Music and Maths Investigation. Sine waves and harmony on the piano.

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I have chosen to investigate the mathematical properties of music; to be more precise, the piano. Since an early age I had a great interest and appreciation for music, in addition, I had decided to play the piano and now I have been playing it for 11 years. I enjoy listening to various types of composers and musical artists varying from classical pieces to modern pieces. However, during my last year in my piano classes, I’ve got a spark of interest in figuring out how composers manage to compose pieces that are appealing to me and the public, especially, when some composers had health problems, which made the journey of creation quite hard for them, in this case, Beethoven created wonderful musical pieces while being deaf. After some personal research, I’ve found out that mathematics has a huge role in determining how it appeals to the public, as mathematics is related to the production of harmony. Thus, the aim of my exploration is to explore and understand how mathematics are related to harmony of music.

All of the aspects of mathematics in relation to harmony will be focused on my main instrument – the classical piano. Although music is made out of all of the aspects such as melody, harmony and rhythm, however, harmony is very intricate and requires an exploration of its own.

This exploration will be split into three parts: the first one will comprise of the graphical representation of the notes and chords and which combinations of notes produce harmonic sound; the second part will explore how the Greek philosopher Pythagoras had contributed to music with ratios that later developed the harmonic series; the third part will delve into the probability of achieving harmony.


The piano is built on the foundation of an octave, which is comprised of 12 semitones ending on a note twice the frequency of the first note’s tone. A semitone is the smallest interval in western classical music. Each tone inclusive produces a unique sound. As a result, each semitone has a representational letter in an octave. The octave the note is located in is represented by a subscript number, for example, C4. On the piano, the scale range of the white notes are C, D, E, F, G, A, and B going from the left to the right. The black notes are named according to what key the music is in and what white note they are related to. They are given the symbols of ♯, which means sharp and ♭, which means flat. Here is an image with the keyboard and its representational letters:

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Each semitone, or note, will produce a sound that has its own frequency. This frequency is a measure of pitch and is recorded in Hertz (Hz) or cycles per second. Each frequency can be graphed in a two-dimensional plane as a sinusoidal wave. We will observe a pattern in harmonic notes and chords through the ratio of periods of each graph. To visualize these frequency patterns, we will graph a single note, in this case, we will use A4, the most commonly used note for a general tuning standard for musical pitch, which has a frequency of 440 Hz. The equation for A4 is:

The graphing of a sine function allows us to visualize the oscillation of the string. 2 gives us the period interval which represents one second, 440 represents Hz and is our variable for time. To observe the important aspects of the graph and to find the correct period ratios, we must scale down each equation by the same factor.

We have divided the Hz value (let R = Hz) in the sine function by 440 in order to create the function for note A scale to . I have chosen A4 as my main function as it will be used to build the harmonic chords of the A4 note. We will use the same transformation for each note’s equation in order to graph and observe the proper ratios.

Thus here is the graph of the equation of note A4:
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Before measure and explore our first harmonic chord, we must write down a few basic clues about the wavelengths. The wavelength or period of the function is directly related to pitch: the shorter the wavelength is, the higher the pitch is and vice versa.

Here is a graphical representation of A major. The notes are A4, C♯, E and A5

Here are the equations used to create the graph of A major. They were determined by using the pitch (Hz) values in the general equation of .

A4: C♯: ...

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