First Anniversary

I almost did not notice. At the middle of this month, I completed my first year of keeping a weblog entitled Blogtrek. I noticed it evolve as I kept contributing to it for the past year. I was made aware of blogs from a friend at church who kept one. I found the idea of keeping some sort of journal online intriguing so I went with it. At first I put in a lot of blogs about how the day went and things I thought about ("Fireflies", for instance). Then it evolved into more political talk, such as my commentary on PETA's protest at Victoria's Secret's show. As of late I have been putting in more personal stuff in, culminating in my review of Chopin's Opus 28 Preludes. Today's by the way is "Floating", Opus 28, Number 23, in F major. I found that it takes time to compose these blogs, and it does make my thoughts about things, such as what I will do after I retire, sharper. So I now continue into my second year of Blogtrek, blogging where no one has blogged before…

Floating: Chopin's Prelude in F Major

Have you, as a child, blown bubbles and watch them float away? or perhaps you launched a paper airplane and watched it soar in the air currents, or maybe a balloon or a bunch of balloons? Don't you wish you could float away for an afternoon? That is what Chopin's Prelude Opus 28, Number 23 reminds me of. Its up and down strains remind me of floating around. The music is fairly organized - an F section, and then one in C, then one in F again, then to the sub-dominant in B-flat and back to F again. But it still gives me the impression of disordered drifting, as it continues to soar. Each section is about a fifth higher than the previous until the prelude is twinkling the highest keys of the keyboard. Then it drifts down like a balloon that has burst or hit a downdraft back into the middle of the keyboard. What happens next is really interesting. Chopin closes with an F arpeggio in both hands first together up, then treble up-bass down, and right at the turn point Chopin throws in an out-of-key E-flat. That colors the entire ending. It feels incomplete, as an F7 chord wants to resolve to B-flat, yet this continues right up into an F. But it does fit in with the floating nature of the piece, and one might consider this to be another blue note that Chopin has thrown into his music, an elusive out-of-chord note reminiscent of his Posthumous Prelude in A-flat, with a B-Flat in the final A-flat ending. As far as playing it, it is not too hard, although it feels awkward with the fingers overlapping the thumb on the descending notes. A good song for a lazy spring day.

Flood

One thing I have been noting is the increase in the number of cloudy and rainy days we have had. This past month have had two or three spells of it and not much sunshine. The sun finally came out today, on Memorial Day. That's the day that the pool is supposed to open in my development - officially anyway, with a kickoff party. But there was no party today. Rainy after rainy day has saturated our water table. Last night, a thunderstorm came and it simply poured and poured here, a solid wall of water. When I got up in the morning I had found that 3.75 inches of rain had fallen overnight, and further, I had recorded 0.83 inches the previous day for a total of 4.58 inches. This caused massive flooding in our development. The creek rose up about 6 to 8 feet, swamping the little park that our development has. Further, the water came over the level of the swimming pool, putting dirty water into the pool and damaging some of the buildings. I went over there today and they tell me the pool will be closed for a week. I suppose they have to pump it out, clean it out, and then fill it with fresh water and chlorinate it. I don't know what they would do about the damage to the buildings - about 6 inches of water there. In any case, if someone tells me if we have a drought now, I will show them our development with its flooded park and swimming pool. It's OK to sprinkle our lawns now, except for one thing. The lawns clearly don't need any sprinking.

Tumult: Chopin's Prelude in G Minor

UNFAIR! NO MORE WAR! XYZ COMPANY UNFAIR TO UNION WORKERS! DUMP THE HUMP! USA GO HOME! Can you feel it? This is the prelude of the protest movement, of social unrest and of demonstrations. Another Chopin piece that resembles this prelude is his Revolutionary Etude. Yes, Chopin's Prelude in G minor, Opus 28, number 22, is the tumultuous one, hence my name for it. Its tempestuous phrases build up to a climax similar to that of Prelude 18's, then it fades away, as though the energy of the protestors was suddenly zapped. This prelude differs from Prelude 18 (in F minor) in that 18's phrases seem like isolated fits, while 22's is more continuous, incessant, and demanding. It is this incessant pounding in the middle that reminds me of a protest movement, especially one that turns into a melee or a revolution. The piece jumps octaves all over the place, making this one a hard one to master, although with some practice I found that I could do it. But it is a good one for a protest group to play before going out into the streets.

