How about some mathematics eh? Fixed Points

. Everybody is familiar with the idea of a mathematical formula. A formula looks like just a bunch of incomprehensible (or worse: comprehensible) symbols with unknowns, like x and y and so on. You put in actual values in for x and y etc, and bingo, you get a number!! The formula is simply a converter that converts values of the variables into a number. The idea of a function is very similar: it is something that converts particular values of variables into a number. The simplest case is where there is only one variable, say x. Values of x are converted into numbers, usually called y values. Some functions can only use x values in a specific set. This set of allowed x values or inputs is called the domain. (Think of a coin operated machine that only accepts certain coins.) The entire collection of possible numbers that the function gives out, in other words all the possible outputs, is called the range of the function. So the domain is the collection of all possible inputs, and the range is the collection of all possible outputs. What if the range is actually a subset of domain? For example, what if all the incoming x values lie between 0 and 10, and the function gives outputs between, say, 2 and 7? Then there is a possibility that the output might be exactly the same as the input for one or more x values! Consider the function f(x) = 3x^3 - 2x, shown at right. (We're using the convention that x^3 represents x times x times x, or raised to the power 3.) When x is any one of 0, 1 or -1, f(x) = x. Those three values are called the fixed points of the function. The Brouwer Fixed-Point Theorem says that if f is a continuous function from the interval [0,1] into the interval [0,1], then f must have at least one fixed point! To explain: the interval [0,1] simply means all the numbers between 0 and 1 inclusive. Continuous is a little harder; it means that the function f must only vary so that small changes in x result in small changes in the output. In particular, as x varies gradually from 0 to 1, f is not allowed to take jumps in its output values. The Contraction Mapping Theorem says that if there is a positive number less than 1, say k, such that the distance between outputs from two values u and v are less than k times the distance between u and v, then the function has a unique fixed point.

