Medieval Harmony

Philippe de Vitry (1291-1361) was one of the most prominent figures in medieval music. He was the author of an important music theory text, Ars Nova, which introduced new rhythmic schemes and musical notation. He had a deep knowledge of philosophy, rhetoric, and mathematics.

In many ways, de Vitry's interests and accomplishments reflected the Pythagorean view that music is a subdivision of arithmetic, as shown, for example, in the simple mathematical relations between pitch and length of a string (see Circles of Dissonance). His work honored the dictum of the Roman philosopher Boethius (480-524) that "music is number made audible."

One of de Vitry's observations concerned numbers that can be written as a power of 2 multiplied by a power of 3. In modern terms, such numbers would be expressed as 2ⁿ x 3^m, where n and m are integers. He called them harmonic numbers.

Here's a table of all harmonic numbers less than 1000.

De Vitry focused on pairs of harmonic numbers that differ by 1: 1, 2; 2, 3; 3, 4; and 8, 9. As it happens, those particular pairs correspond to musically significant ratios, representing an octave, fifth, fourth, and whole tone. He wondered whether there are any other such pairs of harmonic numbers.

In 1342, Levi ben Gerson, also known as Gersonides (1288-1344), proved that the original four pairs of harmonic numbers are the only ones that differ by 1. Here's how his proof went (using modern notation).

If two harmonic numbers differ by 1, one must be odd and the other even. The only odd numbers are powers of 3 (top row of chart, above). Hence, one of the two harmonic numbers must be a power of 3 and the other a power of 2 (first row and first column of chart). The basic task then involves solving two equations:

2ⁿ = 3^m + 1 and 2ⁿ = 3^m − 1.

Gersonides had the idea of looking at remainders after division of powers of 3 by 8 and powers of 2 by 8. For example, 27 divided by 8 gives a remainder of 3. For powers of 3, the remainders all turn out to be 1 or 3, depending on whether the power is even or odd. The remainders for powers of 2 are 1, 2, 4, then 0 for all powers higher than 2.

For 2ⁿ = 3^m + 1, when m is odd, 3^m has remainder 3, and 2ⁿ = 3^m + 1 must then have the remainder 4, so n = 2 and m = 1. That gives the consecutive harmonic numbers 3, 4. When m is odd, the equation gives the consecutive harmonic numbers 1, 2.

For 2ⁿ = 3^m − 1, when m is odd, 3^m has remainder 3, so 2ⁿ = 3^m − 1 has remainder 2; as a result, n = 1 and m = 1, to give the consecutive harmonic numbers 2, 3.

The final case, when m is even, is a little trickier and requires substituting 2k for m, then solving the equation 2ⁿ = 3^2k − 1 = (3^k − 1)(3^k + 1). That gives the consecutive harmonic numbers 8, 9. QED.

Interestingly, the theorem about harmonic numbers proved by Gersonides has connections with Fermat's last theorem and with the abc conjecture, a topic of some interest among number theorists.

 

Originally posted January 25, 1999

Circles of Dissonance

Much of today's music rests upon ancient tradition going back thousands of years to the time of the Greek mathematician and mystic Pythagoras.

The Pythagoreans observed that tones an octave apart are pleasing to the ear. In modern terms, one can say that consonance arises when the frequency of one tone is precisely twice the frequency of the other. Hence, a ratio of 2:1 produces harmonious tones. Similarly, musical intervals involving tones in the ratios of 3:2 (a fifth) and 4:3 (a fourth) are also pleasing.

Out of these ratios, the Pythagoreans and others put together sequences of tones to create various musical scales spanning an octave. The so-called major scale consists of seven notes, which nowadays are designated by the letters C, D, E, F, G, A, and B, or sounded out as do, re, mi, fa, sol, la, and ti. The chromatic scale includes five additional notes—the sharps and flats (black keys of a piano).

The chromatic scale consists of a progression of half steps from one note to the next, while the major scale is an irregular combination of whole steps and half steps (whole, whole, half, whole, whole, whole, half).

