Musical Idea 15: Musical Yaw

01.19.2014 § 4 Comments

Today’s musical idea takes the ideas introduced in my very first entry in this series and addresses the consequences of choosing duration as one of the musical aspects that gets played with.

Duration warrants special attention because things get a little wacky when you try to rotate some crazy plastic type of time while the ultimate-inevitable type of time is marching onwards as it always does. 

Rotate timbre and diegeticity and voxicity etc. against each other to your heart’s content, and time will pass with no concern — but try to rotate time itself in there and you have to make some special considerations.

To keep thing as simple as possible, I will be sticking to relating time to pitch, since pitch is a really easy musical aspect to conceive of metaphorically as vertical, perpendicular to the horizontal aspect of time.

If you’ve seen sheet music before, then you’re familiar with the basic idea that moving left and right denotes moving forward and backward in time through the music, while moving up and down denotes moving up and down respectively in pitch. Now imagine you’re listening to Beethoven’s Fifth, the famous part with all the “bum-bum-bum-BUM… bum-bum-bum-BUM…”‘s. So if we were to just very basically sketch this music on such a sheet, an iteration would look like three dots in a horizontal row, equally spaced apart, representing the first three “bum”‘s, followed by another dot, equally spaced to the right from the first three, but transposed downwards a bit, representing its being lower in pitch, the “BUM”.

If you were to rotate this situation 90 degrees clockwise, then instead you’d be looking at, from left to right, first the “BUM”, and next, the three “bums”, stacked vertically. If this were performed, then, it would sound like a low note, followed by a higher in pitch chord of three notes.

But would anyone know that you had rotated Beethoven’s riff 90 degrees yet? Probably not. The only way to prove it, to make the rotation part of the music, would be to illustrate intermediate states between one orientation and the other. Achieving movement through such intermediate states is another way of stating today’s musical objective.

Let’s frame this with something you might be already familiar with: the concepts of inverse and retrograde. Inverse is when you flip a melody upside down: low becomes high, high becomes low. Retrograde is when you flip a melody backwards: end becomes start, start becomes end. Usually we think of these two operations as flips, because we tend to do them 180 degrees exactly and never as anything more or less. But for a moment, let’s try imagining them more fluidly, and three-dimensionally, as rotations, even if we are still ending up doing them in intervals of 180 degrees.

So inverse, then, would be like sticking an axis horizontally through your sheet music, and rotating halfway around, like it were a pig on the roast. Retrograde would be like sticking an axis vertically through your sheet music, and rotating halfway around, like it was a rotating door. The type of rotation we just performed in the previous paragraph would be the third type, the one that would complete the picture: sticking an axis straight into the sheet music, impaling it perpendicularly, and rotating it like a record on a turntable. You may notice that if we had gone all the way 180 degrees on this axis that we would have just ended up with another familiar structure, the compound of the other two, the inverse retrograde.

It’s only when we start fooling around with non-180 degree rotations (even 90 degree ones, which are not that strange in the scheme of things!) that we get unfamiliar, funky things happening. For 90 degrees inverse you would basically just end up with all notes as the same pitch, along whatever pitch you stuck the horizontal axis. For 90 degrees retrograde you would basically just end up with a single chord consisting of all of your melody’s notes: all notes at the same point in time, at whichever point in time you stuck the vertical axis.

If the music was a plane we were looking up at as it flew left to right across the page, then inverting would be what they call “roll” (do a barrel roll!), halfway turning it upside down; retrograding would be what they call “pitch” (not to be confused with the musical type) — nose-diving and coming back up upside down as well but backwards, and this third, new type of rotation would be what they call “yaw”: turning around by aiming to the sides, rather than ending up upside down. Let’s use “yaw” from now on to describe this third type. Unfortunately since the word “pitch” could get confusing, especially since all the examples I plan to use in this article as I already said will compare duration only with pitch, let’s just use “rotational retrograde” for that one.

We still haven’t really gotten to the main thing I want to address here today, though, which is what happens when you try to do such rotations while the music is playing, that is, rather than just comparing siloed things that could be played, but attempt to move between them live. Now this wouldn’t be so crazy for the inverse rotation, actually, and that’s because rotating on this axis has no effect on duration/time — the rotations are totally perpendicular to time because the direction of time is perfectly parallel to its axis. You can rotate this axis as fast, slow, uniformly, compartmentally, orderly, randomly as you like and it will have no effect on the ability to basically move forward through the music. 

But for the other two rotations, this is not so, because their axises are not parallel to the flow of time. For retrograde, if you place a single axis somewhere in your song for the entire song and begin rotating, then the parts of the song that are at the extremes in distance from the axis are going to be moving really quickly. What’s most important here is that it’s going to be pretty difficult to make anything of the song. If the play cursor just keeps going as if it were a normal song while the entire thing spins this way, half the time the play cursor won’t be over any music at all, and when it is it’ll be incomprehensible. Might be fun for a throwaway effect I guess.

