# Mathematic Principles

Plane on Which Music Lies

As mathematical functions are graphed on visual planes, all music is plotted on the plane of time. This plane is defined by the time signature (meter). The top number of a time signature defines the number of beats in the measure and the lower number tells what type of note each beat gets. These beats can be subdivided evenly within each measure. There are two eighth notes in a quarter, four sixteenth notes in a quarter and even eight thirty-seconds in a quarter note. The brain logically can anticipate and expect these subdivisions in music like when one adds fractions with different denominators. Every note usually stays aligned with time. This logical progression leads to some complex rhythms called polyrhythm. These rhythms do not follow the logical alignment of meter and therefore draw the listener to it.

Rhythm is the canvas on which the human mind can illustrate music. Part of this action relies on feeling the subdivisions within a measure. This explains why odd numbers complicate our perception of musical time. For example a measure with four beats can easily be discerned into two equal groups, while five beats in a measure complicates the brains ability to adjust to the division. To compensate for the unequal divisions we count five beats as a measure of three beats followed by a measure of two beats.

Meter is the microcosm in which we can define the pace of the music and division while phrasing is the big picture. Phrasing is how the music can play out into a grand drama or a leisurely stroll. Just as a sentence is spoken with emphasis communicates an idea, music that has proper emphasis (achieved through rests or style of play) will communicate ideas.

Twelve-Tone Equal Temperament

In the twelve-tone equal temperament (12-TET), the octave is divided into twelve semitones, or half-steps. This is the standard temperament that has been used in western music for the past 200 years. J. S. Bach invented well temperament when he wrote The Well-Tempered Clavier. Through this piece, he showed the musical potential of well temperament. Well temperament has the twelve notes in an octave be tuned in a manner that will allow any major or minor key to be played while not sounding out of tune to the human ear. The twelve-tone equal temperament is an example of well temperament.

Frequencies, Intervals, and Cents

The intervals between different notes in music applies to math through ratios and equations. Remember that the twelve-tone equal temperament divides the octave into twelve equal half-steps.

The ratio of frequencies between two adjacent half-steps is the twelfth root of two. This happens because the equal tempered scale divides an octave, which has a ratio of 2:1, into twelve equal sections. Therefore, the semitone has to equal the ratio 2:1 when multiplied by itself twelve times.

The interval between two adjacent semitones is 100 cents, which makes 1200 cents equal to one octave. Cents is a unit of measure used to describe musical intervals. The value of the twelfth root of two is approximately 1.0595.

Frequency of a note
To find the frequency of a note in the twelve-tone equal temperament (12-TET), this equation is used:

In this mathematical formula,

means the frequency, which is measured in hertz, of the note you want to find the frequency of.

stands for the frequency of a reference pitch, which is the A above middle C, having a standard frequency of 440Hz.

n and a refer to the numbers assigned to the certain pitches. These numbers come from consecutive integers matching up to designated consecutive half-steps.

An example of this would be C4, which is middle C, receiving the number 40, because it is the 40th key from the left side of the piano. To put this equation in action, here is an example of finding the frequency of middle C:

producing approximately

Using this formula, you can find the frequency of any note.

Another Formula to Find Frequencies
Frequency of a Higher Note can be found with this formula:

where f is the frequency of a note (this will be the reference pitch) and f before is the frequency of the higher note you are trying to find, and n is the number of half-steps higher

which can also be expressed as:

Frequency of a Lower Note can be found with this formula:

where f is the frequency of a note (reference pitch) and f before is the frequency of the lower note, and n is the number of semitones lower

which can also be expressed as:

Calculating Intervals and Cents
To express the interval between two musical notes, we can use start from the pitch of the two notes, and use this formula to do the calculation: