harmonic circle

04.13.2015 § 2 Comments

I have created a novel aural illusion whereby the end of a given octave of the harmonic series is made to seem like it reconnects seamlessly with its own beginning. I propose calling this illusion “harmonic circularity”, modeling its proposed name after the widely accepted names for key components of the effect: pitch circularity and tempo circularity. An instance of my new illusion could then be called a harmonic circle, and I have included one such harmonic circle for your listening pleasure below.

This particular harmonic circle is made from the 8th octave of the harmonic series (harmonics 128 through 255). I’ve found the 8th octave to be the ideal for appreciating the effect: it has enough steps to concealingly spread the pitch and tempo circularities across, while not too many steps such that the harmonic relationships between the intervals become imperceptible.

Here is a high level summary of how I achieve this effect (more detail to come):

  1. Establish a pitch circular environment.
  2. Choose an octave of the harmonic series and loop it, so that pitch seems to climb forever. The ending connects to the beginning because in pitch circularity one octave higher is back where you started.
  3. Recognize that the logarithmic nature of the harmonic series poses a problem to the seamlessness of the connection between the end and beginning of the looped octave: intervals at the beginning are twice as wide as those at the end. Correct for this with a proportional tempo acceleration, so that by the end of each octave two notes pass in the space that two pass at the beginning, thereby linearizing the rate of pitch climb.
  4. But now there is an obvious seam between the ending and beginning in that suddenly the tempo drops by half on each loopback. Correct for this by gradually fading out every other note, so that by the time the tempo is 2x, only half the notes are actually being heard.
  5. Finally, a subtle tweak to complete the illusion and impart stylistic consistency.

 

 

Step One: Pitch Circularity

If you already understand pitch circularity, you can safely skip this section. However, I recommend refreshing yourself on it, or at least seeing it through the lens I see it, since themes to its structure will recur in later steps.

Pitch circularity is the property of a sonic space in which pitch can be made to seem to continually rise (or fall) while getting nowhere — over time seeming to stay in about the same place. This aural illusion is achieved by fading out frequencies as they get too high (or low), while fading in frequencies to take their place from the opposite end of the pitch continuum. Listen below to a Shepard scale, a simple showcase of a pitch circular environment.

Most people perceive frequencies related by the simplest possible harmonic ratio, 2 (also known as an octave), to be so similar that in some senses they are essentially the same note, or “pitch class equivalent,” and in traditional Western musical notation we even refer to these frequencies with the same letter (A, B, C, D, E, F, or G). This is the interval, therefore, used between frequencies that take each other’s place in pitch circular environments. When a pitch begins to get too high, we begin replacing it with another pitch having half the frequency; when a pitch begins to get too low, we begin replacing it with another pitch having twice the frequency.

octave_relationships

frequencies related by the ratio 2:1, each an octave apart

 

The fading in and out of frequencies as they get higher and lower is handled by an “amplitude curve”, which is simply a graph relating pitch to loudness. In the middle of this graph, where pitch is moderate, loudness reaches its maximum; at the extreme edges of this graph, where pitch is very high or very low, loudness approaches zero. Such an amplitude curve could be drawn as a simple triangle connecting with line segments the pitch-central loudness-maximum to its two pitch-extreme loudness-zeroes. However, the most effective amplitude curves smooth out the process into a shape like a normal distribution’s bell curve. This way, there are no sudden changes in the derivative of the graph, also known as its slope.

derivative_of_bell_curve_is_smooth

The derivative of a bell curve is smooth and continuous; the rate of change has no sudden changes…

 

derivative_of_triangle_is_discontinuous

…whereas the derivative of a triangle-shaped “curve” is shaped like a square wave; frequent discontinuities where the rate of change flips suddenly from positive to negative.

