About Math 'N Music Notes A story is told about the famous physisict Albert Einstein, who was an accomplished amateur violinist. A charity concert he gave was reviewed by a critic, who said about his performance: "He did tolerably well, but I cannot understand how his name comes to be so well known all over the world!" The fact that the critic wasn't aware that he was an amateur musician says something about the quality of Einstein's musicianship! Musicians and mathematicians both recognize the intimate connection between math and music. Many of the best mathematicians played music in their leisure time and wrote books about music theory. Many of the great composers dabbled in mathematics. A look through a book on music theory turns up the names of many famous mathematicians. Each spring for the past two years Northwood Elementary School PTO has put on a Math and Music Night. The evening begins with a half hour band and orchestra concert, which is followed by an hour of math activities. The instrumental music directors Helen Park and Mary Peterson have made an effort to include in their programs discussion of the mathematical aspects of the music they are performing, and I have created music-related activities for the mathematics part of the evening. This is one of them. This is actually a collection of mathematical-musical facts on a poster, together with a hands-on physics activity which demonstrates the connection between music and ratios, something discovered by the Pythagorean school of ancient Greece. (The story about Einstein is from "Music and Mathematics" by H. S. M. Coxeter in The Mathematics Teacher March 1968 p. 312) The activity All advanced string players know that it's possible to make an eerie sound on the violin by touching the string lightly at certain points while bowing. They call this "playing harmonics". In the activity associated with this module the children find harmonics and understand some of the physics of harmonics. They locate the points at which harmonics sound, measure the distance from the end of the string to the point they touch and compute the ratio of total length to distance to the end. The shorter length should always evenly divide the total string length. The equipment I put together was a simple plywood box with a violin string. I used an old violin string supplied by one of the Suzuki violin teachers in Ames. The string was tied at one end to a screw and at the other end to a turnbuckle so tension could be applied. A couple of tiny scraps of wood serve as bridges (ends of the string). A 36 cm long scale (in Printable Materials) was glued to the box under the string so that it was easy to read the finger position. The bridges were placed (not glued) at the ends of the scale (at 0 and 36 cm). This deluxe version, the plywood box, works a bit better than the quick and easy version--a cardboard box and a rubber band attached at the ends with paper fasteners. The quick and easy one does work also and will do in a pinch. A 36 unit long scale is ideal because 36 is divisible by 2, 3, and 4--all the harmonics you can hear occur at integer locations on the scale. Here's a photo of the equipment. The biggest difficulty with this activity is background noise: one has to listen very carefully for the harmonics. The "Notes" Mozart and the Golden Ratio The golden ratio is a number which appears in many contexts: nature, art, music, mathematics. This is how it is defined. If you cut a line segment into a short part and a long part, the ratio of the long part to the short part is golden if it is equal to the ratio of the length of the original segment to the longer of the two parts of the original segment. With a little algebra you can figure out that the ratio is equal to (1 + 5 1/2)/2. This ratio was studied by the ancient Greeks: it is discussed by Euclid in the Elements. Artist Salvador Dali and composers Bela Bartok and Claude Debussy used the golden ratio quite deliberately and explicitly in their work. We know because they wrote about "the divine ratio". We know from his sister's writing that Mozart loved mathematics and number games. Did he consciously use the golden ratio? He didn't say, so we'll never know for sure! Book: The Golden Ratio: The Story of Phi, the World's Most Astonishing Number by Mario Livio Article: "Did Mozart Use the Golden Section?" by Mike May in American Scientist 84:118 Website: Fibonacci Numbers and The Golden Section in Art, Architecture and Music by R. Knott of the University of Surrey Galileo The picture, of course, brings to mind Galileo's famous experiment in which he drops two balls of different weights from the Tower of Pisa in order to show that they will fall at the same rate. This was not his only experiment in gravity. He did experiments on balls rolling down an inclined plane, concluding with a law of acceleration: the speed of an object accelerating due to gravity increases equal amounts in equal times. How did he manage the precise timing needed for these experiments? Clocks wouldn't answer--they measured seconds precisely, not fractions of seconds. Galileo's father was a professional musician and experimented with the science of harmonics. He had piano wires strung across the rooms in his house for pioneering experiments on the scale of equal temperament. Galileo himself was an accomplished musician and carried on his father's work on the musical scale. For his gravity experiments historian Stillman Drake speculates that he had assistance in his experiments from family members or friends playing fast music at a very precise tempo, which would produce the accurate relative timing which he needed to measure . Sources: "The Role of Music in Galileo's Experiments" by Stillman Drake in Scientific American 232 p. 98-104 Galileo's Daughter by Dava Sobel (The first chapter describes Galileo's upbringing in a musical household.) Website: Vincenzo Galilei by Albert Van Helden The picture comes from The Leaning Tower of Pisa Gallery http://www.endex.com/gf/buildings/ltpisa/ltpgallery/22nov02/ltpgallery22Nov02.htm by Gary Feuerstein of Endex Engineering, and has been slightly modified for the display. Permission to use this photo for the display is greatly appreciated. Music is Like Clock Arithmetic In clock arithmetic (mod 12) you add 5 + 9 to get 2 mod 12. (see the Clock Arithmetic module). In music if you start at F, the fifth half step above C, and go up chromatically 9 more half steps, you reach D, the second half step above C. Octaves sound so much alike that they have the same note name. Common Multiples in Music This is a very simple treatment of rhythmic ratios. When you move away from Western European music, you can reach a mathematically rich rhythmic complexity! Janet Sharp, Anthony Stevens, and Becky Nelson published a wonderful lesson in ratios in African drumming, in which a class of elementary children learn some basics of African rhythms, experiment with drumming, and construct the concept of LCM from the rhythmic counterpoint. The lesson is in Teaching Children Mathematics vol 7 no. 6, p. 376. Supereven Numbers in Music Our perception of pitch is logarithmic. Taking a pitch up an octave always represents a doubling of the frequency. Linear perception of scale, exponential frequency increase. Ratios in Music The Pythagorean school attached mystical significance to harmonies and the associated frequency ratios. They were thought to govern everything from music to the motion of the planets. Pythagorus is supposed to have said: "There is geometry in the humming of the strings. There is music in the motion of the spheres." Astronomer Johann Kepler carried this even further & constructed a musical universe with a structure based on the Platonic solids, in which the planets play tunes based on these pleasing harmonic intervals (as well as discovering the physical equations of planetary motion). The ratios of pitches are simply a result of physics, which math night attendees can investigate in the harmonics activity. The frequencies shown on the poster are the frequencies of the pitches of the scale of just temperament; the scale that results from the physics of harmonics (also known as overtones). These are not the frequencies you would find on a properly tuned piano, which have been adjusted so that music sounds the same in all keys, but the frequencies you would hear from, for instance, the overtones on a trumpet. Sources: "Music and Mathematics" by H. S. M. Coxeter in The Mathematics Teacher March 1968 p. 312. A History of Mathematics: an Introduction 2nd ed. by Victor Katz A Bibliography of Recreational Mathematics by William Schaaf contains the quote attributed to Pythagorus. Written 7/15/03 by Janet A. Dixon Revised 1/10/06