April: Chopin's Prelude in B-flat Major

Chopin's Prelude in B-flat Major, Opus 28, Number 21, is dominated by patterns in which two notes in unison or close to each other move farther and farther apart from each other, usually in the bass. It seems like this prelude is continually opening up throughout. When I hear those chords, I imagine a flower blooming in fast motion. So this is the Opening Up Prelude. I call it "April" because "April" means opening up; it is similar to the Spanish word abrir, meaning to open. The prelude is not too hard to play although there are some stretches here and there.

To me this prelude symbolizes the never-ending quest for truth and meaning, the fourth principle of my religious group, Unitarian Universalism. It starts with the opening up portions in the bass. To me this symbolizes Discovery, the things we find out about our existence that we have not known before. It is a never ending quest of constant opening up. Occasionally a discovery comes that is so dramatic that it seems that we found The Answer. This appears in the prelude as the booming, blaring middle part in the key of G-flat. It sounds like the Ultimate Trumpet above saying "This is The Answer!!". It is so perfect in harmony and its chords are so loud and clear that we feel that we have found the answer for everything. But look again. This passage is almost all on black keys. It seems that we have ignored a lot. At best it is only a partial answer. So we look at it again and get doubts about it. In the prelude this comes out as the G-flat 7 part, which comes in like a question mark. That F-flat makes the chord unsettled. It is not even the usual type of flat. People think of flats as being black keys, but this one is a white key, the one normally called E. It makes us wonder. Do we have the answer? We begin to have doubts. The revelation begins to lose some of its sparkle. Then the prelude reverts back to the April part where everything opens up. It concludes with the opening-ups still continuing. To me these musical blooms remind me of a part of The Truth opening up to us. Sometimes they strike us like a G-flat thunderbolt, only to have a 7th note take the wind out of the bolt. So what do we do with these blooms? Treasure them while they last. Take advantage of each one, but remember that there is more to life than just these few blooms.

Sorrow of a Nation: Chopin's Prelude in C Minor

Chopin's Prelude in C Minor, Opus 28, Number 20, is the most famous of Chopin's Opus 28 preludes. It is part of the repertoire of every moderately advanced youthful piano student and is played frequently at public gatherings, especially funerals and memorial services. It has even become part of a popular rock song in the 1980s. It is in the form of a funeral march, starting out loud and then going soft before building up at the end. It features huge chords which descend the depths of the piano and a melody which practically sings. To me, this mournful sorrow expresses the sorrow of one of Chopin's home countries, Poland. So I follow Joseffy's commentary and call it "Sorrow of a Nation".

Poland has not had a happy history. It was a kingdom at one time in the middle of the second millennium, but Germans, Austrians, and Russians subsequently invaded it and caused it to cease to exist for hundreds of years. It finally gained independence in 1918, only to be divided up again between Russia (Soviet Union) and Germany in World War II, followed by two generations of Communist rule. It is now a democracy, but for most of its history it has not been a good place to live, including Chopin's time, when it was occupied mainly by Russia. I believe that Chopin had Poland in mind when he wrote this song, and he aspired desperately for the day when his country could be its own nation again. In playing it, feel the sorrow of Poland and this will give you a good idea of the dynamics that should be used. In the first stanza, a stanza that runs G7, C7, F-minor feels like it should resolve to the C major again, especially since G major follows it. But it goes instead to C minor - make sure to play E flat instead of E. A magnificent tragic prelude.

Puzzle-Piece Syndrome

I ran into a syndrome which is ubiquitous throughout human existence. It occurs when a concept or entity appears in two different forms, and then is regarded by people as being two different objects. An example might be "annual leave" and "PTO". A person knowing what annual leave might be bewildered as to why she loses some annual leave because of something called PTO. Then she finds that PTO is "paid time off", and is the same as annual leave. Another possibility is a transaction between two check accounts. It shows up as $436.74 in one account but as $992.33 in another, because it got included with some other deposits. These don't appear the same, so one records these both into Quicken, and so the $436.74 deposit into the receiving account gets counted twice, resulting in an overdraft. I call it "Puzzle piece syndrome" because it reminds me of the situation in which there are two clumps of put-together puzzle pieces. Then one finds pieces that fit into these that cause them to come together to form one piece.

