Music of the Christmas Season

All the music that floods the air during the season (especially the shopping season, gotta love it) brings out strong reactions from listeners, usually bad reactions. 

Let me start at the beginning. As a child, my earliest Christmas memories were the most common carols and hymns:
Adeste Fideles (“O come all ye faithful“, from a medieval Latin poem, evidently by a British monk during the Roman occupation);
Stille Nacht, hurriedly composed by Francis Grüber one night when the organ was out of commission, and they had to have carols accompanied by a guitar;
“Ding dong merrily on high”, played on our ancient His Master’s Voice gramophone (phonograph). [Note: “Ding dong merrily on high”appears to be sung to a French (dance) tune, Branslé Officiél.] 

There were also local carols, hardly ever heard outside a 100-mile radius of where I grew up. As I grew older, I found myself participating in actually singing these carols, and since my mother was the local musical impressario, I began to learn unusual carols, taken from the pre-eminent scholarly carol book of the time: The Oxford Book of Carols, edited by the venerable Martin Shaw. To this day, some of those carols remain relatively unknown, and as such the book remains an excellent source for ‘new’ carols. 

Eventually I got old enough to sing in the school choir, and I came under the influence of the celebrated carol festival of King’s College, Cambridge. This influence, felt throughout the Episcopal Church, had both good and bad consequences: the use of boy trebles and male altos, the use of the organ to accompany everything, the use of Victorian harmonies, and a strict metrization of the carols, and the canonical nine readings from the bible. The choir was moderately small, consisting of sixteen trebles, and about four each of the other parts. The director of the choir for many years had been Sir David Willcocks, and his personal influence over the Christmas Carol Industry has been enormous. 

The next, and final, step for me was to be given responsibility for an entire service of carols. I became interested in where the carols came from. They were medieval Christmas dances, for the most part, since Christmas had replaced the ancient midwinter festivals that kept up the spirits of the dwellers in northern climes during the dark, dismal days of December.  (Say that ten times real fast.) 

Romanticism has many aspects; two of these are exoticism and eroticism. Though in common parlance, the word romance is most often understood in its erotic sense (romance novels, etc), the exotic aspects of romanticism is almost as important (e.g. Orientalism, Science Fiction, Egyptology.) Thus, carols fascinate us especially if they are from ancient times (remoteness in time) and from far away (the East, South America), or unusual in some way (Latin rhythms, calypso) or accompanied by unusual instruments. (For instance, consider “Mary’s boy child,” made popular by Harry Belafonte, and Nina and Frederik.) So when people look for variety in Christmas music, they look at these alternatives, since the spirit of romanticism is complementary to the nostalgia of Christmas. 

This post has been influenced by my friend Ileana, who wrote to me deploring the fact that most stores and radio stations keep playing the same songs, until they sound too hacked to be tolerable. The ‘variety’ offerings in the CD market are very dull, and unavoidably so, since the variety offered is relative to the compiler, and, most of all, constrained by the necessity to be marketable.  This means that the contents of the CD must appeal to the widest possible audience, which in turn means that they must be dull, by definition. Dullness in taste is apparently a survival trait in the human species. (The individual who survives to propagate his/her genes is the one who has not been bored to death before he or she can do the propagation.) Wouldn’t it be lovely to have a Christmas collection that one could play through the season, through the quiet hours before midnight of Christmas Eve, to the first few carols of Christmas Day itself, and more lighthearted songs for later in the season? I would like to see a collection along these lines: 

