This week, we’re brushing up on our times tables to explore the vertex between math and music. We believe you're greater than or equal to the task for a show we're calling “Do The Math.”
Get ready to divide and conquer our mathematical playlist below:
- Bach's Musical Offering to mathematicians – J.S. Bach’s A Musical Offering, a piece written as a gift for King Frederick II of Prussia, contains so much exacting complexity that it’s almost mathematical in its precision. In fact, mathematicians are often drawn to this work just for this reason. For instance, the American Mathematical Society was able to write out the canons from A Musical Offering as mathematical functions. And Bach’s “Ricercar a 6,” the most complex movement from the suite, plays a central role in the popular 1979 book Gödel, Escher, Bach: An Eternal Golden Braid by IU Professor Douglas Hofstadter. This fascinating book compares the musical precision of Bach to the logic-bending worlds of artist M.C. Escher and the puzzles of mathematical logician Kurt Gödel to explore the philosophy of intelligence. However, Bach himself had no proclivity towards the numerical arts. His son C.P.E. Bach once said that his father was, quote, “like myself or any true musician, no lover of dry mathematical stuff.”
- Elgar's "Enigma" is pi? – Edward Elgar’s reputation in his native England was cemented after the 1899 premiere of the Enigma Variations. These 14 variations on an original theme were composed as musical portraits of Elgar’s close friends and acquaintances, with each movement’s title being the initials of that friend. The primary“Enigma” at the center has to do with the meaning of that original theme. Although the mystery has not been definitively solved, the prevailing wisdom is that the theme might be counterpoint to some other mysterious melody. Other theories abound. Some have even speculated that there may be a mathematical explanation, that the “Enigma” is the numeral pi. There’s not much evidence to this, except for the fact that the scale degrees of the opening of the “Enigma” melody are 3-1-4 (B-flat, G, and C in G minor), as in 3.14...
- The mathematical precision of isorhythmic motets – Music and math were directly linked for hundreds of years before the disciplines drifted apart in the Romantic Era. In the medieval and Renaissance periods especially, geometry, arithmetic, astronomy and music made up the Quadrivium, the four tenets of philosophical study. In music composition during this time period, this relationship manifested itself as the isorhythmic motet, a musical work where specific rhythmic patterns and melodic motives are related by mathematical ratios. Dufay’s “Nuper Rosarum Flores” is one of the most famous isorhythmic motets. The work was first performed for the consecration of the Cathedral of St. Mary of the Flower in Florence and for the famous dome constructed by architect Filippo Brunelleschi. Some scholars have speculated that maybe the mathematical ratios governing the piece were somehow tied to the architecture of the building itself, whereas others have speculated that the ratios were constructed to emulate the biblical depiction of the Temple of Solomon.
- Bartók and the Golden Ratio – It was music theorist Ernő Lendvai who discovered that certain mathematical principles—namely the Fibonacci Sequence and the related concept of the Golden Ratio—seem to dictate the structure of Béla Bartók’s work Music For Strings, Percussion and Celesta. Fibonacci numbers are a sequence of specific numbers where each number is the sum of the previous two, so 1, 1, 2, 3, 5, 8, 13, 21 and so on. Bartók (likely) used the Fibonacci numbers to determine the opening rhythmic pattern in the xylophone part of this third movement. There’s a property of Fibonacci numbers, where the ratio of any two adjacent numbers, the further along the sequence you get, approaches the so-called “golden ratio,” or about 0.618. This mathematical ratio shows up in nature—you’ll find it governing the structure of nautilus shells and pine cones. And the golden ratio can also be used to describe the structure of the first movement of Music For Strings, Percussion and Celesta.
- Brahms and the mathematical insult – At the turn of the 20th century, the music of Johannes Brahms from just a few years before was already considered by many to be too old fashioned and too conservative. Take this string sextet, written in the 1860s for Brahms’s friend, violinist Joseph Joachim to perform. At the time, it was a lovely contemporary piece of chamber music, but forty years later, a London critic reviewing a performance of it described it using the harshest epithet he could hurl: “mathematical.” This reviewer called Brahms’s sextet a piece of “musical trigonometry” that Brahms wrote, quote, “apparently in order to prove how excellent a mathematician he might have become.” The reviewer goes on to say that instead of feeling art in the piece, quote “you feel an undercurrent of irrational numbers, of quadratic equations, [and] of hyperbolic curves.”
- Xenakis's and geometry, statistics, set theory, game theory, etc... – Iannis Xenakis’ first professional occupation was that of an assistant to the famed French architect, Le Corbusier. When Xenakis began to compose, he had little musical training. He considered starting his music education from scratch, but composer Olivier Messiaen dissuaded him by encouraging him to use his knowledge of architecture and mathematics in his music. Xenakis’s approach to composition was entirely novel, and entirely mathematical, as a result. His first major work Metastasis was composed by graphing geometric spaces onto the parameters of pitch and time. His work Diamorphoses applied statistical distribution to music, Nomos Alpha applied group theory, Eonta applied set theory, Analogique applied Markov chains, and Stratagie applied game theory. These concepts, while familiar to most high-level mathematicians, were brand new to most composers.
- Milton Babbitt and the family business – Milton Babbitt wrote that understanding advanced music was akin to learning mathematics, philosophy, and physics. His association with these complex fields was as much a part of his heritage as it was academic; Babbitt’s father and brother were both professional mathematicians. Milton himself intended to study the subject, but he switched to music while at university (although he did teach mathematics briefly at Princeton, in addition to music). Unsurprisingly, he quickly became interested in the serial music of Schoenberg and the Second Viennese School, who based their approach to music composition on abstract mathematical processes. Babbitt took this connection between math and music to a new level. Whereas Schoenberg generally confined his twelve-tone formulas to musical pitches, Babbitt applied it to all aspects of musical composition, including dynamics, duration, register and expression markings. Babbitt believed his approach was a direct descent of the Second Viennese School, and thus referred to his style as Total Serialism.
- Michael Harrison and just intonation – As we've discussed, the crossover between mathematics and music dates back many, many centuries. In the writings of Plato, music was grouped not with the liberal arts of grammar and rhetoric, but rather with the more scientific studies of arithmetic, geometry, and astronomy. The study of music was concerned primarily with a study in tuning, understanding the perfect ratios that exist between perfect intervals. In our modern equally-tempered tuning system, these perfect ratios are made slightly irrational to allow for greater movement between keys. However, some composers continue to write using these mathematical pure ratios, a tuning system called “just intonation.” Much of the music of the award-winning contemporary composer Michael Harrison uses just intonation, creating harmonies that ring with beautiful clarity while evoking ancient sounds, like in this piece Revelation. But to achieve that sound, Harrison had to retune the piano to achieve the desired effect.
- Tom Lehrer's "New Math" – In 2010, many states adopted new educational standards known as Common Core. While these new standards, backed by meticulous research, helped students gain a better understanding of reading, writing, and arithmetic, they were criticized mostly by flummoxed parents who realized that suddenly they were no longer able to help with their child’s math homework. This idea of an older generation being perplexed by the younger generation’s approach to education is a tale as old as time, as evidenced by the 1965 Tom Lehrer song “New Math.” Although Lehrer was mostly a musical parodist, he was no slouch when it came to mathematics. He got a bachelor’s and master’s degree in math from Harvard University, but gave up on his doctoral degree in math around the time he released this song. Lehrer ended up teaching the subject at both MIT and University of California, Santa Cruz.