The musical modes are not the result of theoretical calculations. They are an ancient part of music. Here we'll explore more deeply what the ears of the mind hear, and how musical intervals are not just sounds, but are footprints of musical ideas.
Since his student days at Tuebingen, Kepler would have often heard the music of Orlando di Lasso, and Josquin des Prez, among others, in whose music one hears a yearning for a true, mature polyphony. The ongoing development of polyphony, of the singing together of several unique voices in a unified composition, strongly affected Kepler's concept of the potential of harmonics, as a language for the expression of more and more complex ideas. It should be no wonder, then, that Kepler would naturally turn to harmonics when seeking an apparatus of expression for his discovery of universal gravitation: an idea which, having no prior existence in the realm of human knowledge, had no precedent in any of the current scientific languages.
“Therefore, the motions of the heavens are nothing but a kind of perennial harmony (in thought not in sound) through dissonant tunings, like certain syncopations or cadences (by which men imitate those natural dissonances), and tending towards definite and prescribed resolutions, individual to the six terms (as with vocal parts) and marking and distinguishing by those notes the immensity of time. Thus it is no longer surprising that Man, imitating his Creator, has at last found a method of singing in harmony which was unknown to the ancients, so that he might play, that is to say, the perpetuity of the whole of cosmic time in some brief fraction of an hour, by the artificial concert of several voices, and taste up to a point the satisfaction of God his Maker in His works by a most delightful sense of pleasure felt in this imitator of God, Music.”
For Kepler, music was not sound, but something higher than the senses, closer to the category of a mode of thought which man comes to by discovery, not as a matter of preference, taste, or passive pleasure. Harmonics itself, then, cannot be the substance of music: it is merely the substrate, receiving the impression of that which animates it, its footprint.
However, the science of harmonics was a battleground. The developments in polyphonic music required a more complex harmonic structure than did a single melody with a simple accompaniment, hence, by Kepler's time, harmonics had become, once again, an active field of development. Kepler himself took up an intensive study of the state of development of harmonics, reading all of the musical texts he could get his hands on: from Boethius and Ptolemy, to his contemporaries, Seth Calvisius in Leipzig, and Vincenzo Galilei and Gioseffo Zarlino in Venice.
While Kepler certainly draws heavily from his sources, it's notable that his own derivation of the harmonic system, from the constructible polygons, is unique, as is his non-mathematical definition of a harmonic mean. His subsequent method of deriving 14 musical modes as an elaboration of the basic system of the octave, is also unique.
For comparison, look at the work of Zarlino. In his 1558 treatise, the last section of which is dedicated to the musical modes, Zarlino ends up with the same basic system of the octave as Kepler later, but he comes to it by a different pathway. Like Kepler, he starts with the octave, the first consonance, and derives everything else from it, but, instead of basing his divisions on the constructible polygons, Zarlino uses number. He divides the octave to create two new intervals: first dividing it by the harmonic mean, then by the arithmetic mean, reminiscent of Plato's approach to the musical system in the Timaeus dialogue. Unlike Kepler, Zarlino uses the harmonic mean of arithmetic, so, taking the whole string of length 6, and its octave, of length 3, the harmonic mean between 6 and 3 is 4. A string length of 4 makes a proportion of 4 to 6, or 2/3 with the whole string—the interval of a fifth. It makes the interval of 3/4—a fourth—with the half-string, or octave. To find the arithmetic mean, he calls the whole string 4, and its octave, 2. Three is the arithmetic mean between 4 and 2, which makes 3/4 with the whole string—the interval of a fourth—and 2/3, a fifth, with the octave.
So, both types of means divide the octave into a fourth and a fifth. In fact, the harmonic and arithmetic means are the inversion of one another. This is also true when dealing purely with numbers and arithmetic, but here, with sound, you can grasp it intuitively. Moving a fifth up from the whole string brings you to the harmonic mean, while moving a fifth down from its octave is the arithmetic mean. Said otherwise, the harmonic mean divides the octave into a fifth, then a fourth, while the arithmetic mean divides it into a fourth, then a fifth.
Continuing on, Zarlino then takes that fifth and divides it both harmonically and arithmetically, which gives 4/5, the hard third and 5/6, the soft third. Then, dividing the hard third harmonically and arithmetically yields the major tone, 8/9, and minor tone, 9/10.
These are all of the notes and intervals necessary to build the same octave system as Kepler. But, by deriving the scale from the knowable, constructible polygons, Kepler consciously keeps up front that it is the mind which is constructing and perceiving these relationships.
As for the modes, whereas Kepler builds them by taking the sequence of smallest intervals, the semitone, diesis, and limma, starting from each note of the system, and exhausting their possible combinations, Zarlino starts from the larger intervals—the fourths and fifths. For example, taking the full system, the fourth from A to D is divided by these melodic intervals, and sounds like this:
It can be combined with the fifth from D to the octave a, to complete the octave from A to a. This yields what Zarlino calls his “second mode”.
In this way, combining all of the different types of 4ths and 5ths to form complete octaves, he derives 12 different modes, starting on different notes of the scale, all different in character.
It should be clear by now that the difference of the two, Kepler and Zarlino, is in what each begins from as the causes of the consonances, and how the musical system is then constructed. Zarlino shows that the entire system can be derived by taking the harmonic and arithmetic means of the different intervals. Kepler criticizes that approach, and instead uses only those proportions which come from knowable divisions of the circle with a polygon, and their derivatives. But both end up with the same notes. Why would such different pathways generate the same system? This raises something we've been pointing to all along: that the notes, the sounds, aren't really the thing of substance here. They are just the residues, the artifacts of a higher, unseen, or unheard organization, which is also reflected to greater or lesser degree in circular action, and in the relationships of number and means. All of these have a relationship to higher principles of organization, which we see and hear mapped into these different domains of our senses, but which can only be known themselves within the mind.
