Boris Li

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Mathematician, Waterloo PhD candidate.

Publications | Mathematics | Bridge | Music | Blog

Blog

I post my many thoughts on different things here, most of which are not related to mathematics. See below for my unadulterated thoughts.

How to Play Bridge

In this article, I shall teach you how to play the game of bridge. Follow this guide and you will learn the rules of the game!

Elements of the Game

Gather four players, ideally around a square table. Shuffle and deal a deck of cards, such that each player has 13 cards. Let everybody sort their cards by suits.

For now, let the dealer play (lead) the first card. Playing clockwise, each subsequent player must each play a card, subject to the following two rules:

  1. if you can, you must follow suit; that is, play a card in the suit that matches that of the lead; and
  2. if you are unable to follow suit, then you must choose a card to discard.

Among the players that did not discard, whomever plays the highest card (with Ace high, Deuce small) wins the round of cards, which is also called winning the trick. The player that wins the trick shall then lead the next trick, and the same rules about following suit then applies, until all 13 tricks have been played.

At the end of the game, the winning tricks are combined between you and your partner, which will be the player sitting across from you. The pair that wins the majority of the tricks is said to have won the game (for now).

Trump suit

Shuffle and deal again (ideally with the player left of the previous dealer as the new dealer). For now, let the dealer designate a suit as the trump suit. Cards in the trump suit are powerful, so you should choose a suit that you have a lot of cards in. The person who dictated the trump suit is called the declarer, and their partner the dummy. The declarer will now play both hands in the partnership and attempt to make as many tricks with their desired suit as trump.

Let the player left of the dealer lead the first card. After this initial lead, the dummy will lay down all their cards facing the declarer, organized by suit, and with the trump suit on the declarer’s left side. Play now continues very similarly as before, with the declarer dictating to the dummy which card to play when it is their turn:

  1. if you can, follow suit; and
  2. if you cannot, you may now choose to play a card in the trump suit; or
  3. discard a card just like before.

If a trump card has not been played, then the winner of the trick is decided like before; however, if a trump card has been played this trick, then whomever plays the largest trump card wins the trick, and must now lead the next trick.

Scoring

In bridge, the declarer actually needs to declare both the number of tricks that they think they can take at the minimum and the trump suit that they want to play in (with no-trump being an option). The combination of these two things is called a contract.

Obviously, when you declare a contract, you should be claiming that you can take at least a majority of the tricks, so the lowest level of contract, the 1-level contracts, mean that you are willing to take at least 7 tricks. In general, to make an $n$-level contract, you are saying that you are able to take at least $n+6$ tricks, with $n$ ranging between 1 to 7. For example, if you are willing to take 9 tricks without a trump suit, the contract will be a 3-level no-trump contract, commonly denoted 3NT.

The score of a contract depends both on the level and the suit.

  1. For a minor contract (clubs and diamonds), you get 20 points per level;
  2. for a major contract (hearts and spades), you get 30 points per level; and
  3. for a no-trump contract, you get 40 points for the first level, and 30 points per subsequent level.

When the score of the contract is below 100 points, the contract is considered partscore, and entails an extra 50 points if you make the contract. When the score of the contract is at least 100 points, the contract is considered game, and entails an extra 300 points if you make the contract. Moreover, a 6-level contract is called a small slam, and a 7-level contract is called a grand slam, which also have bonuses attached to it.

If you make overtricks, which are the extra tricks taken on top of your contract, then in a minor contract, you get 20 points per overtrick; similarly, in a major or no-trump contract, you get 30 points per overtrick. On the other hand, if you do not make a contract, you lose 50 points per undertrick to your opponents.

As we can see from the above, playing in a major or no-trump contract is generally better than playing in a minor contract. This will factor into our decision for deciding trump suits.

Auction

The final element of bridge, the auction, is what makes bridge different from other trick-taking games.

In bridge, in order to determine the trump suit fairly, both partnerships have the opportunity to influence the contract via the auction. Just like any other real-life auction, players will sequentially bid, and the highest bidder wins the auction. However, instead of bidding with money, players bid contracts.

For two contracts of different levels, the contract with the higher level is considered larger. For two contracts of the same level, the contracts are ordered, from largest to smallest, no-trumps, spades, hearts, diamonds, and then clubs.

