Connections Between Mathematics and Music

The two disciplines have been interlinked throughout history since Ancient Greek academics began their theoretical study; since antiquity, mathematicians have often been music theorists. The fascination that mathematicians have with music will then be discussed.

MUSIC THEORISTS AND MATHEMATICIANS : ARE THEY 

ONE  IN THE  SAME ?

For about a millennium, from 600 BC, Ancient Greece was one of the worldʼs leading civilizations. The ideas and knowledge produced at this time have had a lasting influence on modern western civilizations. The “Golden Age” in Greek antiquity was approximately 450 BC, and much of what constitutes western culture today began its invention then . Brilliant Greek academics contributed a wealth of knowledge about music, philosophy, biology, chemistry, physics, architecture and many other disciplines. With the Ancient Greeks came the dawn of serious mathematics. Before their time, mathematics was a craft . It was studied and used to solve everyday problems. For example, farmers might implement mathematical tools to help them lay their fields in the most economical way possible. In Greek antiquity, mathematics became an art. It was studied purely for the sake of knowledge and enjoyment. Philosophers and mathematicians questioned the fundamental ideas of mathematics.

Pythagoras, Plato and Aristotle were three very clever academics, and very influential figures when detailing the historic connection between mathematics and music . Pythagoras was born in the Classical Greek period (approximately 600 BC to 300 BC) when Greece was made of individual city-states. A dictator governed the island on which he lived, so he fled to Italy. It was there that he founded a religion (often called a cult) of

mathematics. Pythagoreans , the followers of his religion, believed mathematical structures were mystical. They had elaborate rituals and rules based on mathematical ideas. To the followers, the numbers 1, 2, 3 and 4 were divine and sacred. They believed reality was constructed out of these numbers and 1, 2, 3 and 4 were deemed the building blocks of life . Pythagoras was instrumental in the origin of mathematics as purely a theoretical science. In fact, the theories and results that were developed by Pythagoreans were not intended for practical use or for applications. It was forbidden for members of the Pythagorean school of thought to even earn money from teaching mathematics . Throughout history, numbers have always been the building block of mathematics .

Plato was a Pythagorean who lived after the Golden Age of Ancient Greece. Plato believed that mathematics was the core of education . He founded the first university in Greece, the Academy. Mathematics was so central to the curriculum, that above the doors of the university, the words “Let no man enter through these doors if ignorant of geometry” were written . From antiquity, many famous Greek mathematicians attended Platoʼs university. Aristotle, the teacher of Alexander the Great, is an example of a famous student of Plato. Aristotle was a man of great genius and the father of his own school. He studied every subject possible at the time. His writings had vast subject matter, including music, physics, poetry, theatre, logic, rhetoric, government, politics, ethics and zoology. Together with Plato and Socrates (Platoʼs teacher), Aristotle was one of the most important founding figures in western philosophy. He was one of the first to create a comprehensive system detailing ideas of morality, philosophy, aesthetics, logic, science, politics and metaphysics . A natural question now arises: why are these ancient figures so important in understanding the relationship between mathematics and music? The answer is simple. It was these early Greek teachers and their schools of thought (the schools of Pythagoras, Plato, and Aristotle) who not only began to study mathematics and music, but considered music to be a part of mathematics . Ancient Greek mathematics education was comprised of four sections: number theory, geometry, music and astronomy; this division of mathematics into four sub-topics is called a quadrivium . Itʼs been previously stated that the ideas and works of the Ancient Greeks were influential and had had a lasting effect throughout history. Those of music and mathematics were no different. The four way division of mathematics, which detailed music should be studied as part of mathematics, lasted until the end of the middle ages (approximately 1500 AD) in European culture. 

The Renaissance (meaning rebirth), a period from about the fourteenth to seventeenth centuries, began in Florence in the late middle ages and spread throughout Europe. The Renaissance was a cultural movement, characterized by the resurgence of learning based on classical sources, and a gradual but widespread educational reform. Education became heavily focused rediscovering Ancient Greek classical writing about cultural knowledge and literature . Music was no longer studied as a field of mathematics. Instead, theoretical music became an independent field, yet strong links with mathematics were maintained . It is interesting to note that during and after the Renaissance, musicians were music theorists, not performers. Music research and teaching were occupations considered more prestigious than music composing or performing . This contrasts earlier times in history. Pythagoras, for example, was a geometer, number theorist and musicologist, but also a performer who played many different instruments.