(1) Traditional hymns tastefully performed: O little town of Bethlehem, O come all ye faithful, Brightest and best of the suns of the morning, etc. And let’s not forget Quem Pastores Laudavere. (A slightly more renaissance-sounding version is here.)
(2) Traditional French carols, e.g. Il est né le divin Enfant. Charpentier has immortalized several of these in his Midnight Mass for Christmas.
(3) Traditional German carols: Es ist ein Ros entsprungen (sung here by the choir of the same St Thomas’s Church School at which Bach worked when he was alive); Wie schön leuchtet der Morgenstern (the clip is of an elaborate setting by Bach, his opus No. 1, movement 1.); Vom Himmel hoch; as well as chorales made famous by Bach: In dulci jubilo, Break forth o beautious heavenly light; modern German and German-Swiss and German-Austrian carols, including Silent Night (Stille Nacht), and German versions of lovely Latin carols: O du fröhliche, o du selige; etc.
(4) Medieval English carols: Lully lulla, thou little tiny child; He came all so still; Lullay myn lyking; ...
(5) Medieval Basque, Catalonian, Spanish and Portuguese carols.
(6) Carols from the Middle East, and Eastern Europe: the Cuckoo Carol, the Zither Carol.
(7) Modern carols from other lands, tastefully selected. An example is Arirang, a carol created by Malcolm Sargent from a Korean Lullaby, and Ariva-rararo, a carol in a collection by D.T. Niles, from a Tamil lullaby. Cantique de Noel, by Adolphe Adam (sung by Joan Baez), (sung by Roberto Alagna).
(8) Modern English carols, e.g. The Little Road to Bethlehem (Michael Head), one of my favorites. Another lovely arrangement is by R. R. Terry, of Myn Lyking, with a string accompaniment by John Rutter.
(9) Last, but not least, Celtic carols, and carols of Ireland. Searching for an Irish carol, I stumbled upon the Cherry Tree Carol, which I had always thought of as English, but what do I know? 

If I do not see you before Christmas: a merry Christmas, a happy holiday season to all my readers, and any others who happen to stumble upon these pages! 

Arch, blissfully listening to Schlafe, mein Liebster

History, Literature and Music

All through school, I just could never get my head around History. It started out inauspiciously enough, with plain ol' history, which was the usual stuff about kings and tribes and war, and on a good day, pestilence. Then what should to my agonized eye appear but: The History of the Kings of Israel. (Israelly important to know this stuff?) This was when I first realized that there was History, and then History of this, and History of that. History of Fashion, History of the Theater, History of the Bible, History of Medicine, History of Methodism. It dawned on me that plain ol' History was merely the history of politics. 

When did I first become interested in History? It was not all at once; it happened obliquely. I was given a book of Letters of Composers, a fabulous book that everyone should not only read, but own. It started with the letters of the composers of earlier times (all in translations) and proceeded to the letters of Tchaikovsky, and he wrote some interesting letters. Schumann, as you could expect, wrote up a storm, but Mozart gave him a good run for his money. Once I had read this book from cover to cover, suddenly I had an actual mental image of what these guys were like. It was startling for a kid of sixteen to realize that they were quite distinct personalities. (OK, so I was a pretty retarded sixteen year old.) Then I happened to get my hands on an enormous book called the Oxford Companion to Music, which had actual engravings of the masters, enormous full-page portraits of Bach, Handel, Mozart, Brahms, Wagner, and so forth. And Mendelssohn. I was beginning to take Mendelssohn seriously. I had heard the famous violin concerto, of course, but I had assumed that old Felix was a two hit wonder, at most. But as I grew more interested in his stuff, I began to take note of the pictures available of him. About this time, too, my high school principal had given me a book to read called Johnny Tremain, which was all about Paul Revere and the revolution. I began to get interested in people such as Benjamin Franklin, George Washington, and William Jefferson. Abraham Lincoln followed soon after. Years later, it was as I was watching Little House on the Prairie on television that it struck me: half the guys were dressed in knee-length breeches, like Franklin and Washington and Bach and Handel, while the more affluent storekeepers dressed like Lincoln and Mendelssohn. By this time I had seen a portrait of Oscar Wilde, and he looked as like Felix Mendelssohn as made no difference to me. [The knee-breeches were going out of style, and the full-length pants were new, and initially more expensive. Eventually, people wore pants most of the time, while they used their old breeches for farmwork, etc. You see this too in the Little House on the Prairie tv series, where breeches were worn for work, and pants for Church, for instance.] So it is that my insight into history, which was based on the idea of contemporaries, was sparked by my interest in fashions. 

Soon afterwards, I became interested in the middle ages, the early renaissance, and ancient Greece and Rome, all spearheaded by my interest in clothing. (I'm by no means a natty dresser; for me fashion is the study of the clothes of other people than myself.) What people wore explained a great deal about what they did on a typical day: fighting, or keeping the weeds down, or bringing in a catch of fish. 