The dealer starts the bidding, and proceeds clockwise. At every turn, players can either:

  1. pass;
  2. bid a contract, larger than the previous bid if there is one;
  3. double an opponent’s bid, or redouble an opponent’s double. We shall not speak of doubles and redoubles for now.

When a bid is placed, and then there are three passes in a row, the contract is set. Between the partnership that won the contract, whomever bids the suit of the contract first is considered the declarer. Play then continues as before.

If there are four passes in a row at the beginning of bidding, then the contract is considered passed out, and the hand is not played.

Conclusion

These are the rules of bridge! Hope you have fun with the game, and if you are interested in strategies, see my elementary bidding guide.

A New Look

After years of neglect, I have finally gotten some encouragement (thank you Maddie) to revamp this website completely, and I have ported over some old posts. Moreover, I have included some backdated posts that were originally written for personal consumption.

Hope you enjoy this new look!

More Fugues

I have three more prelude and fugues that I really like, and that I would like to share with all of you.

Shostakovich, Prelude and Fugue in D-flat major from 24 Preludes and Fugues

This is what I like to call the “we wish you a Merry Chri- dammit” piece. Of course, the prelude is interesting in its own right, but the fugue subject itself uses 11 tones out of the 12 available ones, making it sound almost serial (and atonal), but through careful manipulation of the material, it ends very triumphantly in the home key D-flat major.

Godowsky, Prelude and Fugue on B-A-C-H for the left hand

Honestly, just an impressive little fugue, given that it is written for a single hand, which you might not even notice if you were not told.

Franck, Prelude, Chorale, and Fugue

Instead of a normal prelude and fugue, Franck inserts a chorale in between the two, making it the emotional center of the piece. Of particular note is the cyclic structure of the piece, where the composer brings back the chorale at the end of the fugue.

Instead of fully describing this specific piece, I will attempt to describe the style of writing and soundscape that Franck possesses. As a composer, Franck is famous for his choral and keyboard works, written mainly for the piano and the organ; however, his organ works are often described by organists as too pianistic, and his piano works are often described as highly influenced by organistic harmony. Moreover, most of Franck’s music employs what is known as “cyclic form”, which is the use of not just motifs, but of whole ideas and passages across different movements, most notably at the beginning and at the end.

On Fugues

We’re going to start another series of musical notes, this time focusing on counterpoint, and more specifically fugues. We are going to focus on the structure of the fugue itself and counterpoint writing, so I am going to attempt to do a full analysis for each of the pieces.

Bach, Fugue in C minor from WTC I, BWV 847

What better way to start with discussing fugues than the preludes and fugues of the Well-Tempered Clavier. This is a simple fugue in three voices, which is the textbook standard for the simplest fugue possible.

Definitions

  • Exposition: The beginning section of the fugue, where every voice gets a statement of the thematic material in alternating subject/answer format.
  • Episode: The intervening material between full entries of the fugal material, often composed of fragments of thematic material itself.
  • Stretto: The technique where the thematic material (usually the subject) enters in close proximity (in terms of time), achieving the effect that the subject collapses on itself.
  • Mid-entry: A restatement of the thematic material in other keys, often used as a developmental tool to explore different harmonies.
  • Recapitulation: A final restatement of the thematic material, in the original key, and concludes the fugue.

Bach, Contrapunctus I from the Art of Fugue, BWV 1080

This is the standard example of a simple fugue in four voices, so every entry of the subject can now potentially enter four times, instead of three like the last fugue. There isn’t much to analyze because the structure is relatively simple: in the video, the boxes and the running cylinders highlight the subject/answer, while the cylinders without the boxes represent the fragmented versions of the theme.

There are many more four-part fugues that are worth studying over this particular one, since this has little to no harmonic variation; however, I included this one mostly because the Art of Fugue motif is going to be featured multiple times in this article, so we should be familiar with it.

Ravel, Fugue from Le Tombeau de Couperin

A three voice fugue that has distinctly the mark of Ravel on it. I could talk a lot more about the Tombeau as a whole, but allow me to focus on the fugue itself today. Most importantly, the secondary motif also acts as a countersubject. Notably, Ravel plays with these two elements in a variety of ways, for example, inverting them from the descending motion in the beginning, to the ascending motion as stated in the middle of the piece.