Those steps can be associated with particular numerical ratios, which in turn correspond to ratios of frequencies. Thus, in a system known as just intonation, if C = 1, then D = 9/8, E = 5/4, F = 4/3, G = 3/2, A = 5/3, B = 15/8, and C (one octave higher) = 2. Notice that the scale includes the particularly harmonious Pythagorean ratios.

A violinist playing in the key of C would produce notes with frequencies corresponding to these ratios relative to the frequency of C. However, if the violinist modulates to, say, the key of G (in which the scale starts with G instead of C), then A is no longer 5/3 times the frequency of C, but 9/8 times the frequency of G and, as a result, 27/16 times the frequency of C. Thus, in this scheme, the pitches of a violinist's notes are not fixed, but vary with the key in which he or she is playing at the moment.

Roger Penrose has come up with a visual, remarkably accessible way of illuminating this curious aspect of musical scales. Penrose is best known for his theoretical work on black holes, quantum gravity, and aperiodic tilings. He is also a strong advocate of the notion that mathematics is fun and a firm believer in the value of recreational mathematics.

Penrose's idea is to convert the frequency ratios of the major scale into visual form as points along the circumference of a circle. Because going through one octave is equivalent to going once around the circle, the pattern just repeats itself with each successive octave. The result is a kind of pie chart for musical tones.

The position of C is established first, then all other notes are placed according to the following formula, where the angle determines the size of the segment in the pie chart relative to the position of C:

angle = 360 ✕ log (ratio)/log 2. 

For example, for D, the ratio is 9/8 and the angle is 360 ✕ log (9/8)/log 2 = 61.18. So the point for D is 61.18 degrees away from the position of C along the circle's circumference.

Pie chart representation of the major scale for the key of C.

Suppose you change to the key of G. The consequences of such a shift can be seen by making a copy of the original circle representation on a transparent plastic sheet, then placing it atop the original with C on the transparent sheet lined up with G on the original. Every other note on the transparent copy coincides with a note on the original—except A. The discrepancy is clearly visible.

Discrepancy (shaded area) between A in the key of C and A in the key of G. 

Thus, A tuned according to the C major scale would not be in exactly the right place to give true harmony in the key of G. Similar anomalies can be found in other keys simply by looking at various placements of the transparent sheet.

Penrose's device offers a way for anyone to see the harmony and dissonance that musicians can readily hear. It serves as a marvelous introduction to the mathematical relationships underlying musical scales, keys, intervals, chords, and harmonies, and a peek into the mysteries of intonation and temperament.

Originally posted November 24, 1997

Prime Listening

The human ear has remarkable capabilities—so remarkable that we generally accept as routine such formidable tasks as recognizing a voice, picking out a single word in a cacophony of cocktail-party chatter, or hearing a flute's sweet tone in the midst of an orchestral romp.

Our sense of hearing can integrate disparate sounds into a harmonious whole or detect subtle nuances buried in noise. It has amazing powers of pattern recognition. Indeed, by using sound to represent masses of data or jumbles of numbers, we can take advantage of that capability to identify regularities or make subtle distinctions.

Mathematician Chris K. Caldwell has developed a scheme for listening to sequences of primes—to hear both simple patterns and perplexing irregularities found among those numbers.

"Multimedia allows the use of sight and sound, so why not use sound?" Caldwell asks. "After all, don't we use our ears to detect patterns as we listen to the car motor for a problem or shake a box to determine its contents?"

The MIDI (Musical Instrument Digital Interface) specification often used in computer programs to represent musical notes assigns a number to each note on a keyboard. According to this recipe, middle C is 60, C-sharp is 61, D is 62, and so on, for a total of 128 notes. One could use that correspondence directly to "play" primes. For example, 67 would then be G.

There are infinitely many primes, however, so Caldwell's strategy is to divide each prime by a fixed number, then play just the remainder—a novel application of that standard number-theory tool known as modular arithmetic.