Similar issue for the yaw. Assuming that a traditional song is length enough that it’s shaped more like a long strip than an equilateral sheet (I mean, sheet music is shaped like the latter, but that’s because you cut the strip up and paste it in pieces to fit on this more practical shape), if you rotate that around an axis stuck straight down into it, and treat pitch the same in either direction way off the strip as it would have been up and down, then you are quickly going to be hurtling your notes into inaudibly low and high territory (I suppose that raises the issue of whether you could do something fun to pitch interval distances off in this territory, or even in general if like you want to treat pitch logarithmically or not when relating to duration, but I just don’t think this is generally interesting enough to get into now).

For rotational retrograde and yaw, we must thus recognize, in order to write functional/interesting music, that there are really two types of time: the universal one that no matter what must always pass while listening through a song, and second, that which is practically understood as duration within a piece of the music that is subject to being toyed around with and transformed into other things. In other words, the latter type is the type where you can declare, “I’ve rotated the music by 30 degrees so durations are going to be sqrt(3)/2 what they used to be,” while the former type is the type where you must concede, “I am not Father Time, and therefore I cannot alter the flow of time, which will, as it always does, proceed at a rate of 1 second per second.”

So one way to pull off rotational retrograde and yaw would be to break your music down into many little pieces and have an axis of rotation for each one. To create a sense of an ostinato gradually rotating from how it started to 90 degrees from that, say, repeat it six times, each time rotating it by 15 degrees. I find this method, let’s call it “Beading,” ugly and clumsy, especially in how you’d have to just take a stab at the timings of how you piece them together at the seams, and how you can hear the sudden changes in angling at each repetition. I’d prefer to somehow hear this occurring smoothly. That’s pretty tall order though, when two conceptions of time are fighting each other. The rest of this article will be examples of ways to attempt to do so.

1. Cursor Lean

Suppose there is a “play cursor”, as there actually is in most music sequencing programs: a vertical line which proceeds horizontally from left to right across your sheet music, pronouncing the notes as it crosses them. Let’s say that this is what we rotate, not the notes themselves. And the rotation is independent from its movement — you can just have it continue moving forward at a set tempo like it would have before, or you can have the tempo vary or go backwards for all I care, but you just don’t worry about whether sliding the play cursor along (whichever point of rotation you like) has any relationship to the rotation.

An introductory example of this style would be to just have the play cursor lean forward a bit as it’s going along, essentially causing the higher pitched notes to get a temporary tempo boost while the lower pitched notes get a temporary tempo slowing — and stay a little ahead and behind, respectively, until the cursor decides to lean back upright again, reversing the high’s boost with an equal and opposite slowing and the low’s slowing with an equal and opposite boost. This is already some next-level Nancarrow shenanigans for real, but you could experiment with some more extreme leans, though: lean so far forward that the play cursor is horizontal, hitting only a single pitch, but every instance of it in the entire song, and holding it, then keep leaning until the cursor is basically coming back from as if it had leaned way back the other way — since it doesn’t matter whether the cursor is upside-down or not.

Or you could consider it important that the cursor is upside down by attributing the pitch as a property of the cursor rather than the notes. In other words, as the cursor leans forward, we notice that the highest notes in our music are hitting the cursor at new points along its length that they hadn’t reached before, and if we imagine continuing the rate that pitch was rising on the normal expanse of the line that it hit notes with before, then it’s going to start pronouncing them higher and higher until they’re inaudible (and vice versa for the lower notes). And in this case once the line is upside down, the notes that used to be low will be hitting the cursor on its high parts, so they’ll be the high ones now.

2. Donut Packing

This one requires a lot of steps to imagine, so bear with me. Start out with your music on a lengthy strip. The width of the strip is the range in pitch of the notes, and the length of the strip is the stretch of music that you want to play with, let’s just say it’s twelve bars for now. Now let’s take the two ends of this strip and connect them together. Not like a bracelet, though. This strip is made of flexible enough material that you can just lay it flat on a table and warp it into the shape of a washer, so that the inside bordering the center hole compresses a bit while the outside stretches a bit.

Now imagine the play cursor slides across this. Since we picked a twelve bar stretch, the metaphor of a clock can help us describe how it sounds. The first music we’ll hear will be the ninth bar, and since it’s centered on being sideways, we hear what would be its lowest notes first and then what would be its highest ones. Quite soon after we start hearing the ninth bar’s notes, we’ll start hearing both the tenth bar and the eighth as well, since they’re pretty much directly above and below the ninth bar (respectively), just a little to the side since the ring shape is in the process of becoming two horizontal parts. By the time we’re to the twelfth and sixth bars, we are horizontal (for an instant, at their midpoint) — the twelfth bar will sound nearly normal, while the sixth will be pretty much its own retrograde (warped slightly from its curvedness); if the twelve bars we’ve put into this shape are simply the same bar repeated over and over, then here we’d basically be counterpointing it against its own retrograde at this point. We keep going until the second and fourth bars converge on the third, then we’re done.