Yet another dimension was opened up when I began to be interested in the History of Mathematics. It so happens that the History of mathematics is less important to mathematics as such than the histories of other subjects might be to them. Mathematics at work is in an eternal present, which has subsumed all foregoing understanding of the subject. Still, it is intriguing to learn what the fathers of the subject dreamed up to accomplish something that is done today in quite a different way. As one learns about mathematicians of the past, their costumes and portraits, too, establishes each of them as a contemporary of a musician or a politician, or an author or philosopher. These isochronous layers, to coin a phrase, serve to make sense of the motivations and accomplishments of all these important figures of the past, in whatever field they worked. It is similarly interesting to analyze the stories of the Bible in terms of the level of development of the culture in which they were set. Abraham, for instance, was a stone-age nomad, and his sacrificial knife is often depicted as a flint implement. By the time Solomon came around, they were bronze-age, and presumably there are biblical sources that will establish the precise chronological and cultural facts, in terms of religious specifications of how things were to be done. (I'm keeping a straight face here with some difficulty, but that's why it's called a religion, because it doesn't make much sense.) The Romans, of course, were ultimately deposed by iron-age barbarian warriors, if my memory serves me right, and at least the current thinking in the world of speculative fiction is that the greatest legendary weapons of ancient heroes were probably forged from meteoritic iron. 

As always, the anthropological implications of the various lifestyles, from hunter-gatherer, to nomad, to farmer, inform the stories one reads. Thus History reveals itself as the ultimate application of the principle that connections are the path to assimilating information into knowledge. To the consumer, History is simply a matter of structuring facts and events into some semblance of cause and effect. (To the purveyor of history, the Historian, it is much more: it is a matter of persuading one's readers of one's prejudices, and is a subtle skill. But since they make money out of it, they can do their own damn writing and not look to me to do it for them.) 

To end, The Mists of Avalon, by Marion Zimmer Bradley presents the legendary King Arthur of Britain as one of the earliest kings of the post-Roman era. Her point of view makes amazing sense of the clash of the pagan and the Christian influences in the story. To a Briton, the Arthurian legend is probably connected with various landmarks, the legend of Joseph of Arimathea, Salisbury Cathedral, and the Holy Grail legend, which would serve to place it in a vaguely historical context. For me, though, it was Mists that provided that invaluable service. (Many critical reviews of Bradley's novel focus greedily on the fact that it was supposedly a feminist perspective on the Arthurian legend. In fact, Bradley does far more than simply provide a feminist perspective: she presents the action from the point of view of Christian expansionism, a potent force which complicated the gender issues in many ways. But from our vantage point in the 21st century, it is no longer necessary to use feminism as a crutch to bolster an analysis of a literary work. Feminism should illuminate everyone's thinking, rather than that of a few isolated feminists.) 

Hardest of all, in studying history, is to understand the mind-processes of the protagonists, carefully allowing for the fact that the intellectual environment of another age is simply different from ours. Genocide, for instance, is abhorrent to us. But that attitude is the result of history. As we go backwards, we must judge genocide by Hitler differently than genocide by Genghis Khan, or the Crusades, or the Aryan Invasion of Asia, or Cortez in Mexico. Though abhorrence is inevitable, the same abhorrence for the same crime at different historical times simply becomes an obstacle to understanding history. 

Archimedes, who could be wrong

Dreaming: Wagner's Songs

Unlike most other composers who wrote for the voice, there are few instances of Wagner's writing available for the concert stage; if you want to hear just a single item for a tenor, for instance, you have to hear what is delightfully termed a bleeding chunk from one of the operas. (One of the opera, we should say; opera is the plural, after all.)

There are a few notable exceptions, such as The Prize Song from Die Meistersinger, sung here by the celebrated Canadian tenor, Ben Heppner. (This is the song of a youth, smitten with love. In his defense, we must note that Ben Heppner has been singing this role since he was a young man, with great success.) Here is another rendition by Peter Anders, in which the orchestra has been captured better than in Heppner's live version.

