Music and Mathematics Harmony

Pythagoras was quoted as saying, “there is geometry in the humming of the strings, there is music in the spacing of the spheres.” As poetic as this sounds, the famous Greek mathematician was actually making a direct statement about the connection between mathematics and music. In modern times, math is a subject generally associated with left brained individuals and music is a subject generally associated with right brained individuals. What most don’t realize is that the subjects go hand in hand, and have been intertwined as early as the times of Greek antiquity; “great minds took such pains to include music in their worldview and indeed saw music as the organizing principle of the universe” [rogers]. Viewing music and science as profoundly linked was the dominant and accepted way of thinking in Western and non-Western philosophers of the past.

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Mathematics and music are both subjects that require an abstract way of thinking and contemplation. Both subjects require recognizing and establishing patterns. It is important to note that although the subjects on a whole are more similar than usually given credit for in this day and age, they are also interdisciplinary. Math has historically been used to describe and teach music, and vice versa. Mathematics can be found etched in common musical concepts such as scales, intervals, wave frequencies, and tones. J. Ph. Rameau, a French musicologist of the eighteenth century, said it best in his Traitd de l’harmonie rdduite d ses principes naturels (1722): “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 which I have acquired in music by practising 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.” [Papadopoulos]

In Greek antiquity, it was common knowledge that the schools of Aristotle, Plato, and Pythagoras deemed music a part of mathematics. A Greek mathematical treatise would typically be comprised of four topics: Number Theory, Geometry, Music, and Astronomy. Mathematics and music were strongly linked until the Renaissance, when the two subjects diverged – theoretical music becoming an independent field. Pythagoras is recognized as the first music theorist. His greatest discovery dealt with the relation of musical intervals with ratios of integers. The story is: on a trip through a brazier’s shop, Pythagoras took note of the different sounds being produced by the hammers on anvils. He realized that the pitch he was hearing depended only on the weight of the hammer. The place the hammer hit the anvil, the angle it was hit, the magnitude of the stroke – none of these factored into the pitch. This lead him to ponder about the relationship between two notes produced by two different hammers. In classical Greek music there were intervals of the octave, fifth, and fourth. He recognized that the consonant musical intervals the hammers were creating corresponded, in terms of weights, to the numerical fraction 2/1, 3/2, and 4/3, respectively.

Pythagoras discovered that musical intervals, and hence all harmony, are based on mathematical ratios, ratios that also, amazingly, appear in astronomy [rogers]

Thus, Pythagoras thought that the relative weights of two hammers producing an octave is 2/1, and so on. As soon as this idea occurred to him, Pythagoras went home and performed several experiments using different kinds of instruments, which confirmed the relationship between musical intervals and numerical fractions Papadopoulos

musical theory constructed by Pythagoras. Two sounds from the same taut string are said to be consonant when they are pleasing to listen to simultaneously. In the Greek cultural arena of that period such sounds are produced by lengths of string that are inversely proportional to the numbers 1, 2, 3, and 4. These compose the famous Tetraktys (1 + 2 + 3 + 4 = 10), a diagram of figured numbers symbolising pure harmony, the “vertical hierarchy of relation between Unity and emerging multiplic Perrine

Pythagoreans considered a collection of vases, filled partially with different quantities of the same liquid, and observed on them the “rapidity and the slowness of the movements of air vibrations.” By hitting these vases in pairs and listening to the harmonies produced, they were able to associate numbers to consonances. The result is again that the octaves, fifths, and fourths correspond respectively to the fractions 2/1, 3/2 and 4/3, in terms of the quotients of levels of the liquid.  Papadopoulos

 rich musical evolution flowing from the Greek roots into the Latin world and right up to the fourteenth century of our era. In St. Augustine’s De Musica, written at the end of the fourth century, rhythms are also classified according to their proportions (the proportional notation used today came much later). Then in the ninth century,  Perrine

