Osculating Circles, Serenade, Reverse Video, and all that Jazz (Not)

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Well, dear readers, I’m blowing my own horn here once again, mostly because I want to drum up some viewers for a little YouTube video I put up which, I think (but I’m biased), is a particularly nice piece of work, and because I think there’s so much to talk about.

On second thoughts, I’m not going to talk very much about all that has to be talked about.  At time of writing, the Republican dominated Congress has voted to Sequester, which is a system of automatic and arbitrary cuts in the Federal budgets and services settled upon a couple of decades ago.  The branches of the Federal Government that are cut are those considered inessential, such as Food Stamps, the Federal Parks (certain forest preserves and recreation areas, for those of you who are so lucky as to live abroad), and such things that do not actually discomfit actual Members of Congress, some of whom are among the most vile individuals on the face of the Earth, both ignorant and vicious, while others are merely team players, who have found themselves on the wrong team.

The reason for the Sequester is that the new Health Plan put in place by Democrats and President Obama to provide reasonable health care for most members of American society (and still provide a handsome profit for Insurance Companies) needs a certain amount of government funding to get started.  (As will all insurance schemes, you have to provide a certain amount of startup money.)

Here is an explanation of what is going on through one possible interpretation:


And why do the Republicans want to not fund the health plan?  One reason, supposedly, is because it provides government funding for birth control.  The Pope has declared that religious preoccupation with birth control is disproportionate to its importance.  But certain (very small, but vociferous) sectors of American society are outspoken about the Principle that the government should not participate in birth control.  Ironically, the USA funds birth control in foreign countries.

There has been an onslaught of media advertising that provides a mixture of misleading information and outright lies about the new health plan, urging private individuals to stay away from it.  But it appears that large numbers of people are signing up nevertheless, which means that government funding is not as important as it might have been if people did not sign up.  The American people seem to have weighed in on the side of the new health plan, called the Affordable Care Act, officially, and Obamacare unofficially.  If it goes on to be a great success, the next step for the Republicans, who are now in the majority (the Act was passed in the previous session of Congress, which was dominated by the Democrats) will be to make some minor changes in the law, and relabel it the American Apple Pie Act, or something like that, and take credit for its success.  The changes might be as little as allowing condoms, but not contraceptives, or something equally stupid.  When private companies use marketing tricks to deceive customers it is irritating, but when Congress plays with words and tinkers with the truth, it is truly disgusting.  I sincerely hope the Democrats never indulge in Marketing.

Osculating Circles

An osculating circle is a simple thing.  Suppose we have a curve C, and some point P on the curve.  If you pick three points on the curve, you can easily make a circle pass through those three points.  Now if you make the three points all move to P, the circle gradually becomes the osculating circle at P, which is a circle that matches the shape of the curve perfectly just at the point P.  The radius of this circle is called the radius of curvature of the curve C at the point P.

In the figure we have a curve, in pale blue, unfortunately, and three points F, G, and H.  We also show the circle through those points, and how it was made, using perpendicular bisectors, and so forth.  Now imagine F and H moving towards G.  As you can see, the circle will not change very much, and becomes the osculating circle at G.  This can be done at any point, really; at flat points of the curve, such as F, the circle will be large; at point where the curve turns rapidly, such as A, the circle will be very tight.  In fact, it will be precisely the circle along which you would be driving, briefly, if the curve was a road.

I forgot to say: wherever the curve turns sharply, the Osculating Circle will be small, wherever the road curves hardly at all, the Osculating Circle will be enormous.  The smaller the radius, the tighter the curve.  (Each point has its own osculating circle.)  The video was made by tracing the osculating circles; in other words, the circles remain as the point moves on, so that there are hundreds of circles, which were set to fade gradually (but don’t seem to fade at all, unfortunately). [Added later: I figured out why there was no fading; I had forgotten to click in an important box.]

In the video below, the point A moves along the curve, taking the osculating circle with it.




Meanwhile, down at the farm, my friend is a horn player, and I had visited their home back in the Fall of 2010 --Rally to Restore Sanity, remember?-- and played them a MIDI version of my Serenade, my only original composition --except for an early Chorale-prelude.  Just the other day, my friend said that she had begun playing in a wind quintet (Flute, Oboe, Clarinet, French Horn, and Bassoon), and may she play my Serenade with her gang, just for fun?

A typical woodwind quintet:
the Berlin Philharmonic Wind Quintet

To have my Serenade really played live!  OMG!  Of course, I had to make it worth playing by a quintet, which meant that it had to be rewritten in true 5-voice form.  (The piece I had written was in four parts, which is the easiest thing to do, with the bass being doubled.  That doesn’t really count as 5 parts.)

For the first time I found myself actually engaging with issues of instrumentation: how high can a bassoon go, how low, how long can they play without taking a breath?  Suffice it to say that I had to rewrite the harmony extensively to keep it more or less the same, and to make sure all the parts were within the ranges of the instruments, and that each of the parts was fairly interesting to play.  So that’s what you’re hearing, going on behind the osculating circles.  No, not a live performance, but the new rewritten composition, played by MIDI sequencer.  If and when the live performance actually takes place (possibly even this weekend), and if the parts are all in feasible ranges, and if they happen to record the performance, I will post it here!!!

[Added later:  The Quintet was performed, but it sounded so bad that they never recorded it.  So I have to be content with the version played by software :( ]

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