Richard Wagner: Der Ring Des Nibelungen -- The Other Ring Trilogy

. Richard Wagner grew up under the shadow of Ludwig van Beethoven, that immortal musician who influenced almost everyone in the world of classical music of his time. Beethoven taught his fellow-musicians to fearlessly think big: not necessarily big in the literal sense, but to think of music that was great, emotionally bold and unrestrained, powerful enough to last long after the composer was dead, music that would leave the audience forever changed. Gone were the days when people lived as pale shadows under sufferance from their feudal lords; each person theird his god-given right to exist and to glory in their existence. A person's artistic conscience was answerable to no one. The watchword was freedom: from bondage, from oppression, from artificial strictures, and from tradition. In Wagner, this spirit took the form of an imperative to seek an artistic medium that encompassed everything: graphic art, sculpture, music, song, poetry, dance, and of course, theater. This universal Art he named Gesamkunstwerk, which means something like "Universal Art Work" (or effectively in modern parlance, multi-media art concept). And it was built on the foundation of Opera. Wagner began writing operas from the time he was a youth, until in the middle of his life he decided to take on the subject of the Ring of the Nibelung, (the Nibelungs being a mythical race of dwarves, who mined gold), which he felt gave him the opportunity to address many major issues of good and evil, and expediency and responsibility, and inflexibility and compassion. The first opera in the tetralogy (actually written last) was Rhinegold. This brilliantly written opera serves as a prologue to the action of the remaining three operas: A Dwarf, Alberich, steals the magic Rhinegold from the Rhinemaidens. Meanwhile, the Giants have built Valhalla for the gods, and need repayment. The trickster god Loge (Loki) steals the magic Ring forged by the Dwarves of the Rhinegold, and gives it to the Giants. For the moment, everyone is happy, except the Dwarves and the Rhinemaidens. (Back then the Dwarves had their own eminent domain principle, except for when people stole things from them!) [The painting at right is a depiction of Alberich and a Rhinemaiden by Arthur Rackham.] Wagner's peculiar invention was the concept of the Leitmotif: a tiny musical fragment that is highly recognizable, which would serve not merely as the basis for mood music, but very specific aural allusions to persons, feelings, incidents, objects, principles and events. So when in Star Wars the score recalls Darth Vader, or The Empire, or The Rebellion, or The Force, we must remember that Wagner, decades earlier, invented the means to do this in a way perhaps a little too subtle to succeed in modern Cinema. Modern Cinema has embraced the idea of broad musical themes that refer to persons, places or events, but not at the miniscule, atomic level of Wagner's Leitmotifs. (The score for Gone With the Wind, by Max Steiner, for example embodied this modified Leitmotif principle.) Now Wagner's ideal was to present the entire saga of the Rhinegold using all these media of music, art, dance, poetry and drama. He started with his own script (libretto), written all in verse of an appropriate epic meter, and set about creating a score that had the highly imaginative principle of the Leitmotif at its heart. The opening scene of the entire four-opera tetralogy (some 13 hours of music-drama) opens under the surface of the river Rhine. Before the curtain goes up, the Overture to Das Rheingold begins simply with a huge B Flat chord. Deryck Cooke, a well-known British musicologist who has spent a large proportion of his life in studying the Leitmotifs of Der Ring des Nibelungen, suggests that this theme is the basic theme of Nature. (Note: Wagner himself has written at length about his own Leitmotifs for the Ring; I simply use Deryck Cooke's description because I'm more familiar with it.) Gradually the Nature theme transforms into a more 'churning' theme, that of the River Rhine. Here you get the first, simplest example how Wagner developed musical ideas via transformation of these atoms of meaning, to represent related ideas by related atoms of musical melody and harmony. (He adds that the pentatonic theme of the Rhinemaidens, from whom the Dwarves stole the gold in the first place, as characteristic of nature elementals in the story, such as the Wood bird, who appears in the third opera, Siegfried.) Later, in subsequent operas, Wagner introduces variants of the opening Nature motif to represent the World Ash Tree, from which Wotan cuts himself a staff, or spear. A similar, but minor-key theme represents Erda, the earth-goddess. Here is her first appearance, her minor-key theme a counterpart to the major theme of the River Rhine. (After warning Wotan of sad consequences of his actions, she foretells the downfall of the gods, and under her words is an ominous descending, major version of her rising, minor theme.) Here is a sequence of scene fragments from Rheingold, staged by the San Francisco Opera. In this clip we have (1) the Rhinemaidens frolicking, shortly before being robbed of their gold, (2) Wotan, the King of the gods, (3) Wotan being talked at by his consort, (4) the two giants, Fasolt and Fafnir talking about not getting paid for their construction work, (5) The Dwarf Alberich, (6), (7), (8) scenes I don't recognize, and (9) Wotan leading the way over the Rainbow Bridge into the (finally paid-for) Valhalla. The giants, who are given the Ring, fight among themselves, and one kills the other. The survivor, Fafnir, turns himself into a dragon, the better to guard the Ring. (The fight among the Giants is echoed in Smeagol killing his brother Deagol in Tolkien's Ring Saga.) The gold is cursed, and has an appropriate theme, which one hears in the orchestra at critical moments when the Ring influences events around it, usually ending in death, war or destruction. The second opera (the first of the real operas after the prologue, Rhinegold) is Die Walküre. In it, Wotan puts into motion his devious plan to retrieve the Ring from the Giants, and, maybe, return it to the Rhinemaidens. Wotan arranges for a man and a woman to be born, who would be the parents of a great hero. But the plans go wrong, and the woman (Sieglinde) ends up married to the wrong man. But Wotan arranges for the man he intends should be Sieglinde's lover to stumble into her home during a storm. They fall in love, and run away together while the husband lies sleeping. The betrayed husband follows, and there is a mighty battle where both men die, but luckily Sieglinde already bears the child who is to be the hero. During the duel, the valkyrie Brunhilde aids the lover, in defiance of Wotan's explicit instructions, because she has pity on the pregnant woman and her lover. (This is a poignant moment, because Wotan, in his heart of hearts, supports Brunhilde's action, while his instructions to Brunhilde were in obedience to his wife Fricka's command. Wotan's wife Fricka is the deity who guards the sanctity of marriage, and is naturally appalled at Wotan's plot to have this lover come between a man and his wife.) Now Brunhilde faces Wotan, and after a lengthy accusation of treachery, he condemns her to a magical sleep, after which she has to marry the mortal who wakes her, which takes away her immortality, of course. Losing immortality is a major deal, as we have learned from Tolkien's Ballad of Luthien Tinuviel. The opera ends with the magic fire music, representing the magical ring of fire Wotan arranges to be placed around the sleeping Brunhilde, to ensure that only the exceptional hero can wake her. Sieglinde has already been smuggled away from the scene of the battle by Brunhilde, and hidden away in a cave in a forest, near which it happens that the Dragon Fafnir has his lair. She is given shelter by the brother of Alberich the dwarf, and gives birth to the hero Siegfried. Siegfried, the third opera, recounts how the young hero, brought up by the dwarf in the forest out of sight of humans, forges his father's broken sword, kills the dwarf, kills the dragon Fafnir, gets the Ring, and proceeds to discover the sleeping valkyrie Brunhilde, who wakes and falls in love with him. Siegfried gazes at her in amazement; he has finally found a human woman, the first one he has ever seen. But the Ring causes trouble, and Götterdämmerung, the last episode, recounts how when the couple arrive at a nearby court, Siegfried is offered another bride, and amid the cross-currents of jealousy, suspicion and avarice, Siegfried is murdered, Brunhilde in grief rides her horse into the flames of the funeral pyre, Valhalla catches aflame from the burning pyre (!!), and the Ring falls into the Rhine, where it belongs. [The painting at right is a depiction by Arthur Rackham of Brunhilde riding her horse Grane into Siegfried's funeral pyre. Equine actors in operas have famously been reluctant to do this bit.] Wagner appears to have looked for a story in which the true hero is presented with a series of dilemmas, in which the choices were invariably inevitable but regrettable. To that extent, Wagner's choice of this tortured storyline is understandable, though the inevitability of the choices are occasionally questionable. Wagner's hero is, of course, not Siegfried, but Wotan. While, on the surface he appears to manipulate everything, he is simply the puppet of blind fate, the way Wagner seems to have seen life. To conclude, the entire cycle of four operas (three operas and one opera-length prologue) is driven by the lyrics sung by the actor-singers, the fantastic scenery (nowadays too often substituted with modern interpretations, such as the California Gold Rush, etc), and the amazing score, played by Wagner's enormous orchestra, featuring special instruments for the purpose, such as special tubas to represent the the dragon Fafnir. The story of the Ring of the Nibelungen is a variant of a German-Norse epic, the Nibelungenlied, written by an author whose identity remains a mystery. Wagner conflated this story with that of another legend, the Vollsung Saga, to portray a net of deception and disappointment intense enough to motivate his dramatic requirements. The sequence of betrayals is painful enough on which for Wagner to build the ultimate destruction of Valhalla, and the return of the Ring to the Rhinemaidens at the end of the whole drama. Today, to modern eyes, the melodrama seems too unreal and too long-drawn to be taken seriously. But granted that it was going to be realized in some medium, it is hard to think of a better one than Wagner used, or a more effective execution, if one takes into account the amazing musical creativity and planning that Wagner undertook, and achieved. With the medium of Cinema available today, such projects can be implemented with much greater success, as we see in the Lord of the Rings project with J.R.R. Tolkien's book and Peter Jackson's movie trilogy. Since these were created by an author rather than a composer, the literary characterization is certainly more successful. Understandably so, since Tolkien had not only all German mythology and epic poetry for his sources, but Wagner's work as well. The musical score of the movie, however, takes a secondary place. While it is entirely possible that the score could have been written with the fine attention to detail of Wagner's Ring Cycle score, the needs of modern cinematic production procedures and the level of sophistication of movie-going audiences (compared to opera audiences) all dictate a simpler approach to movie scores. The music community is divided over Wagner's music, especially his Ring cycle. Many musicians and aficionados regard Wagner's music overblown, heavy and self-indulgent, and one can see why. (Many composers after Wagner absorbed a lot of his ideas --Verdi, Mahler, Humperdinck-- but managed to keep their music more agile and their orchestras more compact, thus avoiding the worst of the criticisms that followed Wagner everywhere.) Still, we in the 20th and 21st centuries are accustomed to being indulgent with artists whose egos and ideas would probably dwarf those of Wagner. Perhaps Wagner's greatest mistake was that of being born too early! In later posts, I plan to look at other operas by Wagner that are greater successes, artistically, and which reflect a more mature personality in the composer: Parsifal, and Die Meistersinger von Nürnberg. Archimedes

A History of the famous ADVENTURE game

Back when I was in grad school, hanging out with the nerds across the way at Carnegie-Mellon, I learned about the computer game called Adventure (The nerds at my school were an unfriendly bunch.) It started out with a line of text on your screen, which said something like:

You are on a narrow road. Ahead of you is a gate. On your right is a swift-flowing stream. On the ground in front of you is a keyring and a banana. A chicken is here, crossing the road.

GO: SL

? I do not understand "SL".

GO: Slaughter the chicken

OK.

GO: Look

You are on a narrow road. Ahead of you is a gate. On your right is a swift-flowing stream. On the ground in front of you is a keyring and a banana. There is a dead chicken here.

GO: eat

? Eat what?

GO: The chicken, idiot

? I do not understand "idiot".

