Musical Idea 12: Aharmonics

12.26.2013 § 2 Comments

Imagine if next to your favorite home audio system’s volume knob you had a pitch knob, too. Give your pitch knob a quarter turn clockwise, say, and everything coming out your speakers goes up an octave. Another quarter turn and you’re two octaves higher. Another and another and another and soon all you’re hearing is a high-pitched screech (and your grandmother’s glassware shattering). Another and another and another and the sounds produced are inaudible to humans (but your doggies are losing it). Dial things all the way back counterclockwise and you’ll shake your guts up with ultra-low vibrations before all goes inaudible the other way. Fun!


Changing speed (left) versus changing pitch, or tempo (right). Changing speed simply stretches the waveform out, so that the sound occurs more slowly in proportion to the amount at which at every moment along it it is lower in pitch. Changing pitch or tempo changes only one or the other (frequency along the sound, or durations those frequencies are held, respectively) at a time, which requires much more complex math to achieve. Here, change in tempo is seen (though I wrote “pitch”, whoops): the macro shape of the waveform is preserved through the stretch, but more frequent waveform oscillations are introduced to keep the pitch about the same as it was originally, even though it happens more slowly.

Note that your knob varies only the pitch, not the speed. Changing speed means changing both pitch and tempo together. Contrast pitch with speed by considering what changing speeds on your record player is like: when you toggle from 33 revolutions per minute to 45 on the turntables, both the pitch and the tempo of your music increase, and the pitch is increasing precisely because of the needle being dragged across the waveform more quickly. Your pitch knob works differently. Isolating the effect on pitch alone without affecting durations requires more complicated maths — so be thankful for your pitch knob. It is special.

Now you’ve seen a performance of John Cage’s Imaginary Landscape No. 4 scored for radio and — inspired by Cage’s use of frequency knobs in his composition — want to try your hand at it. Admittedly, any comparison between composing for the radio knob and for the pitch knob is a stretch; both relate to Hertz, sure, but there’s not much else in common. For starters, Cage’s instructions are only about when to play things and where to play them from, not what plays and how. Radio station bandwidths are sources of sound, not effects on sound, as your pitch knob controls. And not only does Cage have nothing to do compositionally with whatever materials he ends up manipulating, nor do these materials have much to do with him, I figure that this independence is a big part of his piece’s point.

You, on the other hand, have quickly tired of superficial games you’ve invented and played with existing sounds out there, e.g. twiddling the pitch knob back and forth to keep the ground bass of Pachelbel’s Canon at the same pitch. You’re ready to write music specifically designed for you to play out of this uniquely equipped sound system and have fun messing with, that is, music which exists expressly to provide opportunities for you pull off pitch knob stunts.

Er, actually, let’s cut to the chase already. The physical pitch knob on your home audio system has been just an introductory metaphor. You’re ready to compose music in which the overall pitch demonstrates its power to move uniformly, and demonstrates this not as some mere accident, but as a compositional technique fully integrated with the rest of them.

If I were you — that is, about to start composing music using such “tonal movement,” as I might call it — I would begin with a basic example, perhaps the most basic conceivable example possible. Say let’s start with a steady, stepwise, chromatic descent. In other words, play a note, then play the next lower note, then the next lower note, then the next lower note, and so on, giving each note the exact same duration. Now throw in the tonal movement in opposition. Suppose we have the overall pitch continuously rising (as if one were smoothly turning the pitch knob) at a constant rate which exactly cancels out the stepwise descent. The result of combining these two efforts would be an endless repetition of the same note, over and over, for by the time it was ready to step down an interval to the next lower note, it has ascended by precisely the amount necessary for such a step down to bring it right back to where it started. Make sense?

Sure it does — but it also sounds kind of boring, right? It is boring. Really. In fact, if someone played the final product for you without prefacing it, you’d have no insight into the complex underlying mechanisms. All you’d be able to ascertain would be a mindless broken-record bendy note.

