The world of our hearing is a continuous space of sound. But within that, we'll discover that what seems like a space where pitch extends linearly is actually structured around an inherent unit: The Octave.
For most of us, our daily lives are full of sound. While not a conscious process, as we develop in infancy, our brains have learned to organize what we hear into discreet, audible "objects", to which we usually give a name. Much of the time, these audible objects function as an adjunct to the world of our vision, sounds which accompany the objects that we see. What most people today never have the opportunity to realize, is that movement within, and the organization of the audible world is actually quite different from that of our vision. The investigation of that difference throughout the material of Book III will lead us away from the senses themselves, and into an investigation of mind. Let's begin with some experimentation.
Take an everyday sound. As you hear the siren, what you hear is not a succession of discreet pitches, from the lowest to the highest. Rather, you hear the siren pass through an infinite number of pitches: a continuum of sound. It's the same with speech. Even though the range of pitches in ordinary speech is small compared to the siren, still, my voice doesn't move from one discreet tone to the next, to the next. Instead, as I speak to you, with every word, the sound of my voice travels through an infinite number of tones between the lowest and the highest.
And so, it seems that sound is a continuous domain which, we can hypothesize, extends infinitely, even if it quickly passes outside the range of human hearing. When I talk about "extending infinitely," what most readily comes to mind is probably a visual image, something like this:
This is a simple abstraction from our visual perception, of what we call space. But, is this how our perception of sound operates, too?
Listen to this example. [[SOUND GOES HERE]] What did you hear? I played two pitches together. One stayed the same, the other changed continuously from low to high. Listen to the two again. Was what you heard the same throughout? While the moving pitch was constantly rising, which I can represent visually like this:
...not only were there changes in the quality of what you heard, but there were several moments where it sounded like we arrived back at the starting point--like the two pitches became one again. So what we heard wasn't linear extension, but something that can be represented more accurately like this :
The tones which have this relationship to each other are typically called octaves, or, as Kepler calls them in the first Chapter of Book III, "identical by opposition." In order to produce these tones by the division of a vibrating string, I would have to measure out lengths which are either double or half of each other.
The relationship is also expressed in the human voice, as when a man and a child, or a man and a woman sing the same melody.
These octaves, or points of identity, mark out a basic, inherent unit in the space of the harmonic system, a point of return, or sameness. This sameness is something which is immediately recognizable to the mind, as if by instinct--not something which requires a long calculation. For the Pythagoreans, this phenomenon, which Plato in the Timaeus dialogue calls the "forceful harmonization of sameness with difference," presents the concept that in the physical world, an infinitude of diversity can be derived from a single, homogeneous One.
In the next video, we'll see how, through pursuing a principle of harmony, we can begin to derive the rest of the musical system.
Book III, Introduction
Book III, Chapter One