Because Wagner wrote his operas in a seamless style, not wanting it broken down into "musical numbers", as are Broadway musicals, for instance, and certainly Italian Opera, it is hard to slice out a "song" from the continuously evolving melodic texture of the opera. He found the choppy style of recitative, aria, bridge too artificial. In fact, Wagner wanted to create a single art-form that encompassed art, music, poetry and drama seamlessly all in one. In a Wagner opera, the protagonists declaim their heroic utterances as if they were instruments in the orchestra, in contrast to earlier styles of opera, where the orchestra backs down to the role of an accompaniment when an aria takes place. (Note that Wagner's artistic ideas have the potential to be realized in Cinema. My Fair Lady, for instance, despite its sectional structure, comes close to this almost seamless unity of visuals, text and music.) As a result, the vocal music of possibly one of the greatest composers for voice is unavailable for enjoyment in the concert hall. Almost.

It so happened that Wagner had an affair with one Mathilde Wesendonck, the wife of a banker. She was a poet, and she gave him a number of her poems, to set to music. Wagner did this, and there is a set of five songs --Lieder, as they are called in German-- and they are beautiful. [A portrait of Madame Wesendonck is at above right.] The songs were originally for voice and piano, but were later orchestrated in Wagnerian style by Felix Mottl, and were recorded by such famous Wagnerian mezzo sopranos as Kirsten Flagstadt, and clips are available on YouTube. Much later, a light scoring of the Lieder was undertaken by Hans Werner Henze, for chamber orchestra and voice. A beautiful example is by Marjana Lipovsek. (I'll try to upload a clip one of these days.) A videoclip of Der Engel (The Angel) sung by Silvana Dussmann is available on YouTube.

[Added later: Der Engel sung by Waltraud Meier.]

Arch

[P.S. : None of the links work, because of an "upgrade" to the blogging software!  Gotta love it.]

BBC Composers of the Year for 2009

Apparently the BBC has decided to select four composers for the year 2009: Purcell, Mendelssohn, Haydn, and Handel.  

Purcell -- Henry Purcell (1659-1695) was the earliest of these composers, and the one who can be considered most British of them all. (There appears to be nothing anniversarial about his selection for the year 2009.) He is clearly one of the most brilliant composers of the time, and the British are fond of declaring that it was just bad timing that prevented Purcell from having as large an impact on western music as Bach himself. This is probably true; just as Bach did, Purcell delighted in writing beautiful dances for orchestra, (some of them featured in Jane Austen films, e.g. Pride and Prejudice), and in the pleasure of complex harmonies. Ayres for the Theatre is a collection of dance music and concert dances intended to be played between the acts of plays in London. [Here is an excerpt from the music for Abdelazer, immortalized as the theme in Benjamin Britten's Young Person's Guide to the Orchestra.] Just like Mozart some years later, Purcell loved the theatre, and spent so much time there that he was locked out of his house one night by his wife, and subsequently died of pneumonia (or something similar they had at that time. Pneumonia had not been yet invented. Ancient diseases, as you know, dear reader, were not as good as modern ones.)

Note: a popular march called the Prince of Denmark's March has long been attributed to Henry Purcell, though it has since been established that it was composed by Jeremiah Clarke. This error puts hardly a dent in Purcell's amazing output. One of Purcell's greatest works is a masque called Dido and Aeneas, which contains some lovely music. He also wrote incidental music to Shakespeare's Tempest, as did Thomas Arne and others.  

Handel -- Georg Friderick Händel was a contemporary of J.S. Bach, and lived in Halle, in Saxony, not far from where Bach was born and lived most of his life. In contrast to Bach, Handel traveled south, through Italy, finally arriving in Britain. While he traveled, he absorbed all the music he heard, and in fact his musical writing was a wonderful synthesis of German and Italian styles, with the Italian sensibility predominating. (Bach admired this style, and adopted it as far as he could in his arias, with moderate success. Apparently his German tendency to work a theme extensively got in the way of the leaner profiles required by the Italian style. Don't quote me on this; I'm not an expert.) We know Handel best for his oratorio, Messiah. He wrote brilliant operas, though, whose quality exceed that of Messiah by many estimates, as well as other oratorii, such as Israel in Egypt, which I have heard nominated to be his greatest work. The Watermusic Suites and the Music for the Royal Fireworks are delightful, grand works for royal celebrations, the kind of thing the British aristocracy love so much. Handel's music deserves to be heard more than it commonly is.  

