Table of Contents
Part I: Sound Reasoning
1. Sound Reasoning: A New Way of Listening
2. How Music Makes Sense
3. Listening Gallery: How Music Makes Sense
4. Musical Emphasis
5. Listening Gallery: Musical Emphasis
6. Musical Form
7. Listening Gallery: Musical Form
8. Expository and Developmental
9. Listening Gallery: Expository and Developmental
10. Overall Destiny
11. Listening Gallery: Overall Destiny
12. Time’s Effect on the Material
13. Listening Gallery: Time’s Effect
14. Summary: A Quick Guide for Listening
15. Making Music Modern
16. Listening Gallery: Making Music Modern
17. Conclusion: What is Music Trying to Express?
15. Making Music Modern
A tension exists between the enduring aspects of the human condition, rooted in our biological make-up, and those aspects of our experience that are impermanent, transitory and rapidly progressing. Physically, we have evolved very gradually. Our maturation process, our inner urges, our life cycle have endured for thousands of years, deeply connecting us to our ancestors from the distant past. Over time, we have “stretched” ourselves biologically—we are taller and live longer– but our essential nature and basic physiognomy have remained the same. On the other hand, in almost every other respect—socially, scientifically, technologically, etc. –the transformations have been far-reaching and dramatic. A caveman from ten thousand years might recognize our bodies; but he would not recognize our world.
One of the purposes of art is to explore this tension between the enduring and the progressing.
Thus, each era of art makes a unique and irreplaceable contribution, illuminating for us a particular moment in humanity’s on-going development.
Whether in ballet, theater, fiction, poetry, architecture or film, the educated public acknowledges and celebrates the continuity of artistic creation and its perpetual innovations and discoveries. Mavericks such as William Faulkner, T.S. Eliot and e.e. cummings in literature, Martha Graham and Merce Cunningham in dance, Pablo Picasso and Mark Rothko in art, Frank Lloyd Wright and Frank Gehry in architecture, Harold Pinter and Edward Albee in theater—to name just a few—all have found an enduring and devoted public: We wait for Godot, we are dazzled by Gehry’s forms, are awed by Picasso’s fractured portraits.
In contrast, progressive modern music of the past one hundred years has struggled to find an audience. Many major musicians consider it possible to live a full professional life without performing the music of their own time. Orchestral programming routinely favors the traditional repertoire. A large community of prominent performers, theorists and historians avoid the creative work of the last century, treating it as an aberration. To many listeners, Western concert music as they know and love it ended, for all practical purposes, at the turn of the 20th-century.
As a result, something deeply meaningful is lost. No one speaks with greater passion and eloquence than Beethoven about the tension between the enduring and transient parts of our selves. But he does so for his own time. Our own era is more heterogeneous than Beethoven’s, more unstable, and more imbued with ambiguities. Beethoven’s world did not have a conception of the unconscious; now psychologists describe most of our mental activities as being beyond our direct awareness. In Beethoven’s world, science depicted the natural world as a giant, predictable machine; in our time, we understand that unpredictability is built into the fabric of the cosmos. In Beethoven’s world, news traveled slowly; in ours, the stock market is updated by the minute on home computers. The New York Times once ran a headline, ”Did Music End With Mozart?” As long as our world is developing, as long as our vision of life is evolving, no composer will ever have the last word.
In this module, we will study the ways in which progressive modern music differs from classical music. We will then use the conceptual and listening tools that we have developed in earlier modules as an entryway into the modern repertoire.
The Shock of the New
A little over three hundred years ago, Sir Isaac Newton created the first mathematically coherent explanation of the universe. To Sir Isaac Newton, nature behaved like a well-regulated, predictable machine. Give Newton comprehensive information about the universe and he could have predicted the future. Famously inspired by a falling apple, Newton’s laws are confirmed by our direct perceptions and agree with our common sense. We still launch satellites into orbit using his method of calculation. But Newton’s view of a predictable universe turned out to be deeply flawed. Perhaps the most the fundamental scientific discovery of the 20th-century was the recognition that ambiguity is irrevocably built into nature.