Definitions

  • Inversion: The act of mirroring a theme/motif across some pitch of symmetry. Note that there is a more specific notion of inversion in counterpoint, which we shall discuss later.

Beethoven, Finale from 29th Piano Sonata, “Hammerklavier”

This monster of a sonata is one of Beethoven’s greatest works from his late period, alongside his 9th symphony and the late quartets, which mostly embrace the Baroque style of music writing, and paired with Beethoven’s penchant for rhythmically-driven motifs. Of the baroque forms, the most complicated one of course is the fugue, which we have an example of here.

The three-voice simple fugue starts at 2:48, and the subject is comprised of two main elements:

  1. the leaping 10th and the trill, which grabs the attention of the listener by interrupting the almost perpetuum mobile texture throughout the movement, represented here by the connected circles;
  2. the descending sixteenth note figure and its subsequent scales, which contributes to the perpetuum mobile texture, represented here by the tumbling squares.

While there are many full re-entries of the subject material, none of them can truly be considered a full mid-entry, because only one voice ever enters at a time. That is, until 4:37, where all three voices present the subject, but in augmentation, and occasionally in inversion too.

The next mid-entry is very much hidden from the casual listener, at 5:57. The subject is presented in retrograde, which without the trill grabbing your attention, it is not immediately obvious that there is counterpoint happening.

Normally, this is when a fugue ends, with a final entry, but Beethoven never ends a piece the normal way; he drags it out, and teases the audience with fragments of a code multiple times before he actually presents it. Hence, at 8:24, Beethoven presents us with a small interlude in the form of a fugato, allowing the listener to take a small break from the unceasing pace of the fugue. After that, he presents a small fugue with three entries, before giving us the real coda, bringing back the peaceful chords that started the prelude.

Definitions

  • Augmentation: Lengthening the duration of each note, such that the subject is slowed. diminution: the opposite of augmentation, which shortens the duration of each note, such that the subject is quickened.
  • Retrograde: Running the subject in reverse. rarely used technique, as it is very difficult to write a subject that works both ways.
  • Fugato: A small, fugue-like passage within a bigger piece. It is often written not up to the standards of a full fugue, but still includes hefty amounts of counterpoint. Beethoven loves his fugato, usually inserting it between larger fugues.

Liszt, Fantasia and Fugue on B-A-C-H

In hindsight, after a few listens, the academic quality of studying this fugue isn’t much, but as with all Liszt, it sounds very impressive.

The Fantasia is comprised of the B-A-C-H motif introduced over and over again, at multiple registers and speed, often simultaneously. The fugue is a very much chromatic one, with a very unclear tonal center.

Liszt originally wrote this for the organ, which sounds musically much more “full”, but the piano version is a better one to study the fugue. I suggest you look that version up for the better listening experience.

Bach, Fugue in D-sharp minor from WTC I, BWV 853

This is (hopefully) last of the simple fugues that we are going to cover. With the exception of retrograde, Bach uses almost every technique that we have discussed in the past few weeks. The entry at 1:00 is stretto, and there is already mild amounts of augmentation and inversion (at 1:08 and 1:24 respectively) immediately after. The first mid-entry is purely inverted, and includes a stretto including all three voices. The second mid-entry is a stretto between the subject and its own augmentation. Try to catch the rest of the techniques used!

Mozart, Kyrie Eleison from Requiem

Largely considered to be Mozart’s last work, he envisioned this to be a grand requiem mass, with fugal endings for each of the sections, such as the Kyrie fugue and the Hosanna fugue in Sanctus. Sadly, he was unable to complete this work before his untimely death at the age of 35.

The Kyrie appears as a double fugue, with both themes introduced simultaneously. The Kyrie eleison theme, highlighted yellow, uses the cross motif (trace out the first four notes); while the Christe eleison theme, highlighted green, provides the driving motion for this entire movement. The rest of the fugue is easy to follow, with alternating entries of the two themes until the second mid-entry, where the Christe eleison theme starts appearing in stretto.

Beethoven, Grand Fugue

Originally composed as the finale to his 13th quartet, it had been described by Beethoven’s contemporaries as “an indecipherable, uncorrected horror” (Spohr, 1826). As a result, Beethoven composed a more digestible finale to the quartet (not before he called his audience “cattle! asses!”), and decided to publish the fugue as a separate work.

