician and put music written in 13/32 or 41/16 in front of them, they'll roll their eyes but figure out how to play it. But give them something with a few bars of 5/6 – a much simpler fraction – and they may flatly insist that it is impossible. Even when they are accepted, such time signatures are often called "irrational," which is horribly inaccurate. I call them "non-dyadic" time signatures, since fractions with a power of two in the denominator are referred to as dyadic rationals. But no time signature? No problem, as long as you don't have to sync up with anyone else.

You may have noticed that, ever since I started talking about incomplete tuplets, I have not included any score examples. That is because, as I alluded to above, incomplete tuplets are difficult (but not impossible) to notate in Finale. I'll gladly go through that kind of effort to get what I want in my compositions, but I didn't think it was worth it for the purposes of a few illustrative examples. But now I can show you some incomplete tuplets in action, by giving you the score to Fanfare for Tuesday:

Fanfare for Tuesday (PDF)

I've held off so long on giving you the score because I wanted to explain the ideas behind this piece, in terms of both rhythmic conception and notation, before having you make sense of the notation. I suggest you try to follow along in the score while listening to the recording. I'll try to give you a few guideposts:

• At about 0:27 in the recording, at the beginning of the third line of the score, I first break out of the quarter-note grid with three eighth notes followed by three triplet eighth notes. This is part of the first passage incorporating all the major beat subdivisions I use in the piece, going from eighth notes to triplet eighth notes to sixteenth notes to quintuplet sixteenth notes.
• 0:45, the beginning of the fifth line: the first incomplete tuplet. I grouped it as 2 triplet eighth notes followed by three triplet quarter notes, but I feel it more like 4 triplet quarter notes, with the first note subdivided into two parts.
• 0:55, beginning of sixth line: A tricky little passage incorporating a ***footnote 4***nested tuplet.
• 1:15, last line of the first page: Five sixteenths, five quintuplet sixteenths, four triplet eighths, and two eighths. That's a lot of gear-switching.
• 1:23, beginning of second page: Shifting from a quarter-note beat to a dotted-eighth-note beat for a few beats in a row. I don't usually stay in one of the shifted beats for that long. I also don't think I played it fully accurately.
• 1:51, page two, end of third line: One of the things I've consistently enjoyed doing in this piece is juxtaposing subdivisions of the beat into 2, 3, and 5 parts, as I do here. I've long been fascinated by the rhythmic relationship of 3 against 5, and when you throw in 2 as well, you start invoking the Fibonacci sequence and the golden ratio. I am often skeptical about aesthetic claims involving the golden ratio, but here I think it gives the changing rhythms more of an organic feeling, while the relationship of 4 to 5 (sixteenth notes to quintuplet sixteenth notes) feels more mechanical.
• 2:11-2:30, 5th and 6th lines of page 2: The busiest passage of the whole piece. Twenty seconds without any long notes (quarter note or longer). I also use dotted sixteenth notes for the first time, first as a quaternary subdvision of a dotted-quarter-note beat, and then as a building block for a ternary beat. It's tricky to play accurately, as dotted sixteenth notes are very close to triplet eighth notes in length. And at the end of the passage, a gradual slowdown of triplet eighth notes to eighth notes to dotted eighth notes represents a nested 3:2 relationship.
• 2:30, seventh line of page 2: I suddenly throw in some multiphonics – I'm playing the lower note while singing the upper note. I hope this came as a surprise when you first listened to the recording without the score.
• 3:03, beginning of third page: Septuplets. Hard to get right when my head has been so full of triplets and quintuplets the rest of the time.
• 3:34 to end, last line of page 3: Due to a combination of the choice of interval, the soft dynamic level, and the mute, these last multiphonics come of more as timbral colorations of the lower notes rather than as chords.

And that’s Fanfare for Tuesday. There are just a couple of more things I want to say before I let you go. First of all, I want to point out that all of the discussion of rhythm above, both in general and in Tuesday, have been solely concerned with horizontal relations – that is, relations between successive points in time – in a monorhythmic context. Not surprisingly, there are many more options for rhythmic complexity when you have more than one simultaneous rhythmic line. Earlier works like Trinkle Dance and Recombinant may sound more complex – and they probably are – because of these polyrhythmic relationships.