(You have been informed, at the initial instructions level, that you can type in a character or a word or a sentence, such as "pick up key," or "go forward" or "turn left" or just "L".) At right is an image from the Windows version, available for download at http://www.rickadams.org/adventure/e_downloads.html. (The image is evidently from M.C.Escher!) The game proceeds in response to what you type. It is essentially a quest to explore an enormous cavern with many chambers. You could arm yourself with various tools and weapons, which you had to use against gnomes and elves, etc. The link above gives a detailed history of how this ancestor of the games Zelda, etc etc came to be. Arch

What's Wrong with American Education?

. I was recently talking to some of my friends, and we were deploring the fact that our education system seemed so ineffective. Rather than simply plug my own views here, I thought I'd ask as many of my friends as would spare the time, to tell me what they thought the problems were. Unfortunately, only two of them responded. (Nobody likes writing assignments, right?) One of them wrote right back. "What education? I don't see any going on, to be honest," she said. She is not directly involved in education, and only sees its effect in the younger people with whom she works: "And where does some of the information and opinions given come from – does learning stop when you walk out of the classroom door in the summer? Were you given the freedom – or, did you demand the freedom – to question what you were taught? Did you challenge ideas, or did you blindly accept, because it was easier and took less energy?" Here is a summary of her assessment of the education that she's seeing:

1. Poor language skills, writing skills, mechanics (spelling, punctuation)
2. Eager acceptance of wild ideas; uncritical attitude towards information and opinions
3. Religious interference with teaching science
4. That external tests (Federal and State administered) steer the curriculum
5. Too many students per teacher
6. Poor preparation of teachers; teachers are poorly educated themselves, and propagate their ignorance.
7. Guns and violence in the classrooms create an atmosphere unfriendly to learning

The second response was from a friend who was a teacher, who is battling their school principal, who is determined not to allow any sorts of enrichment activities. She writes: "The only way we can take out kids on any field trips is if we raise the money ourselves, so there has to be a fund raising component of the job as well as the teaching component. That's another thing wrong with American education, and maybe the most important one. I'm sure even from the beginning it was obvious that funding schools with local property taxes would only perpetuate the inequities of the society."

1. Teachers (and Principals) are trained with a lot of education theory rather than actual education content. The theories are a matter of fashion, in this year, out the next. Substantive education on subject matter is considered unimportant.
2. There are too many public-school principals who were athletic coaches, and not really academics in any way. (My own principal thinks that homework should not be given at all.)
3. The only way I can teach the material is to make the kids stay after school and work with me, because they do not get that kind of reinforcement at home.
4. The only way kids will get better at math is to do math. The principal does not get this.
5. Teachers are responsible for fund-raising, for extra programs. This is not the case in richer school districts; money for extra programs is readily available there, obviously.

My own opinions:

1. Education appears not to be a value in society; a diploma is.
2. There is no national curriculum; state curricula vary widely, and often reflect extreme agendas.
3. Teaching is not taken up by the best, since the profession is not respected. So the teaching profession is populated by an odd mixture of idealists and bumblers.
4. Political pressure is applied not to embarrass poorly performing students, who are often minorities.
5. Mathematics education is particularly weak, because many teachers have been badly taught as children themselves.
6. Finally, there appears to be a common view that education is something that benefits me and my family. It is a community effort only for the sake of convenience, because I just don't have the money to start my own school.

Disparities in resources: As you can see, the causes of difficulties with education emerged to transcend cultural and social effects, and are actually political in nature. The education system is set up so that richer people live together, poorer people live together, and schools in the richer districts have better resources than schools in the poorer districts. Education as a value: Unlike in the Third World, where education is considered something useful in its own right, here it is seen as a minor stumbling-block. In big Corporations, for instance, it is the lower echelons that must be educated and trained to actually do the work. The higher-paying upper-echelon occupations require less education. The more a youngster realizes this, the less incentive he or she has to get an education, which dooms him or her to a middle-level occupation. Furthermore, students are often cheered up by the family if they come home with bad grades, by telling them that good grades are not important after all. In the pursuit of self-esteem, (academic) success in school takes a back seat. In addition, teachers are vilified for giving the bad grades in the first place. Support at home: In the poorest inner-city homes, students understandably find little support for doing homework, etc. Some of the most wonderful inner-city youth projects include spaces for students to simply sit and do homework quietly. But even in middle-class homes in the suburbs, there is often little support for students with their homework. Some parents are so exhausted when they return from work that they simply cannot help. Some parents do not know the material. Some students are overwhelmed with extra-curricular activities that trump the homework. Most of all, doing homework seriously is simply not a priority in many homes. Education as a social activity: There are two radically opposed points of view concerning education. View 1: I educate my child as well as I can, because I want him or her to have the edge when competing with others for employment. View 2: I educate the entire younger generation, because among these kids might be someone who finds a cure for cancer (or finds a way that I can own and operate a dozen cars without harming the environment.) It's not that those of us who hold View 2 have less faith in our offspring. I firmly believe that those individuals who are more evolved can see that community-orientedness is a winning evolutionary strategy, whereas individualism is not. Education does enable solving the problems of society, but it also makes life richer, makes work more meaningful, and makes leisure more interesting. Being interested in a better life for the children of one's neighbors to the degree of being willing to fund it is an enlightened condition to be in. In these days of economic hardship, only this attitude will help us survive with sanity and optimism intact. Archimedes

Johann Sebastian Bach

. Johann Sebastian Bach, one of the greatest names in European classical music, was born on March 21, 1685. Unfortunately, this does not mean that J.S. Bach was born 324 "years" ago from today, within 24 hours of the Spring Equinox. (The Spring Equinox was yesterday, March 20th, at around eleven in the morning, and falls on March 20 or 21 in most years.) The reason for this is that at that time (1685), the calendar in use in Lutheran Germany was still an older one --the Julian Calendar-- which, for reasons having to do with the Leap Year adjustment, had rotated to the point where the Spring Equinox fell much earlier than the standard date in March. What is going awry here: is the Equinox being unpredictable, or was the Calendar faulty? As far as the world is concerned, especially physicists and farmers, the Equinox is basically the Clock Of The Earth; everything else has to be adjusted so that the Spring and Fall Equinoxes continue to fall on roughly the same dates every year, so that the seasons remain in their places relative to the Calendar. A more detailed description of the problems are given at the end of this post.