For example, if the divisor, or modulus, is 7, then for the primes {2, 3, 5, 7, 11, 13, 17, 19, 23. . .}, one would play {2, 3, 5, 0, 4, 6, 3, 5, 2,…}. Because those particular notes on the MIDI scale would be too low in frequency to be audible, Caldwell adds a constant, such as 55. Hence, the first prime, 2, is played as the note A. There are seven possible notes.

Primes modulo seven

In listening to the primes modulo 7, you find that all seven notes are played, but the lowest note occurs just once. That lowest note is the prime number 7. Any other number that leaves a remainder of 0 when divided by 7 must be a multiple of 7, so it can't be a prime. A theorem of Peter Gustav Lejeune Dirichlet (1805-1859) on primes in arithmetic progression guarantees that all the other notes are heard infinitely often when one plays all the primes.

Trying various prime and composite divisors reveals other patterns, and one can check which notes are played infinitely often, which ones are played just once, and which ones are never played for each modulus.

It's possible to use the duration of notes to represent the gaps between prime numbers. There's no gap between the first two primes, 2 and 3. There's a gap of one between 3 and 5, a gap of one between 5 and 7, and a gap of three between 7 and 11. According to the prime number theorem, the average gap length up to a certain prime equals the value of the natural logarithm of that prime.

Caldwell makes the note representing a particular prime last a time proportional to the size of the gap to the next prime in the sequence, divided by the natural logarithm of that prime.

Caldwell's "Prime Number Listening Guide" offers his weirdly tuneful "primes modulo 41 with prime gap tempo" and other primal sounds and even a way to make your own prime music.

There may be more to come. Caldwell has started to develop a program to play the factorizations of sequences of numbers. "The low primes are used to determine what percussion instrument to play, the next larger primes are mapped to one instrument, and the larger ones (modulo some base) are mapped to a second," he says. "If the integer is divisible by a power (greater than 1) of the prime, then that prime's 'note' is played louder."

You may one day hear more of the concert theorems that mathematicians have so painstakingly gleaned from their studies of integers and the primes.

Originally posted June 22, 1998

Mozart's Melody Machine

Music publishing was a thriving trade during the latter part of the 18th century in Europe. Publishers vied with one another to print the works of the latest "hot" composer. Many of them looked for novel ways to entice new customers into their music shops.

One such ploy was to publish systems that would allow any amateur to compose music without having to know the techniques or rules of composition. The London music publisher Welcker, for example, issued a "Tabular System whereby the Art of Composing Minuets is made so easy that any person, without the least Knowledge of Musick, may compose ten thousand, all different, and in the most pleasing and correct Manner."

Many of these schemes involved using dice or other randomizers to select musical fragments from an array of choices. Composer Johann Philipp Kirnberger (1721-1783), a former pupil of Johann Sebastian Bach (1685-1750), suggested the use of dice for this purpose in his book The Ever-ready Composer of Polonaises and Minuets, published in 1757. About two decades later, Austrian composer Maximilian Stadler (1748-1833) put together a set of musical bars and tables for generating minuets and trios with the help of dice.

One well-known example of such a scheme is the "Musikalisches Würfelspiel" (Musical Dice Game), first published in 1792 in Berlin. Attributed by the publisher to Wolfgang Amadeus Mozart (1756-1791), it appeared a year after the composer's death.

The idea was to compose a 16-measure "waltz" by rolling dice to decide which measures to select from a large pool of choices. In the "Musikalisches Würfelspiel," the measures are numbered from 1 to 176, and the numbers are arranged in two charts, each consisting of 11 rows and eight columns. To select the first measure, a player would roll two dice, subtract 1 from the total, and look up the corresponding row in the first column of the first chart to determine the appropriate measure number. Subsequent rolls of the dice decide which measure to select from each successive column to complete the melody.

