It is Mind or the human intellect by the judgment or instinct of which the sense of hearing discriminates pleasant, that is consonant proportions from the unpleasant and dissonant, especially if he ponders carefully that proportions are entities of Reason, perceptible by reason alone, not by sense, and that to distinguish proportions, as form, from that which is proportioned, as matter, is the work of Mind.
Book III, Chapter I
Think back to Book I, where Kepler establishes which proportions are knowable to us in the plane, even if it's by a long series of operations, and which are truly unknowable, meaning that what we see as a “planar” figure is the shadow cast from a different geometry (click here to return to the Heptagon pedagogy). The inconstructibility of the heptagon was a sort of crack through which our minds could peer to catch a glimpse of the domain, which was actually generating what our eyes could see. However, even those various constructible figures fell into several different classes of knowledge, making them distinct and incommensurable with one another. The question then comes to mind: what sort of generating principle unites and orders these constructible, yet incommensurable figures, and their resultant proportions, into a single system? Here begins our study of harmony.
In the study of harmonies, we're not examining each proportion separately, but we're studying the relationships which arise among the proportions, and the creation of a single, organized system, our musical system, from them. We began to examine this in Book II.
As Kepler demonstrates in Book I, man can come to know a geometrical proportion through a process of reasoning and construction alone, since a relationship exists not in the things which are being compared, but only in the mind, which compares them. For example, it is quite impossible to see a quality like incommensurability by looking at two lines, but it can be known by the mind. Despite that, one may quickly realize that some senses are better equipped than others to help the mind perceive a thing like a proportion.
Try a simple experiment. Merely looking at the two lines in the diagram below, can you tell which is exactly half of the longer line, and which is just a little bit off? Now, taking your monochord, have a friend make two divisions: one of exactly half the string, and one a little bit longer or shorter than half. See whether through your sense of hearing alone you can tell which is exactly half and which is not. Click the play button under each string in the diagram to see if you can better hear which is the true proportion of 1:2.
Let's take as our unit the interval between the two tones in the proportion of 1:2, since in geometry this proportion is of the first degree of knowledge, being a cut directly through the center of the circle, and therefore equal to the diameter. The construction of this proportion creates a geometrical figure in which all of the elements (side, star side, angles and area) are one. It is also the figure which divides the circle into two identical parts, and is therefore the relationship to which Kepler gives the name “identity of sound”. Now we have two extremes which define a basic unit, between which there is a continuum of sound. (Click Here:
As the lines rotate, the square is turned into a changing rectangle. When the lines reach the diameter, the edges, angles and area are all unified with the diameter, making it a unique polygon.
So what relationship do the other classes of proportions bear to this unit? Divide your monochord string into four equal parts. Plucking two of the four parts would be the same as ½. Plucking one of the four parts generates a tone which bears the same relationship to ½ of the string that ½ bore to the whole. The remaining three parts also generate a consonant, or pleasing relationship to the whole string. (Click Here:
If you play the two parts (¼ and ¾) together, there is also consonance (Click Here:
In fact, one division at this particular place produces a three-fold consonance (Click Here:
The division forms a complete consonance with every part of itself, and because ¾ falls in between 1 and ½, it functions as a sort of mean between the two extremes. This causes Kepler to name it a harmonic mean, or a harmonic division (more on this here).
Before proceeding, the reader should perform the same test with each of the proportions from Book I. (Click here for sound files of the divisions—this is no substitute for doing it yourself!!)
So we see by trusting the guidance of our hearing that each of the knowable proportions which were the bases of classes of figures in Book I produce a harmonic division of our musical unit, giving six harmonic means in all (Click Here:
Now, what about the figures which were unknowable, like the heptagon? What do they produce?
(Click Here for an Example:
) More on this in the next part.
Kepler answers us: “...the causes of such intervals' being harmonious should have a mental and intellectual essence, that is that the terms of the consonant intervals are properly knowable, but those of the dissonant intervals either cannot be 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, and have in no way matched the archetype.” (Book III, Chapter I, page 139)