One of the most unfortunate aspects of growing up in a post-industrial, no-future generation is that, for the most part, your life becomes deprived of beauty. Now, you might say that beauty is in the eye of the beholder, or that beauty, being subjective, is different for everyone. I say, that by investigating musical harmony, a touchy subject for many, you find one of the most provocative proofs of not only the universality of beauty, but of the science of the mind, more generally.
But, in order to discover this for yourself, it is necessary that you build a monochord, an instrument with a single string, and perform the following experiments with a group of friends; you won’t be able to do this by yourself. This will require you to use your bel canto voice, so if you have been missing the choral sessions provided by the LYM, you may be at a disadvantage. But, since these sessions are ongoing, that problem can be easily remedied. One more piece of advice before moving on: to get the utmost from these experiments the group should be mixed with men and women, preferably at least three of each.
In Chapter 2 of Book III, Kepler defines a harmonic division of a string as a division which produces parts which are both consonant with each other as well as with the whole string. For example, divide the string into two parts which are in the proportion of one to two. Have one group sing the tone produced by the whole string. Another group should sing the tone produced by the shorter part of the division. The last group should sing the tone of the longer part.
How many harmonic divisions are there? What determines the limits of such divisions? Certainly, there is no way to find this out by looking at the string or at mere numbers alone, so you will have to put the string into motion and experiment.
If you have divided the string harmonically, you will find that the total effect of singing all three of the tones thus produced is pleasant. If your group includes both men and women, there is an added benefit. Since the voices of men and women differ, the tones sung might be different but, in a way, identical. This feature can be used to further explore each of the harmonic divisions. Perhaps try all men or all women in a group and then rotate the tones sung through the different groups. You can also try mixing the groups. With all of these variations, the outcome of the experiment should not change. The result should still be pleasant. But, you may ask again, how can we be certain that what is pleasant for one, is pleasant for all? I suggest that you keep that question in mind as we try the next set of experiments.
Now take the harmonic divisions and repeat these divisions on the respective strings. For example, taking the string divided into 2/3, repeat this division, producing a string of 4/9. Have the three groups sing these tones. One group should sing the whole string, the second group the first division, 2/3, and the third group the division of the division, 4/9.
What happened? Try the other proportions. This may produce some funny results. You may find it a little difficult to keep the pitch. Why is that? What is occurring in your mind as you do these experiments?
principles of knowability and constructability, demonstrated in Book I, apply here. Taking the string as the perimeter of a circle, there are an infinite number of divisions that can be made with the corners of an inscribed polygon. However, not all divisions are possible. Dividing the string in such a way as was prescribed by our first example can be done by inscribing a triangle, a constructible figure, into a circle. Since the corner of the triangle cuts the perimeter into thirds, the proportion of two-thirds can be arrived at easily by this process.
When you perform the first set of experiments with a mixed group of men and women, you find that the total effect is still pleasant regardless of which vocal species is singing which part. Although, the pitch may be different, the “note” is the same. The variation thus created is equivalent to dividing the string in half, or doubling the sides of the polygon that generated the original division. According to the principles of Book I, this type of process is indeed constructible. Whatever polygon you are dealing with, its sides can be doubled by dividing the angle in half, a process which can go on indefinitely.
However, the latter set of experiments forces a certain problem to the surface. Since an angle cannot be trisected or quinsected, etc., the constructability of the corresponding polygon is impossible, and we certainly heard this. Our example took 4/9 of the whole string, demanding the construction of a nonagon, an inconstructible figure. The relationship between the whole string and 2/3 of it is consonant, as is the relationship 2/3 and 4/9. In fact, they are identical consonances. But, singing all three together becomes very unstable. Just as the heptagon is altogether unknowable, there is no constructible way of getting beyond the first trisection, the triangle, or the first quinsection, the pentagon. Perhaps, in this way we can begin to see how the domain of consonance is finite, yet unbounded.
Throughout Book III, Kepler thoroughly investigates this audio side of the universe. Ironically, he appears to start with axioms and propositions. However, his method is in no way akin to that of Euclid's. These are the same principles, discovered in Books I and II, which in those locations gave us knowledge of what is possible in the visible domain, that also reach into and determine what is possible in the auditory domain, as we shall see in what follows.
Now, I ask, was that which we heard the harmony? Can harmony even be heard at all? Is it the ear that tells you whether something is harmonious or beautiful? Certainly, there is a sensation occurring in the ear. When we hear an awful sound we are impelled to protect our ears. But, what is it that creates the difficulty to hold a dissonance?
For what, I ask, is the proportion of titillation of the hearing, a corporeal thing, to that unbelievable pleasure, which we feel totally within the mind from harmonic consonances?