  The Music of J.S. Bach In a sense it is difficult to describe what is wonderful about the music of J.S.Bach, because he embodies all that is best in a certain kind of music written even today: music in the conventional, tonal style with some elements of counterpoint. The great composers who enjoy popularity today by and large descended musically from J.S.Bach (though some of them may have had greater affinity for Bach's great contemporary, G.F.Handel; at least Beethoven is said to have declared so.) Still, there is much that is miraculous in Bach's music. His immense range of emotions, the great joy and exaltation he was able to express, the intellectual satisfaction in listening to the music, and the sheer delight of playing and singing it. Bach's music can speak for itself more eloquently than I can. Here are some clips of some of the most appealing of Bach's compositions, in addition to those I have provided in posts over the last month or so. The Chorale, Jesu bleibet meine Freude, or Jesu, joy of man's desiring. This is one piece of a 13-movement cantata, no. 147. This recording is in a style considered to be more true to the original performance style. Finally, a clip of the chorale performed with just one singer and one instrument per part, by Joshua Rifkin and his team. The famous Air on the G String is a movement from Bach's Orchestral Suite no. 3 in D major. Here is a recording of it in its original setting. Bach was mostly famed for his organ music in his lifetime. Here is the amazing Passacaglia in C minor, BWV. 582. It is usually followed by a most amazing fugue on the same theme, but the two are not available together on YouTube.

[Added later: here is E. Power Biggs playing the Passacaglia, and most of the fugue. The registration Power Biggs uses is always extremely clear, so that you can hear everything.] Quite different from the fugues --which have a certain weight to them-- are the organ trio sonatas. A trio sonata normally means exactly what it says: a sonata for three instruments. But Bach decided that an organ presented the opportunity for a single player to play a trio sonata: two hands and feet. Here is one of my favorites: the Trio Sonata in E flat, BWV 525. This clip is of the first movement of three. Bach wrote great music for instruments. His second orchestral suite in B minor featured the flute, and here is a lovely movement from that, the Rondeau. Here is the lovely slow movement of the double concerto for violin and oboe BWV 1060. (The original violin/oboe concerto score has not survived, but has been reconstructed from a version for two harpsichords.) I could not resist also adding a link to the last movement, played here by Kyun What Chung and Heinz Holliger. Here is Kennedy and oboist Aisling Casey playing Movement 1 of this concerto much, much faster. (There are links to the subsequent movements at the site.)

Bach's Brandenburg Concertos are justly famous. This is the opening movement of No. 5 in D major.

This is a joyous aria from Cantata no. 68, possibly written for his young second wife, the famous Anna Magdalena Bach. It is sung here by a boy treble, young Peter Jelosits around the early seventies.

Towards the end of his life, Bach compiled three monumental works, the first of which is the Mass in B minor. Here is a duet from the mass: Et in unum dominum.  Another gorgeous movement from this mass filled with gems is Qui Sedes ad dextram patris.

The next masterpiece --one of the greatest works in Christian music-- is the St Matthew Passion. Here is the great opening chorus. It features three choirs: choir 1 and choir 2, and finally a small chorus of trebles in the center, singing the chorale, O Lamb of God unsullied. Halfway through the work is another great chorus, O Mensch, bewein dein Sundre gross.

The last chorus is essentially a cross between a lullaby and a funeral ode to the crucified Jesus. Sleep softly, it says, and alternates between sad mourning and gentle comforting: Wir setzen uns, mit tranen nieder.

The work by Bach that is considered a crowning glory, at least by those who consider him a secular composer, is the great compilation The Art of Fugue, which is a set of some fourteen fugues, the last of which is incomplete. Here is the opening fugue, one of the most perfect fugues ever written.

 To close on a lighter note, we present the last movement of the 5th Brandenburg, one of Bach's jollier creations.

Calendars [Here is a simple and accurate explanation of the idea of Leap Years.] As I said, at the time Bach was born, there were two calendars (at least) in force. The Roman Church, under Pope Gregory, had already established the improved calendar which is more or less the one we use today. The conservative Lutherans in Saxony defied the use of this new calendar on religious and political grounds. (The British, too, were using a different variant of the older calendar.)

For years I was confused as to why there was such a fuss about calendars. It seemed to me that if every year had 365 days, the world would be a better place. This was hardly my original idea; the world had initially gone on the assumption that this was indeed the case. Actually, though it is true that the length of a year is close to 365 days, at the time of Julius Caesar, already people were finding that the shortest day of the year was falling later and later in the year. The reason was, of course, that the calendar year (i.e. 365 days at that time) was shorter than the period between the Spring equinox of consecutive years by about a quarter of a day. Already at that time, they were clever enough to have ways of telling the dates of the Spring Equinox (e.g. Stonehenge, etc), and it was decided that this event should take place on the same date every year. Why? Because farmers needed to plan their crops and their activities, and they needed to know when the seasons would take place. Taking averages, it was found that the year was actually about 365.25 days long, and so leap years were invented. The so-called Julian Calendar only had leap years. (That is, it had leap years every single fourth year without exception.) Now there was an extra day every four years, and if the year had been exactly 365.25 days long, everything would have been all right.

Unfortunately, as Wikipedia says, this estimate was longer by about 11 minutes than the actual length of a year. The actual length of time between one Spring Equinox and the next is roughly 365.242374 days. So now, every so often, we had to take a day off. This was the improvement of the Gregorian calendar.

 Julian Calendar: One extra day every year numbered a multiple of 4. [11 minutes too long per year]
 Gregorian Calendar: One extra day every multiple of 4, but not on multiples of 100.
 Modern Calendar: One extra day every multiple of 4, but not on multiples of 100, but do add a day on multiples of 400, and other adjustments to be announced from time to time.

With these adjustments, we should be fine for some 800 odd years. At the end of last year, for instance, December 31st was lengthened by one second, I was told. That is scary. So, the main point of all the gymnastics with leap years is to avoid the drift of the seasons, where winter and summer creep around the calendar. A pleasant Spring Equinox to all my readers, and a happy birthday to Johann Sebastian Bach. No matter when he was born, it seems reasonable to celebrate his birthday on the date that he would have recognized if he were alive, namely the 21st day of March, in whichever calendar happened to be locally in force!

Archimedes

The Von Trapp Children

. The Von Trapp Family Singers made famous by the movie The Sound of Music settled in Vermont, as many are aware. This video shows the present generation (great-grandchildren of the famous Maria) singing some of their favourite songs. Maria von Trapp, in her youth was involved in the collection of German folk song, and the family has produced several CDs of folk songs from all over the world.

