The musical system of hard and soft melody is extended to multiple octaves. Now any note of the octave can be the basis for the scale—but all of these other scales contain what Kepler names the Impure Intervals.
At this point, we've been able to define a full musical system of one octave, an octave scale, in two varieties, hard and soft. This was done through a process of investigating the harmonic intervals as primary. In other words, movement through the scale does progress stepwise, but these small unit "steps" were not the first intervals to be derived, and were not subsequently added together to generate the larger intervals. Rather, the first interval to reveal itself was the octave, the largest. Next we were able to find the larger harmonic divisions as points of singularity within the octave, and finally from them we found their differences, the smaller melodic intervals; with those we filled out the scale.
Now return to what was demonstrated in the first video. We started with a continuum of sound. We heard that as the pitch proceeds farther away from the starting note, there are moments when we can clearly hear a return, as if we've come back to the original note. This defined the octave. Now, moving in discrete steps through the scale, we naturally hear the same phenomenon: each successive note past the octave is generated by half the string length of the corresponding lower notes, and sounds exactly an octave higher.
So, the system is extended by repeating the same series of intervals both above and below the original octave. This means that the whole string is no longer the only note which has its octave. This opens up the possibility for a complete octave scale to begin on any of the other tones in the system.
Let's take a closer look: for example, beginning from the second, the note I've here represented with the symbol A if the original string is called G, I can play either the hard or the soft scales. The soft scale sounds like this:
While you may have to go back to listen to it a few times, the intervals of the scale that you just heard were not identical to those of the soft scale in the original octave, to those generated as a result of the harmonic divisions.
Let's take an example to make that more clear. Here, from A to C, I move the interval of a third in terms of the steps of the scale. However, when I calculate exactly what the interval is, between the string of 8/9 and of 3/4, what should be a "soft third" is not 5/6, as it was in the original octave. It's an interval of 27/32. You can hear right away that 5/6 and 27/32 are not the same interval if I play them simultaneously.
Here are those two scales together. To make it easier to hear the differences, I've transposed the "A" scale down to begin on the same tone as G, but kept all of its intervals the same.
You can repeat this process, beginning the scale on any note in the system. None will be identical to the original octave, and most will contain what Kepler names the impure consonances. The “undersized soft third”, 27/32, that we just found in the A scale is one of them. Another example is in the scale that begins on the seventh, or F. The fourth between F and the soft B (or, B-flat) in the next octave is not 3/4, but 20/27, and oversized fourth. If I then take the octave from F to f, you can clearly hear that the fourth, 20/27, does not divide it harmonically.
From this, it seems that any scale, or melody, begun on a note other than G will no longer be built on pure harmonic intervals. Or, if we want to keep the intervals harmonic no matter what note we start on, then the notes will always have to shift around, adjusting to keep the correct intervals.
Notably, in his Mysterium Cosmographicum, 23 years earlier, Kepler already recognized the presence of this ambiguity. He demonstrated it there with this simple example: In a two-octave scale, the seventh note lies a fifth, or a 2/3 interval, above the minor third. It also lies a fifth below the octave fourth; each is five steps of the scale away. If the tones in the system have been lawfully generated, then there should be no ambiguity about their pitches; they should be firmly established quantities, like rungs on a ladder, and moving to them whether from above or below shouldn't change their value. In Kepler's example, starting from the minor third, 5/6, and moving up a fifth, would mean that the seventh is 5/9. Starting from the octave fourth, or 3/8, and moving down a fifth gives me 9/16—a pitch which is just slightly lower than 5/9. The difference of these two sevenths is a very small interval, just 80/81, which Kepler names the 'comma'. But why is there a difference at all?
As before, when we try to find stability, or regularity within the system that's being generated, in the process, more and more irregularities come to light. Not only do different areas within the octave have different harmonic qualities and characteristics, as we saw with the very first experiment, but now we see that those harmonies and intervals are not even uniform throughout the system.
While these imperfections might seem, right now, to be frustrating inconsistencies to anyone looking to pin down a fixed musical system—or, more practically, to a musician who would like to play music in more than one octave—the existence of the imperfect intervals are actually an integral part of both the harmonic space that we perceive through the ears, and also the harmonic space of the Solar System. This will lead to the identification of a higher quality of harmony than that found in the intervals alone. We'll begin to address this in the next video, however, it can't really be made clear until the material of Book V.
Book III, Chapter Eleven
Book III, Chapter Twelve