Chapters 1-2
Chapters 3-8
Melodic Divisions
Impure Consonance
Galilei's Convenient Division

©2006 LaRouche Youth L.L.P.
Constructing Kepler's Scale

Constructing Kepler's Scale

What gives us power over the universe? There is no question that we are always interacting with the universe in some way. But what gives us the power to take a prominent role in determining the outcome of our actions and other events? In other words, how do we shape the course of history? The cases of Kepler, as well as of Lyndon LaRouche, make explicitly clear that what separates human beings from any other creature in the universe is our ability to pierce the veil of sense perception and grab a hold of the principles that order the interactions in the universe. Inseparable from this ability is the ability to utilize these principles to the effect of improving the lives of the living and yet unborn. At the present moment, the United States, as well as the rest of the world, is confronted with an existential crisis. Therefore, unless we shed some light on this very question, we may not end up where we intend to.

Composition of classical music exemplifies the intervention of creativity in a social process. We all know that music has a certain type of power over us. However, to adequately understand that power, it is important to understand that classical music, specifically, came lawfully out of a historical process driven by the continuous exploration of the physical universe for a greater means of achieving this effect of improvement in the quality of life. This is clearly demonstrated by the work of Kepler, his predecessors, and his followers. So, therefore let's explore the universe in the way that Kepler did.

In the first few chapters of Book 3, we discovered a quality of consonance and particular proportions that created harmonic divisions of the string, where all the relationships in that division were consonant with one another. Each of these harmonic divisions were like self-sufficient universes, in that each part fit in with the whole. But, how are these universes of harmonic divisions united? So far, all we have is an accumulation of effects. Perhaps we can find a higher intention behind these things. If not, we can’t make the power of music our own.


Navigating the Audible World (chapter 3)

Let’s look at these proportions as musical magnitudes. As we noted earlier, if we took a string and gradually but continuously changed the length of it, we would pass through an array of tones until we came to half of the string, where the pattern of sounds would begin to repeat. So, for that reason, let’s take half of the string as our greatest interval. Lining up these harmonic proportions up according to size gives us the following. Here, 1/2 is the greatest interval, followed by 3/5, 5/8, 2/3, 3/4, 4/5, and 5/6.

How do these proportions relate to one another? Clearly, these proportions are all moments of singularity. But, can we put these singularities together in such a way that we can travel through the audio world? Of course, these are proportions in relationship to the whole string. But they also have relationships to each other. What is the nature of these relationships? Remember, in this world of sound, we are not as interested in the sound, as such. We are interested in principle that binds the sounds, as well as the intervals between them, so that these relationships can be recognized by the mind. In other words, we are investigating the principle of harmony itself. From that perspective, what kind of investigations will lead us to conceptualize this audio world as an integrated whole?

How do we move through this space? What kind of motion is possible harmonically? If we take a closer look at the largest interval, could we find a way of traversing this same space by a different path? Yes. In fact, there are several ways to do this. Taking the three smallest intervals, there are six ways to move from the whole string to half the string via these three smallest intervals. One of those routes is illustrated below.

First take 5/6 of the whole string. By then taking 4/5 of that, another harmonic division, we arrive at the 2/3 of the original string. We can then take 3/4 of that to arrive at our destination of 1/2. Each of these actions are characterized by the harmonic proportions.

The rest of these routes are in the following chart, given by Kepler.

In this diagram the harmonic division of 1/2 is itself divided harmonically. The proportion of 1/2 can be divided in such a way that there result two means, i.e., two other divisions that are consonant with the extremes (the whole string and half of it) as well as with each other. There are six possible ways of dividing this proportion so that all of the divisions are consonant with one another. On that account these divisions are all harmonic.

Kepler says you can think of these as harmonic means. In fact, he comments that it is superfluous to think of a harmonic mean in mathematical terms. As the name implies, the idea of such a mean is conveyed to the mind more eloquently through the ears. This type of process can also be applied to smaller harmonies. 2/3, 5/8, and 3/5 can all be divided in a harmonic way. The only difference being that in the former case, the proportion of 1/2 could be divided so that there results two means. In the latter cases, with the smaller consonances, there can only be one mean at a time. So, this means that we can also traverse the smaller space defined by these other harmonic proportions, moving through the space harmonically.