Haydn -- Franz Joseph Haydn was an Austrian composer of great significance. The year 2009 marks the 200th anniversary of Haydn's death. The vast majority of his music was composed at the court of the Esterhazys, a wealthy aristocratic family. Haydn, in retrospect, was very much influenced by a couple of visits to Britain, at which time he heard the wonderful works of Handel, as well as the British national anthem, which he admired greatly. Subsequently, with the encouragement of one Baron Gottfried van Sweiten (whom you can see in the movie Amadeus, having been a popular visitor at the Viennese court), Haydn composed his last masterwork, The Creation (Die Schöpfung), which is claimed to be one of the last of his works he heard performed. (When I see that German title, I keep thinking: when the going gets tough, the tough go Schöpfung.) [Here is a clip of Nun beut dir Flur das frische Grün, a lyrical aria that expresses the delight in the verdant woods.  Note 2018/4/9: Blogger has eaten all the video links that had been inserted in these posts, and any you are seeing in these early posts were repaired by me just today, and only a bare minimum have had the treatment.] Haydn also wrote the national anthem of Austria, whose tune is still used in the present German national anthem, and is a model of restrained patriotic musical rhetoric. (The clip features modified harmony.) One must not forget to mention a couple beloved concertos for the 'cello, as well as a trumpet concerto, also played as a concerto for recorder, oboe or flute. And finally, it was Haydn who perfected the classical string quartet, much admired by Mozart. Mozart was born after Haydn was born, and died before the latter, and the two men were musically much influenced by each other to the good. And Haydn was one of Beethoven's teachers. In Haydn's vast output are also numbered a hundred symphonies, a genre that he helped develop, along with the sons of J.S. Bach, in particular Johann Christian Bach. In recognition of his contributions to music, Haydn was granted an honorary doctorate by Oxford University on one of his journeys to Britain.  

Mendelssohn -- Felix Mendelssohn-Bartholdy was born to a Jewish family that converted to Lutheranism in the 18th century. 2009 marks the 200th anniversary of his birth. A renaissance man, Felix, together with his sister Fanny, were adept in numerous areas, including painting, literature and music, and had wide interests in all areas of Art. Felix began composing at an early age, and one of his most celebrated compositions originated at around the age of 15, or possibly earlier. The Octet for Strings is a lively work for eight stringed instruments -- four violins, two violas, and two cellos. (Some modern performances add a double bass, as far as I know not in the original scoring.) Its Scherzo, the third movement, is justly famous. Two of Mendelssohn's string quartets are gorgeous, and delightful to play. As did other German composers before him, Mendelssohn traveled through Italy, and eventually to Britain, and Scotland. He was much impressed by the natural beauty of Scotland, which he celebrated with the Hebrides, or Fingal's Cave overture. He wrote several overtures, all still in the orchestral repertoire, and numerous chamber works, and vocal pieces, such as On the Wings of Song. [Clip: Victoria de Los Angeles.] His overture to A Midsummer Night's Dream is one of his most famous compositions, in addition to the magical Violin Concerto in E minor, and his last few symphonies. Long before these last symphonies for full orchestra, he had written symphonies for strings only, to be performed at private musical evenings in the Mendelssohn home with their family orchestra. The British have a particular fondness for the composer, even if his music is not considered as British as that of Handel and Purcell. Still, his oratorio Elijah was a regular offering in Britain through the last century.

A lovely aria from J. S. Bach

I have heard this aria (solo vocal movement) many times, but it was only recently that I began to really notice it. This is a recording of Andreas Scholl, on YouTube. I looked up the piece on Amazon, wanting to buy an Mp3 of it, and one of the more beautiful renderings of it is by a French mezzo soprano called Guillemette Laurens.