Mathematics abounds with instances of Puzzle-Piece Syndrome. For example equations such as x² + 2x - 3 = 0 looks like a series of symbols, and a parabola looks like a curve that a home run makes. But when one plots the equation on paper, the equation turns into a parabola. Understanding the Universe may be one giant case of Puzzle-Piece Syndrome. Magnetism enables a piece of metal to pick up nails and tacks. Electricity causes sparks and makes a current flow through a wire. These don't look at all the same. But they are. To demonstrate that, coil an electric wire about a metal core and the core will pick up nails and tacks when the wire is plugged in. Or, move a piece of metal back and forth across a powerful magnet. This will light a light bulb. In other words, electricity and magnetism are the same. In the same way, Einstein finds that gravity, which makes me fall down from a ladder, and inertia, which makes me fall forward when I trip on a badly constructed sidewalk, are the same. Later on, physicists find that gravity and electromagnetism are the same, so it appears as though understanding the Universe is like solving one big jigsaw puzzle. We have many pieces in place but not all, so we continue to find ways of combining things, of finding instances of Puzzle Piece Syndrome. So it is more than just incomprehensible acronyms and overdrawn check accounts that come from Puzzle Piece Syndrome. It's the very understanding of our Universe that comes from it, for Life indeed is One Big Jigsaw Puzzle without End.

Excelsior: Chopin's Prelude in E-flat Major

It took me a long time to figure out what to call Chopin's Opus 28, Number 19 in E-flat. It is a waltz or gavotte of some sort, but it did not seem to have any unusual features. Recently when I played it I noticed that I was having to stretch my hand longer and longer in the treble until eventually I was having to span two octaves with each hand, making this a prelude that is difficult to learn to play. The theme of higher and higher has appeared before, as lower and lower in Opus 28, Number 15 ("Dream") and in Posthumous Etude Number 2 ("Mercury"). I thought of "Higher and Higher" as a name for this prelude but that is the name of a pop song, and I wanted something more interesting. Then I thought of it: Excelsior, which means "ever upward" in Latin and is the motto of my home state, New York. As such it is an inspiring prelude, with its striving to hit loftier and loftier heights, and it fits neatly between the fits of anger of its predecessor and the tragedy of its successor.

Virtuoso: Chopin's Prelude in F Minor

Have you ever wanted to get up on a stage and sing a few broad and loud phrases to a sophisticated audience? That is what this piece suggests to me. Chopin's Prelude in F Minor, Opus 28, Number 18, suggests such virtuosity; hence the name, "Virtuoso". It starts with an angry argument that ends in a harsh, off-balance discord, based on F-G-flat. It repeats this argument and then goes into an arpeggioistic fit, followed by the same arguing line, this time in the sub-dominant key of B-flat minor. It then proceeds through a bunch of quarreling phrases, including some with a banging E double-flat in four octaves, and then it descends into the bass, where it mutters and fades into the final chords. It takes a bit of practice to get the running lines down, but once you do, you get a sort of satisfaction out of playing the phrases and then banging the keys when the downbeat chord occurs, sort of like punching a punching bag. It's a good piece to play if you want to boast about your piano-playing ability, or if you want to get some anger off your chest. Take your frustrations out on this piece instead of on your neighbor for raking onto your lawn, and you will be calmer and better able to deal with the crisis. This is one of his better preludes.

Autumn Idyll: Chopin's Prelude in A-flat Major

This is a magnificent prelude. Opus 28, Number 17 features luscious harmonies and harmonic transitions, a sonorous melody, and a majestic bass that at times reminds me of a huge organ at a cathedral, not to mention the 13 chiming bass A-flats in the last part of the piece that reminds me of a bell tolling out 13 o'clock. To me it reminds me of one of these gorgeous days in the autumn (or in the spring) in the park under the trees, sunny with moderately cool winds, that entice you to spend the entire afternoon out in the park sitting and admiring the nature that is around us. It is one that I learned when I was a child. The most interesting part was that with many of the chords, the left hand overlaps the right, so that if you are not careful, you stumble all over the place on the keyboard when you play it. It starts out with the melody in 6/8 time, or maybe a fast 3/4 waltz. It does have some unexpected transitions, such as E-flat to D-flat instead of the more usual B-flat 7. This tripped me up when I played it for my piano teacher - I kept wanting to play D natural instead of D-flat. It then goes four keys flatwise, and thus into sharps, into the key of E. The harmonies then go from one key to another - C-sharp minor, E7, D-sharp major, C-sharp major and so forth, finally winding up in an alteration between E-flat and A-flat minor. We then return to the main theme, but this time with the volume of an enormous church organ, as the bass hits the lowest keys of the piano. It then goes into a variation of the alternate theme, then goes back into a plain A-flat ending with its bell-like series of A-flats in the bass. This is a piece I want to hear over and over again, and I hope to hear it at a concert sooner or later.