A further investigation of the means, as they arise in the domain of music is presented in Book V.

Of the harmonic proportions smaller than 1/2, only 3/5, 5/8, and 2/3 can be further divided harmonically. In each of these cases, there are two possible ways of inserting one mean between the extremes. Here, they are all demonstrated.

But, this is the extent of this method of investigation. We have reached a limit. The smallest intervals, 5/6, 4/5, and 3/4 cannot be further divided. And if we were to keep it at that, the only music we could compose would have to be pretty bouncy. This may not be fitting for all occasions. We must try to pursue another route to navigate this musical world. Perhaps we should add some smaller intervals that will give us more freedom to move about. Although these new intervals will not be harmonic divisions, we shall call them melodic if they help us to produce melodies. But we must find the intervals that are appropriate to achieving this end.


A Common Unit? (Chapter 4)

In mathematics, it is often useful to find a common denominator between terms. Maybe if we examine the proportions that arise between these intervals, we might find a unit of interval which is common to all, a common denominator. And so, in this way the ancients tried to establish this common element, of which all of the intervals were composed. Let us see.

Each of these circles have been divided according to the polygons responsible for each harmonic division of the string. To find the interval between the two harmonic divisions, it is necessary to further divide the circle. For example, to find the interval between 2/3 and 3/4, combine the square and triangle. At each vertex of the triangle draw in a square. This divides the circle into twelve parts. 9 of these parts are equal to 3/4 of the whole, whereas, 8 parts are equal to 2/3. Therefore 2/3 is equal to 8 of the 9 parts that make 3/4. The difference between these two divisions is shown in bold.

By this method we find these secondary proportions 9/10, 8/9, 15/16, 24/25. But if we are to find a common unit, we must continue this process and see where it leads. The intervals that exist between these are:

Between 9/10 and 8/9 there is 80/81, called a comma
Between 8/9 and 15/16 there is 128/135, called a limma
Between 9/10 and 15/16 there is 24/25, called a diesis

To hear these tertiary intervals, compare the secondary intervals in the previous diagram.

Now, in these intervals we still have not found what we were looking for. But if we continue to subdivide, as Kepler does in the text, the proportions that we find get more and more hairy. I don’t think we will ever find it. There is a sort of incommensurability here. Not only that, but we are moving further and further away from the harmonies that we intended to unify, into an area where the change of tone is becoming altogether imperceptible. So, Kepler affirms that, “thus to seek to establish such a smallest interval which is common to them is inappropriate, since smallest and greatest are observed not in qualities but in bare quantities, whereas to divide consonances as consonances is to destroy a sort of consonance, and in its place to establish either other kinds of consonance, or dissonant melodic intervals, or even downright unmelodic intervals.” This is an important ontological point that Kepler is making. The universe is ordered top-down by principles, and not assembled from the bottom up. These consonances are a species. In the same way that you cannot put together the right combination of carbon, oxygen, nitrogen, and other elements to make life, you will not find the essence of harmony in its parts.

However, looking back over what we have, these secondary intervals that arose between the consonances, which Kepler calls melodic, are sufficient to aid in navigating this auditory terrain. These are made up of the tertiary intervals.

The major tone, 8/9, is made up of a semitone, 15/16, and a limma, 128/135
The minor tone, 9/10, is made up of a semitone, 15/16, and a diesis, 24/25

All of the consonances can be traversed by a combination of major tones, minor tones, and semitones. Therefore, if we take the semitone, limma, and diesis, as our smallest units we can use the secondary melodic intervals of 8/9, 9/10, and 15/16, generated from the consonances themselves, to move about this auditory field without having to make too big of leaps.

(Click for a summary of the names and order of perfection of the melodic intervals)


Naming the Consonances (Chapter 5)

Here is what we've attained thus far;

Between the outer terms, 3/5 and 4/5, we find the same relationship as the whole string to 3/4 of it. Here we see that this space is made up of one major tone, one minor tone, and a semitone. Since these three intervals provide a pathway to this consonance, a consonance which, as we saw earlier, is not made up of any other, it was designated the diatessaron, (taken from the Greek, meaning "over four") because these three intervals separate four tones. Adding a major tone to the 3/4, the diatessaron, we arrive at the consonance of 2/3. So, we shall call that one a diapente ("over five"). Continuing in this way we find two intervals which can be named the diahex ("over six"), since the diesis was not one of the proper melodic intervals. One, 5/8, separated from the diapente by a semitone. The other, 3/5, is separated from diapente by a minor tone. Similarly, removing a minor tone from the diatessaron brings us to 5/6, the minor third, whereas removing a semitone brings us to 4/5, the major third. Half of the string was termed diapason meaning over all, because it is the interval that encompasses all of the others.