Carolingian policy in educational and ecclesiastical matters defined new practices. It encouraged the use of neumes that indicate the inflexions of the voice, but not the pitch of the sounds. The names Do, Re, Mi, Fa, Sol, etc., appeared with Guido d’Arezzo in the eleventh century, derix4ng from the syllables at the beginning of the stanzas (voces) of a hymn addressed to St John the Baptist, written around 770 A.D. The notes (claves) are also designated by letters, a practice that is still in use today in Englishspeaking countries (La = A, Ti = B, Do = C, . . . ) and in Germany (with some specificities). Finally, polyphony created new needs for harmonic mastery, the response coming from Philippe de Vitry in the fourteenth century with his Ars Nova: in this work he defined new musical notations as well as new ways of combining rhythms. However, this culmination of the pythagorean musical base that had developed over many centuries eventually degenerated in the following century because it proved to be inadequate for responding to the new aesthetic trends that were appearing as well as the practical needs of musicians Perrine

 Music makes use of a symbolic language, together with a rich system of notation, including diagrams which, starting from the eleventh century (in the case of Western European music), are similar to mathematical graphs of discrete functions in two-dimensional cartesian coordinates (the x-coordinate representing time and the y-coordinate representing pitch). Music theorists used these “cartesian” diagrams long before they were introduced in geometry. Musical scores from the twentieth century have a variety of forms which are close to all sorts of diagrams used in mathematics. Besides abstract language and notation, mathematical notions like symmetry, periodicity, proportion, discreteness, and continuity, among others, are omnipresent in music. Lengths of musical intervals, rhythm, duration, tempi, and several other musical notions are naturally expressed by numbers.  {Papadopoulos}

Logarithms The arithmetic of musical intervals involves in a very natural way the theory of logarithms.Pythagoras defined the tone as the difference between the intervals of fifth and of fourth.  The point now is that the fraction associated to the tone interval is not the difference 3/2 – 4/3, but the quotient (3/2)/(4/3) = 9/8. It is natural to define the compass of a musical interval as the number (or the fractions of) octaves it contains. Thus, when we say that two notes are n octaves apart, the fraction associated to the interval that they define is 2 n. The definition of the compass can be made in terms of frequency, and in fact one usually defines the pitch as the logarithm in base 2 of the frequency. (Of course, the notion of frequency did not exist as such in antiquity, but it is clear that the ancient Greek musicologists were aware that the lowness or the highness of pitch depends on the slowness or rapidity of the air vibration that produces it, as explained in Theon’s treatise [12], Chapter XIII.) The relation of musical intervals with logarithms can also be seen by considering the lengths of strings (which in fact are inversely proportional to the frequency). For instance, if a violinist (or a lyre player in antiquity) wants to produce a note which is an octave higher than the note produced by a certain string, he must divide the length of the string by two. Thus, music theorists dealt intuitively with logarithms long before these were defined as an abstract mathematical notion. (It was only in the seventeenth century that logarithms were formally introduced in music theory, by Isaac Newton, and then by Leonhard Euier and Jacques Lambert.) The theory of musical intervals is a natural example of the practical use of logarithms, an example easily explained to children, provided they have some acquaintance with musical intervals. {Papadopoulos}

Today the Music of corpuscles and solitons is taking the place of the Music of spheres and mermaids. Considerations of the multiple infinitely small (chaos?) are replacing those on the single infinitely great (the cosmos?). The bifurcation took place at the end of the eighteenth century, at the very moment when musicians were being pushed into the category of artists, whose role was to provide pleasure for the present, and mathematicians into the category of scientists, building the society of the future.  Perrine

Rogers, G. L. (2016). The Music of the Spheres. Music Educators Journal, 103(1), 41-48. doi:10.1177/0027432116654547

 Papadopoulos, A. (2002). Mathematics and music theory: From pythagoras to rameau. The Mathematical Intelligencer, 24(1), 65-73. doi:10.1007/BF03025314

Perrine, S. (2005). Mathematics and music. a diderot mathematical forum. The Mathematical Intelligencer, 27(3), 69-73. doi:10.1007/BF02985844