And lastly, I would like to apologize for going on so long. If you made it this far, you no doubt spent significantly more time reading this explanation than you did listening to Fanfare for Tuesday, and I probably spent more time writing this than I did composing Tuesday in the first place. But in trying to explain why this little piece was so significant to me, I had to summarize the past 20 years of my experience in learning a tradition of notated music spanning over 500 years, so I guess 5,000 words isn’t too bad in that context. In any event, I promise that next Tuesday’s feature will not be nearly so wordy.

Footnotes

¹It may be argued that there is a lower conceptual level, consisting of the absolute and relative durations of individual sonic events, which our brains perceive to a certain degree of accuracy. However, I do not consider this an organizational level of rhythm, or more properly, I consider to be on the level of sound rather than music, at least for the purposes of this discussion.
²This is not meant to be a slur against conductors. In top-level orchestras, the primary function of the conductor is not to mark time but to indicate subtler details like shadings of volume, attack, and other elements which create the "character" of a musical passage, beyond what is strictly notated in the score. The musicians of the Boston Symphony Orchestra are quite capable of keeping time on their own, for the most part.
³Usually, this does not actually indicate a change in the pulse. In most cases, one of the two subdivisions would predominate and we would perceive a steady pulse which is temporarily displaced by the other subdivision, but if multiple subdivisions appear consistently and regularly throughout the work, (as they do in Fanfare for Tuesday, not coincidentally) then we might perceive a constantly shifting pulse.
³The other tool you need to understand a very small amount of my pre-Tuesday music is that tuplets can be nested: you might start with a group of triplet quarter notes -- three of them in the space of a half note -- and then replace one of those notional quarter notes with three triplet eighth notes. It looks something like this:

⁵Or in a grouping with other note values⁶ (quarters, sixteenths, etc.) which together add up to the same length as three notional eighth notes:

⁶Or rests with the same note values, for that matter.
⁷The sentence fragment is the better analogy of the two. After all, people utter sentence fragments all the time, and we have no trouble figuring out what they mean. And the concept that incomplete tuplets represent is, at least to me, as natural as a sentence fragment, as explained in the succeeding paragraph.
⁸Actually, my initial draft was in pencil, so getting the notation right was easy-peasy. But I then needed to get the score into Finale for editing and publishing.

It's been over two weeks since my last%20recital, and I still haven't written about it. I'm sorry to have kept you waiting, but that's because I've been so busy working on the next recital, which is zomg tomorrow. I'll tell you about the upcoming recital in a second, but first let me tell you about the last one. My recital on August 30th, which I named "Anarchist Nut," went quite well. In particular, the premiere of the Holy Tango Songbook was excellent, with guest performers Lorinne Lampert and Stephen Williams doing a great job singing and playing piano, respectively. Also, Francis Heaney gave a number of readings from his delightful Holy Tango of Literature, which may be worth the price of admission all by themselves. For my own part, I faltered a bit in the two older pieces on the recital -- I somehow managed to make mistakes I had never before made, not in several hours of practicing -- but held my own on the new works. Several people have told me that they couldn't hear the mistakes, which is understandable since they don't know what was supposed to happen at those points, but a little discomforting to me as a composer, because that means that the audience was reacting to something other than what I had written. Yes, that's a bit egotistical of me, especially for something so trifling as a couple of wrong notes out of however many pieces, but I am hardly the only egotistical composer in the world. I've been working on editing the recordings from the recital, and I hope to have them up for sale sometime next week. To whet your appetite, here are Lorinne and Stephen singing "Skinny Domicile" (after Emily Dickinson) from the Holy Tango Songbook. And, lest I forget, I want to give a hearty thank you Lorinne, Stephen, and Francis for their invaluable assistance in putting on this recital. So, now we come to my next recital, which is, as I mentioned before, tomorrow. This one is called "Musical Hydra," as I will be playing a lot of new music on a lot of instruments -- specifically, bass trombone, flute, and piano. In addition to my own works, I will be playing pieces by fellow Tufts graduates Beau Kenyon and Warren Weberg, as well as my advisor John McDonald. "Musical Hydra" will also be recorded, and I hope to get those recordings online sometime in the next two or three weeks. But why wait for the recording when you can come hear it yourself? Again, for more details please see the event listing.