While the piece is called the great or grand fugue, it is actually a double fugue, with a fugato nestled between the two fugues; and as a result, it has three big themes, one for each of the fugues and the fugato.

The piece starts with an overture, which introduces us to all the themes in reverse order: firstly, a simple statement of theme 2, then demonstrating its use in fugue 2, and then demonstrating how it is used in the fugato (including a small snippet of the fugato, highlighted in yellow), before stating it as the countersubject of fugue 1.

We then get to the first fugue, by introducing theme 1, highlighted in red. It is comprised of jumps of large intervals, and is presented in iambic rhythm; theme 2 is presented in syncopation as a countersubject. In each subsequent entry, Beethoven introduces a new rhythmic device for us to play with. The 1st mid-entry arms us with triplets, which when played against the iambs, is mildly 3 against 4. The 2nd mid-entry is filled with dactylic rhythms, and theme 2 is now played off a main beat, leading to what I call “rhythmic armageddon”. The final entry is presented in diminution, with each of the themes stated in different rhythms again, before breaking down into an uncertain G-flat major chord that each voice slowly arrives to.

The fugato theme is then presented once plainly, then contrapunctally in three entries, in increasing harmonic depth. The theme is then played in unison, before fading out into nothing.

We are then surprised by the appearance of an interlude, preceding the actual second fugue. Beethoven then pits theme 2 against itself in 3 different ways, presenting the subject in constant stretto. As with the first fugue, the 1st mid-entry arms us with triplets, and theme 1 comes back into the counterpoint in the 2nd mid-entry. This all accumulates into the final entry, which is played in constant fortissimo, before leading us back into the fugato theme.

Beethoven usually likes to extend his codas, teasing his audience with the ending before it actually really comes along. In true Beethovenian fashion, we get a section of uncertainty, before jumping right back into the interlude; then another section of uncertainty, before jumping into a snippet of fugue 1 and a snippet of the fugato, and the overture then returns with theme 2 played in 4 octaves. After one more section of uncertainty, theme 1 is then placed lightly atop a simple statement of theme 2, before the grand fugue ends in a glorious B-flat major chord.

Hamelin, 12th Etude “Prelude and Fugue” from 12 Etudes in All the Minor Keys

A harmonically complex double fugue, he plays with two main fugal themes (introduced at 2:44 and 3:36). There isn’t much to remark, other than the fact that there is so much technique involved in this modern fugue.

Bach, Contrapunctus VIII from the Art of Fugue, BWV 1080

First of the triple fugues here, the three themes are developed within the same exposition, resulting in a relatively short fugue. The first theme is given by the hollow rectangles, which is particularly chromatic among baroque themes; the second theme is introduced at 0:56 during the first mid-entry, highlighted by the rhombi, and accentuates the chromaticity of the first theme. And of course, no fugue in the Art of Fugue is complete without its famous theme, of which a variant is introduced at 2:03 as our third theme, highlighted by the cylinders.

Bach, Contrapunctus XI from the Art of Fugue, BWV 1080

I have always meant for this fugue to immediately follow VIII, mostly because all three themes from VIII have been inverted here. Amazingly, this creates a vastly different fugue with a different texture. The three themes are highlighted with the same shapes, so I’m pretty sure you can identify them.

Bach, Fugue in C-sharp minor from WTC I, BWV 849

We are going full steam ahead, so we have another triple fugue, but this time in 5 voices. By modern speculation, the three themes are now known as the crucifix, the crown of thorns, and the pounding nails. The three themes are explored within the same exposition, so they are very tight-packed in a sense.

I decided to choose a recording on viols instead of on the piano mostly because it sounded a lot more haunting, but maybe you can listen to both and give a comparison.

Bach, Contrapunctus XIV “Fuga a 3 Soggetti” from the Art of Fugue, BWV 1080

Literally, this contrapunctus is called “fugue with three subjects”, and as the name suggests, this is a triple fugue with 3 subjects, but unlike the previous triple fugues, the 3 subjects here are introduced in separate expositions: we have a first fugue with 1 subject, a second fugue with 2 subjects, and then a third fugue with 3 subjects.