The Theory of Relativity
Einstein’s Theory of Relativity stipulates that the speed of light is constant for all observers. One startling consequence of this is that simultaneity and cause and effect are not absolute, but relative to one’s perspective. It is possible for one observer to report two events as happening at the same time that another observer sees as happening in sequence. Thus, according to the Theory of Relativity, there is no definitive “reality,” no commanding perspective that overrides all others. Instead, nature allows for multiple, and even contradictory, points-of-view. Decades of experiments have confirmed Einstein’s theory.
Ambiguity also intruded into quantum mechanics, the study of sub-atomic particles. To give a speeding ticket, a police officer must know both a car’s location—in order to identify it—and its speed—in order to determine whether it is breaking the law. The Heisenberg Uncertainty Principle stipulates that an observer cannot measure both the position and speed of a sub-atomic particle with exact certainty. Thus, it would be impossible to give a speeding ticket in the quantum world. Why? If the police officer were to accurately measure the location of a sub-atomic particle, he would have to sacrifice knowledge of its speed. On the other hand, if he were to measure how fast the particle were traveling, he could not know its position. Nature would continually confound him; his information is doomed to be incomplete.
It is not just the outer world that is saturated with ambiguity. Sigmund Freud was the first scientist to deeply explore the concept of the unconscious—mental processes that lie beyond our direct awareness. These range from metabolic processes like breathing to the complex motivations that underlie every day decisions. A century of research has established that most of human thinking is unconscious. Various experimental methods have been devised to explore the unconscious, from dream analysis to word association, Rorshach tests, brain scans, and more. Yet deciphering our unconscious thoughts remains elusive. Thus, not only must we must accept the ambiguities of the natural world, we must acknowledge it within ourselves.
Nature’s Ambiguities and Daily Life
Nature’s ambiguities generally lie outside our direct perception . Relativistic effects only become pronounced at near the speed of light. The contradictory, unresolved behaviors of sub-atomic particles dissipates as objects get larger. Unconscious thoughts, by definition, lie outside our immediate awareness. Thus, it is possible to be largely oblivious to the ambiguities inherent in nature. However, one hundred years of scientific research has established that ambiguity imbues the world around and within us.
Ambiguity in Art
As ambiguity became heightened in science, so too did ambiguity become heightened in art.
All great works of art leave questions open: Is Hamlet mad or just pretending to be? Is the Mona Lisa smiling? 20th century artists didn’t need to make their art ambiguous—it already was. Instead, they strove to amplify art’s ambiguity. Painters created abstract images that did not refer explicitly to observable reality. Writers created non-linear narratives that shifted around in time or were told from multiple perspectives. How did composers heighten the ambiguity in music?
Heightening Musical Ambiguity
Because it is non-verbal and often non-representational, music is particularly ambiguous.
During a pre-concert radio interview, a radio announcer commented to the conductor that a section of a Bruckner Symphony was one of the composer’s most “optimistic” passages. To which the maestro replied soberly, ”Actually, I find it quite pessimistic.” Abstract music will always resist easy interpretation.
And yet, as the following discussion will make clear, classical composers put a high value on clarity and resolution. Progressive 20th-composers shifted the balance much more strongly towards the uncertain and the unresolved.
Individualized Musical Languages
“U tita enska aka ca vik i totar i tari”
Speaking in a personal language—no matter how thoroughly imagined and consistent—automatically heightens ambiguity. The sentence above—an example of Skerre, a language invented by linguist Doug Ball—would take a long time and a great deal of analysis to decipher. Language functions most conveniently in a community where everyone shares a similar vocabulary and syntax. Because music does not have fixed definitions, linguistic parallels are often misleading. Nevertheless, the shared materials, methods and formal methods of the “common practice era” helped to make the music more accessible. Listening to one common practice era work helped you understand how to listen to others.
The following excerpts by Franz Schubert and Johannes Brahms were written seventy years apart.
If Schubert had been alive to hear Brahms’ work, the music would no doubt have been intelligible to him.
During the 20th-century, the common practice era came to an end. Composers intensified the individuality of their musical voices. The following works for speaker and ensemble were written within several years of each other:
A few decades later, the following string quartets were written very close together.
Finally, the following works for two pianos were written at nearly the same time.
Listening to the Carter does not help teach you how to listen to the Cage. Listening to the Reich does not help you with the Boulez. Each work much be considered on its own terms.
The personality of individual musical languages were established in a myriad of ways. Some composers, such as Harry Partch, invented their own instruments. (Partch gave his instruments such fanciful names such as Cloud-Chamber Bowls, Diamond Marimba and Chromolodeon.)
