Musical Idea 22: Pitch Circularity

03.01.2014 § Leave a comment

If you’re not yet familiar with pitch circularity, it is basically an aural illusion in which pitch seems to continuously rise or lower but without getting anywhere. It is achieved with a few simple steps:

  1. Consider the spectrum of frequencies, then apply a bell curve of amplitude across them, such that both the higher and lower pitches are quieter than those in the middle.

  2. Whenever an entity ascends or descends in pitch, it gets quieter and quieter, then, until it fades into inaudibility.

  3. sameltimeously, a new entity fades in, moving in parallel to it, but rather than disappearing into quietude on the same side of that louder middle band of frequency, it is emerging from the quietude on the other side.

Typically, 2:1 — the octave — is chosen as the frequency ratio between the two entities’ pitches, because this is the simplest harmonic relationship possible, and many people perceive pitches related by it as essentially the same note. Therefore, by the time one entity has climbed an octave from where it began, the other entity will have climbed to that exact spot and attained the exact amplitude as the first had had at that time: the process has returned to its start, and can repeat infinitely. It is as if pitch, rather than existing on a line, now existed on a circle: hence pitch circularity.

This illusion is amusing unto itself, and perhaps its most common application is also the simplest: the Shepard-Risset Glissando, presenting stacked octaves of sine waves either rising or lowering in pitch through a Gaussian function.

But pitch circularity has seen some light in music. Georg Friedrich Haas uses it in several of his compositions. When human performers are attempting to pull this effect off, it may not be as precise. If a particular voice happens to be doubled the composer might call for them to cooperate with each other such that when one is fading out the other fades in, but the gist is gotten across well enough through a mass of sound in which each individual disconnected in time perpetually minds her upper and lower registers, striving to sneak in as imperceptibly as possible from either direction whenever they are doing so.

I haven’t heard near enough pitch circular music for my tastes, though. I feel certain that there must be untapped potential in melodies and harmonic progressions. Pitchwise maneuvers which were ruled out in the past due to the fact that the voices would depart the playable or audible range too quickly can be permitted with pitch circularity. Or in counterpoint one voice could descend and another ascend from unison, but they miraculously arrive back together again.

Plus, there is a lot of potential beyond the typical constraints of pitch circularity.

  • Pitch circles with circumferences other than the octave. Suppose one must climb two octaves before returning.

  • Circularity achieved using stacked equivalences other than the octave. In Bohlen-Pierce I’d expect you’d use the tritave.

  • The circumference and the interval stacked can be independent from each other.

  • The stacked intervals don’t all have to be the same size. They can be random or follow a pattern (harmonics, perhaps).

  • Pitch curvedness / dynamic. Somewhere between straight and circular. That is, suppose your music climbs two octaves, but when it’s over, you’re not back where you started as you would be in normal pitch circularity, but only one octave higher. In other words, the center of the amplitudinal bell curve over frequency rose one octave as your music rose two.

  • Heck, you could have even had the bell curve move in opposition, so that as the music seemed to rise in pitch, the pitch overall actually lowered.

  • Some entities can be in one circle, others in a completely different one. Or jump ship from one circle to another.

How about it, science?

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