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Mathematics Marches On

In the 1100s in Paris, an innovative experiment was brewing. 1  After five hundred years of Gregorian chants which featured a group of people singing a single melody, attempts were made to make music more interesting by dividing the singers into two groups and assigning to each a different melody!

1No, not beer – the monks preferred champagne.

The idea was simple and brilliant – the hard part was deciding what notes to give the second group. The first group, of course, sang the original melody. To give you an insight into the way this was, and is, done, we must speak briefly about music theory.

Now the entire subject of music is very mathematical, as you shall see. The basic notes of Western music consists of the C scale, whose notes c, d, e, f, g, a, b, c ascend in pitch. No doubt, you have sung this at some point and you might even have used the names do, re, mi, fa, sol, la, ti, do, as immortalized in the movie The Sound of Music, in the amusing song which describes “do” as a female deer. 2  These notes are, by the way, the white keys on a piano keyboard.

Notes are written today as circles on a staff of five lines. The location of the circle indicates the pitch – the higher the note, the higher the pitch. This is a profoundly mathematical idea! The range from the initial c to the final c is called an octave, the oct for the eight notes between the two c’s (inclusive). The octave is also a measure of distance between the two notes. If you sing the notes, one after another, they will sound very similar. Other pitches are supplied by raising or lowering these pitches by a small amount, denoted by sharps (#) and flats (♭).

After the scale is finished, it starts again, until we reach the c two octaves above the one with which we started. This process occurs in both directions (up and down). The reference point is middle c, easily located in the middle of the piano keyboard. A piano has 88 keys – an impressive range indeed. The notes on the extreme left side sound very deep, while the notes on the extreme right sound shrill. This system did not evolve overnight. It took centuries. Notation in the Middle Ages was very primitive and spanned little more than the octave.

2Many musicians use the solfege note si instead of ti.

Next problem – the duration of the note, that is, the length of time it should be sustained by the singer or instrumentalist, is indicated by the way the circle is drawn. The unit of length is the so-called quarter note, denoted (no pun intended) as a filled-in circle with a stem. The composer can state the time value of a quarter note with reference to a metronome – a device that makes noise at constant time intervals. A quarter note can be shortened to half its value by adding a small flag to its stem, thereby changing it to an eighth note. Note that  is half of. The next subdivision, which you may have anticipated, is the sixteenth note, written with two flags attached to its stem. One can continue this bisection process by adding more flags, each flag halving the previous time value.

In the opposite direction, a note whose duration is twice the value of a quarter note requires leaving the circle unfilled. If, in addition, the stem is removed, the result is a whole note – whose duration is four quarter notes. (Four quarters equals one whole, logical?) How does one obtain a  note, that is, a note of duration the same as three quarter notes? One widely used method places a dot after the unfilled circle of a half-note. This tells us to augment the value of the note by 50%, or, in other words, to add on half its value.

Another major issue is meter. Do we wish to group the beats in two’s, three’s, four’s, and so on? Most rock songs use four quarter notes per measure. We call this signature “four-four” and display a brief sample in  Figure 7-1 . Measures are separated by thin vertical bars. Notice the “four-four” time signature at the beginning of the melody. A waltz usually has a “three-four” time signature, signifying that each measure contains three quarter notes.

What other qualities of music are measurable, and hence, are mathematical characteristics? The volume of a piece of music some times varies. This is measured with letters, such as p forpianissimo, meaning very quietly, and f for fortissimo, meaning very loudly. This is not adequate for indicating a gradual change in volume. This is achieved using pairs of diverging lines for an increase and converging lines for a decrease in volume, as illustrated in  Figure 7-2 .

Figure 7-1

Figure 7-2

Finally, how do we measure the distance between two notes? How far away are c and g? Since we must count five notes to get from c to g, we call the interval between them a fifth. The notes c and f determine a fourth, since we must count four notes to get from c to f, and so on. The octave, fifth, and fourth are among the early intervals used in the twelfth-century attempts in creating a second melody to be sung simultaneously with the first. These early attempts were quite crude by later standards and over the next two hundred years were greatly improved by the inclusion of thirds and sixths. In the fourteenth century, composers started writing a bit of secular music, some for wealthy patrons, though the majority of works were religious in nature. In addition to the inclusion of pleasing intervals such as thirds, composers started writing pieces with three or more simultaneous voices. The emphasis was often on creating independent voices – accomplished by syncopating rhythms and varying the direction of the melody, that is, one voice ascending and the other descending or one voice sustaining a note while the other melody contains several shorter notes. This kind of writing is called counterpoint and is very difficult.