Some, like Mario Davidovsky, pioneered the use of electronic sounds. In Davidovsky’s Synchronism No.9, live and recorded, electronically transformed violin sounds are intertwined.
Some, such as Charles Ives, blended familiar music in unusual ways. In this excerpt from his String Quartet No. 2, Ives creates a musical “discussion” in which American folk tunes from North and South are quoted in opposition to each other.
Some, such as Lou Harrison, incorporated influences from other cultures. This excerpt from Harrison’s Song of Quetzalcoatl uses many exotic percussion instruments.
Others, such as Elliott Carter and Milton Babbitt, developed sophisticated, very carefully constructed musical methods. In this excerpt from Carter’s Variations for Orchestra, ensembles within the orchestra are characterized uniquely—the winds, for instance, are soft and slow-paced—and then layered on top of each other in a complex counterpoint.
Now, over a hundred years after the end of the “common practice” period, there is an enormous proliferation of musical styles. The break-up of the musical community in favor of much more personal musical languages greatly heightened ambiguity.
Absence of Pulse
A steady pulse or “backbeat,” so crucial to pop music, jazz and much world music, provides continuity and predictability: You tap your feet to the beat.
A steady meter divides musical time into a fixed cycle of beats. Classical ballet and ballroom dancing depend on a steady meter.
Removing the steady pulse or meter disrupts the musical continuity and makes events much harder to predict. There are two main ways to accomplish this: One is to make the pulse or meter erratic.
The second is to remove the sense of pulse and meter altogether, creating what Pierre Boulez has termed “unstriated time.” In the following example from Boulez’s Eclat, the solitary, sporadic events seem to float freely, unanchored by meter or pulse.
Weakening the sense of pulse or meter heightens ambiguity by removing an important frame of reference.
It is frequently remarked that classical music is constantly creating expectations that encourage us to guess what will happen next. In expository sections, when the music is striving for maximum clarity, many of those expectations will be met. For instance, listen to the opening of J.S. Bach’s Prelude in E-flat from the Well-Tempered Clavier, Book I. Can you predict what happens next?
The upper register continues with fast motion The lower register answers the upper with fast motion Both registers move in slow values.
Now, listen to the actual continuation.
The first few exchanges between upper and lower registers created the expectation that the lower register will continue to imitate the upper. Sure enough, the lower register answers in fast motion, confirming our prediction.
A surprise occurs when one outcome is strongly anticipated but another one occurs. Ambiguity arises when multiple outcomes are all equally expected or no good forecast can be made. Listen to the opening of the second movement of Igor Stravinsky’s Three Pieces for String Quartet. Can you predict what happens next?
Which of the various gestures that Stravinsky has introduced follows next? How sure are you? Here is how the music actually continues:
This time, you were likely to have much less confident of your answer. In the Bach example, a pattern was established: the upper register was repeatedly answered by the lower. Stravinsky does not establish a consistent pattern, making any predictions much more uncertain. When we cannot confidently forecast what will happen in the future, ambiguity is heightened.
In football, the quarterback announces the play in the huddle; then the offense steps up to the line of scrimmage and runs the play. In music, expository statements establish the identity of a musical idea; developmental passages put the idea into action. Most classical music operates like a football offense: an idea is first introduced, then put into action.
In a no-huddle offense, the quarterback calls out the plays at the line of scrimmage. Teams use the no-huddle offense to speed up the pace of the game and confuse the defense. This creates a much more ambiguous and hectic situation. It is harder to defend, because there is less time to analyze formations. Analogously, in music, when exposition is abbreviated and development intensified, ambiguity is heightened.
In the most extreme cases, a modern work may consist exclusively of development. This is as if a team were to spend the entire game in a no-huddle offense! In such cases, the identity of the underlying material may be very difficult to perceive.
Lack of Literal Repetition
We establish our identity through our name, our driver’s license, social security number, credit cards, personal belongings, habits, tastes, family and friendships. In music, the most forceful and clear way to establish identity is through literal repetition. Literal repetition is the strongest way to make a musical idea recognizable.
Buddhism challenges the concept of identity, considering it an illusion. We may cling to the emblems of an enduring self; but they are no more substantial than sand castles. The only permanent truth is “impermanence.” This finds a powerful correlation in one of modern music’s most radical innovations: The elimination of literal repetition. Removing literal repetition weakens any sense of a stable “musical identity” and heightens the music’s sense of impermanence and flux.