The resulting tunes, described by the publisher as waltzes, have a recognizably Mozartean flavor, and the pieces conform to the harmonic and compositional requirements of Viennese minuets of that period.

In principle, the system produces 11¹⁶ (or 45,949,729,863,572,161) possible waltzes. "The number is so large that any waltz you generate with the dice and actually play is almost certainly a waltz never heard before," mathematical games columnist Martin Gardner once remarked in an article about automated music composition. "If you fail to preserve it, it will be a waltz that will probably never be heard again."

Because many of the bars listed in the eighth column of each chart are identical, the number of different waltzes that the system generates is actually less than 11¹⁶. Indeed, the scheme is now sometimes simplified so that a player has only two choices for the eighth and 16th bars, determined by whether the dice toss is odd or even. In this case, there are 11¹⁴ x 2² (or 1,518,999,334,332,964) different versions.

Interestingly, because two six-sided dice are used, measures in certain rows of the chart come up more often than those in other rows. For example, because a roll of 7 has the highest probability, measures in row 6 would occur in a melody most often. It isn't clear whether the original inventor of the scheme was aware of this skewing of the choices. You can overcome this bias and make each choice equally likely by instead using a 12-sided (or dodecahedral) die or some sort of random-number generator.

Most scholars reject the publisher's claim that Mozart himself devised this particular scheme. Nonetheless, there are indications that Mozart enjoyed mathematical puzzles. He also had a lively sense of humor and was fond of playing around with names. And he had a passion for gambling—a major preoccupation at the time (along with drink) among the men of both Salzburg and Vienna.

One Mozart manuscript actually includes what might be considered a musical game, though not played with dice. On both sides of the sheet, Mozart wrote down long strings of measures, grouped into two-bar melodies, each labeled with a letter of the alphabet and a number (1 or 2). However, other than supplying a "worked-out" example at the end of each page, he gave no instructions on how to proceed.

Hideo Noguchi tried to work out the game's rules. He speculated that Mozart's starting point was the name of an acquaintance, such as Francisca. The idea was to add "z" to the end of the name, rewrite the letters in alphabetical order, alternately assign the number 1 or 2 to each letter in succession (with certain refinements), then return to the original spelling: f1 r2 a1 n1 c2 i2 s1 c1 a1 z2. A player would then select the appropriate measures in the required order from the labeled groups to come up with a signature tune.

Hideo Noguchi described his findings in his paper "Mozart—Musical Game in C K. 516f," which also includes facsimiles and transcriptions of the Mozart manuscript pages.

"Since there is no limit to the number of names and the number of letters one may choose, the number of [possible] compositions [is] infinite," Hideo Noguchi remarked. He invited Mozart lovers throughout the world to play this musical game using their own names to hear from the music that results where they stand in the Mozartean realm.

Originally posted August 27, 2001

Music, Math, and Education

In the introduction to his 1996 book Emblems of Mind: The Inner Life of Music and Mathematics, Edward Rothstein, then music critic for The New York Times, described how his education and interests encompassed both music and mathematics.

He wrote: "Before setting out to make my way in the music business, I was in training to become a 'pure' mathematician. Such esoteric subjects as algebraic topology, measure theory, and nonstandard analysis were my preoccupations. I would stay up nights trying to solve knotty mathematical problems, playing with abstract phrases and structures.

"But at the same time, I would be lured away from these constructions by another activity. With an enthusiasm that could come only when critical faculties are in happy slumber, I would listen to or play a musical composition again and again, imprinting my ear and mind and hands with its logic and sense. Music and math together satisfied a sort of abstract 'appetite,' a desire that was partly intellectual, partly aesthetic, partly emotional, partly, even, physical."

Rothstein went on to say that such an experience is by no means unique to him. He noted that music and mathematucs have been associated throughout history.

Pythagoras and his followers saw numbers as models of everything in the physical world, and they identified music with number, noting its scales, tempos, and other regularities. Johannes Kepler envisioned planetary motions as the "music of the spheres." Galileo Galilei speculated on the numerical reasons why some combinations of tones are more pleasing than others. Leonhard Euler considered the same problem in a treatise on consonance and whole numbers.