But that’s okay — we’re just getting started. To illuminate the tonal movement, it’s clear now that we’ll need to switch something up, either on the side of the tonal movement itself, or the notes its applying to. At least one or the other will need to deviate in a way that distinguishes itself long enough and continuously enough that one can’t mistake it for anything else.

Let’s try the former first (naturally), doing something with the tonal movement.

    1. Maybe we’ll start off as simple as changing the constant of the rate of the tonal movement. Let’s try halving it. This way, the pitch drifts up one note for every two notes it steps down, rather than one for one as it had been doing.
      1. Well, I would say the tonal movement is still not yet sufficiently independent here to assert itself. Just ask someone not indoctrinated what’s going on and they’ll probably just tell you that they’re hearing a descent by half steps, each note bending up a whole step. We’ll need to do more than this to bring out what’s happening behind the scenes, and tap its full potential, so that it shouts out loud “look at me”.
    2. So let’s say that rather than turning the tonal movement knob at some constant rate, we’ll vary its rate. For example, make the rate sinusoidal (the rate, not the movement itself; the movement’s derivative would be a sine wave). This way, sometimes the pitch drifts up relatively quickly and sometimes the pitch drifts up more slowly. Depending on how we transpose and amplify this sine wave, maybe the pitch even stops drifting at moments. Hell, maybe it even drifts down a bit at times, so long as it still nets upward.
      1. “So long”, that is, that we’re still assuming an attempt to cancel out or at least resist the downward stepping of the notes. Perhaps you reflected on how in the example we just gave for changing the constant rate of tonal movement, the resultant pitch movements no longer canceled each other out. If the tonal ascent is moving slower than the descent of the notes, then the combination cannot continue indefinitely; the descent is winning, in a sense, and pitch will gradually get lower until eventually it goes gut-rumblingly inaudible. Of course, not every musical process is capable of continuing or meant to continue indefinitely, so there are still certainly times and places for such imbalance… but one not even need venture into imbalance with the notes, yet, to find fertile territory for experimentation with this variable-rate tonal movement.
        1. Start by determining at which pitch interval the tonal movement ultimately cancels out the notes, no matter what sinusoidal or chaotic behavior it exhibits within this interval.
          1. In our initial example, it was one of our scale’s specific intervals, that is, its atomic, chromatic step, at which it canceled out. We previously decided this interval was too basic to be of interest, since the result is that the entire thing repeats every friggin’ note. Now, in this initial example, the rate of tonal movement was steady as well, but I don’t think that making this second-long bend wobbly by varying the rate of tonal movement within such a short timespan helps increase the interest much, now matter how you vary it. You just gotta’ give a little more time for something interesting to develop before repeating it. So another basic suggestion which might give a little more time for something interesting to happen before repetition would be to have the tonal movement cancel out the motion of the notes each time they make it an octave. I say this since the octave is the most fundamental pitch interval in music.
          2. But we could have it happen every 17 steps, or any other arbitrary number of steps.
          3. Or we could have it cancel out at a non-whole number of steps, like every 3.5 of them.
          4. Or it could even happen at some completely irrational number of steps each time — though without an easily perceptible ratio between the durations of the processes, if you expected your listeners to perceive its approximate relationship, you’d have to keep whatever irrational number you picked constant. (At this point it becomes clearer that with the stepwise descent moving at a constant speed, we’re really considering this unbound to any pitch interval and more simply to a duration of time.)
        2. Speaking of inconsistency, if you did use all simple ratios between durations in the notes and the tonal movement, then you could certainly get dynamic with which durations you used for the tonal movement to cancel out the notes each time. In other words, in each moment the tonal movement is behaving unpredictably, but overall the tonal movement is going at an overall constant rate which is the same (the opposite) constant rate as the descent of the notes, but it makes a point to align perfectly back at a certain pitch at the precise moment a new note hits, but at an unusual number of notes in between each time. Like say the rate makes eleven big sine undulations each time before aligning, so it just keeps making different polyrhythms.
          1. These numbers of notes could in turn either be predictable, like 3,5,3,7,4,3,5,3,7,4,etc. or unpredictable, like 3,7,2,4,2,3,3,5,4,2,7, etc. (I suppose this would be the meta-meta-meta? The changing of the changing of the changing of the pitch…?)
        3. Whatever you choose for what happens with the duration, you can play around with what happens inside the windows of realignment each time.
          1. I mean, on one hand, there could be no patterns between them, just total chaos other than those realignments.
          2. Or maybe it’s just the same snippet of chaos each time (stretched if need be, if you’re doing the variable durations thing), breathing meaning into it.
          3. Or maybe whether it’s chaos or not, you’re varying the amplitude of the rate change from iteration to iteration — I mean, make the fasts faster and the slows slower (probably more pronounced when the durations are the same).
      2. But don’t forget that there are certainly endless finitely repeatable processes that don’t involve the tonal movement and note movement being in balance.
        1. In examples here we’ve seen them in multiple of each other, but even that is not necessary.
        2. Or they could be in imbalance from lining-up to lining-up but combining line-up units balances out.

