Introduction Harmonies and Solids Chapters 3 and 4 Ellipse Proposition 13 Prop 13 Calculator Chapter 5 Harmony of the World Chapter 9 A Dissonant Harmony

Music, Means, and the Ellipse

In the 12th proposition in Chapter 3, Kepler states that the mean motion of a planet is found by taking half of the difference of the arithmetic and geometric means of the extreme motions and subtracting that from their geometric mean.  Engaged in countless hours of examining this small proposition, ignoring ridiculously formalistic footnotes by the translator that don’t explain anything, a number of questions, some remaining unanswered, came to the fore.

What do you Mean?

Exactly what kind of a mean is the mean motion?  The Greeks knew of at least three different kinds of means.  All of these means are best thought of as singularities in a process.  The effects of these means are described in terms of music in the Timaeus dialogue. Taking the whole string on a monochord, as is done in Book 3, and dividing it in half defines the boundaries of a process where the end has something of similarity to the beginning.  For example, if you were to ask a man and a woman to sing a note, would they sing the exact same note?  More than likely, something ironic would occur.  If a tenor and soprano were to sing the note that is relatively in the same “place” in their vocal range, they would be singing two different tones that share a certain quality of sameness.  The relationship thus created is the same as that created by dividing our string in half.  This is commonly called an octave.

The Greeks further investigated the space in between these boundary points. If you divide the space between these in half again, taking three-fourths of the whole string, a different note emerges. In this process, the string is changed by the same amount from both extremes but, in opposite directions.  Because the differences between the mean and the extremes are equal, this was called the arithmetic mean. However, although the differences in length may be the same, the musical difference between the tones is not.  What you may call the distance between the tones changes.

The quality of change in tone between the whole string and the arithmetic mean is a different quality of change than the arithmetic mean and half of the string.

As we are already finding out, auditory space, greatly abused in our culture, has its own characteristics that must be explored if one is to navigate it.  Otherwise, the science of musical composition will remain forever unknown.  As with any type of space, such as auditory or visual, the sensual impressions are simply the effects of principles and should not be confused with the principles themselves.  However, if these principles are universal, it should not be surprising if we find a coherence between the effects perceived in these various types of spaces.  Perhaps they are, after all, only one space.

Now, the Greeks discovered that this latter musical interval is characteristic of another type of process.  The harmonic mean emerges when the change from each of the extremes is the same proportion of each.  At this moment the differences between the mean and the extremes are in the same proportion as the extremes themselves.

The proportion of the motions from each of the extremes is the same as the proportion of the extremes.

This division of the string produces the inverse of the arithmetic division. How did this happen? Let’s try to find the principle that produced this affect.  How are these sets of relationships that where generated by the arithmetic and harmonic divisions related to one another?

The intervallic relationship between the whole string and arithmetic mean is the same as the relationship between the harmonic mean and half the string.  Similarly, the interval between arithmetic mean and half the string is the same as the one as between the whole string and the harmonic mean. In the animation the tones have been played together to aid the reader.

The key here is proportion. As you shorten the string in an arithmetic way, or at a constant rate, the proportion between the whole string and the shortened length changes in another way. What you hear is this latter proportion. If the proportion between two strings is the same as the proportion between two other strings, no matter how large or small, the difference in tone will always be the same. So, even though in the case of the arithmetic division, the distances of the mean from the extremes is equal, nonetheless the proportion is not. And this is what you hear. Between these two types of means, the proportions between the strings have an inverted relationship.

There is a third type of process, by which the string can grow or shrink in constant proportion. See what happes when you apply this to the string. In other words, divide the space between the whole string an half of the string in such a way that the intervallic relations between the mean and the extremes is the same.

Since pairs of strings that are in the same proportion produce the same musical intervals, the Greeks utilized this type of transformation to move about the musical realm. This process is called geometric. And this is how the Greeks filled out their diatonic musical scale.  So the Greeks found had to find the right proportions that would allow them to move throughout the octave. Between the arithmetic and harmonic means, there is the proportion, 8/9.

The Greeks constructed the major mode by using the interval of 8/9.  Taking this twice gets you to a point where another, smaller, interval, 256/243, must be admitted to get to the arithmetic mean. Since the harmonic mean is the inverse of the arithmetic mean, this process can be repeated from the harmonic mean to get to half the string.  The minor is then constructed by repeating this whole process from half of the string, in the opposite direction.

Ellipse

So, which mean is Kepler talking about when he says the mean motion of the planet?  Is he referring to the speed that the planet must travel if it is to complete the same circuit in the same amount of time moving constantly?  If so, how would we find that speed?  The speed is changing at every moment.  We must find how it is changing.

Because of Kepler, we know that the speed of a planet is the inverse of its distance from the sun.  Therefore, any inquiry into to the speed of the planet, or the process that defines how it changes speed, demands that we look at the distances.  Since we also know that Kepler discovered the shape of the orbit to be elliptical, this should be no problem.  We just have to know the way that the distances change on an ellipse; but, which ellipse?  Don’t the planets travel on different ellipses?

Let’s taker a closer look.  If we take the perihelial distance as a unit, and the speed of the planet at this position as maximum, what can we come up with?  Since, the speed changes inversely to the distance, then at double the distance, the planet must be traveling at half the speed.  And at triple the distance, it must be one third of the speed.  And so forth.

But here we have a few problems.  These relative speeds occur at different places on these different ellipses.  And the speed changes along these elliptical arcs between the isolated moments.  If all of these arcs differ from each other in each ellipse, can we know how long they are?

Even if we took equal angles, whether from the center or the focus, although we might be able to tell the distance and speed at that moment, we still could not determine the length of these elliptical arcs, or for that matter, what is happening between these moments.  This is not a problem of one ellipse in particular, but all of them.  But, luckily Kepler can help us generalize this process of change.