Lack of Resolution
In classical music, a dissonance is a tendency tone that is considered unstable. A dissonance demands continuation: It must resolve to a stable tone, called a consonance.
Classical music makes an essential promise: All dissonances will resolve. Sometimes, resolutions are delayed; or new dissonances enter just as others are resolved. Eventually, however, the music will reach a state of repose and clarity.
In progressive modern music, dissonance is frequently intensified and sustained way beyond classical expectations.
In addition, there is a new paradigm: Dissonances no longer must resolve. Stability and clarification are no longer guaranteed.
Nowhere is the clarity of classical music more strongly established than at the end of a work. There, the music summons its greatest powers of resolution. Beethoven’s Symphony No. 5 ends with an emphatic affirmation of stability.
The absence of resolution at a work’s close guarantees greater ambiguity. In the following example from Pierre Boulez’s Dérive, a stable sound is sustained by the violin. The other instruments dart towards and away from this sound, never wholeheartedly coinciding with it. The effect is much more precarious than in the Beethoven example.
There is nothing that we can do to make Boulez’s ending sound as secure as Beethoven’s: It is inherently more ambivalent.
In music theory, dissonance is a functional term. To listeners, though, “dissonant” is often a value judgment, typically meaning “harsh” and “unpleasant.” Those attributes, though, are subjective and carry strong negative connotations. I would prefer a different description. Acoustically, a stable sound is more “transparent:” It is easier to identify its inner constituents. A sound with a lot of dissonance is more “opaque:” The greater the amount of dissonance, the harder it is to analyze and interpret the sound.
It is easy to understand, then, why modern composers might heighten dissonance: Not necessarily to make the music more strident but rather to increase the ambiguity by making the sounds harder to aurally decipher.
In a family-style restaurant, everyone sitting at one table is fed the same food. As the platters are brought to the table, the guests choose their own portions; yet they are bound together by sharing the same meal. If someone were to ask about the menu of the day, there would be a clear and united answer.
The word harmony describes the notes that are sounding at the same time. In classical music, no matter how many instruments are playing, they will share the same harmony. As one harmony leads to another, the instruments will move together, partaking of the same notes. In addition to a steady pulse, harmonic coordination is the primary way that classical music coheres. Harmony is the reason that the instruments “sound good together” even when they are playing independent lines.
At a salad bar, each person creates his or her own meal. One person might make one trip to the buffet; another might visit repeatedly, each time choosing different items. The diners no longer cohere in the same way: It would be impossible to know from one person’s plate what someone else was eating.
In music, the absence of harmonic coordination may create great ambiguity and complexity. Harmonic independence makes is much harder to get a “comprehensive” overview of how the instruments fit together. The third movement of Luciano Berio’s Sinfonia dramatizes this effect. In this movement, the Scherzo from Mahler’s Second Symphony is played continuously. On top of it, an elaborate collage of music and text is layered: graffiti from the walls of the Sorbonne, quotes from Samuel Beckett, excerpts from classical and modern music. Strong clashes arise because the collage elements do not agree harmonically with the Mahler.
Harmonic independence does not mean that modern composers do not care how independent lines sound together. They do care, but they are trying to create ambiguity rather than clarity. Giving each instrument its own “plate of food,” which may complement others in intricate ways, leads to radically new resulting sounds.
Weak Rhetorical Reinforcement
When the winner is declared in a typical Presidential election, streamers and balloons fall down from the ceiling, supporters cheer, cameras flash—all reinforcing the decisive outcome.
In classical music, united emphasis or “rhetorical reinforcement” is a primary means of creating structural clarity. In Beethoven’s Symphony No. 5, the third movement continues into the fourth without a break. The boundary between the movements is marked by strong rhetorical reinforcement: The dynamics, texture, meter and speed all change at once to herald the opening of the fourth movement.
The Election Night 2000 offered a different picture: No balloons fell, people milled about in a state of confusion, television announcers nervously shuffled their papers. Indeed, the country managed to peacefully sustain the uncertain outcome for the seven weeks that followed.
