Living in the 21st Century provides us with the opportunity of looking back on Kepler’s work from the standpoint of the development that was to come out of it. The greatest of these developments includes Bach, Mozart, and Beethoven. Looking back to Kepler through the eyes of these composers and in turn looking at them through the eyes of LaRouche provides an interesting vantage point!
Although Bach took the work a step further, Kepler was already on a particular pathway of investigating the role of the musical comma in musical composition. In the concluding chapters of Book 3, Kepler examines his system of tuning further. It was already explained how the major and minor modes are generated by this system, but what other modes are possible? Looking at this question from the advanced standpoint of Bach, Mozart, et .al., the glory of the comma shines forth. But before we get there, let’s look at what Kepler had to say.
Once we have established the octave, encompassing both of these primary modes, we could extend it to a second octave. But, when we did this, some unexpected ironies popped up. Namely, the extended system revealed to us a number of impure consonances. These seemed to somewhat interfere with the perfect harmonies from which the first octave was generated. However, as we will see, these seeming deviations from perfection may turn out to be part of the intention of the Creator after all.
Finding out what is possible
If one were to compose in a mode other that G major or G minor, what kind of difficulties would arise? Could we play such a composition on the same instrument, or would we need a new instrument for every mode? What about transposing from one mode to another? And, what were to happen if we wanted to utilize the power of several modes at once?
Taking the system of two octaves, we have all that we need to create a full octave starting on any note. For example, the diagram depicts how one could start from A, rather than from G. However, the arrangement of the melodic intervals in this octave differs from the arrangement that starts on G. The different ordering of the semitones, limmas, and dieses allows for several possible orderings of the melodic intervals. In fact, these melodic intervals can be ordered in several different ways within the octave, depending on the octave that you are in. The order of these tertiary intervals in each octave is unique but, the question is whether they are all adequate for composition? These variations situate the impure consonances in different ways. What effect does this have on the geometry of the octave as a whole?
Kepler puts the series of tertiary intervals for each octave into the following matrix. The starting note for each of these is given at the bottom of the table. The series of tertiary intervals is then given, beginning from the tonic and moving up the scale.
Let’s take the lower tetrachord in the octave. This is the set of the first four tones in the octave, which are separated by three intervals. In the case of the minor and major modes, this tetrachord is a perfect diatessaron, which contains a major tone, a minor tone, and a semitone. These intervals have a unique arrangement in these modes. The major mode has in its lower tetrachord the arrangement of major tone, minor tone, and semitone, in that order. The tetrachord of the minor mode has the major tone in the lowest position but, it is immediately followed by the semitone. In this case, the minor tone has the highest position. However, these three intervals can be arranged in 4 other ways that would still produce a perfect diatessaron, laying a foundation for other possible modes to be developed. These occur in the octaves of A, C, C#, D, and F#. In all of these cases, the fourth created from the tonic remains perfect.
In this diagram from page 209 in the English translation, the six arrangements are shown. The long line depicts the major tone, the medium line the minor tone, and the short line the semitone. The major and minor modes both have G in the lowest position.
The fifth tone from the bottom is a major whole tone above the fourth, when the consonances are perfect. This is possible in the cases of C, C#, D, and F#. In the case of A, the fifth is only a minor tone above the fourth, and so it is undersized by a comma, and therefore it is melodic, but imperfect.
But, what about our imperfect diatessaron?
What melodic intervals can these contain?
One law that Kepler states about
composition of tuneful melody is that three whole tones in a row in the lowest part of the scale, whether major, minor, or a combination of
the two, is altogether unmelodic. For, this would produce a tone that is between 702/1000 and
729/1000 of the whole string. This entire region is very unstable. For this reason, no melodic mode is possible in the octave of D#, since it would necessarily have two major tones and a minor tone in the lowest position. The fourth tone from the bottom would therefore be roughly in the proportion 711/1000 with the starting tone.
Having two semitones among the three intervals, again, is unmelodic and would make our diatessaron so small that it would be just greater than a major third, in fact, by less than a diesis. For a similar reason, there cannot be two semitones among the melodic intervals making up the diapente. Because of the 2 semitones in the lowest positions in the octave of G#, it is impossible to create even an imperfect fourth or fifth, and therefore no melodic mode can exist within that octave, either.