On the music side, Johann Sebastian Bach sometimes treated the composition of canons and other types of music as exercises akin to solving mathematical puzzles. Frederic Chopin described the fugue as "pure logic in music." And twentieth-century composers have applied sophisticated mathematical theory in their works.

In his 1993 book The Music of the Spheres:Music, Science and the Natural Order of the Universe, music critic Jamie James examined and pondered the history of the concept of a musical universe—a cosmos envisioned as a stately, ordered mechanism both mathematical and musical. Music and science were once intimately intertwined and united by a grand vision, he pointed out. But music, like much of the most fundamental art and literature of our culture, has now been relegated to the obscure margins of the curriculum.

"All art, including music, was a much more serious matter before the self-conscious aestheticism of the late nineteenth century took root," James argued. "It is a recent notion that music is a divertissement to be enjoyed in comfortable surroundings at the end of the day, far removed from the hurly-burly of life's business."

The intriguing interplay between mathematics and music, which goes back to antiquity and possibly much earlier in human history, hints at a deep and fundamental connection.

In concluding his book, Rothstein commented that music and mathematics share not only the clarity of their expression but also their beauty and mystery.

He wrote: "Our attempt to comprehend music and mathematics, to understand their workings and their purposes, is ... a model for our coming to know at all—a model for our education, for the ways we make distinctions and connections.

"We begin with objects that look dissimilar. We compare, find patterns, analogies with what we already know. We step back and create abstractions, laws, systems, using transformations, mappings, and metaphors. This is how mathematics grows increasingly abstract and powerful; it is how music obtains much of its power, with grand structures growing out of small details."

See also "A Fractal in Bach's Cello Suite," "Cosmic Reflection," and "Waltz of the Cycloids."

American Mathematical Society links on mathematics and music, including the PBS video "The Majesty of Music and Math."

Möbius Pop

Möbius strips can show up in some unexpected places—including the lyrics of a pop song.

Canadian singer and songwriter Nelly Furtado showcases this amazing, one-sided, one-edged object in her song "Hey, Man!" The song leads off her debut album Whoa, Nelly!, which was released in 2000.

The surprise comes in the following portion of the chorus, repeated several times throughout the song:

We are part of a circle

It's like a Möbius strip

And it goes 'round and 'round

Until it loses a link.

The song is credited to Nelly Furtado, Brian West, Kevin Charles Volans, and Gerald Eaton, but it's not clear who contributed the Möbius strip simile to the lyrics. West and Eaton, who once performed as part of a Toronto-based group known as The Philosopher Kings, produced the album.

Cosmic Reflection

The universe began with a bang 13.7 billion years ago, so it was no surprise that a terse bang accompanied a video illustration of the event. This sonic boom was itself just one fleeing moment in a narrated, 40-minute symphony, titled Cosmic Reflection, which debuted on Nov. 2, 2009, at the Kennedy Center in Washington, D.C.

Composed by Nolan Gasser, the symphony was his tribute to the Fermi Gamma-Ray Space Telescope in its first year in orbit. In words, pictures, and music, the composition told the story of the universe, offering a striking lesson in modern astrophysics and cosmology, from the big bang to dark energy and cold death.

Written by Classical Archives founder Pierre R. Schwob and physicist Lawrence M. Krauss, the narration presented the known facts and theoretical speculations simply and clearly—offering a model of accessible exposition. The poetry was in the music, played crisply and impressively by the Boston University Symphony Orchestra, and in the video images provided by the NASA Goddard Space Flight Center: twisting tendrils and flickering rhythms; swinging galaxies and swirling tones; flashing stars and melodic fragments; cataclysmic collapses and percussive effects.

The large audience, blending the sensibilities of the arts, science, and engineering, gave the performance a standing ovation.

