Archive for October, 2013

Porgi Amor

Uncategorized | Posted by Brian PCF
Oct 31 2013

The Countess’s Aria presents us for the first time in the opera, a seemingly uncomplicated character. Everyone so far has secrets and schemes. All of the characters want someone who wants someone else or wants something that someone else has, but the Countess just wants her husband, the man that should be hers.

The first things we notice about the Countess’s aria are the long intro. She has a full 17 measures of instrumentation that lays the mood for the piece before she begins her solo. What we here in that opening is strings: violin and viola, followed by clarinet, french horn, and bassoon. In the first few measures, the higher pitched violin and the lower pitched viola play together with no instrument taking the lead. The same goes for the first few measures when we have the higher pitched clarinet, with the lower pitched french horn and bassoon. But soon the lower pitched horns and viola sound softer and play less often, while the clarinet and violins continue to play louder and more often. The instrumentation seems to back up what we find later through the vocals: the Countess love lingers, but the Count’s no longer interested in what he’s already caught.

Next we notice the key: the same E-Flat we found in Cherubino’s aria in No. 6. His aria was “allegro vivace”, meaning “joyful, very lively tempo.” The Countess in “larghetto”, meaning “fairly slow tempo.” Also noteworthy is that Cherubino was on stage for much of Act I and never alone. The Countess is on for a mere 48 measures (the smallest part so far) and is singing by herself. Cherubino can’t really decide exactly who he wants in his aria, and his demand is that he either desperately wants love or he will “talk alone of love”.

The Countess words match the accompaniment: simple and to the point. The Countess wants what was seemingly promised (and perhaps at one time had), the love of her Count. And if she doesn’t get it, she would rather die. The instrumentation throughout her part is lacking in flourish. Many quarter notes in a very simple 2/4 time signature. Moreover, she just repeats the same refrain again and again.

This is seemingly the simplest and most straightforward character we’ve been introduced too, which makes for an interesting element. Every other character has been maneuvering and presenting one face to the audience, and another face to the other players. It is perhaps the most intriguing way to add complexity to the plot: add a simple, genuine person. The other characters won’t know how to handle her.

Complete video here.
Complete score here.

On Euclid’s Definitions and Postulates

Uncategorized | Posted by Brian PCF
Oct 20 2013

The structure of Euclid’s definitions and postulates assumes a movement from simple to complex. From a beginning of the point, “that which has no part” we finish the postulates with a vast (possibly infinite in our modern understanding of the word) ability to create and explore points, lines, plane surfaces, and perhaps an approximation of our entire physical universe.

The definitions seem to build on each other in a seemingly logical form. We start with the point and line, and then expand on what a point and a line can bring about: straight lines, surfaces, plane angles, figures, circles, parallel lines. Literally, we can create from this ad infinitum. A basic proposition of Euclid seems to be that we can represent anything with this system of points and lines.

Interestingly though, we define these basic structures in the first two definitions in our Euclidean universe not by their properties, but by that which they are not: a point is “that which has no part”, a line “breadthless length”. We create representations of our physical universe from these definitions, but they can’t exist in our physical universe. Something that has no part, or no breadth or no depth can’t exist in reality as we understand it based on our senses.

Moreover, everything that comes after these basic definitions is relative to one another. There is no existence beyond the point and line in and of itself, simply (or perhaps more complicated, as we’ll see) is the idea that there are fundamental properties based on these relative relationships.

The first relative definition is the straight line, “a line which lies evenly with the points on itself.” Let’s assume that this means that “evenly” here means an angle of zero as we would define it. Euclid requires a much more flexible definition here as he doesn’t want to define a line with an angle zero. It seems from his Definition 8, “a plane angle is the inclination to one another of two lines in a plane which meet one another and do not line in a straight line” that he wants to have the ability to put as many points or lines on a single straight line as he needs. So a seemingly simple idea, but with an infinite amount of complexity that we can build in as needed.

We continue to see complex and relative figures emerge. Even something seemingly simple like a boundary, “that which is an extremity of anything”, or that of a figure, “that which is contained by any boundary or boundaries” depends upon some other thing for its existence and is hence relative.

Right angles are a fascinating example of this idea of relativity in that they are not only created by another right angle (definition 10), but are all exactly the same no matter what (postulate 4). So a right angle can’t come into being as a singular right angle, it must be born as a twin.

Circles are the next figure that require a relative relationship. They cannot exist in and of themselves, but require a center. The center also requires a circle, otherwise it is just a point. This and the right angle are all strikingly similar to “the swerve” of Lucretius: atoms colliding with one another to create the entire physical universe. While Euclid is contriving his universe, that idea of collision between basic atoms or ideas is still fundamental to the Elements.

Parallel lines are very similar to the idea of the right angle. They need each other to exist and they are fundamentally identical in that they are “breadthless length” and they share a relative angle.

So while nothing can exist in Euclid’s universe without points and lines, those can only exist by seemingly simple definitions that speak more to what these things are not versus what they are. Moreover, everything built from these points and lines is defined by their relative relationship with one another. There are no new things in Euclid, only these basic ideas of points and lines and how they interact. So the baseline is simple, but from this we can grow the whole of mathematics and approximate everything we see in the physical world.

[Using the Heath translation]