The Harmonic Divisions

The continuous space of sound can be organized into unit octaves, but there is more differentiation to be found within it.   Here, we will discover by physical experiment, and then geometric construction, the 7 harmonic divisions of the octave.

In the last video, we began investigating the world of our hearing, of sound. What seemed, at first, to be a space in which extension, or movement is simply linear, as it appears to be in visual space, we began to uncover that sound has a cyclic quality to it, which defined an inherent unit: the octave.

Here is that demonstration again:

When we hear only a single tone, the extension upward in pitch seems to proceed linearly, with each successive pitch in the continuum proceeding as if in a straight line, farther and farther away from the starting note. But when we add another dimension to the experiment, a second tone, creating an interval, a different characteristic of the space is exposed. This quality of sameness of two different tones defines an inherent unit in the harmonic system, the interval which Kepler at first calls "identical by opposition", and later an "octave".

Now, let's examine the space in between—the continuum of sound which is bounded by the two notes of the octave interval. Watch this experiment:

What's happening here? The string is plucked, and as the finger is lightly moved up the string, barely touching it, certain regions come to life, giving out a sound almost as if on their own, while other regions remain inert. Again, there is no change in what the finger is doing from place to place, such as applying more or less pressure, but the response of the string is quite different from moment to moment. Go back and watch the experiment again. This phenomenon is probably familiar to string players, like guitarists or violinists, the tones usually called overtones, but it also demonstrates that what at first sounds like a uniform space, is really not uniform at all; it has within it inherent differentiation.

Let's investigate further.

If I mark off each place where the string gave out a clear sound in the experiment, and then play them, I get this:

Each of these divisions, we found by a simple physical experiment. However, what you just heard also corresponds to the consonant divisions of a string, and thus the consonant intervals, which were known long before Kepler's time. In other words, they correspond to divisions of a string made using whole-number proportions, and each produces a sound which is consonant, or pleasing when played together with the whole string. For example, the division where a very strong resonance occurs, is near the division of 2/3 of the string. Sounding 2/3 of the string with the whole string is consonant. Here are the rest of the consonant divisions:

Each of these divisions produces not just one consonance with the whole string, but are what Kepler refers to as the harmonic divisions, harmonic meaning that the single division produces a three-fold harmony: the larger part with the whole is consonant, the smaller with the whole, and the two parts with each other, so that if all three tones created by this single division are sounded together, the result is completely consonant with itself.

For the sake of contrast, the division of 5/9 is not a harmonic division. This is also the case with a division which is a just a little bit longer or shorter than a whole number division. Though these imperfect divisions are not harmonic, we will encounter them again, later in Book III, and in again Book V.

From this experiment, we find that there are six unique harmonic divisions between the whole string and the half1.

The Venetians, such as Francesco Zorzi, tried to obfuscate real scientific method, and thus the human mind's actual relationship to the universe, by attributing the causes of these harmonic divisions, and why there are only six—seven, including the division of 1/2—to numerology, or mere symbology, ascribing a mystical property or power to the numbers themselves. For instance, one of the properties of three is that it's a “perfect number”, meaning that it is equal to the sum of its divisors, 2 and 1. Nine and 27 are the square and cube of 3. These are interesting relationships in the domain of number, but these numeric properties don't translate to a physical harmony. Another example is the number 7, to which Zorzi among others, ascribed a power to because of its religious association, but, again, not only is 7 a dissonant division of the string—this association gives the mind no basis for knowledge.

Kepler, as Cusa before him, pursued causality within the physical ontology of mind. Since the human mind is a physical force, a source of change and development within the physical universe, and is an image of the Mind of the Creator, then the causes of physical phenomena must be locatable within what is knowable to the mind of man—within the realm of his hypothesis—not in the mystical properties of the interpretation of mere numbers, as such. This is something which Kepler repeatedly attacks throughout the book.

“...and after [the causes of the consonant intervals] have been sought for two thousand years, I shall be the first, unless I am mistaken, to reveal them with such accuracy...For since the terms of the consonant intervals are continuous quantities, the causes which set them apart from the discords must also be sought among the family of continuous quantities, not among abstract numbers, that is in discrete quantity; and since it is Mind which shaped human intellects in such a way that they would delight in such an interval (which is the true definition of consonance and discordance) the differences between one and the other, and the causes of such intervals' being harmonious, should also have a mental and intellectual essence, that is that the terms of the consonant intervals are properly known or are unknowable. For if they are knowable, then they can enter the Mind and into the shaping of the archetype; but if they are unknowable (in the sense which has been explained in Book I) then they have remained outside the Mind of the eternal Craftsman...”

Think back to Book I. There, Kepler demonstrates rigorously what he means by knowledge, or that a phenomenon is knowable. What is unknown can be brought within the boundaries of the mind by bringing it into relationship with what is known, by a constructive action of that mind.

A musical interval, then, which is not a sense-perceptible object, like a tone itself, but a proportion or relationship between two tones, cannot exist in either of the sounds themselves. It is a relationship of them created within the mind. Therefore, if an interval is to be pleasing to the mind, then it must represent a relationship which is constructable by that mind. The reason why 4/5 is consonant, while 5/9 is not, has nothing to do with the properties of the numbers as numbers, such as 5 being a prime number, or 9 being the square of 3. Rather, Kepler locates their consonance or dissonance in their quality of knowability; in that certain proportions are derived from a constructable division of a circle with a polygon. The square and pentagon which create the divisions of 4 and 5 are constructable, and knowable, and therefore, consonant—the nine-sided figure is not.

The cause for the set of harmonic divisions which form the basis for the musical system is to be found in those proportions which can be physically constructed by man.

Knowledge, and harmony are not severed from the physical universe, accessible only by 'code', but are physical and discoverable.


1. You can try another, related experiment on a piano. Lift the dampers on the piano strings by holding down the rightmost pedal. Then, with your foot still on the pedal, strike a key, C, for example. What you'll hear is not only the note 'C' that you played, but certain other strings will begin to vibrate and sound, even though they weren't struck by the hammers, while other strings won't move or sound at all. You can even lightly touch the strings and feel which ones are moving, and which aren't. Those strings which sound most strongly will be the octaves (strings in a 1:2 relationship with C) and the fifth (strings in a 2:3 relationship with C.) Kepler comments on this phemonenon is Chapter One of Book III: “...a string which is set in motion draws another string which has been set in motion into sounding with it, if it has been tightened into consonance with itself, but if it has been tightened into dissonance leaves it motionless. Since that cannot come about by the intervention of any mind, because the sound, the supposed cause of it, does not have mind or understanding, it follows that we can say it comes about by the adjustment of the motions to each other.”


Book III, Chapter One
Book III, Chapter Two

Next: Harmonic Means