While experiencing the event, I couldn't help wondering what a narrated, illustrated symphony devoted to a mathematical theme would be like. Of course, astronomy itself has been a significant driving force in the development of mathematics. Isaac Newton, for one, wrote his immortal Principia Mathematica in answer to the question of what sorts of orbits would occur under an inverse-square force law. I wrote about these connections in my book Newton’s Clock: Chaos in the Solar System. And mathematics has been an invaluable tool for astrophysicists and cosmologists peering into the deepest reaches of the universe. So, mathematics could lay a claim to Cosmic Reflection, too.

What about the story of the prime numbers and the 150-year-old Riemann Hypothesis, a mathematical question for which the Clay Mathematics Institute has offered a $1 million prize for its solution? But that story isn’t complete; there is no proof yet, and no certainty that anyone will ever find a proof. And what images would you use?

Despite the pervasiveness of mathematical thought and its crucial role in underpinning scientific research, I am at a loss. What story, told in words, images, and music, could match the incredible epic that astronomers and physicists have forged from their observations over many millennia of spots of light in the sky?

Waltz of the Cycloids

Mathematical vocabulary isn't something that I normally associate with classical music, so I was caught by surprise recently when an announcer on a local radio station gamely tried to explain the term "cycloid."

Why? The announcer was introducing a piece of music called the Cycloids Waltz (Cycloiden Walzer), written by Johann Strauss Jr (1825-1899). Strauss had dedicated his composition to the "Gentlemen Technical Students at Vienna University" and conducted its debut at their ball in the Sofienbad-Saal on Feb. 10, 1858. The apt title not only came directly out of mathematical vocabulary that would be familiar to the students but also hinted at the whirling movements of a waltz.

A cycloid is the path traced by a point on the rim of a circle that rolls (without slipping) along a straight line.


Galileo Galilei (1564–1642) studied the cycloid around 1599 and gave the curve its name, using the Greek word for "circle" as its main element.

A segment of a cycloid also represents the shape of the curve along which a bead sliding from rest and accelerated by gravity will slip from one point to another in the least time, a problem originally posed in 1696 by Johann Bernoulli (see the brachistochrone problem). Moreover, a bead sliding on a cycloid will exhibit simple harmonic motion, with a period independent of the starting point.

Galileo suggested that the cycloid would be the strongest possible arch for a bridge, and many concrete viaducts do have cycloidal arches. Cogwheels often have cycloidal sides to reduce friction as gears mesh.

In a chapter on the cycloid in Martin Gardner's 6th Book of Mathematical Diversions from Scientific American, Gardner notes that the cycloid has been called the "Helen of geometry," not only because of its beautiful properties but also because it has been the object of so many historic quarrels between eminent mathematicians.

The version of Cycloiden, Walzer, Op. 207, that I heard on the radio was recorded for the Marco Polo label by the Slovak State Philharmonic Orchestra, Kosice. The author of the accompanying album notes remarks that the title page design for the first piano edition of the piece featured a circle, encompassing the name of the work and its composer and the dedication to the students. The designer then surrounded this feature with representations of various tools of the technician (set square, compasses, theodolite, and so on) and depictions of several engineering achievements, such as the steamship, steam engine, blast furnace, and plough. But no cycloid.

Strauss' composition isn't the only one in the musical literature that refers to the cycloid. The Library of Congress collection of American sheet music includes Cycloid Polka, written by Charles Kinkel and published in 1873. Unfortunately, I can't play the piano or find a recording, so I have no idea what this piece sounds like. But I do know there's usually a lot of lively circling when you dance a polka.

A Fractal in Bach's Cello Suite

Johann Sebastian Bach surely did not have fractals in mind when he composed six suites for solo cello several centuries ago. Nonetheless, at least one movement has the repeating structure on different scales that is characteristic of a fractal.

Harlan J. Brothers of The Country School in Madison, Conn., contends that the first Bourrée in Bach's Cello Suite No. 3 provides a clear example of structural scaling. The recursive form of this musical structure can be visualized as a fractal construction called the Cantor set, he says.