Another offering of this aria is by one of my most favorite sopranos, Magdalena Kozena. Unfortunately, it is taken too fast, making the words come out as rather a tirade, whereas it is probably intended to be a reflection. Another treasure hiding in YouTube is a recording by Alfred Deller, one of the earliest of the celebrated counter-tenors, and you can hear why he was so celebrated. Janet Baker, one of Britain's most precious voices, a woman with an austere and disciplined lyric talent, has recorded this cantata (BWV 170) for EMI, and I'm eagerly awaiting its arrival. Unfortunately it is not available on YouTube. I listen to the music, and it is almost as if Bach is trying to explain something so earnestly that he is in pain. The song is about renunciation, that true peace is not obtainable except in god. The music alone seems able to give some inkling of this sentiment. The solo wind instrument is a sort of oboe that was just going out of favor at the time of Bach: the Oboe d' Amore. Now, of course, the instrument has been revived, especially for use in the music of Bach and his contemporaries.

Archimedes

Archimedes meets Myers-Briggs

It might delight my readers to know that your favorite blogger (I know, I know, ok, second favorite blogger) is of type ISTP: The Mechanic, according to Typealyzer, a website that analyzes a blog and tells you what sort of person the blogger is. Yours truly is...

The independent and problem-solving type. They are especially attuned to the demands of the moment are masters of responding to challenges that arise spontaneously. They generally prefer to think things out for themselves and often avoid inter-personal conflicts. The Mechanics enjoy working together with other independent and highly skilled people and often like seek fun and action both in their work and personal life. They enjoy adventure and risk such as in driving race cars or working as policemen and firefighters.

Maybe these guys know more than I know about myself. I am one of the biggest chickens you ever saw; firefighting--I think not. And I am not the artistic type, apparently. But, you know what? I did submit the blog of an artistic friend of mine, and it did come back with "artistic, soft-hearted type," which does describe this particular person frighteningly accurately. So there you have it. Pretty soon the Internet will know exactly how to sweet-talk us into what it wants.

[Added Later:] I decided to independently verify my personality according to the Myers-Briggs Type Indicator. (The subject was initially studied by a mother-and-daughter team in the early 1900s, and developed partially with an eye to placing women in effective wartime jobs.) According to an on-line test by HumanMetrics I am an "ISFP", that is introvert, sensing, feeling, perceiving. This is all very alarming to a trained mathematician such as myself. We do not sense, we do not feel, we do not perceive, we KNOW. Golly gee, this is embarrassing in the extreme. However, I was somewhat mollified by the list of other folks in the same category, namely: Princess Diana, J.K. Rowling, John Lennon, Florence Nightingale. These are four of my most favorite people. [Important note: Florence Nightingale was one of the matriarchs of modern statistics.]

2018-4-10: This just in: I repeated the Typelyzer test, and we have been re-allocated to INTP.  I also submitted Archie's Archives, our companion blog, and that did come in as ISTP.  So I guess I'm more mechanical when doing music on the radio.  No wonder the radio station went on the fritz...

Archimedes, always the cynic. The Mechacynic.

Rachmaninov: Symphony No. 2

Listening to the early morning syndicated program featured by my local Public Radio station, I was startled to hear a very familiar melody. I had heard the announcer's introduction and registered it subconsciously, but had not grasped it completely. The words that popped into my mind each time the melody returned were: "Never gonna fall in love again!" In contrast to the determined sobriety of the words, the tune was quite jaunty to begin with, though it retreated into a rather injured melancholy, only to make a sally again with some slight optimism, and so on. Of course, I had to find out more.

As soon as I could wake myself up enough to get on the Internet and find out more, I discovered that the playlist of this early-morning filler program that was aired before Weekend Edition was not published with the same meticulous care as were those of the NPR programs and those created at the local studios. The only clue I had were the words "Never gonna fall in love again," and I didn't know for certain that it would be "gonna" rather than "going to". As it turned out, I myself knew at least two songs that had many of those words in them. Google gave numerous references, including songs by New Kids On the Block, and so on. What I needed was a source that would play the songs right away. The next stop was YouTube, and within seconds I had a hit with Eric Carmen. This was the song I remembered, a jaunty tune, with rather peevish lyrics that expressed the unworthy feelings many of us experience at the demise of a love affair. The sumptuous sound of the orchestral version I had half-heard while still asleep, with its broad repeating sequences was in weird contrast with the popular song, though the song faithfully retained the harmony of the original. Going back to Google, I was sent to Wikipedia, which had not only an article on Eric Carmen, but on this specific song, with a pointer to Sergei Rachmaninov's Symphony No. 2.