Scurrying: Chopin's Prelude in B-flat Minor

Chopin's Prelude Opus 28, Number 16 runs all over the place and is not the easiest to master. It features jumpy triples of notes in the bass, and a scurrying around in the treble going up and down the black keys of the piano, with an occasional breakout to the white keys to provide the F7 dominants. I tried playing it and although I could get the general action of the running up and down the keyboard, I could not get it perfect, and coordinating it with the jumpy left hand is even harder yet. The entire piece reminds me of someone running around and scurrying all over the place, sort of like a mouse running all about the floor looking for cheese or for a place to hide from the cat. So I call this one "Scurrying". It continues for four pages of printed music scurrying almost non-stop until it drifts down into the bass, sort of like the mouse finding the hole in the wall, and then pum! pum! it's all over; he's in his hole. Chopin does not break any ground with this one with the usual standard harmonies of B-flat minor, E-flat minor, and F7th. But it still is an interesting one to listen to, and it will get your heart pumping.

By the way, I have now posted a web page consisting entirely of Chopin's Opus 28. Not all the links work on it yet, because this page will not go ahead of this blog. After I blog a prelude, I will also put it on the web page.

Storm: Chopin's Prelude in D-flat Major

This, along with Number 17 (A-flat) and Number 20 (C minor) is one of the more well-known pieces in Chopin's Opus 28. This one, Number 15, is usually called the Raindrop Prelude because of the constant A-flat eighth notes pitter-pattering in the bass like a light rain shower. The beginning and end of this piece are tranquil, something like the F-sharp or A major preludes, like a pretty day out in the garden. But that is not what makes this piece notable. To me the most characteristic part of this prelude is the middle part, a troubled C-sharp minor passage that reminds me either of a bad dream, or of the tranquil day represented by the rest of the piece interrupted by a storm, replete with thunder. I therefore call this one "Storm". It is the longest C-sharp minor passage in the Preludes of Opus 28, even longer than the C-sharp minor (Opus 10) Prelude itself. The song starts tranquilly, with a constant series of A-flat eighth notes, but then all of a sudden during a resolution from an E-flat 7 chord, it not only changes mood but name as well: the A-flats become G-sharps as the piece enters the C-sharp storm. The G-sharps keep pattering away, while an ominous series of chords accumulate in the bass. These eventually resolve into bangs, symbolizing thunder. The storm continues to rage, featuring a strange resolution from G-sharp minor to B augmented and back to G-sharp minor again. After the thunder reaches a climax, the rest of the C-sharp minor part develops melody, perhaps symbolizing the rainstorm and finally calming down and going back to the tranquil part as the G-sharps become A-flats again. The piece naturally has to flip-flop from flats to sharps when it goes into minor; else we have a passage in C-sharp Major (rare) or in D-flat minor (impossible). But to me this transition from flats to sharps also symbolizes the subtle and sudden change of moods from happy to sullen and back again.

It is not too hard to learn; the main problem is that many of the chords have overlapping hands - the highest left hand note is higher than the lowest right hand one. This can cause a good deal of stumbling while playing it for the first few times. This prelude is a good one to put into a program such as Cakewalk - I would insert audio thunderclaps to coincide with the musical ones in the middle part. Although I have found stronger storms in Beethoven's Pastoral Symphony Number 6, in Ferde Grofé's Grand Canyon Suite, in Igor Stravinsky's Le Sacre du Printemps or in Andreas Vollenweider's recording from an actual thunderstorm in his album Behind the Garden, I regard this as one of the better preludes of this opus.

Mathematicians done for at 50?