(top left) if tonal movement is gradually up while the music moves punctuatedly down, but at the same rate, the result is a broken record endlessly repeated bending note. (top center) This chapter’s bullet I: when the rate of tonal movement is varied at least a bit, some listeners may start to get the idea, since the upward bending is at a constant rate as is the interval by which each note steps down, yet one process wins out so they distinguish themselves from each other. (top right) Bullet II: when the rate of tonal movement is dynamic (here, sinusoidal), the idea of the continuity of the rate of tonal moment becomes much more clearly perceptible. (bottom) More space was needed to articulate this iteration. The left half shows the usual stepwise descent of the underlying music, superimposed with the all-over-the-place, yet both continuous and ultimately perfectly canceling-out, rate of tonal movement. The right half shows the end result: the dotted line represents that since the rate cancels out the tonal movement, the process can continue without the pitch straying from home. In fact, this example cancels out not only once, but twice, every 6 of the underlying steps.

Okay, well that’s surely enough fun to be had without even altering the mindless, methodical descent of the notes. It’s time to explore this other way of illuminating tonal movement’s presence. I guess you could call it backlighting it. This is by varying the rate at which the notes descend.

Take the opening riff to Metallica’s Master of Puppets, which is basically also a chromatic descent, except that critically it is syncopated: the descent in pitch comes at an irregular rate. If we return to using a steady, continuous rate of tonal movement, but instead apply it to this angular, stop-and-start riff, the tonal movement will stand out in relief to it. The overall pitch will stay about the same, as it did in our initial example, but rather than wobbling in an lame and unilluminating way, the lurching ahead and falling behind of the riff to the tonal movement will create a fascinating effect. Especially if we’re previously familiar with the riff, we’ll feel on one hand the constant descent of the original composition in our mind’s ear’s memory, while on the other hand our audio-trigonometry will be perceiving the tonal movement.Let’s break this down. Here’s the riff transcribed:



where B = 11 and A = 10; these are the number of steps above the chugging root (x). Rhythmically, I’ve just divided an iteration into 32nds, with 8 of them on each line. Metallica uses the 12 tone equal temperament that most metal and western music in general uses, so each chromatic step is a twelfth of an octave. Descending by these twelfths stepwise and irregularly but quantized to the thirty-secondth divisions of time along which the pitch is simultaneously ascending smoothly and regularly, we get these pitches for the twelve notes of the riff (other than the chugged roots), in ninety-sixths:

B = -1/12+1/16 = -2/96

A = -1/6+5/32 = -1/96

9 = -1/4+1/4 = 0/96

8 = -1/3+5/16 = -2/96

7 = -5/12+3/8 to 1/2 = -4 to 8/96

6 = -1/2+9/16 = 6/96

5 = -7/12+21/32 = 7/96

4 = -2/3+3/4 = 8/96

3 = -3/4+13/16 = 6/96

2 = -5/6+7/8 = 4 /96

1 = -11/12+15/16 = 2/96

Here is a sample:

It’s a little difficult to perceive the snaking of pitch in the sample because there’s just too much going on with the drums and bass clashing across the divide pronouncing the difference badly (couldn’t find a section in the song where they play the riff solo on the guitar with no interference) the timbre changing, as well as the chugging climbing (which is a cool effect, but may distract from grasping this for now). So here’s another sample of just a sine wave so you can better pick it out.

To sum up: in the original, the pitch simply moved stepwise from high to low each time then started over back at the top. But now, it hovers around one pitch right in the middle, varying slightly up and down from it as a result of the upward and downward motions that ultimately cancel each other out not doing so at the same rates during. You can see that the lowest the pitch ever gets is a quartertone, and the highest it ever gets is a semitone.


The dotted arrow represents the constant rise in pitch. The O’s represent the staccato notes, and the O’s with lines extruded from them represent sustained notes. The X’s along the bottom represent the palm-muted “chugged” lowest pitch between the notes of the chromatic descent. You can see the vertical axis shows the 12 pitches of the standard tuning, and that there are 32 total beats from left to right. (bottom) the end effect of this constant rise in pitch and slightly irregular rate of punctuated decreased in pitch are notes which hover around a single pitch, deviating slightly up or down depending on which process is “winning” at that point along the riff.

Alright, wow. So I’ve just been getting started coaching y’all on how to compose using tonal movement. I haven’t really even gotten into any compositional recommendations. I would suggest articulating motives first before the tonal movement, and then apply it in a surprising way. For example, look at the Beatles’s Day Tripper opening riff. Its last note is an octave higher than its first note. Tonal movement could be used to lower this riff by the end until that last note is the same pitch as the first. Rather than have two notes in a row of the same pitch, you could just conflate them. This would do fun stuff to rhythm. And you’d never get it if you hadn’t heard the riff before, in the original or otherwise (just an example; in your non-famous song you’d have to put it in its original form somewhere else).

Now that we’ve gone through plenty of example techniques, it should be apparent that tonal movement will not necessarily lead to harmony as we normally think about it. With the entire pitch space in motion, no one will be striving to grasp the harmonic relationships in the purely vertical sense that we traditionally do, nor will they be meant to. I’ve tried to come up with a good word to describe this situation, because I don’t feel that atonality quite pegs it. This is more than simply obscuring any sense of a tonic among the vertically stable harmonic relationships. I am a member of a community of folks who explore uncommon tunings and chords who refer to their studies as xenharmonics, but what I’m imagining here doesn’t really fit in with the themes there since I’m not trying to get at the roots of why traditional harmonic systems work and then expand upon or twist them up. What’s most fundamental to this idea here is that I’m adding a horizontal dimension to harmony: asking listeners to track harmonic relationships as they’re animated over time. I’ve decided that the only term that really captures this idea for me is “aharmonic”, since while it is still related to harmonics, more than anything else related to it, it obliterates your ability to truly perceive them, undoing what they are essentially, transforming them into precisely what they’re not. I somehow doubt that there will be more than one kind of aharmonic mood, that is, though I feel confident that the aharmonic mood will be different from atonal and from any xenharmonic mood out there.

If you’re not yet convinced that aharmonics is absolutely bonkers, then consider that there is more out there than tonal movement. How about tonal stretching. This is where you get a new knob which scales the intervals. Let’s call it the interval knob. You turn this puppy to the right and all the intervals between notes get proportionally bigger. You turn it to the left and they all get proportionally smaller until everything is the same pitch. Have fun with that one, too.

Stay tuned for my next posts which will be about some advanced aharmonic topics. 



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