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When Graphs Are a Matter of Life and Death

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Where van Langren had abstracted the range of longitudinal estimates into a line, Playfair had gone further. He discovered that you could encode time by its position on the page. This idea may have come naturally to him. Friendly and Wainer describe how, when Playfair was younger, his brother had explained one way to record the daily high temperatures over an extended period: he should imagine a bunch of thermometers in a row and record his temperature readings as if he were tracing the different mercury levels; from there, it was only a small step to let the image of the thermometer fade into the background, use a dot to represent the top of the column of mercury, and line up the dots from left to right on the page. By visualizing time on the x-axis, Playfair had created a tool for making pictures from numbers which offered a portal to a much deeper connection with time and distance. As the industrial age emerged, this proved to be a life-saving insight.

Back when long-distance travel was provided by horse-drawn stagecoaches, departure timetables were suggestive rather than definitive. Where schedules did exist, they would often be listed alongside caveats, such as “barring accidents!” or “God permitting!” Once passenger railways started to open up, in the eighteen-twenties and thirties, train times would be advertised, but, without nationally agreed-on time and time zones, their punctuality fell well shy of modern standards. When George Hudson, the English tycoon known as the Railway King, was confronted with data showing how often his trains ran late, he countered with the data on how often his trains were early, and insisted that, in net terms, his railway ran roughly on time.

As train travel became increasingly popular, patience was no longer the only casualty of this system: head-on collisions started to occur. With more lines and stations being added, rail operators needed a way to avoid accidents. A big breakthrough came from France, in an elegant new style of graph first demonstrated by the railway engineer Charles Ibry.

In a presentation to the French Minister of Public Works in 1847, Ibry displayed a chart that could show simultaneously the locations of all the trains between Paris and Le Havre in a twenty-four-hour period. Like Playfair, Ibry used the horizontal axis to denote the passing of time. Every millimetre across represented two minutes. In the top left corner was a mark to denote the Paris railway station, and then, down the vertical axis, each station was marked out along the route to Le Havre. They were positioned precisely according to distance, with one kilometre in the physical world corresponding to two and a half millimetres on the graph.

With the axes set up in this way, the trains appeared on the graph as simple diagonal lines, sweeping from left to right as they travelled across distance and time. In the simplest sections of the rail network, with no junctions or crossings or stops, you could choose where to place the diagonal line of each train to insure that there was sufficient spacing around it. Things got complicated, however, if the trains weren’t moving at the same speed. The faster the train, the steeper the line, so a passenger express train crossed quickly from top to bottom, while slower freight trains appeared as thin lines with a far shallower angle. The problem of scheduling became a matter of spacing a series of differently angled lines in a box so that they never unintentionally crossed on the page, and hence never met on the track.

A graph from 1878 shows every train between Paris and Lyon in a twenty-four-hour period; the intersections of the lines indicate where and when trains would pass on the tracks.Source: Étienne-Jules Marey, “La méthode graphique dans les sciences expérimentales et principalement en physiologie et en médecine”

These train graphs weren’t meant to be illustrations—they weren’t designed to persuade or to provide conceptual insight. They were created as an instrument for solving the intricate complexities of timetabling, almost akin to a slide rule. Yet they also constituted a map of an abstract conceptual space, a place where, to paraphrase the statistician John Tukey, you were forced to notice what you otherwise wouldn’t see.

Within a decade, the graphs were being used to create train schedules across the world. Until recently, some transit departments still preferred to work by hand, rather than by computer, using lined paper and a pencil, angling the ruler more sharply to denote faster trains on the line. And contemporary train-planning software relies heavily on these very graphs, essentially unchanged since Ibry’s day. In 2016, a team of data scientists was able to work out that a series of unexplained disruptions on Singapore’s MRT Circle Line were caused by a single rogue train. Onboard, the train appeared to be operating normally, but as it passed other trains in the tunnels it would trigger their emergency brakes. The pattern could not be seen by sorting the data by trains, or by times, or by locations. Only when a version of Ibry’s graph was used did the problem reveal itself.

Until the nineteenth century, Friendly and Wainer tell us, most modern forms of data graphics—pie charts, line graphs, and bar charts—tended to have a one-dimensional view of their data. Playfair’s line graph of Navy expenditures, for instance, was concerned only with how that one variable changed over time. But, as the nineteenth century progressed, graphs began to break free of their one-dimensional roots. The scatter plot, which some trace back to the English scientist John Herschel, and which Tufte heralds as “the greatest of all graphical designs,” allowed statistical graphs to take on the form of two continuous variables at once—temperature, or money, or unemployment rates, or wine consumption—whether it had a real-world physical presence or not. Rather than featuring a single line joining single values as they move over time, these graphs could present clouds of points, each plotted according to two variables.