Classical Music Niche Areas You Might Like: The Renaissance and Earlier

. Previous entries in this series:

• Doctor, I want to learn to select classical CDs
• Next Steps getting into classical CDs
• Classical music niche areas you might like: The Romantics

For nearly the first 15 centuries of its existence, the Roman Catholic Church dominated the politics and culture of Europe. Austere musical forms were developed in these times, mostly having to do with chanting for services (or Masses, which is to say the communion service of the catholics). No matter how luxurious the lives of the clergy themselves (especially in Rome), they preferred the lay people to live lives of ascetism, and music and dance were considered extravagances, and close to sinful. Despite this, though, music survived and flourished; dance music and folk song (songs not attributable to a particular composer, and handed down over the generations) was learned by children in their homes. Once the Church lost its hold on society a little after the end of the first millennium, as with all the arts, music had a flowering that produced an enormous volume of amazing works: songs, dances, and, a new invention: simply music just for listening. At first, monks in monasteries notated these new pieces of music for their own illicit amusement, and then gradually lay people learned how to do it, and musical pieces spread freely from court to court, for the entertainment of the nobility, and minstrels and bards spread ballads and chansons in all the taverns they visited. Until the invention of the printing press, it was difficult to collect and preserve notated music. In the 18th and 19th centuries, there were interested individuals collecting folk music all up and down Europe and even in America; this was the only way to find it, before it was lost. Often only a few aged folks knew a song, or a variant of it, and the work of the collector was just in time to prevent the complete loss of the piece. It is thanks to these people that we have much of the knowledge of folk music that we do today. Michael Praetorius (1571-1621) of Germany was such a collector, and his collection of renaissance dances entitled Terpsichore is one of the best-known. Not everyone has the genes to enjoy music of the distant past, which could feature instruments that sound strange to our ears, to say the least. But once bitten with the early music bug, it becomes a delight to listen to it more and more. Here is a dance from Terpsichore. Note the drum that adds so much to the first part, and the quite complex rhythms. It compares quite well to rhythmic music of the modern day. (Obviously, once you go to the page on the YouTube website, you can easily find numerous links to similar music, as well as a lot of information about the clips. Here are a couple more dances, performed by another group.) In the early part of the 20th century, the brilliant Italian composer Ottorino Respighi wrote a series of pieces including The Pines of Rome. Among these is a work titled Ancient Airs and Dances, in which he presents old folk tunes orchestrated to preserve some of their original modal character in some degree. Bear in mind that Respighi (1879-1936) is a modern composer, even if he found his inspiration in renaissance melodies. In the late 1500s, in England, a wonderful musical tradition was established. The music of Orlando Gibbons, William Byrd, John Dowland, Thomas Arne and Thomas Morley and others led to the music of the brilliant Henry Purcell. There are yet other names in this list: Giles Farnaby, John Bull, Thomas Tallis, John Taverner, Thomas Weelkes, John Wilbye, and Thomas Campion, some of whom specialized in vocal ensemble music, in particular, madrigals. A dance of the Renaissance was the pavane, slow and dignified, while other dances were more energetic, such as the galliard. Dowland wrote several allmains, which is probably an Anglicization of the French word Allemande, which means simply "German", describing a dance considered to be of German origin. Here is a fragment of the King of Denmark's Galliard, by Dowland, played by the King's Noyse. Next, here is a fragment of a pavane by William Byrd. (This sort of music is rare on YouTube, so I must make do with the samples given on Amazon.com.) Much of the earliest music played today is sacred music. Palestrina was a early renaissance composer who composed for the Church in Rome the greater part of his life. This is the Kyrie of one of Palestrina's masses. In England, Thomas Tallis wrote a famous motet Spem in Alium, for 40 voices. (It is written for 8 groups of 5 soloists each, not a choir as such; the music has 40 lines. I believe there were actually larger scores than this.) The Venetian composer Giovanni Gabrielli wrote for voices and brass instruments; unforgettable music that was written for performance at St Mark's Church in Venice, which has choir galleries on opposite sides of the nave, so that antiphonal effects (super stereophonic effects) were possible. Here is one of Gabrielli's famous canzonas for brass and organ, featuring the late beloved E. Power Biggs, the British-born organist who immigrated to the USA shortly before WWII. Note that this is performed on modern brass instruments, in contrast to the instruments in use five hundred years ago, which were significantly different. Orchestra of the 17th Century is a modern ensemble that attempts to reconstruct ancient instruments. Here is one of the canzonas played by them. The Renaissance was followed by the Baroque, and one of the most important composers who bridged the two periods was Heinrich Schutz of Germany, who worked at the court in Dresden, at which they attempted to create music sung in German, adapting Italian models of song-writing, to suit the sounds and rhythms of German speech. Here is an excerpt from Schutz's lovely representation of the Christmas story, The History of the Birth of Jesus Christ. A recent phenomenon is that of Hildegarde of Bingen (1098 – 17 September 1179), a nun of the 11th century whose musical compostions were startlingly original, within the context of medieval plainchant. This is A feather on the Breath of God. The group called Sequentia has realized many of Hildegarde's compositions. (It is a mystery how accurately these people can reconstruct Hildegarde's tunes. I suspect some of it is speculative.) Almost in the Baroque period is Henry Purcell, one of the most beloved of British composers. Here is the Rondeau from the suite for Abdelazar, a play. (Purcell was crazy about the Theater. His wife locked him out of the house on one occasion when he was late coming home from the theater, and he died of pneumonia.) This is popular music of the time; Purcell also wrote operas and sonatas, which were considerably more serious in style. Purcell's music has a glow to it that is irresistible. More examples will be provided presently! Archimedes

Doing My Bit for The Environment

. Though everyone ought to be conscious of how their daily life affects the environment, those among us who are lucky to have a scientific background have a special responsibility. On the face of it, environmental decisions are easy: if you have a gas-guzzler car, you should trade it in for a small hybrid. Or should you? There is a minor complication; if it isn't such a big gas-guzzler as all that, and you don't really drive it much, maybe you should keep it. Especially if it is about to be trashed, and you hardly need to drive it at all, you may be saving the planet from a half-ton of scrap metal. On the other hand, if the engine is so shot that you burn oil all the time, and you spew toxic fumes from your exhaust continually, maybe you should trash it. On the other hand, if some kid buys it for cheap, he or she might drive the car around a lot more than you do, spewing ever more noxious fumes around the county. This is a tricky decision. Refrigerators are among the trickiest. Unlike cars, they're not easy to sell, since most people want a brand new fridge. Should you get that super-efficient refrigerator and trash your antique Frigidaire? At least, we know, they're supposed to recover the refrigerant (the Freon, or whatever it is these days; probably a kind of coca cola) before they toss the thing, but one wonders whether they actually do it. (At our school, for instance, we learned that the janitorial staff was simply trashing all the aluminum cans we were so conscientiously putting in special recycling bins. I think we made them stop.) At any rate, once the refrigerant is sucked out, the shell will be trashed, but of course it contains a ton of plastics and paint that will probably pollute the landfill; I don't know. Paints often have toxic components. I have read somewhere that magazines were printed with inks that were at least mildly toxic at one time. Perhaps this is not presently the case... MOST IMPORTANTLY, incandescent bulbs, as you know, not only use up electricity, they heat up the rooms. In the winter this is probably fine, but in the Summer, it makes no sense to heat your house with the lights, and then cool it with your AC. A possible solution to this problem, even if less than perfect, is the fluorescent bulbs. These come with screw-in bases, so they can be put right where the old 75-watters used to be. They use up only a tiny fraction of the wattage of a regular incandescent bulb. [Added later: "Bright idea makes a big comeback: Conservation" is an article from back in 2006 that makes even more sense today.] Now another consideration cuts in. If you keep turning off your fluorescent bulbs as you leave the room, they burn out faster. But if you keep them burning, they last longer, but eat up electricity. If you keep turning them off, they have to be replaced more frequently. What to do? In one case, you're using up energy, and bleeding a tiny bit of heat into your home all the time. In the other case, you're spending money, using up the energy it takes to manufacture a bulb each time you toss one out, and putting a fluorescent bulb in the landfill. I'm not absolutely sure, but I think the phosphors inside the bulbs contain tiny bits of heavy metals which are not good for the landfills. So it appears that, particularly if you have installed a very low-wattage bulb (e.g. a 13-watt fluorescent bulb is as bright as a 60-watt incandescent bulb, even if not as warm-colored) it may be better to simply leave it on permanently. I have done this, and a bulb has lasted for around two years. Furthermore, fluorescent bulbs gradually get brighter as they warm up, so it pays to keep them on. My front and back porches have 13-watt (fluorescent) bulbs that burn 24 hours a day. The lobby light at the top of the stairs, and the light on the piano, too, are left on continuously, and are 27 watts, and 9 watts respectively. (One of these days, they'll make household lights out of LEDs, and we'll have to learn all about how to use them.) Disposing of used batteries and engine oil properly, alone, can make a huge difference in the degree to which you're personally burdening the environment. Similarly with used paint. So though none of us are perfect, it seems the right thing to make one more choice carefully each time you think of the environment. Arch