One of the results of the newly emerging music, with its pretty harmonies, was a heightened awareness of the beauty of this world and an increased desire to study it. In a sense, this typified the spirit of the Renaissance. It should be noted that the Ancient Greeks developed music theory to some degree, and it is from them that we inherited our scales and letter notation. The Greeks used letters like α and β to represent notes over a thousand years before the Parisian musicians of 1100 did. They used dots over letters to denote short duration, and dashes above them to indicate long duration – a practice still adhered to today.

The art of the Middle Ages was as primitive as the music of the times. As the drawing in  Figure 7-3 illustrates, the canvas did not capture what the eye sees. The circle over the angel’s head should be an ellipse, and the man in the background should be smaller than the one in the foreground.

Figure 7-3

This typifies the way paintings appeared before the introduction of perspective in the Renaissance. With the influx of classical Greek geometry, it was not long before artists such as Filippo Brunelleschi (1377-1446), Leone Battista Alberti (1404-1472), Piro della Francesca (1420-1492?), Albrecht Dürer (1471-1528), and Leonardo da Vinci (1452-1519) began employing techniques such as a vanishing point (or points) on the horizon line, to which receding lines appear to converge.

No doubt, you have observed the seeming convergence of railroad tracks as they recede into the distance, as if they meet on the horizon. On the other hand, the railroad ties remain parallel – though they appear to get closer to one another as they get further away from the observer. Vertical telephone poles along the route remain parallel but seem to get shorter.

The Renaissance began in Florence, the cultural center of fourteenth-century Europe. Artists and intellectuals began to study the great literary, mathematical, and artistic achievements of the ancient glorious civilizations of Greece and Rome. Works were produced in the style of the classical masters. Dante (1265-1321) wrote The Divine Comedy in a style reminiscent of Virgil. Bocaccio (1313-1375) and Petrarch (1304-1374) wrote prose and poetry modeled after ancient works.

Much can be said about the spirit of the Renaissance, typified by Leonardo Da Vinci. Though his friends did not refer to him as a “Renaissance man,” he certainly was the prototype. He was an artist, scientist, inventor, architect, and mathematician, among many other things. His curiosity encompassed all of nature and then some. He saw geometry in the things he drew and was one of the first to see the role of mathematics in the scientific study of the universe.

In the middle of the fifteenth century, the development of Johan Gutenberg’s “movable block” printing press ushered in a new era in the transmission of knowledge as dramatic as the computer revolution some five hundred years later. By the end of the century, mass production of texts on a host of subjects enhanced the distribution of knowledge to the far reaches of the continent. In 1494, one of the first printed mathematics text Summa, by Luca Pacioli (1445-1517), contained a summary, as the title suggests, of virtually all of the mathematics of the times.

In 1453, the Eastern Roman Empire fell to the Seljuk Turks, which resulted in a mass exodus of Byzantine intellectuals to the Italian city-states where the Renaissance was in full swing. They brought a slew of Greek manuscripts along with them, thereby introducing many classical Greek works in philosophy and mathematics into the boiling cauldron of intellectual and artistic activity in Europe.

Finally, the discovery of the New World in the 1490s added to the exciting air of empiricism. There was an increased willingness to explore, take risks, and challenge authority – of the church or of ancient authorities such as Aristotle, often referred to as “the philosopher.” After all, the New World wasn’t mentioned in the Bible, so revelation was not the final word in the acquisition of knowledge. Of course, the Reformation wars between Protestants and Catholics in the following century did much to further compromise the authority of the Vatican and its ability to retard free inquiry. The successful challenge of papal infallibility encouraged empiricism and ultimately gave scientists a safer environment in which to conduct their business.

A practical outgrowth of the Age of Exploration was the need for mathematics to design better ships and to provide adequate navigation for ocean voyages.

The theoretical mathematics of the fifteenth and sixteenth centuries consisted mostly of improvements in algebra, such as the development of formulas that solve cubic equations. A cubic equation is similar to a quadratic equation but includes a term of the form ax3. In other words the general cubic is ax3 + bx2 + cx + d = 0. The formula involves the coefficients of the terms and is quite lengthy.

A part of the weakness of the algebra of the ancient world was the absence of notation. There was no symbol for the basic operations of adding, subtracting, multiplying, dividing, or calculating powers and roots. They didn’t even have an equals sign! An even more serious problem was the use of different symbols for the unknown quantity and its square. How is one supposed to factor the difference x2 − x into x (x − 1) if we use one symbol for x (the unknown quantity) and another, say S, for its square? We can’t see the factoring if we have S − x instead of x2 − x. This problem of notation took hundreds of years of gradual development until it was fairly good – just in time for the incredible seventeenth century – the so-called heroic century of mathematics.