Brothers' findings appear in the paper "Structural Scaling in Bach's Cello Suite No. 3," published in the March 2007 issue of the journal Fractals.

Examining only the written score, Brothers focused on the phrasing in the first section of the first Bourée. Musical phrasing refers to the way certain sequences of notes are naturally associated with each other, Brothers says.


Brothers detected repeated use of the pattern AAB on different scales, where each B section lasts twice as long as each A section.


Analysis of the first 16 measures of the Bourrée from Bach's Suite No. 3. Courtesy of Harlan Brothers.

For example, the piece starts off with two eighth notes and a quarter note (m1), repeats that pattern (m2), then continues with a phrase (m3) that is twice as long. The same pattern of short, short, long (s1) is repeated (s2), followed by a longer sequence (s3).

Analogously, the first eight measures are repeated, giving two "short" sections that are followed by a 20-measure "long" section.

"Interestingly, although Bach wrote the piece with a repeat symbol at the end of this 20-measure section, anecdotal evidence suggests that some cellists choose to perform it without the second repeat," Brothers noted in his paper. "Performed in this fashion, the Bourrée Part I exhibits a full four levels of structural scaling symmetry."

The structure of Bach's music resembles that of a classic type of fractal known as a Cantor set. Start with a line segment. Remove the middle third. Then remove the middle third from the remaining pieces, and so on. The result is a "Cantor comb."


Four levels in the creation of a Cantor comb. Courtesy of Harlan Brothers.

The hierarchical nesting of the AAB phrasing in the first Bourrée produces a similar pattern.

"The fact that Bach was born almost three centuries before the formal concept of fractals came into existence may well indicate that an intuitive affinity for fractal structure is, at least for some composers, an inherent motivational element in the compositional process," Brothers concluded.

Brothers has set about establishing a mathematical foundation for the classification of fractal music and correcting widespread misconceptions about fractal music. His efforts have revealed that musicians have been composing a form of fractal music for at least six centuries. One example is a type of canon in which different voices repeat the same melody or rhythmic motif simultaneously at different tempos.

Music can exhibit a wide variety of scaling behavior, Brothers says. He has himself written a number of compositions illustrating such properties. And he is keen to have others find further examples of scaling symmetry in what he describes as "the rich and vast body of musical expression."

Tom Lehrer's Derivative Ditties

Way back in the 1960s, one of my favorite TV shows was a satire program called That Was The Week That Was (also known as TW3), which aired weekly on NBC. It was the U.S. version of a late-night British series that was the talk of the country in the early 60s. The U.S. edition featured Elliot Reed, Alan Alda, Phyllis Newman, Henry Morgan, singer Nancy Ames, and, later, David Frost.

One of the program's highlights was its opening song, with lyrics that changed every week to reflect current news events. The show's resident songwriter was Tom Lehrer, who tackled and parodied such subjects as pollution ("Pollution"), Vatican II ("The Vatican Rag"), race relations ("National Brotherhood Week"), education ("New Math"), and nuclear proliferation ("Who's Next?"). A selection of these songs (uncensored) subsequently appeared on a record album titled That Was The Year That Was.

The fact that some of Lehrer's songs have mathematical elements reflects his background. Lehrer earned B.A. and M.A. degrees in mathematics from Harvard University and even entered Harvard's doctoral program, though he never completed his degree. In 1972, he came to the University of California, Santa Cruz, often teaching an introductory course titled "The Nature of Mathematics." He also taught a class in musical theater and would, on occasion, perform songs in his lectures.

In 1974, The American Mathematical Monthly published three Lehrer songs devoted to higher mathematics (mainly calculus). The May issue featured "The Derivative Song," sung to the tune of "There'll Be Some Changes Made," composed in 1922 by Billy Higgins and W. Benton Overstreet. It was originally performed as part of The Physical Revue, a show presented by the Harvard Physics Department in 1951 and 1952.