Classical Music Niche Areas You Might Like: The Romantics

. The story so far: I first described some pieces, or works, that you might be interested in. (I tried to keep them to smaller works, less than about 20 minutes long, for the most part, though I wasn't trying very hard.) I focused on chamber works (small groups of instruments) and concertos (works for a featured solo instrument and orchestra), which are, for the lack of a better word, more focused pieces, so that they're a bit more accessible, in some ways. Following up on that post, I gave you links to some of the works mentioned (I should have given the links in the original post, I suppose, but perhaps it was as well that I didn't, in order not to steer you towards any particular recordings.) I also gave you an idea of how to follow-up on a liking you might have discovered to particular composers. That's how I got into classical music: by looking for works by the composer of something I really, really liked. (In my case, it was the famous chorale "Jesu Joy of Man's Desiring," which is a clumsy English versification of Jesu bleibet meine Freude, a chorale from Bach's Cantata number 147.) You now really have a pretty good map for the music from about 1750 to 1890, spanning the end of Bach's life to the end of Mendelssohn's life. The so-called classical period falls in the early part of this period, and then we go into the early romantic period. We have already met some composers in the romantic period: Beethoven, Mendelssohn, Dvorak, Tchaikovsky, and even our friend Mozart, according to some people, e.g. Charles Rosen, reputedly. (For those of you who actually make money as authorities who decide which composers are allowed to be classified as Romantics, note that this is a not-for-profit blog.) The Romantic movement in music (and we're still talking about classical music, even if the word "classical" is used to describe a certain period) is one where the composers broke the rules, used bigger and more expressive orchestras, wrote works of longer duration, tried to express more acute and complex emotions (though Bach might have objected to this statement if he were alive) and were inspired by literature. This isn't surprising, because the literature of the 1800 was an explosion that simply could not be ignored. In keeping with my plan to suggest shorter works for you to "taste", I'm going to introduce you to a variety of composers. It is very likely that many of you who were left cold by music of the classical era we've looked at so far, will find something here. (EVEN IF YOU DON'T, THERE IS YET MORE!!!) One kind of thing to try is Overtures. Overtures are relatively short things, the opening of an opera, or just simply an overture for fun. We've already met Mendelssohn, and his Midsummer Night's Dream Overture is most definitely romantic, but if you've tried Mendelssohn and not liked him much, don't waste your time. Wagner is squarely in the romantic classification. Here is the Overture to Lohengrin. This will blow your mind. unfortunately, Wagner did not write a great deal like this, but his operas Lohengrin and Parsifal live in the same sound-world, if I might be allowed the use of that phrase. This recording is by Otto Klemperer. The music is supposed to, at least in some degree, represent the pure love of Elsa for the knight Lohengrin. Another possible piece to try is Wagner's Siegfried-Idyll, though it is so untypical of Wagner that you might be frustrated by trying to follow up. It is actually chamber music, as played here, since there is only one instrument per part, a total of some 13 instruments in all, very restrained orchestration for Wagner, let me assure you. Or try the Overtures to Tannhäuser or Die Meistersinger, both easily found, or even purchased for a couple of dollars from Amazon.com. Tchaikovsky wrote some lovely overtures, including the famous 1812 overture. (It's hard to appreciate this one because it's so familiar.) Bizet: The Overture to Carmen. This one is likely to be a hit with anybody, as is the opera as a whole. This overture style is called the Potpourri Overture, for obvious reasons. Mozart's Overture to The Magic Flute (Der Zauberflöte), is classical, but has those shimmers of romantic feeling that throws our methods of sorting styles into a cocked hat. Don Giovanni, if possible, is more romantic still. Verdi's Overture to Aida might make you a convert to Italian opera. Also check out A Faust Overture by Liszt, and the overture to Rosamunde (Schubert), and the symphonic suite titled Sheherazade by Rimsky-Korsakov. Belshazzar's Feast is a work by William Walton with which I'm not personally familiar, but I mention it here as a possible listening choice. Leonard Bernstein (Candide) and Benjamin Britten (Noye's Fludde, or Peter Grimes) Suites and Variations are another possible choice. The Enigma variations by Edward Elgar, and the Pomp and Circumstances marches are well loved, especially in English-speaking countries. One of the most well-known variations is Nimrod. The Wise Virgins, a ballet suite by William Walton is based on Bach tunes, and is not a good representative of the romantic style. (Still, here is a fragment, just because I love it so much!) Other suites are by Percy Grainger, Aaron Copland (Appalachian Spring), Ralph Vaughan Williams (The Lark Ascending, a single movement and not a suite, actually). Pictures at an Exhibition is a must-hear, an orchestration (by Ravel) of a piano work by Mussorgsky. Ravel, of course, wrote the amazing Bolero, a ballet movement originally. However, his suite Pour le Tombeau le Couperin brings us to the impressionist style of Debussy, Ravel and Faure, and from there to the minimalists of the present day: Philip Glass, Henryk Górecki, Arvo Pärt, John Tavener and others. [To be continued...]

Health Care look like the Post Office?