Their appearance is instantly familiar. As Alberto Cairo puts it in his recent book, “How Charts Lie,” scatter plots got their name for a reason: “They are intended to show the relative scattering of the dots, their dispersion or concentration in different regions of the chart.” Glancing at a scatter allows you to judge whether the data is trending in one direction or another, and to spot if there are clusters of similar dots that are hiding in the numbers.

A famous example comes from around 1911, when the astronomers Ejnar Hertzsprung and Henry Norris Russell independently produced a scatter of a series of stars, plotting their luminosity against their color, moving across the spectrum from blue to red. (A star’s color is determined by its surface temperature; its luminosity, or intrinsic brightness, is determined both by its surface temperature and by its size.) The result, as Friendly and Wainer concede, is “not a graph of great beauty,” but it did revolutionize astrophysics. The scatter plot showed that the stars were distributed not at random but concentrated in groups, huddled together by type. These clusters would prove to be home to the blue and red giants, and also the red and white dwarfs.

In graphs like these, the distance between any two given dots on the page took on an entirely abstract meaning. It was no longer related to physical proximity; it now meant something more akin to similarity. Closeness within the conceptual space of the graph meant that two stars were alike in their characteristics. A surprising number of stars were, say, reddish and dim, because the red dwarf turned out to be a significant category of star; the way stars in this category clustered on the scatter plot showed that they were conceptually proximate, not that they were physically so.

But if you could find clusters of dots in two dimensions, why not three? Friendly and Wainer discuss a three-dimensional scatter plot that improved our understanding of Type 2 diabetes. In 1979, two scientists, Gerald M. Reaven and R. G. Miller, plotted blood-glucose levels against the production of insulin in the pancreas for a series of patients. Along a third axis, they added a metric for how efficiently insulin is used by the body. What emerged was a three-dimensional structure that looks a little like an egg with floppy wings. It allowed Reaven and Miller to split participants into three groups—those with overt diabetes, those with latent diabetes, and those who were unaffected—and to understand how patients might transition from one state to another. Previously it had been thought that overt diabetes was preceded by the latent stage, but the graph showed that the only “path” from one to the other was through the region occupied by those classified as normal. Because of this and evidence from other studies, they are now considered two separate disease classes.

If three dimensions are possible, though, why not four? Or four hundred? Today, much of data science is founded on precisely these high-dimensional spaces. They’re dizzying to contemplate, but the fundamental principles are the same as those of their nineteenth-century scatter-plot predecessors. The axes could be the range of possible answers to a questionnaire on a dating Web site, with individuals floating as dots in a vast high-dimensional space, their positions fixed by the responses they gave when they signed up. In 2012, Chris McKinlay, a grad student in applied mathematics, worked out how to scrape data from OkCupid and used this strategy—hunting for dots in a similar region, in the hope that proximity translated into romantic compatibility. (He says the eighty-eighth time was the charm.) Or the axes could relate to your reaction to a film on a streaming service, or the amount of time you spend looking at a particular post on a social-media site. Or they could relate to something physical, like the DNA in your cells: the genetic analysis used to infer our ancestry looks for variability and clusters within these abstract, conceptual spaces. There are subtle shifts in the codes for proteins sprinkled throughout our DNA; often they have no noticeable effect on our development, but they can leave clues to where our ancestors came from. Geneticists have found millions of these little variations, which can be shared with particular frequency among groups of people who have common ancestors. The only way to reveal the groups is by examining the variation in a high-dimensional space.

These are scatter plots that no one ever needs to see. They exist in vast number arrays on the hard drives of powerful computers, turned and manipulated as though the distances between the imagined dots were real. Data visualization has progressed from a means of making things tractable and comprehensible on the page to an automated hunt for clusters and connections, with trained machines that do the searching. Patterns still emerge and drive our understanding of the world forward, even if they are no longer visible to the human eye. But these modern innovations exist only because of the original insight that it was possible to think of numbers visually. The invention of graphs and charts was a much quieter affair than that of the telescope, but these tools have done just as much to change how and what we see. ♦

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The Musical Mysteries of Josquin

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The singer and composer Josquin Desprez traversed his time like a diffident ghost, glimpsed here and there amid the splendor of the Renaissance. He is thought to have been born around 1450 in what is now western Belgium, the son of a policeman who was once jailed for using excessive force. In 1466, a boy named Gossequin completed a stint as a choirboy in the city of Cambrai. A decade later, the singer Jusquinus de Pratis turned up at the court of René of Anjou, in Aix. In the fourteen-eighties, in Milan, Judocus Despres was in the service of the House of Sforza, which also employed Leonardo da Vinci. At the end of the decade, Judo. de Prez joined the musical staff at the Vatican, remaining there into the reign of Alexander VI, of the House of Borgia. The name Josquin can be seen carved on a wall of the Sistine Chapel. In 1503, the maestro Juschino took a post in Ferrara, singing in the presence of Lucrezia Borgia. Not long afterward, Josse des Prez retired to Condé-sur-l’Escaut, near his presumed birthplace, serving as the provost of the local church. There he died, on August 27, 1521. His tomb was destroyed during the French Revolution.