 

In other words, the rate of change in amplitude of a voice in a pitch circularity environment as it changed frequency would never itself suddenly change; the rate of change should change gradually. Bell-shaped amplitude curves are more effective for pitch circularity because the illusion relies upon concealing its underlying mechanisms, and gradual changes are more difficult to perceive than sudden ones. Here’s what that Shepherd-Risset Glissando would have sounded like with a triangular, non-curvy amplitude “curve”:

The simplest implementation of pitch circularity is to begin with a single pure frequency and center an amplitude curve on it. As the pure frequency begins to rise in pitch it also begins to descend in amplitude down the right half of the bell toward a point where it simultaneously reaches one octave higher in pitch and silence in amplitude. In the meantime a new frequency having appeared at a point one octave lower in pitch and silent in amplitude has been climbing the left half of the bell toward the point the first frequency originally occupied at the top of the bell where it is central in pitch and maximal in amplitude.

too_few_voices_pitch_circularity

However, if our intent is to conceal the mechanism by which the pitch circularity is being achieved, this simplest implementation will be unsuccessful, for a couple reasons.

  1. Texture. The sonic texture when the sound is mostly consisted by a single pure frequency at the top and center of the bell will be easily distinguished from the rest of the time when the sound consists of two quieter frequencies an octave apart on opposite slopes of the bell.
  2. Isolation. With only two total frequencies being heard, it is readily apparent that one is fading in while the other fading out.

Here’s what this simplistic version sounds like:

In order to effectively conceal the octave-stacking mechanism that enables pitch circularity it is necessary for the sound at all times to consist of a larger number of stacked octaves, with the amplitude bell curve spanning not only two octaves as was described before, but all the way across the many-octave range.

  1. Texture. This corrects for the textural issue because it is far more difficult to pick out the moment of convergence (when one frequency is dead-center on the amplitude curve and two others are silent at its extreme ends) when the net minus one frequency it results in is out of many frequencies rather than only a total two. Said another way, superparticular ratios start at infinity and approach 1 as they increase in size ( x + 1 / x -> 1 as x goes to infinity), meaning that the numerator and denominator become more alike.
  2. Isolation. This also corrects for the issue with being able to perceive individual fading frequencies because the process of fading is now distributed across many laps around the pitch circle, which are all overlaid with each other (the stretch from halfway up the bell to the tip-top is concurrent with the stretch from the very bottom to halfway up).

Here’s what this slightly denser sounding but more obscure version looks like:

While it is still possible to listen closely and figure out what is going on and feel where the moment of convergence occurs, in general the technique described here is sufficient to produce an environment where pitches can seem to climb or descend infinitely while staying in place. That said, if you’re interested, I have described a technique whereby the pitch circular effect and its artifacts could be even further obscured: recursive Shepard-Risset glissandoing.

 

 

Step Two: Enter the Harmonic Series

I have been fascinated by pitch circularity for many years, but it only recently occurred to me to wonder how it could be applied to the harmonic series.

As you may know, a musical harmonic series simply maps a sequence of consecutive integers to multiples of a base frequency. Here’s what that looks like:

harmonic_series

Because pitch space is logarithmic (remember, each octave is related not by a constant amount, but by a factor of 2), each successive step up a harmonic series results in a smaller perceived difference in pitch. This shrinking of interval size can be readily perceived by listening to the first 16 notes of a harmonic series (the first four octaves):.

And here’s what the first part of that looks like graphed on a non-logarithmic graph:

logarithmic_nature

Looking at it the other way, each successive octave in the harmonic series holds twice as many pitches as the previous octave.

firstfiveoctavesHarmonicSeries20100410thanksAndrewHeathwaite

Diagram courtesy of the amazing Andrew Heathwaite.

 

As can be seen, the pitches in each octave “bunch up” toward the top, the intervals between them approaching the smallness of the first interval of the next octave (which in turn is the largest interval within that octave). In other words, it is not at all the case that the interval size instantly halves at a punctuated moment of octave ascension; the interval size is always gradually changing and irrespective to octaves.

