Music, Mathematics, and Music Education
D. L. Stieg

The importance of musical training in an effective and well-rounded approach to general education has been demonstrated repeatedly in numerous recent studies. A definite connection has been established between musical training and a greater aptitude for mathematics and science. An interesting and telling corollary to this connection is given by the fact that most successful people, understood in a broader sense than in commonly accepted measures of success like wealth or fame, are competent amateur musicians. It is now a commonly accepted fact that playing music and doing mathematics are very similar, but the exact nature of the connection between the two is nevertheless seldom discussed.

Music historians and music theorists have weighed in on the interconnection of music and mathematics by means of a traditional emphasis on Pythagorean principles and the theory of intervals. Pythagoras was an ancient Greek who devised exact mathematical proportions to describe pitch and to describe the properties of a vibrating string. These mathematical formulations were used extensively in the development of Western musical instruments, including most of the instruments still in existence today. The theory of intervals is also closely related to mathematics, since numbers are used to express the relative size of the interval or distance between two notes (for example, the interval of a 6th is larger than the interval of a 4th). The stacking of notes above one another in 3rds to form chords is the theoretical foundation of the Major-minor system, which is the harmonic basis of Western music. But while the importance of Pythagorean mathematics and the intervalic basis of Western harmony are clear enough, the more fundamental truth is that music itself is mathematical. To be more specific, quite apart from the mathematical properties of components of the theory of music, music itself is geometric and algebraic.

The geometry of music can most easily be observed in the linear quality of melody, the basic unit of all music. It is not by accident that melodies are commonly referred to as melodic lines. Monophonic (one note at a time) parts of ensemble arrangements are also commonly referred to as lines (for example, the tenor line of a choral arrangement, or the bass line of a rock arrangement). The three characteristics of a line (a start point, a direction of movement, and an end point) are equally descriptive of melodies and monophonic parts, and in a wider sense are descriptive of the development of musical ideas in general. In music there is always a sense of linear motion and forward movement, and a sense of resolution as a result of that movement. It is the sense of purposeful forward movement that distinguishes a melody from a random sequence of notes, and this same sense is an essential component of the elusive definition of musicianship. Competent musicians understand intuitively that melodies and music are not merely sequences of notes, but rather are an expression of ideas and purpose that is framed and defined by forward movement, and are therefore fundamentally linear in character.

The linear quality of music suggests the representation of rhythm along a single line. Rhythm refers to the timing and duration of notes. Rhythm in turn is a function of meter, which is the progression of evenly timed beats about which music is fashioned. For most music, the beats of meter are grouped into measures containing a uniformly equal number of beats, most commonly four beats per measure, and less commonly three or six beats per measure. By representing the evenly timed beats of meter with evenly spaced marks along a line, the rhythmic timing of the notes of a melody can be indicated spatially and in geometric proportion relative to this sequence of evenly spaced beat marks.

Since most of the notes of simple melodies occur on the beats (coincident with beats), the spatial placement of these notes in vertical alignment with the corresponding beats effectively indicates their rhythmic timing. Apart from occurrence on the beats, the rhythmic timing of notes is determined, and can be represented in geometric proportion, according to how beats are subdivided. Apart from occurrence on the beats, the most common incidence of rhythmic timing is halfway between beats, which results from a binary division (division into two halves) of the beat. Quicker rhythms sometimes require the binary division of one or both half-beats, thus producing a quartering of the beat (in staff notation, a quartering of the beat is indicated by the use of 16th notes).

The altogether different rhythmic timing of notes for music in compound meter (9 or 12) results from a ternary division (division into thirds) of the beat, and can likewise be represented in geometric proportion along a line and in relation to beat marks. Music in 9 is actually in 3, but with ternary division of the beats, and music in 12 is actually in 4, again with ternary division of the beats. Since music in 6 is actually in 2 with ternary division of the beats, 6 is also a compound meter. Even complex rhythms like triplets (3 against 2) or duplets (2 against 3) are characterized by geometric proportions that can easily be represented along a line in relation to evenly spaced beat marks. The duration of notes can likewise be indicated by the use of marks or symbols that can be placed along the line at the exact points in time at which notes must be terminated.

A melody is defined by the pitches and rhythms of the notes of which it is comprised. Like rhythm, pitch can be indicated with relative ease along a single line. It is therefore possible to graph a melody, in much the same manner in which an algebraic function is graphed, by indicating rhythm on a horizontal axis and pitch on a vertical axis. The resultant graph would have little practical purpose, except possibly to show the basic contours of the melodic line, which in any case can be readily observed in the staff notation of a melody. But the fact that a graph of a melody can be constructed, however, effectively demonstrates that melody is multidimensional, and can be defined mathematically in terms of the intersection of the two dimensions of rhythm and pitch. In a broader sense, music itself is multidimensional, at least in the Western world, because it consists largely of the intersection of the two dimensions of melody and harmony.

The value of music education is implied in the multidimensional character of music, a defining trait that is also a defining characteristic of mathematics. Multidimensional thought, because it is the basis of analytical thought and critical reasoning, is of great importance in most other areas of learning as well. Interestingly enough, another important aspect of the value of music education is given by the fact that, in at least one important sense, music and mathematics are completely dissimilar. Mathematics is a result of the application of theory, and is essentially abstract in nature. Music, on the other hand, is not in the least bit abstract, and the theory of music comes after the fact and not before, as in mathematics. The notation of music is by definition abstract, since notation is the representation of music in written form, but the connection between abstract and actual is much more easily understood in music that in mathematics.

The value of music education, then, resides in the fact that musical training promotes multidimensional thought and a better understanding of abstract concepts. But in music education, the pathways along which learning takes place are different than in other academic disciplines, because an intuitive understanding of music is normally achieved well in advance of a consideration of the theory of music. Music is ubiquitous in modern society, and most people are exposed to music for at least several hours a day. As a result, most people already have a fairly well developed sense of melody, and at least a rudimentary understanding of harmony, before they even undertake the study of music. Music education therefore largely consists of translating an intuitive understanding of music into a theoretical understanding of music, and it is therefore far easier to develop multidimensional and abstract patterns of thought in the study of music than in other academic disciplines. The relative ease with which this can be accomplished in the study of music is the key to the value of music education.