The murkiness of his existence notwithstanding, Josquin attained an enduring renown of a kind that no previous composer had enjoyed. In 1502, the Venetian printer Ottaviano Petrucci, the chief pioneer of movable-type music publishing, issued a volume of sacred motets, with Josquin’s four-voice setting of “Ave Maria . . . virgo serena” (“Hail Mary . . . serene virgin”) at its head. The piece must have cast a spell, and the beginning shows why. The highest voice, the superius, sings a graceful rising-and-falling phrase: G C C D E C. Each of the lower voices presents the motif in turn. After it arrives in the bass, the superius enters again on a high C, forming an octave pillar. A second phrase unfurls in similar fashion, then a third, with the voices staggered so that only two move together at a time. Eventually, the scheme changes, the texture thickens, and the descending order of vocal entries is reversed. About a minute in, all four voices coalesce to form a gleaming C-major sonority. The entire opening gives the illusion of breadth and depth, as though lamps have been lit in a vaulted room. Music becomes a space in which you walk around in wonder.

Interest in Josquin was strong enough that Petrucci released three volumes of the composer’s masses—settings of five sections of the Roman Mass (Kyrie, Gloria, Credo, Sanctus, Agnus Dei). Posthumously, the flood of publications only increased, to the point where an observer wryly said, “Now that Josquin is dead, he is putting out more works than when he was still alive.” Extravagant claims were made. The humanist Cosimo Bartoli described Josquin as the Michelangelo of music; Martin Luther called him “the master of the notes.” In subsequent centuries, performances of his works all but ceased, yet his name remained one to conjure with. In 1782, the historian Charles Burney declared that Josquin had achieved “universal monarchy and dominion over the affections and passions of the musical part of mankind.” For August Wilhelm Ambros, in 1868, he was the first composer in history “who makes a prevailing impression of genius.” In the twentieth century, the early-music movement brought Josquin’s scores back to life, and the revival continues five hundred years after his death. The Tallis Scholars, the best known of Renaissance vocal ensembles, recently completed a recorded survey of eighteen masses attributed to Josquin. Such groups as Stile Antico, Cappella Pratensis, Blue Heron, and the Huelgas Ensemble are participating in a Josquin festival in Antwerp in August. The “Ave Maria” is a staple of choirs around the world.

With Josquin began the cult of the great composer—a mind-set that has left a distinctly ambiguous imprint on classical-music culture. And his rise to superhero status brought with it a curious paradox. Although commentators across five centuries have agreed on Josquin’s preëminence, his works can easily be confused with those of other gifted contemporaries. Two anecdotes from the early sixteenth century illustrate what might be called the Josquin mirage, in which the lustre of his name warps musical perceptions. Baldassare Castiglione, in his treatise “The Book of the Courtier” (1528), made note of the composer’s snob appeal in aristocratic settings: “When a motet was sung in the presence of the Duchess, it pleased no one, and was considered worthless, until it became known that it had been composed by Josquin Desprez.” The opposite fate befell a piece by Adrian Willaert, one of Josquin’s most accomplished successors. When Willaert first came to Rome, he found that the papal choir was singing one of his motets, under the impression that it was by Josquin. When Willaert corrected the mistake, the singers lost interest in the work. Such stories help to explain why attributions to Josquin proliferated after his death: affixing his name to a score was guaranteed to stir interest. The same syndrome has long haunted Renaissance art, where an emphasis on the singular profile of canonical artists has led to violent debates over authenticity and a thriving marketplace in forgeries.

Well over three hundred pieces were ascribed to Josquin at one time or another. In recent decades, musicologists have been culling dubious items from the catalogue. This spring, I followed the work of two leading Josquin authorities, Joshua Rifkin and Jesse Rodin, who are preparing a drastically pruned list of likely Josquin pieces—a hundred and three in all. Some scholars worry that the deattribution process has got out of hand; the half-joking fear is that Josquin will end up disappearing almost completely, like the Cheshire Cat. Thanks to the pandemic-era phenomenon of the Zoom seminar, I was able to watch some of the deliberations, which kept raising bigger philosophical questions: How does an aura of infallibility come to surround a figure like Josquin? What becomes of the music that lapses into anonymity, just as “The Man with the Golden Helmet” seems to have fallen out of the Rembrandt canon?