Now we finally come to the crux of why I was interested in what happens to the harmonic series in pitch circularity. Let’s listen to the 5th octave of the harmonic series, with pitch circularity naively applied.

Something sounds a bit off, right? It’s very easy to pick out the point where it loops back around, is it not? Well, we’re using our grade-A pitch circularity, here, nothing is amiss with pitch circularity itself. The problem resides in the nature of the harmonic series. When we snap back to an earlier place in a logarithmic dimension, resolution changes. The pitch interval sizes suddenly change.

What we have at this point is the illusion of infinitely rising pitch, but rather than a smooth infinite rise, a really jerky rise with one hump after another. The pitches suddenly begin to climb rapidly from step to step, gradually climbing less and less with each step, and then suddenly starting again to climb rapidly again, and repeating that over and over.

bumpiness

For this illusion to work, we need that change to happen gradually. But how?

 

 

Step Three: Vary Tempo

To flatten out those logarithmic humps into a linear climb it will be necessary to proportionally increase the tempo as we proceed through each octave. This way, while the pitch interval between the penultimate and final harmonic is smaller than the pitch interval between the first and second harmonic, the time interval is equivalently smaller. In other words, by the time a harmonic step covers half the pitch as an earlier harmonic step, we’ll have it whizzing by in half the time, neutralizing the logarithmicity. Pitch climb is proceeding linearly.

linearized_pitch_climb

As pitch intervals decrease, duration intervals are decreased proportionally, linearizing the pitch climb (though the notes grow denser… more on that later)

 

This, however, introduces yet another problem! Now we’ve concealed the pitch circularity and the logarithmic rate of pitch climb inherent to the harmonic series, however at the moment we switch back around to the beginning of the harmonic series, there is an extreme change in tempo: it drops by half. This is what that sounds like:

And this is what that looks like in my audio software:

harmonic_circle_step_three_marked_up

 

 

 

Step Four: Rudimentary Tempo Circularity

In order to correct for this, it is necessary to introduce another process whereby every other note is fading out toward silence (this fading out is independent from and additional to any amplitude attenuation caused by the pitch circularity).

With tempo doubled, twice as many notes are occurring at the end of the process, yes, but with half of those notes descending into nothingness, the process cancels out. Here’s what that looks like in my audio software:

harmonic_circle_step_four_marked_up

Finally the two ends of our harmonic series octave sound the same and can be connected. This makes sense, since as can be clearly seen in an earlier diagram, one of the beautiful aspects of the harmonic series, in that each successive octave essentially recovers the previous one while filling in another set of notes in-between each previous note; we’re fading out that new set of notes.

But we’re not quite done yet. You may have noticed that this most recent sample still wasn’t quite as impressive as the final result we listened to at the outset of this discussion (and not just because of lack of reverb finesse). Our tempo circularity isn’t concealed enough at this point.

 

 

Step Five: Philosophical Coherence

The final subtlety to make this effect really come together is to realize that to leave the every-other-note-fading-out-effect in this state would be to commit the same offense that we avoided when implementing pitch circularity: not burying the fading out within enough similar simultaneous fadings, leaving it exposed and easy to pick out.

Analogously to how we stacked many octaves of pitch and drew out the fade out process over them, then, we will actually take the every-other fade-out effect and extend it over multiple iterations of the loop. Notes which are multiples of 4 stay at 100% amplitude throughout, while notes that are multiples of 2 fade to half strength by the end, connecting up with those at the beginning which are odd harmonics, which begin at 50% and fade to zero. And we do this along the shape of a bell curve, again, of course:

final_tempo_circularity

 

 

Conclusion

Which brings us back to where we started:

Thank you for listening and reading! Please feel free to use harmonic circles in your own compositions. Here is a link to my GitHub entry with the final product: https://github.com/DougBlumeyer/Harmonic-Circle

I’ll be back someday soon hopefully with some fun spatial circlings to complement these.

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