. In this video clip, Bill Maher addresses the typical conservative diatribe against allowing the government to run anything. You want [Wells Fargo Bank and General Motors] to look like the Post Office? Yes, actually, says Bill Maher.

Next Steps in getting into Classical CDs

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Last week, I sent you off to get yourself a CD or two, to listen to it, and see whether you liked what you heard. Most lovers of classical music --even if they're generally happy to hear classical music of any flavor-- do have their particular favorite kinds. Here are my thoughts about each piece (or work) that I suggested in the previous post, just to give you a (possible) idea why you did or did not like the work, and to supply connections to others. I'm going on the initial assumption that you would probably like pieces with similar attributes, but of course you might like something completely different a lot better! Anyway, for what it's worth:

Chamber Music

I sent you off looking for Mozart's Clarinet Quintet. This is a piece I really like, and I know I'm not alone. The second movement, particularly, is just gorgeous: tender, contemplative, melodious, delicate. In contrast, the Rondo movement, with its main Minuet-like theme, is witty and jolly, though it stays within the more or less dignified mood that Minuets are generally associated with. (One episode is very much like a Laendler, the dance featured in the Sound of Music, in which Julie Andrews begins dancing with one of the boys, and ends up dancing with the Baron himself.) Mozart also wrote several wonderful string quartets, including this String Quartet in G major.

My favorite composer is Bach, and I encourage you to try listening to some Bach chamber music.
Here is a sonata for violin and harpsichord (being played by a recorder and harpsichord instead);
Here is a trio sonata from The Musical Offering, one of the most beautiful movements in that collection; another movement, played by the Kuijken family (notice the viola da gamba, which looks rather like a cello with way too many strings). This is Baroque music. The word Baroque is used to describe a highly ornate style developed around the sixteen hundreds, extending to around 1750. This period is sufficiently removed from the present for the music to sound "dated", which is something to recommend it to some tastes, and a setback to other tastes.

Haydn developed the String Quartet, a sort of very austere sonata for the four very closely-related instruments of two violins, a viola and a cello. Here is one of them (nicknamed "The Lark"). The plinking sound in some places is gotten by using a technique called pizzicato, where the strings are plucked instead of bowed.

Brahms wrote several amazing sextets (6 instruments). Mendelssohn wrote some gorgeous music for strings. His Octet for Strings is one of the miracles of the modern world. Another genre is the Piano Trio (Piano+violin+cello). A famous work in that genre is the Archduke Trio of Beethoven.

Concertos

The J. S. Bach Concertos: 1041, 1042, 1043. It's hardly possible to dislike these. Here is Anne Sofie Mutter playing the last movement of the 1041. The rhythm is a gigue (or jig), and boy, does she play it fast. Who said Bach can't be exciting?

The Triple Concerto, 1044 has a wonderfully philosophical middle movement, played by the soloists alone (sometimes with a cello backing the bass line). Triple concertos have a wonderfully different feeling to them, because the soloists form a team, and the success of the concerto depends as much on the feeling among the soloists as it does on their virtuosity.

[Added later:
Georg Friderik Händel, a German from Halle who settled in England, has written not only some of the most listenable and famous music played today. His more serious music is in the genre of Opera, which he loved. His lighter works, however, have immediate as well as lasting appeal. This piece is one of my greatest favorites: the Alla Hornpipe from one of the Watermusic Suites. (This music always reminds me of a royal wedding; it is so grand and festive!) Handel's work is utterly musical, even if a little on the solid, foursquare side, far less complex, and a little more forward-looking that that of Bach, tending towards the so-called galant style that was coming into fashion in the middle years of the 18th century. Even if I neglected to mention Handel in the first post in this series, he is by no means a mere footnote in the panoply of the music of the last four centuries. For example, look at his concerto for organ.]

Mozart: The 5 violin concertos. These are lighthearted, melodious confections that are immediately attractive to the beginning listener, but lack the depth that will keep one coming back to them. The A major (no. 5) is one of the most memorable. If you're destined to be a Mozart lover, though, these will signal that fact. The grace of Mozart's string writing is indescribable. This is Monica Huggett, one of the most respected violinist of the last few decades. Here is the first movement from Concerto No. 1 in B Flat major, being played by an all-women orchestra, except for the director and soloist! (Why not a female soloist, too?)

[Added later: One of the most beloved pieces by Mozart is the famous Sinfonia Concertante for Violin and Viola. This is an utterly romantic piece which is a nice combination of brilliance and musicality.]

The Piano concertos are entirely another matter. They are serious works, with more to say than the violin concertos, and were intended as vehicles for Mozart himself. Here's Keith Jarrett playing the first movement of the K488 (no. 23 in A major). Another clip features Zoltan Koscis, playing the concerto while also playing along in continuo style, something that is coming back in fashion for Mozart concertos. The last movement is fast and exciting, and one of my favorites.

Mozart's Clarinet Concerto, K 622 is a universal favorite. Here we hear a performance of the slow movement, evoking great tranquility and almost otherworldly stillness. (The slow movement of the Quintet also has a similar feeling in many spots, though the more intimate writing makes the effect more romantic.)

Beethoven: Here's Concerto no. 4. As you can see, the piano itself plays the opening notes, an enormous innovation. (In Concerto no. 3, you see how the piano stays silent until almost the middle of the movement, after which it makes its entrance and dominates the remainder of the movement.) These are works with great expressiveness, even if there is a great deal of display of bravura on the part of the soloist.

The violin concerto is a favorite among violinists, for the most part. Here is Itzhak Perlman playing (most of) the first movement.

Mendelssohn's Violin Concerto: this is a highly approachable concerto, and will probably make friends for violin concertos in general, and the violin as an instrument, than for Mendelssohn himself. Mendelssohn's music is gaining in importance, but is still on the fringes of the classical repertoire. (This clip is annotated, giving several items of trivia about the work, which keeps up the interest!)

Tchaikovsky's Violin Concerto is interesting because of its uncompromising nature, the almost Beethovenesque willingness to make "ugly sounds", which one forgets in the sheer energy of the work. It is one of the most virtuosic pieces for violin and orchestra, and very, very Russian in flavor. Here is a very youthful Sarah Chang playing the first movement.

Brahms: Violin concerto, 2 piano concertos, and a double concerto for violin and 'cello. The violin concerto is on the monumental side; the first movement goes on for around 20 minutes, most of one side of an LP record. This recording is by Henryk Szeryng. Brahms, though he had his lighter moments, was a serious fellow; to this day, it is a puzzle why I, of all people find his music so satisfying. It's as if Brahms is serious enough for all us, so that we can take a break from it. The piano concerti are simply wonderful, too. (In this excerpt you see how long a shadow Beethoven cast. I would say Brahms is chaneling Beethoven here, if it weren't so impertinent.) And the double concerto is also a gem; probably more beloved by Brahms lovers than by the general classical music world.

