When we hear music, we are never perceiving a single tone alone, but we are always hearing—with the ears of the mind—the space that those notes are a part of. The musical modes will help us explore this.
Take a familiar melody.
Now, listen to it again.
One more time.
The was the same, recognizable melody, right? But each time you heard it, there was also something distinctly different, there was a different feeling to the piece. That difference is a difference in the mode in which the melody was cast.
Here, we'll present the musical modes as a continuation of the development of the musical system, first as Kepler treats them. In the second part, we'll look backwards to one of Kepler's sources, Giuseffo Zarlino, and then, forward to J.S. Bach.
Before we get to Kepler, listen again to the example. I played a melody in its original mode, and then in two other types of mode, or scale. The boundaries of each are the same, each spans the octave, and each has the same number of notes. However, the space within that octave is organized differently by each. In the first case, we have this set of intervals, and in the second, this set.
When you hear one, and then the other, what your mind is perceiving is not simply the notes that you heard, but the space, or territory of that entire octave, and how these various modes are uniquely encompassing the notes by their individual higher unity. The whole octave is in the mind, implicitly, the whole time, acting to create the context from which the music can take shape.
You don't believe me? Listen to this example, and when it stops, I want you to hum or sing what the next note should be.
So the whole octave is implicitly heard in the mind, by instinct, even when it's not heard with the ears, and the natural ability to perceive a difference in mode comes from the mind's ability to hear what the ears cannot. This points our attention to which of the two is really primary.
With that, return to Kepler. I showed in the last video that beginning a scale on any note other than that of the original whole string gives a series of intervals which is not the same as the original scale. You can hear it best here, in the "A" scale, or the scale beginning on the second note of the system.
Kepler, playing more the part of the theorist here than the practicing musician, begins a systematic analysis of what possibilities for variety can be derived as an elaboration of the harmonic system, derived from the knowable divisions--as we've just developed it fully in the preceding videos. He begins by laying out this matrix with a listing across the bottom of each note of the scale, and then in each column above, the order of the intervals, if one began the scale on that note.
The symbols he uses, S, D and L stand for semitone, diesis, and limma, which are the three smallest intervals of the system. You can hear right away that they sound very similar, almost indistinguishable. The difference is more perceptible than you might think, but we'll get to that soon enough. You can see that the order of the intervals, listed vertically, doesn't change from column to column, we're merely shifting the starting note up by one interval. To be more clear: in the first column, I start on the note “G”, then move a limma to the next note, “G#”, then a semitone to the note “A”, and then another semitone to the note “B-flat”. When I start on “G#” as the basis for the scale, here in the next column, I'm already a limma higher than “G”, so the first two intervals are semitones, from “G#” to “A” and from “A” to “B-flat”.
The diesis and limma, two of the three different “half steps”, are not melodic intervals, however, they combine with the semitone to yield the major tone, 8/9, and the minor tone, 9/10. Those two intervals along with the semitone itself are the three melodic intervals that we hear in any scale. Kepler systematically goes through each series of smallest intervals, and exhausts the possible ways they can be combined to yield the 7 melodic intervals of an octave. For example, starting on C, we can combine the semitones, dieses and limmas in this arrangement, or in this one, yielding what Kepler designates the sixth and seventh modes, both of which begin on C.
Kepler rejects some, such as G#, as not having any possible modes built on them. The series of smallest intervals cannot be combined in a way which yields a scale that would not violate a sense of musical aesthetics, such as the absence of a proper fifth.
In all, there are fourteen different modes, as derived by Kepler—all of them variations upon the original soft and hard scales. Remember the six skeletons of the octave, from earlier? These came about from dividing the octave harmonically with two harmonic means. Three of the skeletons express the soft mode, and three the hard.
The musical modes which we just derived correspond to these skeletons, depending on whether they are of the soft or hard kind, and on where the interval of the pure fourth lies in the scale. For example, Kepler's seventh mode could correspond to the FOURTH skeleton, or the SIXTH ONE—depending on which notes a composer emphasizes in a melody. So the modes are related based on their relationships to the six skeletons, which, again, are derivatives of building a system based on harmony.
At this point, go back to the three, small “half-steps” which were combined by Kepler to create the melodic intervals of the different modes. We noted that it's very hard to hear a difference between them. When I combine the semitone with first a diesis, to form a minor tone, and then a limma to form a major tone, it's likewise very difficult to tell them apart, especially when I play them in succession. Their difference is only the tiny interval of a comma, 80/81. Therefore, you might protest that two modes whose only differences are in swapping a major tone for a minor tone, and vice versa, are really indistinguishable. However, if you think that, you're underestimating the types of subtleties the mind has the power perceive.
“I reply that although the hearing does not distinguish when only three strings of the octave are stuck (the terms of the two intervals [the major tone and the minor tone]), yet when all the strings in one octave are struck, in that way it does eventually distinguish the first born and natural octave of G from the octaves of the other keys. For there are in all the kinds of octaves the same seven melodic intervals; and when they have once been evinced by the striking of the strings, they at once stick closely in the memory, so that it is easily clear to the hearing in which position in the octave each natural tone begins. On this basis, appreciation of a comma will be implicit in the act of distinguishing between the octaves. For just as the hearing approves the consonances themselves and all the melodic intervals by its effect, though it does not reckon the lengths of the strings, which provide the cause of the consonances, so the same hearing also recognizes the effect of the comma: 80/81, even though it does not reckon it, and therefore does not detect it as a separate sensation in practice.”
At this point, you may be asking what these modes have to do with Kepler's discovery of gravitation. Listen again to the example from the beginning.
Taking a melody, just one musical line, and casting it in different modes alters its quality—you understand the idea differently when it's stated in one mode, rather than another. With each mode having such distinct characteristic, how, then do we go about harmonizing multiple modes, at once? This is exactly the problem that Kepler has to solve, as an astrophysicist, to understand the construction of the Solar system. So Kepler studies the musical modes in a very particular way, with a very particular aim: of establishing as fully as possible, the variety available in the language of harmonics, and establishing the level of subtlety and nuance of that, that a mind can hear and respond to. He needs to do this, because he's going to use this harmonic language, as we'll see, to express the motions of the planets in Book V.