There is nothing fake about that aura: Josquin was an astonishing composer, one whose contrapuntal dazzlements can make Bach look clumsy. But he dwelled within a comprehensively astonishing community of creative artists. To explore Renaissance choral music is to enter a forbidding forest of names: Dunstable, Power, Binchois, Dufay, Busnois, Ockeghem, Regis, Faugues, Compère, Weerbeke, Agricola, de Orto, Obrecht, Isaac, de la Rue, Mouton, Brumel, Févin, Richafort, Ghiselin, Gombert, Pipelare, Martini, Clemens non Papa, Morales, Willaert, Lassus, Palestrina. Every one of them wrote music worth hearing. The period bears witness to the emergence of composition as an art: Josquin becomes the patron saint of an essentially new profession that is struggling to gain the level of recognition long accorded to painters and poets. Distinct personalities materialize from the historical mist. We hear the sound of the self, singing toward a kind of freedom.

The term “composer” began to enter general circulation only in the late fifteenth century. The practice of naming the authors of musical works was still catching on. Documents of the period usually call Josquin a cantore, or singer. Yet his rise to fame helped bring about a change in status. In 1502, a courtier to Ercole I, the Duke of Ferrara, wrote a letter evaluating candidates for a musical appointment. One of them, Heinrich Isaac, was “easy to get along with,” the courtier said; another, Josquin, “composes when he wants to, and not when one wants him to.” Also, Josquin asked for two hundred ducats, Isaac for much less. Ercole I hired Josquin.

Composers were a new phenomenon because written music was itself a relatively recent innovation in the long history of the arts. The earliest examples of fully decipherable staff notation, from the early eleventh century, record Gregorian chant; multivoiced sacred music was written down at Notre-Dame, in Paris, in the twelfth and thirteenth centuries. Troubadours, trouvères, and other poet-composers produced a beloved corpus of song, though the words tended to receive more attention than the notes. The most formidable figure of the age was Guillaume de Machaut, who lived from around 1300 to 1377. Celebrated chiefly for his sung poems of courtly love, Machaut also wrote two dozen motets and the earliest mass cycle for which a composer is known. Such large-scale elaborations on canonical texts sustained careers in the following century, as Popes, princes, and other potentates sought to flesh out courtly ceremonies with splendid sounds. The history of written music is inextricable from structures of worldly power, even if the composers occupied a low place in the hierarchy.

Josquin exemplifies the art of polyphony: the interweaving of multiple voices according to strict contrapuntal rules. The primary mandate was to control dissonance—a term that was understood differently in medieval and Renaissance times than it is today. It indicated not just discordant combinations of tones but also problematic relationships between notes. The octave, the fifth, and sometimes the fourth were considered to be “perfect” consonances; thirds and sixths were “imperfect”; other intervals fell into the “dissonant” category. A wariness of thirds partly explains why medieval music can sound stark and strange to modern ears. Thirds are at the core of tonal harmony, defining major and minor keys. In the early fifteenth century, English composers, led by John Dunstable, began using thirds in abundance. Their lush, chord-rich sound became known as the “English countenance,” surprising and delighting listeners on the Continent. English sources are also the first to name composers in large numbers.

“And that, son, is where wealth comes from.”
Cartoon by Robert Leighton

Geopolitics had a hand in what happened next. King Henry V of England, who may have dabbled in composing, won at Agincourt, in 1415, and soon took over northern France. English officials brought with them their favorite choristers; Dunstable evidently served John of Lancaster, Henry V’s brother and military commander. Thanks to Joan of Arc, England’s holdings soon shrank, but not before its music had seeped into northern France and Belgian lands. Coincidentally or not, this region brought forth the next major wave of musical activity. A vast number of fifteenth- and early-sixteenth-century composers, Josquin included, belonged to what is today called the Franco-Flemish School.

Leading the procession was Guillaume Dufay (circa 1397-1474), who brought dancing elegance to exalted masses and streetwise chansons alike. His motet “Nuper rosarum flores” was written for the consecration of Florence’s cathedral, in 1436, its stately sonorities echoing against Filippo Brunelleschi’s octagonal dome. Other mid- and late-fifteenth-century composers expanded the field of possibility. Antoine Busnois specialized in a lucid interplay of motifs; Johannes Ockeghem in opulent, unpredictably flowing designs; Johannes Regis in intricate structures that gather narrative energy from the calculated addition and subtraction of voices. (Josquin may have based his setting of “Ave Maria” on Regis’s motet of the same name.) By 1500, dozens of Franco-Flemish singer-composers had radiated across Europe, establishing a virtual monopoly at certain Italian musical centers, the Vatican included.

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