While not a simple task, writing music can become an easier undertaking if one has mastered the basics of mathematics. They are both universal languages, with each having its own set of concepts, rules, symbols, and terms. As David Hodge says in his article, Timing is Everything, “If you can count up to sixteen and have the capacity to divide things into twos and threes, then you can read music.”
Both musicians and mathematicians use basic concepts. Musicians use numbers, rhythm, shapes, and patterns. Complementary to that, mathematicians use numbers, measurement, geometry, and algebra. They both use numbers to communicate on paper what they hear and see in their minds: musicians by writing their compositions on a staff of music, and mathematicians by recording facts and figures in ledgers and accounting books.
When musicians use rhythm, by assigning notes a particular sequence and pattern in their pieces, they are measuring the notes, and determining the length they will be held. Mathematicians use measurement in various aspects of their work, as well. They measure length and time in working everyday problems such as square footage for carpet, miles per hour for the car, and monthly payments for a mortgage.
Musicians put their thoughts on paper by using the appropriate shapes. There are various types of notes and symbols in music: whole, half, quarter, eighth notes and more, that have their own particular shape and indicate how long a note is to be held; rests that have a distinct shape and are placed on the staff in a particular manner to indicate how long the musician rests before hitting another tone; accidentals that have their unique shape and indicate to raise or lower a tone a specific amount in the composition. Mathematicians use shapes in various forms in their work, as well: circles, squares, ovals, rectangles, and triangles, just to mention a few. They use them to calculate area and volume, and in pie charts and bar graphs. They are vital for their use in geometry.
Music is made up of patterns. Notes are grouped together in patterns to indicate a precise rhythm, tones go up and down in patterns to indicate a specific tune, and volume and intensity raise and lower in patterns to fit the songs. Mathematicians use patterns in various ways in Algebra. As Alex Bogomolny says in his article, “Between Arithmetic and Algebra,” the difference between 3+5=5+3 and a+b=b+a is that “One is a specific fact, another is a pattern valid in a multitude of situations.” He also points out that “multiplication tables are rich in patterns. Put your finger on an entry on the main diagonal. Move the finger one step in either North-East or South-West direction. The entry you'll get is 1 less than the diagonal entry.”
As anyone who has worked with music or math in any fashion knows, both are controlled by strict rules. In music, there are rules of tone, intervals, and tempo. “Tones that sound good when sung at the same time are said to be in harmony with each other or are said to be harmonious. Tones that sound unpleasant when sung at the same time are called discordant” (Luebbert, Tone). Intervals are used to create tunes. “Melodies are formed by skipping from one pitch to another using one of the possible intervals….Different interval sequences are recognized as different melodies” (Luebbert, Interval). The rhythm and beat of music make up its tempo and follow the rules determined by the musicians as they enter the time signature.
Mathematics is ripe with rules and regulations. Two examples of mathematical rules are commutative property of addition and multiplication, and the PEMDAS rule. The rule of commutative property of addition and subtraction simply states that order is not important when adding or multiplying. The total of 3+6 is the same as the total of 6+3. By the same token, the total of 3x6 is the same as the total of 6X3. The PEMDAS rule states that when one evaluates expressions it must be in the order of parentheses, exponents, multiplication/division, and lastly, addition/subtraction. To deviate from this rule will result in a different and incorrect answer.
Musicians work with a variety of symbols. A few symbols are the staff, ledger lines, bar lines, accidentals, clefs, and notes. The staff, made up of ledger lines, is the canvas on which the composer draws his musical picture with the other symbols: the bar lines which break the composition into smaller, manageable units, called measures; the accidentals, which allow for more complex tunes; and the clefs which give the reader of the music direction as to which notes they are referring.
While on the other hand, mathematicians' symbols consist of signs that indicate addition, subtraction, multiplication, equals, greater than, square roots, and many more. They direct the reader as to what functions to perform on which values.
It is evident that a working knowledge of math will assist musicians. If they know addition and multiplication, if they understand intervals, and if they grasp the concept of patterns, they will be well on their way in the music world. By the same token, mathematicians can appreciate a great work of music as they listen for the mathematical designs hidden in the movements and harmonies of the melodies.