In progressive 20th century music, rhetorical reinforcement is often weak or absent. This makes the structural arrival points much more difficult to perceive. In Henri Dutilleux’s Ainsi la nuit…, the individual movements are played without pause. However, the boundaries between movements are difficult to discern because there are conflicting cues.
Perhaps you recognized that the second movement begins with the loud gesture played a little over a minute into the excerpt. However, this gesture does not have a greater perceptual priority than other potential markers, such as the long silences. As a result, you are likely to be far less certain about the formal boundary.
In traditional ballet, music and movement typically reinforce each other: For instance, the music will reflect the change from a solo to an ensemble number. However, when composer John Cage and choreographer Merce Cunningham collaborated, they did not coordinate their work. Music and dance were combined for the first time at the premiere. This made rhetorical reinforcement highly unlikely; if it did occur, it could only be the result of chance. Thus, the method of collaboration guaranteed greater ambiguity.
In his book “Signifying Nothing,” the mathematician Brian Rotman presents an analysis of William Shakespeare’s King Lear. Dividing up his kingdom before his death, Lear asks each of his three daughters to pledge their love for him. His youngest daughter Cordelia’s turn comes:
Lear:…what can you say to draw a third more opulent than your sisters? Speak.
Cordelia: Nothing my lord.
Lear: Nothing will come out of nothing: speak again.
In Rotman’s interpretation, Lear understands Cordelia’s “nothing” in the medieval sense, as a “void,” “death,” the total absence of life and feeling. But Cordelia intends her “nothing” in a more modern sense: She refuses to treat her love as a commodity, to be traded for land. Her “nothing” does not mean that she has no love; only that she will not offer it in exchange for her inheritance. From that misunderstanding, the tragedy of Lear unfolds.
In the same way that medieval thinkers regarded “nothing” as the “absence of creation,” many musical traditions treat silence as the “absence of music.” Silence is almost totally absent from pop music. In classical music, it is used sparingly: It may occur as a “breath” to short phrases or as a formal articulant to large sections. The opening of Mozart’s Symphony No. 40 in g-minor consists of continuous sound until the arrival of the contrasting section, which is marked by silence:
In progressive 20th-music, silence began to be treated as a musical material in its own right. Its musical information is limited: All we can analyze is how long it lasts. But, in seeking to heighten ambiguity, this limitation became a strength. We can read many possible meanings and inferences into silence: It is a hesitation, an interruption, a “trap door” into the unexpected.
To John Cage, silence marked a musical event over which the composer had no control, which could function as a “window” into other sounds. His Imaginary Landscape No.4, is scored for twelve radios. The performers move the frequency and volume dials according to precisely timed instructions. Cage has no control over the resulting sound: It depends entirely on what is being broadcast that day. At one performance, none of the frequencies marked in the score coincided with stations in that location, resulting in a completely silent performance.
The greater the use of silence, the greater the ambiguity.
If silence is the “absence of sound,” then noise is “indiscriminate” or “indistinguishable” sound, in which it is impossible to tell the pitches or what instruments are playing. Classical music is generally purged of noise. Exceptions such as the following are rare:
To progressive 20th-century composers, the inherent ambiguity of noise became very attractive.
Composers incorporated noise in their music in numerous ways. Some brought the outside world into the concert hall. For instance, to create his electronic composition Finnegan’s Wake, the John Cage recorded sounds in the Dublin neighborhood where a scene from Joyce’s novel occurred; he then layered these in a complex collage.
Other composers asked for standard instruments to be played in non-traditional ways. In his string quartet Dark Angels, George Crumb has the amplified quartet run their fingers rapidly up and down their fingerboards, creating a sound meant to evoke the frantic buzzing of insects.
As with silence, the more noise, the greater the ambiguity.
The furniture from IKEA comes in a box, with a manual on how to put it together. There is room for individual touches: But the over-arching goal is to create a piece of furniture that matches the instructions.
Classical music also comes with detailed instructions. A classical score typically specifies the instrumentation, pitches and rhythms, speed, dynamics and articulations. Not everything is marked with equal precision, leaving room for interpretation. However, the purpose of the score is to create a recognizable performance: Much more is shared between interpretations than differs. For instance, compare two performances of Beethoven’s Bagatelle, Opus 126, no.1.
Modern composers sometimes sold their furniture with the barest of instructions. Compare the following two recordings.