Completing the scale: Finding what is missing (Chapter 6, 7, 8)

You may have noticed that our interval of 3/4, which we called a diatessaron, has only two intermediary intervals from the whole string, when it should have three. Also, since the diapason is a diatessaron above a diapente, it appears that there should be a total of seven intervals, making eight tones. What is missing?

Because of the nature of the figures from which these consonances are derived, Kepler says that there must be two kinds of melody. The two thirds (5/6 and 4/5) and the two sixths (5/8 and 3/5) were both separated by a diesis. However, the pair of 5/6 and 5/8 and the pair of 3/5 and 4/5, have between them both the proportion of 3/4, a perfect consonance which unites the terms of the pairs. Therefore, we have the basis for completing two melodic pathways to travel through the octave (diapason).

Here, the basis for the two types of melody are laid out. On the left, the pathway from the whole string to half of it is via a minor third, a minor tone, a major tone, a semitone, and major third. On the right, the pathway is the inverse, a major third, a semitone, a minor tone, and a minor third. The two thirds on both ends of both pathways are, however, still relatively large jumps. To create melodic progressions from these two potential pathways requires that we add a string on both sides of the pathways, in such a way that the thirds are divided melodically.

In the former case, 5/6 takes the place of the third and 5/8 the sixth. These divisions, although they do participate in the pentagonal, participate to a greater extent in the family of figures with expressible sides, making this type of melody more noble. It is thus, called soft. In the latter case, 3/5 and 4/5 participate more properly in the pentagon, a figure whose sides are not expressible. This type of melody is called hard. But, both of our melodies are still missing some notes. What a victory it would be for music if we could find a way to unite these two types of melodies in one instrument and at the same time fill in the missing parts!

So this is what our string looks like at this point;

Both melodies utilize the diatessaron, made up of a major tone, a minor tone and a semitone. The soft melody requires a division of the string at 5/6, which is made up of a major tone and a semitone. The hard melody requires a division of 4/5, made up of a major tone and a minor tone.

3/4 = 8/9 x 9/10 x 15/16
4/5 = 8/9 x 9/10
5/6 = 8/9 x 15/16

Can we make a division on this string in such a way that all three of these harmonic divisions have their proper melodic intervals? The diatessaron already has its semitone. It is still missing the major and minor tones that shares with the hard third. The hard third has a diesis separating it from the soft third. Earlier we noted that the minor tone was made up of a semitone and a diesis. What would happen if we made a division a semitone lower then the soft third? It is after all one of its melodic intervals. What interval would this make with the whole string? To find that, we must lengthen the string of 5/6. Since these tones are related to one another by proportion, we must multiply to get the correct tone. So, multiplying 5/6 by 16/15 produces a string whose length is 8/9.

This turns out to be our missing major tone. Now all three of the consonances have all of their melodic parts. This same process can be repeated for the top of the scale.

But there is one more issue that must be examined before we can say we have the completed scale. We can now navigate the diapason by moving among our melodic intervals. But, if we further add the divisions of these major tones, we can make our melody more in accord with nature. For example, the tone in the seventh position of the octave would be more perfect if it were able to make a perfect diapente with the tone in the third position. So, by adding the divisions of the major tones in the system, this becomes possible. Therefore, if we include all thirteen divisions in the following manner we can provide a route for the greatest variety. That way we can play both modes on one instrument.

Now that we have generated the major and minor modes from the consonances, we have unified these consonances into a system fit for the composition of music. We are one step closer to unlocking the secret to the power of music. But, we are not quite there yet. This will require more work. However, if you stick with the mission to gain insight into the type of creative process that shapes history, not only you, but the rest of mankind shall be rewarded.

Melodic Divisions