You take a function of x and you call it y,
Take any x-nought that you care to try,
You make a little change and call it delta x,
The corresponding change in y is what you find nex',
And then you take the quotient and now carefully
Send delta x to zero, and I think you'll see
That what the limit gives us, if our work all checks,
Is what we call dy/dx,
It's just dy/dx.

The June-July issue featured "There's a Delta for Every Epsilon (Calypso)." It was also performed earlier in The Physical Revue.

There's a delta for every epsilon,
It's a fact that you can always count upon.
There's a delta for every epsilon
   And now and again,
   There's also an N.

But one condition I must give:
The epsilon must be positive
A lonely life all the others live,
   In no theorem
   A delta for them.

How sad, how cruel, how tragic,
How pitiful, and other adjec-
Tives that I might mention.
The matter merits our attention.
If an epsilon is a hero,
Just because it is greater than zero,
It must be might discouragin'
To lie to the left of the origin.

This rank discrimination is not for us,
We must fight for an enlightened calculus,
Where epsilons all, both minus and plus,
   Have deltas
   To call their own.

"The Professor's Song" appeared in the August-September issue. It is sung to the tune of "If You Give Me Your Attention" from the Gilbert and Sullivan operetta Princess Ida.

If you give me your attention, I will tell you what I am.
I'm a brilliant math'matician — also something of a ham.
I have tried for numerous degrees, in fact I've one of each;
Of course that makes me eminently qualified to teach.
I understand the subject matter thoroughly, it's true,
And I can't see why it isn't all as obvious to you.
Each lecture is a masterpiece, meticulously planned,
Yet everybody tells me that I'm hard to understand,
   And I can't think why.

My diagrams are models of true art, you must agree,
And my handwriting is famous for its legibility.
Take a word like "minimum" (to choose a random word), (*)
For anyone to say he cannot read that, is absurd.
The anecdotes I tell get more amusing every year,
Though frankly, what they go to prove is sometimes less than clear,
And all my explanations are quite lucid, I am sure,
Yet everybody tells me that my lectures are obscure,
   And I can't think why.

Consider, for example, just the force of gravity:
It's inversely proportional to something — let me see —
It's r³ — no, r² — no, it's just r, I'll bet —
The sign in front is plus — or is it minus, I forget —
Well, anyway, there is a force, of that there is no doubt.
All these formulas are trivial if you only think them out.
Yet students tell me, "I have memorized the whole year through
Ev-rything you've told us, but the problems I can't do."
   And I can't think why!

(*) This was performed at a blackboard and the professor wrote: /VVVVVVVVVVVVVVV.

An earlier version of "The Professor Song," performed in The Physical Revue, has somewhat different lyrics.

Lehrer continued to return to mathematical themes in his songwriting. In 1993, for example, he composed the song "That's Mathematics!" for a special program, sponsored by the Mathematical Sciences Research Institute in Berkeley, to commemorate the proof by Andrew Wiles of Fermat's Last Theorem. At the Fermat Fest, Morris Bobrow performed not only "That's Mathematics!" but also "The Derivative Song" and "There's a Delta for Every Epsilon." And he rounded out the musical part of the program with a Danny Kaye classic, "The Square of the Hypotenuse," written by Saul Chapin and Johnny Mercer for the movie Merry Andrew. The Fermat Fest program is available as a video from MSRI.

References:

2003. Stop clapping, this is serious. Sydney Morning Herald (March 1).

Lehrer, T. 1974. The professor's song. The American Mathematical Monthly 81(August-September):745.

______. 1974. There's a delta for every epsilon (calypso). The American Mathematical Monthly 81(June-July):612.

______. 1974. The derivative song. The American Mathematical Monthly 81(May):490.

Peterson, I. 2000. Square of the hypotenuse. MAA Online (Nov. 27).

Purdom, T.S. 2000. Still a sly wit, now mostly for himself. New York Times (July 16).