I have been hearing and reading about mathematicians not accomplishing very much after they turn 30. It is the youth of the mathematics world, it seems, that turns out the best mathematics and comes up with the best theorems and algorithms. Perhaps part of this image comes from mathematicians who are forever young because they died young: Eisenstein and Abel, for instance, or especially Galois, who was almost a child when he was shot to death in a duel at the age of 20. There are some exceptions; for example, the people who solved Fermat's Last Theorem were around 40. So what do mathematicians like me who are over 50 do? I say that it is the childlike nature of these young mathematicians that enables them to make great discoveries. They grab for the silly and ridiculous. They imagine things out of this world, such as five-dimensional manifolds. So if you want to be able to prove theorems well into your 60s, maybe the best way is to keep forever young, and keep a childlike attitude to life, as though it were all a game. Those of us who come up with the wildest ideas are those who will be productive into our 50s and beyond. You can see some of that in my writings, where I liken hat games to aviation maintenance and contra dancing to flipping mattresses. So just keep on playing, and maybe we will be productive into our old age. No, mathematicians are not done for at 50 - they keep on going like the transfinite ordinals, which keep on coming without end.

Throbbing: Chopin's Prelude in E-flat Minor

Opus 28 Number 14 is the first of Chopin's Opus 28 preludes in a flat key. It seems simple enough. The right and left hand parts are identical, except an octave apart. I was going to say treble and bass parts, but almost all the way through the piece, both parts are bass. So this prelude grumbles. It is relatively easy to learn, since you do the same thing with your left and right hands. Some of the notes are tricky to read, especially when a D natural greets you after an F-flat major chord. Shouldn't that be an E double-flat? But the hardest thing to get is the dynamics. Follow the crescendos and diminuendos carefully. The song is a continuous rise from soft to loud to soft to loud, giving it a throbbing feel, like a pain from an injury; hence the name. The piece keeps on going with the eighth notes non-stop until thump, it ends with a concluding octave E-flat. Not a good prelude to listen to if you hurt somewhere.

Bush's chances for re-election

I know I covered this subject recently, but Bush's officially throwing his hat in the ring today brings this subject up. So how can Bush's reelection be stopped? One source says the Democrats' best chance is health costs. If so, then this is something new in Presidential politics, for it does not appear among the 13 Lichtman keys. Others say that his anti-terror stands are popular. But what is really going to happen? As I said before, Bush has lost three keys: 4 (long-term economy), 10 (foreign/military failure), and 12 (incumbent charisma). He has gained 3 (incumbency), 1 (mandate), and 7 (policy). It seems to me that the most likely scenario for defeating Bush would be an economic sputtering in 2004 together with failure to achieve our goals in Iraq (set up a democratic, peaceful government) plus something else, like a charismatic Democratic candidate or a scandal hitting the Bush administration. These things need to happen else Bush wins. But see the previous item:

The Texas Democrats

It was an absurd sight. ALL of the Democrats in the Texas legislature running off to the next state like some sort of fugitives to prevent the Republicans from ramming through a redistricting plan that could increase substantially Republican representation in the House next year. To me the foremost thing that it says is that you can't require any more for a quorum to conduct business than you require for a bill or item to pass. For if not, then a group of people greater than an anti-quorum (1 minus a quorum) but less than the bill-passing requirement can run off into the woods and kill the item. Most quorums are a majority, but the Texas legislature has 2/3, making the legend of the Texas Democrats possible.

Could this lead to Bush's defeat next year? It could happen. Texas is now shown to be the laughing stock of the nation, so much so that it affects Bush the President. The economy does not pick up and things in Iraq don't do too well. In the meantime, the US congress considers doing what the Texas legislature did, and the threat of that causes people to examine Bush and his administration more carefully, perhaps unlifting a major scandal. Or maybe the Texas Democrats would secede from the Democratic party and form a third party getting a huge share of the vote. Either of these two scenarios will defeat Bush. Not likely, but look carefully at the consequences of the Legend of the Democrats who Ran.

Dream: Chopin's Prelude in F-sharp Major

Opus 28 Number 13 is a dream. It is a relaxed 3/4 piece with sustained chords suggesting a contented state. The middle section goes into some dreamy harmonies: first into C-sharp major, then into B major, then finally ending on an elusive D-sharp minor/F-sharp major ambiguous chord that does not resolve itself until the final chord. This piece is the last of the sharp key preludes in Opus 28. The next one would be in D-sharp minor, but here Chopin flip-flops into flats into E-flat minor, presumably because the leading tone there would be a natural instead of a double-sharp. The piece reminds me strongly of Chopin's Posthumous Etude Number 2, which is marked by a duple-time bass with bass notes that come up and down in an sine-wave pattern with a 3/4 treble. I call that one "Mercury" because to me it symbolizes the 3/2 pattern of Mercury's rotation and revolution around the Sun. This 3/2 juxtaposition appears also in Opus 28 Number 13, as does the sine-wave bass, in the middle dreamy section. This piece also has some interesting harmonies, such as E natural over E sharp in the middle section, and apparently Chopin at the end wanted to play some chords that are unplayable because they stretch more than a tenth. A good prelude to play just before going to bed. Tomorrow starts the flat preludes.