Whether you're conscious of it, or not, your sense of hearing reflects this. It is not merely passive, but is guided and shaped by the higher hearing of the mind.
Take a simple example. These two modes, beginning on C and on E, both contain the interval of a minor third, or 5/6, between E and G. When I play these notes, not out of context, but as a part of the entire mode, they don't sound the same anymore.
The frequencies of the sounds of the three notes, taken in by the ears, are mechanically identical: a machine would record them as the same pitches. But, in reality, they are not: their meaning, as tones, is heard differently by the mind.
It's the same when you hear an interval, like a fifth. Each time you hear a fifth, even if it's between different sets of notes, it's the same interval.
But what happens when you hear the fifth in the context of the music, rather than abstracted from it?
The characteristic of the mode, the way that the fifth within it is divided by the intervening melodic intervals, and its role within the mode, changes how it is heard, how it is understood. What's being perceived here is not the fifth alone, but the way that the fifth is characterized by its internal organization and the system it's a part of. This can only be heard by the mind. As opposed to the physical ears, the ears of the the mind don't just hear what's sounding at the moment; they hear the whole context: where the notes have come from, and where they are going.
So we see that what is technically the same note or interval is not a fixed, self-evident object. It's also the case that the pitch of a note is not the only thing which determines what you hear. The quality of the note, and of an entire mode will be different depending on which voice or instrument is producing it. For example, this mode, starting from A, falls in the lower part of the alto voice, while in the tenor voice, it sits in the higher registers, so the same notes, sung by different voices, are lent a different quality by the nature and organization of the instruments.
Up until this point, we had only been considering the notes of the harmonic system in terms of their quantities—comparing and ordering them by the size of their intervals. This measurement is one-dimensional, meaning that I can plot the value of a note on a simple number line, as being greater or lesser in pitch, than another. But, from the preceding examples, it's clear that a note is not one-dimensional—each note is part of an entire system, which the mind perceives, and is determined not just by quantity, but also by the various living and cognitive factors involved in the production and perception of sound.
These aren't things which the simple language of arithmetic can express, which Kepler was well aware of and delves further into in the material of Books IV and V. But, this analytical, or maybe better said, theoretical, "on paper" approach, is actually quite useful. Not only will Kepler need to have a more precise mapping of the harmonic system in terms of interval values to express the motions of the heavens, but even much earlier, it became necessary to have a calculated system of intervals as man began to build musical instruments. As opposed to a singer, who doesn't need to know the mathematical fraction of an interval in order to sing it, an instrument builder must know precisely how long to make the string or pipe lengths of the instrument, so that the instrument can replicate the notes of the singer, and play in a variety of modes; therefore, having a science to calculate these intervals became part of the science of the musical world. However, this analysis, "on paper", comes after the fact. Music itself, which is the real source of harmonics, does not flow from the theoretical, it comes from the desire to externalize and communicate something unutterable precisely. Kepler takes care to introduce his theoretical derivation with this idea:
“Certainly, just as it is ordained in all human affairs that in those things which are bestowed on us by nature, use precedes understanding of causes, similarly as far as melody is concerned it happened to the human race that from its very beginning it used without speculating or knowing about their causes the same rhythms and melodies, not only in churches and in choirs of musicians, but everywhere without applying any art, even at crossroads and in the fields.”
Zarlino, too, recognized that the musical modes have their true root in poetry, in the need to communicate precise, and differing qualities of thought and emotion. At the very beginning of his treatise, he says that the inventors of the modes were poets, that for the Greeks, “musicians and poets...were one and the same.”
Therefore, let's take a brief look via the more modern Church modes, since not much is known for certain about the modes of the Greeks, at how a differentiation of mode could have been developed not from abstract mathematical reasoning, but from the requirements of developing music.
For example, in the church, the Psalms, originally written in song or poetical form, were sung to the congregation. Often, even in very early church music, a second voice, or a chorus joined in, either in response to or in echo of the first voice, like a dialogue between two people. We can see in more developed church music, later, like that of Orlando di Lasso, that when multiple voices enter, in echo of one another, to be distinguishable from the first voice, and yet to still be unified and in harmony with it, the second voice starts on a different note, yet stays within the same sequence of tones as the first voice. It's probably easier to understand what I mean by hearing it.
If the second voice started its melody on, say, the second or third of the original scale, it would either be in frequent dissonance with the first voice, or would be very restricted in its movements. If it starts on the fourth, however, or the fifth below, which is the same thing, then the voices have the greatest potential to be in frequent harmony with the greatest freedom of movement. Let's look at these two lines separately for a minute. The tenor's melody starts on E, rises to C, and then finally back down to E, in this mode:
The bass starts his response on A, a fifth below, rising to E, and then back down, sketching out this mode:
Adding a second voice in this way automatically created another mode which is different than, though at the same time related to the first. For that reason, it was named “plagal”, or “lateral”, indicating that it was moved to the side, while the original mode was called “authentic”.
Though this is only a very simple, limited example, it still suggests how an expanding palette of modes could have been developed by, and yet also driven the development of more and more complex musical ideas within the harmonic system, through multiple, distinguishable voices.
However, the real revolution in this was yet to come, as we'll see in the next video.