Of course I had to narrow down the actual movement. Going back to YouTube, I found several clips of this symphony, and I finally hit the jackpot with Movement 3. It was precisely this movement that I had heard; a sweetly earnest assertion of youthful desire, still not in the stage of disappointment expressed by Mr Carmen. The youthfulness of the Rachmaninov movement is actually quite comparable to that of the popular song; but where the symphonic movement is romantic and positive, the song is ambiguous and negative. Still, Mr Carmen has to be congratulated on a song that deserved better than it got, in my opinion. And he deserves a lot of appreciation for pointing me in the direction of Sergei Rachmaninov's Second Symphony.
Archimedes

Leonhard Euler: Mathematical Genius

It is strange, but true, that not a lot of people enjoy reading about mathematics, or doing pretty much anything with it. This post is aimed at the exceptions, and especially those who are on the edges, just becoming interested in the subject.

In the early 1700s, mathematics had just taken off like a jet plane, shortly to be more like a rocket. Newton and Leibniz had just invented Calculus, and Johann Bernoulli, a brilliant Swiss mathematician, had turned his mind to inspect everything that the theory could do, and had asked the big theoretical questions that would drive the engine of mathematical discovery in the next hundred years or so. It was an exciting time in which to be a mathematician.

William Dunham, of Muhlenberg University has made a particular study of Leonhard Euler, one of the greatest mathematicians of all time, born in Basel, and a protegee of the famous Bernoulli. In his book "The Master of Us All", Dunham describes some of the most accessible and interesting of the discoveries of Euler, making a point to underscore the intuitive methods Euler used, which though not acceptable as formal proof today, reveal the amazing mind of Euler.

To the layman, the idea of an infinite series might be a little strange. An infinite series is simply an addition of an infinite number of terms. Generally, a sum of infinitely many terms will produce an infinite number. However, if the numbers are very small, the sum could be finite. Such finite-valued "infinite sums" can be highly useful; many useful numbers can be approximated as "truncated"infinite series (infinite sums of which only a finite number of terms have been added; what is omitted is carefully calculated to be less than an acceptable error tolerance).

Consider the infinite sum 1/2 + 1/4 + 1/8 + 1/16 + .... With a little thought, it can be seen that this sum can never be greater than 1. As Zeno observed, if you walk half a mile, then walk half of the remaining half mile, and then half of that, and keep doing this, you'll never walk more than a mile. In the chart below, the blue bars represent the terms that we want to add up, and the pink bars represent the running subtotals. As you can see, the subtotals approach 1, and rise no higher.


In contrast, the series 1 + 1/2 + 1/3 + 1/4 + ... can be seen to be infinite. Observe the sums climbing steadily. (They do slow down, but not enough to make the sums approach a finite value.)


It is harder to show, but the series 1 + 1/4 + 1/9 + 1/16 + ... , (the sum of the reciprocals of the square numbers, 1, 4, 9, 16, 25, ...) is a finite sum. (See below. The sum is about 1.64.) The story of how Euler showed that it adds up to the unlikely value of (Pi)^2/6 (Pi squared divided by 6) is entertainingly recounted by Dunham in his wonderful book. This derivation is characteristically Eulerian!

The discovery above is topped by a result that connects an infinite series consisting of the reciprocals of the squares of the positive integers on the left, with an infinite product of factors of the form (1 - 1/P^2) on the right:

Bill Dunham's book, beautifully written, combines stylish exposition with absolutely fascinating content. Euler, a contemporary of J.S.Bach, showed a similar brilliance and creativity to the musical genius, and both had similarly enormous influence over the development of their respective areas. Both had many children: Bach had 20, and Euler had 13, and in each case, only a handful lived to adulthood. Finally, both were afflicted with diseases of the eye. While Bach died at the age of 65, Euler lived to be nearly 80.

