Why do pitches separated by an octave sound “the same”?
Answer by Paul King:
This phenomenon is called “circularity of pitch.”
Once a tone has gone up one octave, it seems to be “back to where it started” but “higher”:
As others have mentioned, this effect is derived from the overtone structure of natural sounds. The “richness” of a natural sound comes from several overlaid frequencies, each of which are an integer multiple of the base frequency or “fundamental”, and the reason for this has to do with the physics of how sounds are produced by vibrating objects like strings and vocal cords.
The reason that shifting up one octave “sounds the same” is that the overtone structure of a tone and the same tone one octave higher (all frequencies doubled) is almost the same.
Here is the frequency spectrum of a violin string (the horizontal axis is frequency, and the vertical axis is”power”). The first “bump” is the fundamental and the ones to the right are the overtones:
Shifting this tone up one octave amounts to stretching this spectrum to the right by 2x. When this happens, the spectrum will be almost identical except that every other overtone will be missing. The tone thus sounds almost the same (activates the same frequency-sensitive neurons in the brain), but with a higher “average frequency” and “thinner” due to the missing overtones. This is illustrated here by stretching the above image horizontally by 2x and showing the overtones that line up:
If these two tones are played together, they reinforce each other and will merge to sound like a single note but with a different timbre (different frequency spectrum).
This circular relationship between frequency and pitch leads to the “circularity in pitch judgement” illusion called the Shepard scale in which a chromatic scale of notes seems to rise forever. Audio demo here:
The animation accompanying the audio shows how it works: The frequency spectrum is shifted to the right, increasing the perceived “pitch” (chroma), however the power envelope, and thus the average frequency (height), is held artificially fixed the tone does not actually climb higher. The net effect is this:
Perhaps the creepiest version of this illusion is the never-ending falling tone auditory illusion, here:
To show just how intertwined overtones are with the perception of scale, pitch, and octaves, it turns out that when a piece of music is played on a “stretched scale” (one octave stretched from 2x frequency to 2.2x), the music sounds horribly out of tune and wrong. But if the overtone structure of the notes being played is synthetically stretched by the same amount, the music sounds oddly in tune again.
I think that this reveals a lot about the nature of sense in general. Rather than calling these perceptual surprises ‘illusions’, I would say that they are examples of how conflicts are resolved among multiple levels of sense and sense-making.
In particular, I think that the fact of overtone dominance in tone perception tells us about the Top-Down nature of sensation, where larger wholes or gestalts are interpreted at a higher priority than granular, low level sensation. I think the illusion more likely is in the confidence that we have for our expectations about what perception actually is. When we assume that physics is an observer-independent reality with pockets of privacy containing approximations of that reality, then we overlook the possibility that physics is indivisibly both private and public, universal and proximal. This is the more accurate model in my opinion.
Overtones show us the nested nature of perception where our sensitivity plays an active role on many levels. It’s not just a matter of data accumulating in structures, but of encountering our own local experience of eternity as a rolling ‘here and now’. Like the perpetual floating peak of circular pitch, our here and now is only the most obvious range of a larger phenomenon united by likeness.
Our personal range of awareness yokes together a fugue of sympathetic echoes, both from repeating pasts and the promise of novelty from possible futures. These sub-personal and super-personal ranges are bound by instantaneous space and eternal time, respectively. The more sub-personal you get, the more you are talking about the experiences of organs, tissues, cells, and molecules in spatial relation to each other as bodies, objects, or random machines. The more super-personal you get, the more we refer to timeless themes of inspiration and teleology.
Physics can teach us how to understand the mathematics of ratios and the mechanics of wave, but in its current legacy form, physics can’t explain the physics of ratios themselves, or the mechanisms which drive us to perform the production of acoustic pressure waves. We are dazzled by the perfection of the ratios, but we no longer care what they are actually ratios of.