What's all this about Matrix?

In 1999 I was in a group of two men and two women who wanted to decide what movie to watch one night. The two movies that were offered were Sleepless in Seattle and Matrix. The women were interested in Sleepless in Seattle and it seemed intriguing to me. Matrix, though, was based on an intriguing idea. Look at the world you see. Hear it. Feel it. This is not the real world. It is enough like the real world so you can maneuver in it, get meals, find friends, read and understand integral calculus and so forth. But it is not the real world, but rather, what your brain perceives of as the world. If something could go in there and change the references of your brain cells, might not your brain think it is in an office building in a dot com in 1999 whereas the real world is a burnt-out hulk in 2199? That is the theme of Matrix and that is exactly how it starts out. However, after seeing the movie, I began to feel the women were right. This movie, although it was based on an excellent idea, was far, far too violent. Every time you turn out someone is threatening someone, or pointing a gun at someone, or shooting someone, and one scene shows someone shooting a gun and shows zillions of gun shells falling to the ground. I therefore rated it a bomb.

Now here comes Matrix Unloaded. Unloaded? That's gun talk. It seems to me as though this new movie is just as violent as the predecessor. And I hear lots of excitement about it. But I hear most of it coming from the movie industry and the media. They are really hyping this movie up good. No matter where you turn to, you get matrix. You turn on the news, you get matrix. You turn on Larry King, you get matrix. You turn on Phil, you get matrix. Even if you turn on sports, you get matrix. Matrix, matrix, matrix. And people are lapping it up good, as can be told by the box office figures and the predicted lines to get in the theater. So this movie is going to attract crowds far beyond what one would expect from a movie of this type. Already, the local newspaper rates the story plot a dud. I can predict by how the previous movie turned out and by the hype necessary to carry this movie to the movie fans, that almost certainly this movie is just as much a bomb as Number 1 was. I feel that on a scale of 0-10, it rates 0.8; i.e., a shade less than 1. If I see it, it will be while I have time to kill and it will be from a video tape, not in a movie theater. Matrix indeed. I'd like to invert this matrix and extract its eigenvector.

Agitated: Chopin's Prelude in G-sharp Minor

Here, in Opus 28, Number 12, is a fast and agitated waltz. First there is the rising strain of notes from a G-sharp minor chord, reminding me of a climactic buildup, misplaced at the beginning of a number, not the end. The buildup resolves itself, only to rebuild again. The waltz goes like this for a while, shifting keys a bit and building up to a big climax with its D-sharp minor chords. It then continues on and on, until it reduces to just two notes jumping back and forth. The waltz then resumes and ebbs away, only to have two sprightly octaves pop up at the end. It is not too hard to learn to play, although some of the chords are a bit of a stretch. There is also the problem of moving one finger of the right hand while keeping the thumb on the same note. I suppose one could dance the waltz to this, but it would be a bit more forceful than the usual waltz, because every one of the three beats is accented, not just the first one. As a waltz, this piece has somewhat of a Slavic feel, but that may just be because of the minor key. In any case, it seems to be the direct opposite of the previous prelude in B Major.

Contentment: Chopin's Prelude in B Major

This prelude, Opus 28, Number 11, suggests contentment to me, hence the name. It is a happy and resolved piece. Down under the contented exterior, there are some unexpected turns. These come out if you play the prelude. You stumble all over the place to begin with because the notes are not where you expect them. An arpeggio goes zigzag up and down then it unexpectedly goes up after an up. The pieces of the prelude do fit together into an organized whole. So this can be thought of as a Yin-Yang piece: there is a disturbance in the contentment. It terminates with a one-note solo (as do the preludes in D flat, A minor, and some others) followed by parallel fifths (Fx-Cx -> G#-D#), something the classical musical theorists say should not be done. So Chopin is breaking ground with this pleasant-sounding piece.

By the way, I am going to put all of these reviews of Chopin's Opus 28 on a separate web page and will tell you when it is ready.