OPINION Interfacing Music and Mathematics: A Case for More Engagement Lawrence C. Udeigwe The opinions expressed here are not necessarily those of the Notices or the AMS. Introduction Music is as old as humankind and, since the time of an- cient civilization, there have been knowledge seekers inter- ested in exploring various elements of music. Mathemat- ics, in both elementary and advanced forms, has been used in the analysis of lower-level elements of music such as tempo, pitch, timing, chord formation, and meter. Music- related physical phenomena, such as acoustics, and their implication on the development of musical instruments have readily employed mathematics for analysis. Essential information-age appliances, such as computers and cell phones, that have played a substantial role in the dissem- ination and preservation of music rely heavily on mathe- matics for their implementation and continual improve- ment. These are but a few areas of interface between math- ematics and music, yet the subject of “mathematics of mu- sic,” which I will henceforth refer to as musical mathemat- ics, has been relegated to the fringe and seems to only be touchable to bold algebraists and number theorists who do not mind being labeled hobbyists. The assertion of this Lawrence C. Udeigwe is an associate professor of mathematics at Manhattan College and an MLK Visiting Associate Professor in Brain and Cognitive Sci- ences at the Massachusetts Institute of Technology (MIT). His email address is lawrence.udeigwe@manhattan.edu. For permission to reprint this article, please contact: reprint-permission@ams.org. DOI: https://doi.org/10.1090/noti2610 article is that from the classroom through private and gov- ernment organizations interested in STEM, there is a need to create more room for the exploration of the interface between music and mathematics. There is Historical Precedent for Engagement The Pythagoreans in ancient Greece were the frst re- searchers to link musical scales and the principles of num- bers [Pla74], although there are records showing that the ancient Chinese, Indians, and Egyptians studied the math- ematical principles of sound [Bri87]. Pythagoras’s apoc- ryphal experiment with vibrating strings showed him that if he plucked two strings of equal tension, in which the length of one is of a certain proportion to the length of the other, he would get a certain type of harmonious sound, referred to today as consonance. In particular, if the ratio of the lengths of the two strings is 2:1, one gets an octave, that is, two of the same note with the pitch frequency of the shorter string being double that of the longer string; if their ratio is 3:2, the pitch frequency of shorter one is 1 1 2 times that of the longer one, and their combined sound— simultaneously or temporally apart—is referred to as a per- fect ffth; and if their ratio is 4:3, their combined sound is referred to as a perfect fourth. Pythagoras’s vibrating string experiment ushered the incorporation of musical sound into the philosophical framework of the Pythagoreans, ul- timately helping to shape their central doctrine that “all na- ture consists of harmony arising out of numbers” [Jea68]. FEBRUARY 2023 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 309