The first fugue begins with a slow processional theme, which is used in both its upright and inverted form. Bach, as if teaching all the techniques in counterpoint, allows the voices to enter twice in the exposition, first all in upright form sequentially, and then alternating the inverted and upright forms (first mid-entry 0:48). At the second mid-entry (1:46), Bach demonstrates all of the possible permutations of stretti of two voices, for two upright statements, then two inverted statements, then for one upright and one inverted statement each. At the last entry (2:46), he shows us the stretto of three voices, before wrapping up with a small coda to bring us back to D minor.

The second fugue begins (3:30), in traditional baroque style, with a simple statement of the second subject, which is full of motion as compared to the first subject. After two voices have entered, he teases the listener with the first subject, presented in diminution and with embellishment. After a brief meandering episode, the first mid-entry (4:22) graces us with its presence, entering with the second subject on top, and the first subject as a countersubject. The second mid-entry (4:52) follows soon after, this time with the roles of the two subjects reversed, with the first subject on top. Finally, just like the end of the first fugue, the final entry is presented with three simultaneous statements of the subjects, this time having the second subject and two upright statements of the first subject. Notably, the second subject is presented in the bass register, while the two first subjects are present only in the descant, which allows us to clearly delineate the two themes.

At the exposition of the third fugue (5:32), after 13 full contrapunctii and 4 canons, Bach finally signs his name on the art of fugue, presenting the B-A-C-H motif as our third subject. Just as the first fugue, the first mid-entry (6:01) has the B-A-C-H motif entering in alternating upright and inverted forms, but here, you can also hear the first subject hinted at once in the soprano. In the second mid-entry (6:40), we hear the stretto of all 3 subjects, as Bach teaches us how the 3 subjects is supposed to fit together.

At this point, the manuscript stops, and Bach leaves the art of fugue unfinished. His son, CPE bach, propagates the rumour that Bach died at this precise moment. Every performer at this point has to follow in the master’s footsteps, and compose the ending to a triple fugue. Subsequent entries here present the three subjects inverted, and in stretto, and includes some more modern (chromatic) harmonies, before ending the piece.

Curiously, this is the only contrapunctus out of the set of 14 that does not employ the Art of Fugue theme from Contrapunctus I.

Busoni, Fantasia Contrappuntistica

Our hard work understanding fugues culminates here, with the gargantuan extension to the Fuga a 3 Soggetti that Busoni composed.

Although large in scale, the Fantasia is actually just a prelude and fugue. The prelude itself is an almost faithful transcription of his very own third elegy, written in the form of a theme and variations, which he dubs as a “praeludium chorale”. There are two themes that are featured here, a strong theme A, and a hauntingly beautiful theme B that takes inspiration from a Bach chorale.

After a brief transition, Busoni goes back and trascribes the Fuga a 3 Soggetti. Following its structure, there are 3 fugues: a simple fugue, a double fugue, and then a triple fugue. He applies a slightly more modern flair to these fugues, often making minor changes to double certain voices in octaves for emphasis. Busoni also takes the liberty to extend the third fugue by a massive amount, introducing multiple ways to combine the three subjects.

Then we come to a small break, where we are introduced to a small intermezzo, three fugato variations on the fugal subjects, and then a small cadenza. This leads us all the way back to Contrapunctus I, where he finally states the oft-hinted Art of Fugue subject, and goes into the fourth fugue, the quadruple fugue that forms the centerpiece of the subject.

After the four subjects break down into unrecognizable pieces, Busoni cycles back to the chorale, before thundering out a mutated form of the third subject, the B-A-C-H, and brings the entire piece into conclusion.

Harmony and Dissonance

Harmony is a fundamental building block of music. It creates and evokes certain emotions that are ingrained in us. However, this also means that unusal uses of harmony is often puzzling or uncanny, unless the listener is willing to familiarize themselves with it. We shall, through this tour of a few pieces, hope to better understand late Romantic harmony.

Hindemith - 2nd String Quartet

If i needed to demonstrate neo-classicism in a single piece, this would be my choice. With almost exclusively Mozartian and Beethovenian gestures emanating throughout, it feels almost like any other string quartet written in the past 200 years; yet, Hindemith’s use of consonance and dissonance to drive tension and development frees the functions of chords from their traditional roles of dictating cadences, and instead focuses more on the particular intervals within those chords themselves.

