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Math and Music: An Introduction and Mathematical Analysis

Rhythm and Frequency

To understand the relation between math and music, the primary step is to study the nature of rhythm, frequency and amplitude. Everything around us has its own pattern of rhythm, from the motion of protons and neutrons, to the beats in rock music. Rhythm is determined by the periodicity of vibration of certain object in its surrounding substance, or medium. The vibration is repeated at a constant time length, thus creating wave motion. The repeated portion of the rhythm is referred as a cycle, or one wave. There are two types of wave motion: transverse waves and longitudinal waves. The particles in a transverse wave vibrate perpendicular to the direction in which it is traveling. In a longitudinal wave, however, the particles vibrate parallel to the direction in which it is traveling.

Consider the difference in human voice, animal noises and instrumental sounds. The unique characteristic of sound results from amplitude and frequency. Amplitude refers to the distance from crest or trough of the wave to its equilibrium position. Frequency is a term that describes the number of waves, or vibrations, that pass a given point per second (Stenstedt 28). It is inversely proportional to the wavelength, which is the distance of one wave cycle. Frequency can be evaluated in the pitch of a tone: a higher pitch has a higher frequency. A simple tone has constant frequency and amplitude, and its graph is similar to a sine curve. Tones that are more complicated result from combinations of several simple curves. The smallest pure tone frequency in complex tunes is called fundamental frequency. Integer multiples of the fundamental frequency make the resultant tone musical.

The graph shows the frequency data from A3 to A4 key. Note that A4 key has twice as much frequency as A3 key, because any two keys that are one octave apart have the frequency ratio of 2:1.

Further inspection shows that two adjacent notes are in the ratio of 1.059…

For example:

Freq. of A#3 / freq. of A3 = 233.1 Hz / 220 Hz = 1.059…

Freq. of C4 / freq. of B3 = 261.6 Hz/ 246.9 Hz = 1.059…

Freq. of G4 / freq. of F#4 = 392.0 Hz / 370.0 Hz = 1.059…

The examples show that the frequency of any note is a product of the frequency of the adjacent note before it and the constant number 1.059.

Proof:

Let an octave start from key A. The ratio of frequency between adjacent keys is h.

Then, A# / A = h, A# = A•h

A# = A• h

B = A# • h = (A•h) h = A • h2

C = B• h = (A • h2) h = A • h3

…

Since there are 12 half steps in an octave, A* = A • h12

Which means, A* / A = h12, h = (A* / A) ^ (1/12)

Since A* / A = 2/1,

h = 2^(1/12) = 1.059463094…

The frequency relationships within any octave are as follows:

Intervals

An interval is the scale between two tones. Basically, there are two kinds of intervals: Melodic interval and Harmonic Interval. Musical interval is distinguishable from music...

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