How does the distance change as the planet travels in its orbit?

In the Epitome of Copernican Astronomy, Kepler describes how the distance can be measured.  In the animation the distance of the planet from the sun is made the radius of a circle.  Notice that the circles change size in a non-constant way.  Near the extremes the change is more gradual, while quicker near the middle longitudes.  As the planet travels around the sun, it's distance, and therefore its speed, changes in a non-constant way.  Notice furthermore, that this non-constant change is also different in different ellipses.

Here you can see the distance of the planet from the sun as the radius of a circle.  The distance changes non-constantly, while the change in angle at the center is constant.  The difference between the greatest distance and the distance at a particular moment (AD) is always in proportion to the versed sin ( 1- cos ) of the eccentric anomaly, (AS).  The total change in distance that occurs as the planet travels in its orbit is equal to double the eccentricity, which is what you get by subtracting the perihelial distance from the aphelial distance.

If you were to lineup the distances for every degree of eccentric anomaly, the pink curve is the curve that would result. In the New Astronomy Kepler calls this curve the conchoid. Since the speed is inversely proportional to the distances, you can see the change of speed throughout the orbit by taking the proportion of the distance to the radius of the circle ( the arithmetic mean distance) and applying this proportion to the radius. The resulting curve, in blue, is the acceleration of the planet as it travels through its orbit. These figures correspond to the ones immediately above them.

So, what does this tell us about the speed?  It seems that we are at a loss.  At every turn we keep finding more evidence that these ellipses, despite however similar they appear, are very different.  But, maybe there are universals. Looking at the boundaries, the aphelial and perihelial distances, which determine the greatest and least speeds, can we find some singularities that are true for every ellipse?

What is universal in all ellipses?

An ellipse is the figure produced by the intersection of a cone with a particular plane. The projection of this figure onto a plane that is perpendicular to the axis creates another ellipse. Can you find the cone to which this projected ellipse belongs?

We will use the characteristics of this projection to investigate the properties of the ellipse. In the projection, the apexes of the cones project to two specific points. These points are called by Kepler, the foci, meaning "hearths". This is because of the physical properties of these points. All the light emanating form one os these foci is gathered together again by reflection at the other focus. In our solar system, All of the planets share the sun as one focus, which kepler proves in the New Astronomy.

From the preceding construction, since the radii of these projected circles grow and shrink at the the constant rate, it is easily seen that the sum of the distances from the foci stays constant for every point on the periphery of the ellipse.   This being the case, it is also easily seen, that at one of the extremes, these distances are the aphelial and perihelial distances, whose sum is the total major axis.  Taking half of this, the semi-major axis, produces the arithmetic mean between the two.  Therefore, the point on the ellipse which is equidistant from both foci has a distance from each that is equal to this arithmetic mean.

As the foci change, while keeping the sum of the distances constant, the ellipse becomes more narrow, but the major axis does not change.

From the animation we can see that in every ellipse, this occurs at intersection of the semi-minor axis with the ellipse. What about this length?  If we draw a circle, with radius equal to the semi-major axis around the ellipse and take the distance drawn perpendicular from the focus to the circle, we generate a height equal to the semi-minor axis.

On the left, the two right triangles are equal.  This can be proved by noting that they share the same base and they both have as their hypotenuse the radius of equal circles.  Therefore, their heights, both perpendiculars to their base, are also equal. On the right, you see how height of the of FG, equal to the semi-minor axis is the geometric mean between PF and FA.

This height on the circle is the geometric mean between the two lengths on the diameter that are produced by this perpendicular cut, which in our case, are the aphelial and perihelial distances. Thus, the semi-minor axis is also the geometric mean between the aphelial and perihelial distances.

But what about the actual distance to the ellipse at this position, 90 degrees from the focus?  To find this, can apply a relationship known to Kepler, but discovered by the ancients.  The line drawn perpendicular from the diameter to the circle is cut by the ellipse in the proportion of the minor to the major axis of the ellipse, or as we just discovered, the geometric to the arithmetic mean of the aphelial and perihelial distances.

This line dropped perpendicular to the diameter of the circle form it's perimeter, LP, is cut by the ellipse in the proportion of the minor axis to the major axis.

Since

So, we can apply the proportion from Archimedes to this height on the circle to find the height on the ellipse.

Now, earlier we said that the harmonic mean divides a line in such a way that the differences between it and the extremes are in the same proportion as the extremes.

If we can show that this is the relationship that this height has to the extreme distances, then we have found our harmonic mean in the ellipse.

P : A :: (H-P) : (A-H)  where H is the harmonic mean.

This gives us

AP - HP = HA - AP
2AP = HA + HP
2AP = H (A+P)
2AP /(A+P) = H.

Hence, the height of the ellipse 90 degrees from the focus is the harmonic mean between the perihelial and aphelial distances.

We have now found three singularities in the distances that are universal to all ellipses.  In addition to that, with the use of a circle and ellipse, we can generate all three of these means that were known by the ancients, for any two numbers.

 Now that we have isolated what is universal to all ellipses, can we use this to tell us about the way a planet is traveling?  Well, what we know so far is that the maximum speed is at perihelion and the minimal speed at aphelion.  What about at these mean distances?  Since the speed changes inversely to the distances, we can determine 3 things.  At the geometric mean distance, the planet is traveling at the geometric mean of the speeds.  At the arithmetic mean distance, we find the harmonic mean of the speeds.  And, at the harmonic mean of the distances, we find the arithmetic mean speed!  This inverse relationship should remind you of our divided string.

But, again, these, although they are universal, are still singular moments in a process that is always changing.  And unless we can know what is happening between these moments, how can we grasp what that process of change is?

And so, our question is still unanswered.  What is the mean motion that Kepler is referring to?