What are the intervals between the harmonic intervals? Here we'll derive those, called the melodic intervals, and discover how they fit into the larger harmonic intervals.
So far, we've discovered the six possible harmonic intervals. But, clearly, these aren't the only intervals that we hear in music. Even in the simplest music, rather than this, a succession of harmonic intervals, we're much more likely to hear something like this:
As was said earlier, when I play the whole string, followed by 2/3 of it, I'm perceiving a proportion, a relationship of 3 to 2. Now listen to what's been constructed from the harmonic divisions:
Moving from note to note like this, in a series, the intervals we hear are the relationship of each harmonic division with its neighbors, rather than with the whole string, although the whole string does remain in the mind as a reference point. Now, instead of treating each division as if it were a separate entity, we're beginning to treat them as parts of a unified a system.
Take 3/4 and 2/3. If I play them together, they're clearly not harmonic. We can find out what their relationship is by simply treating 3/4 as the new whole string. Since 3/4 and 2/3 can also be expressed as 9/12 and 8/12, we divide the whole string into twelve parts. You see here that if 9/12, or 3/4 is thought of as the whole string, then 8/12, or 2/3, is 8/9 of it. Moving from 3/4 to 2/3, what we're hearing is a relationship of 8 to 9. Kepler calls this interval the major whole tone, or major tone.
If I do this for the entire system of the octave, I get these intervals between the harmonic divisions. These four are what Kepler calls the melodic intervals, since they are what one would hear when listening to a melody; they don't come from constructible divisions of the circle, hence the cause of their dissonance, but they do arise as lawful derivatives of the harmonic divisions.
This also demonstrates that movement from tone to tone through the audible world is not arithmetic, as is visual space, but is geometric. What I mean is that if I move from 8 inches to 9 inches on a ruler, I've moved 1 inch, and the difference between those places is 1. If I move from 8/12 to 9/12 of a string, however, while the difference of the string lengths is 1/12, what I hear is not a difference of 1/12. It's a proportional difference; a relationship, of 8 to 9.
This may be tricky as you work through the material of Book III, so let's go through another example or two.
If we want to find the interval between 4/5 and 3/4, we treat 4/5 as the new whole string, and figure out how many parts of it 3/4 is. The common denominator of 3/4 and 4/5 is 20, so 3/4 is 15/20, while 4/5 is 16/20, and their relationship, their interval is 15 to 16, which Kepler calls the semitone.
If we likewise figure out the interval between 2/3 and 5/8, or 16/24 and 15/24, we find that it's also a semitone, because the proportion is again 15/16—even though the differences of string length, in this case 1/24, and in the previous case 1/20, are clearly not equal.
The string lengths which produce the semitone are different, but the interval that you hear is the same. This shows, again, that motion, and equality, in this domain, are geometric, not arithmetic; equality is an equality of proportions.
Using the same method, we find that the differences between 5/6 and 4/5, and between 5/8 and 3/5 are both 24/25. This is the interval that Kepler calls a diesis; it's a smaller interval than the semitone, 15/16 1.
Now, the diesis is small enough that the human voice, as Kepler says, almost always has trouble singing it accurately, and overshoots it. Therefore, although it is derived from two neighboring harmonic divisions, it is not suitable for use in composing melodies. This means that in a piece of music, except for rare cases, you wouldn't hear 5/6 and 4/5, or 5/8 and 3/5 in succession.
For that reason, the sequence of harmonic intervals has two variations: one includes 5/6 and 5/8, and the other 4/5 and 3/5. In fact, these two sets of intervals exclude each other for another reason: 4/5 and 3/5 are generated by the class of polygons which is built on the golden section, the pentagon; 5/6 and 5/8 are not. Although they both contain the number 5, the polygons which produce the divisions are the hexagon and octagon. These do not contain the golden section, and have a very distant relationship to the pentagon.
It's important to realize at this point that the melodic intervals are not self evident, something derivable on their own from physical construction, but are a product of comparing the harmonic, constructable intervals. While I can say that a certain combination of melodic intervals "fit into" a harmonic interval—for instance, 8/9 and 15/16 fit precisely into 5/6—the harmonic intervals are not composed of them, as from causes. That would be like saying that because I can look at a man, and say that he has a head, a torso, and limbs, that I could start with just those parts, and build him up from them. Or that when I'm speaking to you now, I first think of the appropriate letters that I want, and make the corresponding sounds—this adding up to words, and then to sentences. Though I can look back after the fact, and say that my sentences are composed of words and letters, it's not the addition of the letters which generated my ideas. The whole is the idea; and the parts come later.
I think you get the point, but it's important to be clear that in deriving the melodic intervals, we're not creating an elementary 'particle', or unit interval which can be added together in various amounts to give us all of the harmonic intervals of the musical system, as if that were the source or beginning of the harmonies. This is underscored by the fact that the melodic intervals are all dissonances—adding dissonance to dissonance is not going to somehow give rise to a harmony, which is of a different quality entirely.
Rather than this bottom up approach, Kepler begins from the top, from the principle of harmony itself; from wholes, as a man is a whole, not a pile of parts. It's only by thinking this way, that Kepler is able to use the harmonic system to complete his discovery of universal gravitation, and present to man an image of a unified, single Solar system. But we'll get to that, in due time...
1. Adding and subtracting harmonic intervals can be counterintuitive. When adding quantities, like lengths of string, addition and subtraction function arithmetically: adding 2/5 inches to 3/5 inches is 5/5 inches, or, 1 inch. With harmonic intervals, which are geometric relationships, increases and decreases are geometric movements, so addition and subtraction become multiplication and division. For example, if I move up octave from the whole string (1/2), and then want to increase that interval by 2/3, I need take 2/3 of 1/2, which means multiplying 1/2 by 2/3 to give 1/3. Subtracting intervals is just the inverse: division. Moving a 2/3 interval down from the octave (rather than up,) I divide 1/2 by 2/3, giving me 3/4.
Book III, Chapter Four