Now, just like we did with the previous variety, we could treat the cursor as the arbiter of pitch, rather than the notes. This might not be as immediately obvious as an option in this case because this washer shape is much taller than the single strip we worked with in the previous variety, meaning that if we assume the strip itself represents a typical pitch range, then to simply continue that up or down to cover the full height of the washer it creates (the height of which, the diameter, will depend on how long a stretch of music you apply this to, that is, what becomes the circumference, and could quickly get out of hand). In other words, supposing we center the cursor’s normal pitch range to center on the midline across this washer, that is, through the ninth and third bars, then by the time we reach the point on the washer with the most extreme heights and depths (the twelfth bar up top and the sixth bar down below), the twelfth bar may likely be way too high in pitch to hear, and the sixth bar may likely be way too low in pitch to hear. If all we really want to achieve is turning the sixth bar basically into its inverse retrograde rather than merely its plain retrograde, then we have to something else to reel its pitch back in.

I would suggest using pitch circularity to accomplish this. That will look like drawing several horizontal lines across the washer, dividing it into sections which all have the same set of pitches, and which circle back around by the time you cross the line. For starters, you’d probably want these horizontal lines made tangent to the inner and outer circles of the washer at their extreme high and low points, in order to place the twelfth and sixth bars in the same pitch space.

I expect that what you’d end up doing is setting these washers side by side to make a song, that is, the third bar of your first washer would butt up against the ninth bar of the next washer. Perhaps the most elementary illustration of the potential of this structure would be to repeat the same single bar twelve times within each washer, while each next washer would represent a development on that bar. But you could also focus more on the structure of your space if you wanted not to iterate upon the content ever and experiment with hexagonal packings of donuts, at various angles, or different sized donuts turning it into a circle packing problem, etc.

3. Snaking

This one also requires a lot of steps to imagine. So take that washer shape we had in the previous example. Cut it in half along its midline, right through the ninth and third bars. Flip it sideways, so that now the bottom half of the ninth touches the top half of the third, and vice versa. Now slide it over to the side so that the two halves of the ninth are just hanging out there at the extremes, and the two halves of the third are reconnected. Your shape now looks kiiiinda like one iteration of a sine wave. The problem remaining is that the two halves of the third don’t match up, since we’ve flipped one of the washer halves. This next step is weird, but try to, without changing the shape of the flipped piece, turn it inside-out, so that the compressed stuff that was in the tighter curve is now on the outside, stretched along the looser curve, and vice versa. Ta da! Now your third bar is intact again. And more importantly, you can take this entire sine-wavy shape and repeat it, because the two ends of the ninth bar will now reconnect in the same way we just reconnected the third. In other words, your music will now snake repeatedly, tilting forward until it’s facing directly downward, at which point it starts pulling up, up, up until it’s facing directly upward, at which point it starts turning down, down, down again until the process starts over, and in between each climb and nosedive it will be moving forward, never upside down, just alternating being higher up and lower down (if you’re using the cursor as arbiter pitch, that would matter).

I mean, there’s no particular reason to disallow this snake from diving so far downward that it goes upside down, if it is your intended effect to encounter “edges” of curves as the play cursor proceeds, kind of like a weird experience of time, knowing that the music must be about to turn downward in a bit, because otherwise how else could it be coming up from that far below all of a sudden?

You could do all sorts of crazy shit with this snake, really, if you permit that. Have it follow the path of various kinds of roulettes… a cycloid or trochoid, prolate or curtate, epi- or hypo-, involute or not.

4. Tiling

Treat a piece of music as an infinite field of itself recurring in all directions, and you’re just moving through it. To prevent hearing infinite music, you would then need to develop a system for filtering out the majority of notes. You could just draw a horizontal band and declare “this is my audible space”, and as the play cursor goes across it, no matter what happens to the rotation of the notes, you only hear the ones inside the band. Again, you could either make the notes themselves bearers of their own pitch, or say that pitch is correlated with the elevation on the band/cursor (assuming the cursor doesn’t need to rotate at all under this conception). I would expect that pitch circularity could really help take the edge off of this, so that notes don’t just cut off when they drift off the band, but fade out.

What’s interesting to consider here (well, as it would be in the first option presented, too) is where the centerpoint of the rotation goes. It could be bound to the cursor. Or it could be at a particular point, say, midway through the song, such that there is a dramatic approach toward this point where everything changes orientations much faster and is thus incomprehensible, and then a slow release as you drift farther away from it again and things go back to rotating manageably perceptibly. Or maybe the center of rotation has a mind of its own, moving around dynamically in front of, below, behind, above, etc. as you move through the song, giving hints when it’s about to “attack” by moving right toward the segment of the play cursor that’s inside the audible band.

Or, once you’re in this sort of tiled plane, you can take this idea of an audible band and do the same stuff we did do it in the previous example, snaking through in whatever spirographic means you can find.

These four techniques (five, if you count the ugly sloppy one) are certainly not exhaustive. I could go on and on with examples of using them in various compounds with each other, for starters. In any case, I’m just trying to open your mind to the possibilities of writing melodies — just as composers in the past wrote them with the intent for them to play with their own inverses and retrogrades — to play interestingly at multiple rotations (not only 90 degree ones!) or along with various rotations of themselves (like, say, a chunk of music designed to be traveled through a stack of itself diagonally). And remember that while for simplicity I described musical yaw using only pitch against duration, you could apply any musical aspect in place of pitch. However you feel like showing your listeners how this is what you’ve done: do it.


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