Hard as it may be to believe, those are actually two performances of the same work: Earle Brown’s December 1952. How can that possibly be? The instrumentation is different. The musical content—the pattern of sounds and silences–is totally different. Not a single detail is the same. The first performance lasts just 45 seconds. The second is actually only an excerpt of a 6-minute performance.
The score for Brown’s work is shown below:
The composer offers no suggestions as to how to interpret the image: All decisions are left up to the performer. Brown’s goal was to provide the impetus for a musical performance but not to impose an outcome. With such ambiguity in the notation, enormous variation in performance is possible.
Ambiguity in notation represents perhaps the greatest extreme reached in modern music. The more the musical text leaves open, the more it moves away from the constructive clarity of the classical era.
Listening to Ambiguity
Tolerating the Ambiguity
In Samuel Beckett’s “Waiting for Godot,” two vagabonds—Vladimir and Estragon—await the arrival of a mysterious visitor, Godot. Godot’s arrival is anticipated, it is hoped for, it is repeatedly heralded–but it never happens. No matter how many times you see the play, Godot will never appear. Similarly, the ambiguities in a modern musical work are built in and can never be removed. Acknowledging this is the first step to a deeper understanding. Listeners are so often frustrated because they expect the ambiguities eventually to be clarified—if only they knew more or could listen more attentively. Doing so does not remove the ambiguities, it only makes them more acute and palpable.
Thinking Clearly About Ambiguity
Once you learn to tolerate the ambiguity, you can begin to discover its source. Are pulse and meter absent or erratic? Is dissonance heightened? Is the continuity unpredictable? Is there minimal exposition? Perpetual variation? Do noise and silence figure prominently? Any or all of these may contribute to the work’s open-endedness.
Considering the sources of the ambiguity will help you relate different pieces to each other and enable you to become more articulate about what you hear.
Ask Comprehensive Questions
When listening to a modern work, the most effective way to surmount the challenges created by an individualized musical language is to ask comprehensive questions that are not style specific.
Each of the questions below is addressed in its own module:
- What is the form of the work? If it is an “A-type” form? If so, what is being prolonged? Is it an “A/B-type form”? If so, how is contrast created and where does it occur? [Musical Form]
- What is the balance of expository and developmental sections? [Expository and Developmental]
- What is the overall destiny of the work? Do you consider it a strong round-trip, weak roundtrip or one-way progression? [Overall Destiny]
- What is time’s effect upon the material? Does any music ever return in its original form? Or is it always subjected to transformation? [Time’s Effect on the Material]
Be Prepared for more Personal Reactions
Progressive modern works often do not strongly direct the listener’s attention: There may not be a clear hierarchy of theme and accompaniment; structural arrival points may be more subtle or evasive. Be prepared for your reaction to be more personal; and be prepared for your perspective to change with repeated hearings, as you focus on different aspects of the work.
In the same way that a Jackson Pollock drip painting will never resolve itself into a clear image, the ambiguity in a progressive modern composition is irreversible. Whether it is now or in fifty or five hundred years, the only way to appreciate such music is to learn to sustain, tolerate and celebrate the ambiguity. There’s nothing that we can do to make the ending of Boulez’s Dérive sound like the end of Beethoven’s 5th. We cannot remove the noise from Dark Angels or make a single performance of Earle Brown’s December 1952 definitive.
In an art form that is already abstract and non-verbal, heightening the ambiguity only increases the feelings of isolation and uncertainty. In addition, music is conventionally taught using concepts and terms specific to the common practice era. This training conditions listeners to certain expectations that modern music often fails to meet, leaving them baffled. To enjoy modern music, you must recognize the integrity of our own experience with the music—you must learn to trust your ears. You must also learn to abandon your pre-conceptions and listen in a style-independent way.
Most of us live comfortably in a Newtonian world, with modern advances in physics only at the periphery of our awareness. In a recent Op-Ed piece in the New York Times, the physicist Brian Greene lamented that, even one hundred years after Einstein’s insights, the Theory of Relativity has not yet infiltrated our daily experience. In life and in music, we often long for clarity. And yet, in so many ways, we are learning how deeply ambiguity is embedded in our experience and how acknowledging and tolerating it enlarges our spirit. Progressive modern music offers one of the safest ways to experience ambiguity. If we can learn to reckon with modern music with an open mind and careful attention, it may help us deal more patiently and constructively with a world filled with contradictions and paradoxes.