Dvorak: here's a tantalizing clip of the Violin concerto. Dvorak inherited the marvelous repertoire of Mendelssohn and Beethoven, as well an inexhaustible fund of melody from his native land. (If we can judge from the music of Dvorak, the Czechs must be musical indeed.)

[Added later:
Schumann piano concerto: Here is the last movement being played by Hélène Grimaud. The whole concerto is filled with simple joy, in great contrast to the life of this most literary of musicians, which was plagued with acute psychological problems (which appear, from their description, as caused by a very severe bipolar disorder). A concerto with a very similar mood, often paired with this one, is that of Edvard Grieg, one of the first Norwegian composers to receive international recognition.]

To proceed:

One thing you have done, we hope, is to get a feel for a composer in the era 1700 - 1900 whose music you can relate to! (If you can't there is still a huge variety of composers you could connect with, so hold on!)

Johann Sebastian Bach: If you liked the Bach excerpts, there is good news and bad news. Bach wrote lots of concertos, about a dozen, in addition to the famous Brandenburg Concertos, a set of 6 that hearkens back to an earlier style of concerto called a Concerto Grosso written for performance in noble houses, which contrasted a small group of expert musicians with a larger group of (probably part-time) musicians who played essentially easier portions which were repetitive and less intricate. By the time Bach got his hands on this form, there were really no easy portions in the works. One of the most famous Brandenburgs is No. 5 in D major, featuring a virtuoso harpsichord part.

In spite of all this, the major portion of Bach's output was for the Church, in the form of the great Cantatas he wrote for performance at Sunday services. If you like choral music, you should begin to look at Bach's choral music, which set a standard for choral music throughout the modern era (even though Beethoven, Handel, Haydn and Mozart made very important contributions, too). Finally, Bach wrote four fabulous Suites for Orchestra which are grand and brilliant, intended to bring the grandeur of Versailles to the smaller German courts of the 18th century. It is hardly possibly to dislike these Suites, which consist of a grand opening movement, followed by a set of a half-dozen old-time dances. Here is an example.

Wolfgang Amadeus Mozart: The music of Mozart is puzzling in its mixture of simplicity and complexity. It is possible to love this music, then become bored with it, and then hear it again and fall in love with it a second time, and be completely baffled that you missed things in it all this while.

The Mozart Piano Concertos: these were written just at the time when composers were beginning to see composition as an expressive vehicle, sort of similar to writing a novel: a way of both becoming famous, leaving a legacy, and showing off one's talent. In the case of Piano concertos, the performer was usually Mozart himself (occasionally a special pupil). In his younger days, Mozart was very possibly quite satisfied to write a piece more brilliant than anything written before. As he matured, though, we see thoughts and ideas in those piano concertos approaching those that came out of Beethoven. These are difficult to put into words; they are a distillation of emotions and experiences that transcend words. Some piano concertos have a certain gravity, e.g. Concerto no. 24 in C minor, played here by Andre Previn, but not sad. In contrast, the A major, no. 23, which is so sunny for most of its length, has spots that simply seem bewilderingly grief-filled. But the grief seems to pass, as if it were simply fleetingly remembered grief.

If you like Mozart, eventually (and I mean possibly months or years in the future) you must look at his operas, most notably the Marriage of Figaro. I'm not giving you a link, simply because you must wait until you really want to move into Mozart opera. Be aware that opera fans aren't often attracted to Mozart opera, but prefer Puccini or Verdi or even Wagner. (And certainly Bizet!)

If you prefer something a little more serious, but without pianos all over the place, try the Mozart Symphonies. The last two or three are the most celebrated, including the amazing "Jupiter" (no. 41) and the well-loved g minor (no. 40). (Warning; the "Jupiter" could be seriously underwhelming at first.)

Ludwig van Beethoven wrote in most instrumental forms, but when one hears the name Beethoven, one usually thinks of his symphonies. Symphonies were invented several times over; first by J.C. Bach, then by Haydn, then by Mozart, and then by Beethoven, in such a way that all subsequent developments of the genre had to prove themselves against Beethoven's nine amazing symphonies. A good one to begin with is the famous Eroica. My personal beginnings in classical music were with the famous c minor symphony, No. 5.

Felix Mendelssohn, as I said, is highly underrated as a composer, a situation which is aggravated by his very early death, at the age of about thirty, of pneumonia, I believe. Mendelssohn wrote a vast volume of chamber works, the violin concerto, string quartets, quintets (at least one, anyway,) the octet, oratorios, incidental music for plays (especially a Midsummer Night's Dream), overtures, and symphonies. He wrote symphonies for string orchestras as a kid (around 15), and later, full orchestra symphonies, more of which are being discovered in recent times. Try his overtures (Fingal's Cave) and his Italian Symphony. All his music has a certain effervescent quality which is very addictive!

Peter IlyichTchaikovsky, of course, is famed for his ballet music. He wrote several symphonies, which are not played as frequently as are his ballets (Sleeping Beauty, Swan Lake, The Nutcracker), and several piano concerti. It's easy to find clips of these on YouTube, to check whether any of them appeal to you. This clip from the sixth symphony presents a very unusual waltz-like movement in quintuple time. It's hard to believe that it could sound so utterly natural.

Johannes Brahms wrote a large volume of piano music, several symphonies, several concerti, an "inter-denominational" requiem mass (for his mother), chamber music, choral music for a girl's choir, and several other miscellaneous pieces, including works for the organ, which are considered excellent. I'd like to suggest a short work called Variations on the St Anthony Chorale, which is a suite of variations on a hymn-tune once attributed to Haydn. (This is thought to be Brahm's very first orchestral work.) Here is the famous Hungarian Dance no 5. The majority of Brahms's works, though, are quite different from this, except for a few pieces such as the finale of the violin concerto which has a Hungarian flavor.

Antonin Dvorak wrote, in addition to his concertos, several symphonies (including From the New World, inspired by the United States of America). He, too, wrote several Slavonic Dances, which are really fabulous, and some really unique chamber works, including the Bagatelles. This one features a harmonium!

[Added later:
If you liked Robert Schumann, you should definitely listen to his piano miniatures, such as Kinderschenen. He also wrote a large number of gorgeous, atmospheric Lieder (art songs). Here is Die Lotosblume sung by Dorotea Roschmann. (I wish she had given us a less dramatic reading of it; it seems to require a more lyrical interpretation...)
[Added still later:  Here is Karita Mattila of Finland singing Die Lotoblume]

Edvard Grieg, unfortunately, is not a composer I know much about. There is a celebrated orchestral suite titled Peer Gynt, who is a popular Norwegian legendary character. You should have no trouble looking up other piano music by Grieg, as well as a number of songs. I know that Solveigg's Song is a favorite of sopranos.]

Archimedes