In the seventeenth and eighteenth centuries, several of the most prominent and significant mathematicians were also music theorists . René Descartes, for example, had many mathematical achievements include creating the field of analytic geometry, and developing Cartesian geometry. His first book, Compendium Musicale (1618) was about music theory . Marin Mersenne, a mathematician, philosopher and music theorist is often called the father of acoustics. He authored several treaties on music, including Harmonicorum Libri (1635) and Traité de lʼHarmonie Universelle (1636) . Mersenne also corresponded on the subject with many other important mathematicians including Descartes, Isaac Beekman and Constantijn Huygens .

John Wallis, an English mathematician in the fifteenth and sixteenth centuries, published editions of the works of Ancient Greeks and other academics, especially those about music and mathematics . His works include fundamental works of Ptolemy (2 AD), of Porhyrius (3 AD), and of Bryennius who was a fourteenth century Byzantine musicologist . Leonhard Euler was the preeminent mathematician of the eighteenth century and one

of the greatest mathematicians of all time. While he contributed greatly to the field of mathematics, he also was a music theorist. In 1731, Euler published Tentamen Novae Theoriae Musicae Excertissimis Harmoniae Princiliis Dilucide Expositae . In 1752, Jean dʼAlembert published works on music including Eléments de Musique Théorique et Pratique Suivant les Principes de M. Rameau and in 1754, Réflexions sur la Musique . DʼAlembert was a French mathematician, physicist and philosopher who was instrumental in studying wave equations. 

WHY ARE MATHEMATICIANS SO FASCINATED BY MUSIC THEORY ?

Mathematicians fascination with music theory are explained clearly and precisely by Jean Philippe Rameau in Traité de lʼHarmonie Réduite à ses Principes Naturels (1722). Some musicologists and academics argue that Rameau was the greatest French music theorist of the eighteenth century [4]. Rameau said:

“Music is a science which must have determined rules. These rules must be drawn from a principle which should be evident, and this principle cannot be known without the help of mathematics. I must confess that in spite of all the experience I have acquired in music by practicing it for a fairly long period, it is nevertheless only with the help of mathematics that my ideas became disentangled and that light has succeeded to a certain darkness of which I was not aware before.” 

Mathematicians have been attracted to the study of music theory since the Ancient Greeks, because music theory and composition require an abstract way of thinking and contemplation . This method of thinking is similar to that required for pure mathematical thought . Milton Babbitt, a composer who also taught mathematics and music theory at Princeton University, wrote that “a musical theory should be statable as connected set of axions, definitions and theorems, the proofs of which are derived by means of an appropriate logic” .

Those who create music use symbolic language as well as a rich system of notation, including diagrams . In the case of European music, from the eleventh century, the diagrams used in music are similar to mathematical graphs of discrete functions in two dimensional Cartesian coordinates . The x-axis represents time, while the y-axis represents pitch. The Cartesian graph used to represent music was used by music theorists before they were introduced into geometry . In fact, many musical scores of twentieth century musicians have many forms that are similar to mathematical diagrams. At the beginning of a piece of music, after the clef is marked, the time signature is marked by a fraction on the music staff . Common time signatures include 2/4, 3/4, 4/4. and 6/8. The denominator of the fraction, is the unit of measure, and used to denote pulse. The numerator indicates the number of these units or their equivalent included in the division of

a measure. Groups of stressed and relaxed pulses in music are called meters. The meter is also given in the numerator of the time signature. Common meters are 2, 3, 4, 6, 9, 12 which denote the number of beats or pulses in the measure . For example, take the time signature 3/4. Each measure is equivalent to three (information from the numerator) quarter notes (information from the denominator). The count in each measure would be: 1, 2, 3. The 1 is the stressed pulse, while the 2 and 3 are relaxed. The time signature 3/4 is common in waltzes . Besides abstract language and notation, mathematics concepts such as symmetry, periodicity, proportion, discreteness, and continuity make up a piece of music . Numbers are also very instrumental, and influence the length of a musical interval, rhythm, duration, tempo and several other notations . The two fields have been studied in such unison, that musical words have been applied to mathematics. For example, harmonic is a word that is used throughout mathematics (harmonic series, harmonic analysis), yet its origin is in music theory . Itʼs been discussed that throughout history, mathematicians have long been fascinated with music theory. This concept will be further developed in the, which suggest mathematics is, like music, a form of art.