Arch

The Youthful Johann Sebastian Bach

Johann Sebastian Bach, one of the fathers of modern harmony, was born in 1685 (a few decades after the Pilgrims landed on Plymouth Rock--though I could be wrong about this) and lived to the age of about 65. Though by modern standards, this was a rather brief existence, because of the limited number of distinct portraits of him that are available (actually just one original portrait, of Bach at the age of about 50) we have an image of the composer as a grumpy-looking, serious, dried-up, aged, authoritarian fellow, a father of 20 children (21, if you count P. D. Q. Bach, which you should not, if you know what's good for you) and the teacher of numerous other young men. From his letters, we also get a plaintive whining tone, which has attached itself to the perception of Bach by those who know of him mostly as an important historical musician, rather than a beloved composer of miraculous, inspired pieces.  

In contrast, for those whose perception of Sebastian Bach (the Bach family was enormous, and to them, he was Sebastian, since there were lots of Johanns in the family) is based on his actual music--rather than his reputation or his letters-- the personality that comes through is entirely different. To us (and I count myself among this blessed number) he was a guy who lived for music, who knew he had far better musical instincts than anyone else whom he was acquainted with, just as confident of his abilities as Mozart was to be a few decades later, but who had little leisure in his busy day in which to dwell on his own brilliance. There were a few things that he could thank God for besides his talent: that he had a wonderful and talented wife, who gave their children a measure of her own gifts in addition to those of their father, and the fact that he earned his living essentially in the service of music. (One shudders to think what his life would have been like if he had to paint houses for a living, for example. P.S: Not that housepainting is an unpleasant occupation. Rather that Bach would have had no access to an organ, for instance.) Among other things, this meant that he usually had at his disposal moderately good musical instruments on which to play. That many of Bach's mature musical works are somber and serious cannot be doubted, even if they were imbued with the amazing melodic, harmonic and rhythmic energy that keeps them an important part of the 21st-century repertoire. It is possible to contrast badly-played Bach with more rhythmically interesting Romantic period composers to the detriment of Bach, but this is not to say that rhythmic sophistication was absent in Bach. Quite the contrary. Some critics give Bach credit only for his harmonic skills; this is also a mistake, since Bach's melodic skills are unsurpassed, even in comparison with such fabulous melodists as Chopin, Mendelssohn and Rachmaninov. The reason I was suddenly inspired to actually write about a composer who is easily my favorite today, of all days, is that I was trying to play one of my favorite organ fugues, opus S. 545 in C major. The catalog of Bach’s works is not chronological, but rather based on categories; the entire set of cantatas were catalogued first, and so on. (The organ fugues seem to have been catalogued by key, so that 543 is A minor, I believe, 544 is b minor, and 545 is C major.) I first heard the C major fugue on NPR early on a Sunday morning, and it seemed to me that I had heard it before. To this day I do not know whether I might have heard it as a child, but it seemed somehow interesting. Having listened to it for years, now, I begin to realize that it is a very youthful composition; it is drenched with youthful optimism, romance, happiness, pride, grandeur, a carefree quality that is utterly and captivatingly naïve and infectious. The writing is redolent with parallel sixths, where the parts go together in the same direction, the highest ‘voice’ going in parallel with the ‘tenor’ or the ‘alto’ voice. (If you need an example, the first few notes of (the verse of) “If I fell” by the Beatles is one.)  

This three-minute fugue is widely available in any collection of Bach organ preludes and fugues; in particular the CBS recording by E. Power Biggs is a good one; it has good pace, and the Harvard baroque organ has a nice clear tone. I have uploaded a MIDI-based sound-file made into a video, which will serve as a minimally acceptable substitute for a good performance. In an actual performance, the organist will vary the tone enough, so that it is a little easier on the ear. In this one, the software gives a uniform organ tone, which you have to endure for the sake of finding out what I’m talking about. Above all (in my opinion--I'm not a scholar, and I don't know, off the top of my head, what year this fugue was composed according to the scholars), this fugue shows Bach as a young man. This work could only have been written by someone in his twenties, in my humble opinion; someone whose native optimism has not yet been dented by the vicissitudes of 18th-century life. Listen and see whether you agree!


Archimedes