# Math and Music

January 29, 2009

Mathematics shares connections with many other concepts that no one would think involves math. Music and mathematics share a very close bond. Many areas of music exhibit various mathematic principles. Many mathematic ideas, such as proportions, patterns, and geometric transformations, share harmonious connections with the beat, tone, composition, and tuning of music.

The first aspect of music that relates to mathematics is the beat. Rhythm, which creates the beat in music, is the basis of music, as are numbers in math. "Rhythm is created whenever the time continuum is split up into pieces by some sound or movement" (Garland, page 6). Examples of rhythm include beats of a heartbeat, crickets at night, and waves crashing onto a shore. Rhythm shows that music is a system of patterns as is mathematics. The pattern of musical rhythm is devised into the music theory. The music theory uses measures and time signatures to set up the rules of rhythm for a piece of music. Measures of music are separated by bar lines and are defined by the time signature. The time signature is a fraction that appears at the very beginning of every piece of music. The top number, like the numerator, tells how many equal beats will occur in each measure. The bottom number, like the denominator, tells what kind of note gets the beat. The standard time signature in music is 4/4. This is what the beats of music are measured by. If a complete measure is filled by one note, it is called a whole note. If a measure is filled by two equal notes, this note is defined as a half note. Musical notes follow this pattern by each note following the previous note and divided by two. The following chart shows the ratio of each note, the note's symbol, and how the note is represented in a piece of music:
(page 8)
When writing music, each measure must equal the equivalent of the time signature ratio. This is key in understanding how music is created. For example, in a 4/4 time signature, one measure may consist of a ½ note + a ½ note + a ½ note because it is equal to 4/4 or 1. A measure containing a ½ note + a ½ note + an eight note is an incomplete measure because it is equal to 5/8 which is not equivalent to 1. No matter how complicated the rhythm of music is, it can always be analyzed mathematically. One of the hardest rhythms of music is playing a rhythm ½ against ½ another. An example of this is when a piano player must play two notes of equal length with the right hand and three notes of equal length with the left. The question that arises is where the notes of two will equal length fall between the notes of three equal lengths. This can be solved by the mathematic principle of least common multiple. Since the least common multiple of two and three is six, the measure could be broken up into 6 equal parts. It is then easier for the composer to tell where the notes should lie in relation to each other. The first of the notes of two equal lengths will occur on the first beat, and the second on the fourth beat (out of the six equal parts). Rhythm, the basis behind music, is easily analyzed and felt through the use of mathematics.

The tone of music is connected to mathematics through sound waves and their properties. Sound waves are longitudinal waves that constantly refract, then compress, like a slinky. Sound waves determine the pitch of every sound and how that sound is interpreted by the human ear. The pitch of a sound is based on the frequency of the wave. Frequency is the measure of vibrations of a sound wave per second of time. The higher the frequency is, the higher the pitch of the sound. The following diagram shows two sound waves and the relationship of their frequencies and their pitch:
(page 29)
The frequency of a sound wave, which in essence is the tone of the sound, also determines what type of sound is created. The human ear depicts different sounds based on their rate of vibrations, or frequency, and by their amplitude, or the height of the compression of a sound wave above an equilibrium line.

Another aspect of music that has a harmonious connection with mathematics is the tune. The tune of the music helps to describe scales and whether two notes will sound good or bad when played together. The easiest way to understand scales and similar notes is by using the piano. The piano contains 88 keys, grouped into 11 octaves with eight notes each. The notes of a piano are A, B, C G. On the piano, for example, one could play the note A in many different octaves. But how does the ear find a similarity between A (A played on the third octave) and A4 (A played on the fourth octave)? This can be determined by analyzing the ratios of their frequencies. Frequency is measured in Hertz (vibrations per second). The following diagram shows the amount of Hertz for every note from A' to A4:
(page 40)
Notice the ratio of A and A4. Note A is 220.0 Hertz, while A4 is 440.0 Hertz, a ratio of 2:1. The human ear interprets these waves as an impact of air every complete wavelength. So every other time the A3 impacts the ear, the A4 note impacts the ear, creating a similarity in the sounds. This concept can also be used to tell whether two sounds will sound good or bad when played simultaneously. The following diagram shows two groups of two notes played together, the first with the usually good sounding C and G, along with the usually bad sounding C and F#:

The normally good sounding duet of C and G has waves that are in proportion together. At certain points their waves impact the human ear at the same time, creating a pleasing sound. The noticeably terrible sounding C and F# have waves that aren't in any proportion, creating a sound unpleasant to the human ear. The person who actually discovered this concept of different pitches resulting from ratios and different sounds based on wavelengths was famous mathematician Pythagoras. Pythagoras experimented with a single string and a bridge and discovered the relationship that if the bridge was twice the distance away from the end-point of the string, the note will be constant but the pitch will be twice as high. Pythagoras' discovery is known as the diatonic scale which is very similar to the major scale used today. These mathematical concepts of the tunes of the notes being in proportion explain why instruments must be tuned. A musician tunes his or her instruments so each note stays in the same proportion as all the other notes.

The final feature of music that is connected to mathematics is the actual composition of music. Musical compositions are highly influenced and are based upon the mathematical principles of geometric transformations. A geometric transformation relocates a rigid geometric figure in the plane while carefully preserving its size and shape (Garland, page 69). The first and most simple transformation commonly used in music is translation. In translation, each point of the shape is moved a defined distance along a course parallel to the path of every other point in that figure. The following three figures show how translation is used in a piece of music:
(page 69)
The simplest use of translation in musical application is repetition. Repetitive translations can be seen in the spiritual song When the Saints Go Marching In:
(page 70)
Another translation commonly found in songs and musical compositions is reflection. Reflection is when a geometric figure is reflected of a line and its mirror image appears on the other side. The musical counterpart of reflection is retrogression and is exemplified below:
(page 73)
These mathematical transformations can be found in all different genres of musical compositions. It is this simple mathematical concept that shapes all songs and every musical piece.

Mathematics and music will always share a harmonious connection. Musical concepts of beat, tone, tuning, and composition all share relationships with mathematical principles such as proportions, pattern, and geometric transformation. In order to understand the beauty between the seemingly different topics of math and music, one must learn to identify their harmony and their strong and powerful bond.