Beethoven - Grand Fugue

Compared to the Hindemith above, there is less harmonic craziness here, but plenty of rhythmic creativity present. In the words of the famous composer Stravinsky, the Grand Fugue is an “eternally contemporary” piece. Multiple motifs and themes feature throughout the piece: most prominently is the chromatic figure that is first introduced as a countersubject in the first section, expanded upon in the middle intermezzo section, and completely upended in the second section via multiple rhythmic augmentations and diminuitions. I could go on for another hour analyzing the structure of this entire piece, but it’s best that you experience this firsthand before anything else.

On a side note, Beethoven wrote plenty of fugues in his later years, and there are in particular a few that are less accessible to the listener, such as this one, the fugue in the last movement of the Hammerklavier, and perhaps the fugue from the first movement of the 14th quartet, which is often said to be Beethoven’s favourite. It is absolutely daunting and emotionally draining to listen to and analyze some of these late Beethoven works, so don’t be discouraged the first time around!

Ravel - Noble and Sentimental Waltzes

In this suite of eight waltzes, constructed almost purely in triadic harmony, Ravel adds his own blend of impressionism and tonal ambiguity. From his pre-war works such as this one, in my opinion, there is a more academic quality to it, as in these modern, post-romantic elements are sprinkled in carefully; on the other end of the spectrum are his post-war works, which are much rarer in quantity, but usually involve a much larger emotional outburst (see the his two piano concerti).

Scriabin - 4th Piano Sonata

As of the time of writing, this is my favourite piece of music that I have found this year. Superficially, it has a dreamy quality that is in stark contrast to the traditional romantic sound; I really could only describe it as feeling like a daydream on a summer’s night, which quite accurately describes the mood right now. Substantially, on the other hand, the music is barely tonal, as in you can ascribe it the key of F-sharp major, but the development ventures far away from that tonal center. Scriabin’s use of (to his contemporaries, relatively) complex polyrhythms is in full display throughout, often pitting 3 against 4 against 5.

Debussy - Violin Sonata

His large major composition and last performance, Debussy’s violin sonata is a fine example of his unique composition style (as compared to other romantic period composers). In my opinion, this sonata is best contrasted with Ravel’s violin sonata, as it starkly contrasts debussy’s heavy use of atmospheric effects, as opposed to Ravel’s more traditional use of melody as a driving element.

Scriabin - 7th Piano Sonata, “White Mass”

When listening to late Scriabin, one should throw away all they know about classical music, and think purely in terms of motifs. In this piece, function of chords are, once again, like the Hindemith quartet no longer part of constructing cadences, or at least, the cadences are definitely not classical. Classical music theory usually dictates that the dominant seventh drives resolutions, while the diminished seventh drives tension; on the other hand, one example in this piece, in as early as the 4th measure, the composer disguises, through chord inversions, a clustered major and minor chord into a triadic chord, supported by an interval of major seventh, which is used as one of the major driving points of tension in the piece.

Like beethoven, Scriabin likes to build his music from complex rhythmic structures; but the composer here brings this even further, sprinkling his infamous quintuplet gesture everywhere throughout the piece. moreover, the diminution and augmentation of the subject material layered together is the exact same counterpoint structure that Beethoven uses in his Grand Fugue (2nd fugue section, about 9 minutes into the earlier recording).

Although atonal, this Scriabin sonata does not lack in melodic elements. This I will leave to the listener to discover, but one can indeed say that this piece is melodically driven, with the powerful motifs from the beginning of the piece coming back multiple times in the recapitulation, in different forms.

When comparing Scriabin’s “White Mass” to his 4th sonata, one can say this sonata is an extremized version of that dreamy quality that draws me to his 4th sonata. However, the moment that I would like to emphasize is the chord close to the end of the “White Mass”, the large, 5-octave, arpeggiated chord representing a revelation; it is not so dissimilar to the end of the 4th sonata, where the chords, in triple forte, come out in full force, driving the music to its finale.

Finally, one can interpret Scriabin’s deviation from tonality as one form of impressionism. We can compare this piece with the Debussy violin sonata, for its use of descriptive, almost sentence-like musical passages. One such example is how both composers like to trail off after certain gestures, as if they are losing, momentarily, their train of thought (this appears multiple times in the 1st movement of the debussy, as early as the 14th measure).