There remain for the imperfect diatessaron the combinations of one semitone and two whole tones, either both major or both minor. Since the perfect diatessaron is made of a semitone, major tone, and minor tone, and the difference in proportion between a major and minor tone is a comma, 80/81, the resulting imperfect consonances would be either greater by a comma in the former case, and smaller by a comma in the latter case. So, we have the possibility for more modes based on this arrangement but, if we look at the matrix, we find that the latter possibility, a semitone and two minor tones does not exist in the lower position of any of the octaves. This leaves 3 possibilities for the arrangement of the tetrachord, with two major tones and a semitone. All three of these occur in the octave of E, and one in the octave of F. In these instances, the fourth, at 60/81, is oversized by a comma, but the fifth is perfect.
To the mathematical fanatic it may appear that more modes are possible, which just do not happen to occur in the matrix. However, the following point must be emphasized. The modes, as defined by Kepler, are not mathematical permutations. They are all generated by the system of the most natural and primary modes, the major and minor, which themselves were generated by a principle of harmony.
So, after examining in each of the octaves their unique characteristics, Kepler concludes that a total of 14 modes are possible according to the arrangement of the first 5 intervals. The arrangement of the intervals in the upper tetrachord adds further variations, raising the total to 24.
But, ask yourself, what is fundamental here? Is it the arrangement of the intervals and tones? Of course not, what would that be saying about the power of music? As mentioned above, each of these modes, or octaves, have distinct features. When a singer is attempting to sing a piece by a composer such as Bach, they are never trying to calculate the correct value of a note to sing. There is a characteristic quality that they are looking for, which must already exist in their mind before they sing any note. The question of how it got there goes beyond the scope of this pedagogical. But according to Kepler,
Although the hearing does not distinguish when only three strings of the octave are struck (the terms of two intervals), 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 octave 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 the comma will be implicit in the act of distinguishing between the octaves.
Kepler Book III
In the chapters that follow, Kepler assigns emotional characteristics to the particular modes. But the magnificence of those such as Bach comes through in their ability not to compose in one particular mode, but several at once, utilizing the discovery of an even higher principle that unifies the idea of the modes in the musical realm. Here, once again, the idea of the musical comma is our shining star.
Beyond the Boundaries
Speaking to a group of Youth in Berlin on June 28, 2006 Lyndon LaRouche had the following to say on the subject of performance
The key thing in music, is you get to a point which takes you beyond the printed score. It's not a configuration of notes you're singing. But you're getting to an irony, which comes out as an apparent dissonance. But, in the hands of a great composer like Bach, the dissonance is never intended to be a dissonance. It's a transformation. And therefore, going through an unexpected transformation, but a lawful one, is the notion of an idea in music…
But then you realize that the music, as you walk away from it, from a good performance, you find out you don't replay the entire composition in your head as the idea of the composition. Now you have a pivotal idea about certain ironies, certain transitions, huh? And these transitions form the idea. You can get an instant recall of the entire composition, from certain features of the composition. And you can find the necessity for the role of each voice in the singing of the composition, within this idea, or two or three ideas which combine as one idea.
Each of these ideas involves a singularity. What might appear to be a dissonance, but because of the way in which it's resolved by the composer, is not a dissonance.
Now, this means that you are going to have to adjust the way you sing, to compensate for this. You're going to have to somewhat flatten; you're going to have to decide what the relative dynamics of voices are; how they lead into each other; how they lead out of the transition. Which means you're going to darken or brighten certain aspects; increase the volume of certain voices; lower the volume of other voices, in order to get this progress, this sense of dynamics. Because the objective is to walk away from the performance, and be able to put the whole composition into your mind, as a single idea. In such a way that the entire composition comes back to you as an extension of this
single idea. That is, the whole composition now has a unique identity, different than any other composition.
Creativity is truly anti-deductive. This is exemplified by the works of Bach, Mozart, and Beethoven. There is no analytical approach that one could take to understand their music. The notes on the score are the remnants of a creative process. But, competent performance involves reliving the ideas that led those notes to the score. To come to an end, listen to this clip taken from the December 23, 2006 LaRouche Show. Here John Sigerson works through a particular passage of Mozart’s Ave Verum Corpus with a chorus of young people from the LaRouche Youth Movement. If you pay close attention, you will notice the slight adjustment that has to be made in one of the voices, which according to the score stays on the same note, in order to get the transition that is required by the movement of the other voices. As you listen, keep in mind that Kepler not only discovered the harmonic principle ordering the entire solar system, but also of the dynamic role of the infinitesimal in determining the change in each individual planet’s orbit.