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Mathematics of the Violin

Junior Year Math Project
by

Thomas Lee

on 29 May 2013

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Transcript of Mathematics of the Violin

Mathematics of the Violin By: Thomas Lee The Basics: Violin's have 4 different strings, G, D, A, E.
When the strings are played they create a fundamental resonance from the vibrations.
To create a note, the strings vibrate at their ideal fundamental frequency, measured in Hertz.
All notes can be created by alternating the frequency Strings and Frequencies: In order to produce a variety of notes, each string has a different frequency. G: 196.00 Hz
D: 293.66 Hz
A: 440.00 Hz
E: 659.26 Hz Side note: Each string is 3 notes apart, thus, between each of these frequencies, 3 different pure notes can be played.
Ex: A: 440.00 Hz
1. B: 493.88 Hz
2. C: 523.25 Hz
3. D: 587.33 Hz
E: 659.26 Hz Octaves: An octave is the relationship between one musical pitch to another half or double its frequency.

A440 and A880
G196 and G392 Equation for Fundamental Frequency
on a Violin Equation for frequency
Measured in Hertz (Hz) Most Orchestras tune to an A440 pitch or "Concert A" Harmonics: Harmonics Cont. Notice, the relationship between the fundamental harmonic and the others.
The 3rd harmonic has a third of the period as the original, 5th harmonic has a fifth of the period, and so on.
Also note that each function creates a sinusoidal function. This function can be expressed by the simple sinusoidal equation: Adding Sounds The Resulting Function: The resulting function is the sum of all the sine functions of the individual notes using the frequencies. So what does this look like?? Where:
|A| : Amplitude
B : Cycles (Period - 2pi/B)
C : Phase Shift
D : Vertical Shift Note: All the sounds are chords composed of a variety of sinusoidal curves, thus, the result is a collection the added sinusoidal functions. Now lets apply it! Music Excerpt: Excerpt: Step 1: Write down notes. 5F#, 5E, 5D, 5C#, 4B, 4A, 4B, 5C#, 5D, 5C#, 4B, 4A, 4G, 4F#, 4G, 4E, 4D-4F#, 4A-4G, 4F#-4D, 4F#-4E, 4D-3B, 4D-4A, 4G-4B There are only 8 notes to find, however, the octaves are important. The frequencies will be different at each note. The number next to the letter shows the octave it is on. A440 is the open string on the violin. Only 10 distinct notes:
5F#, 5E, 5D, 5C#, 4B, 4A, 4G, 4F#, 4E, 4D Frequencies of these notes (in Hz):
5F#: 739.99 5E: 659.26
5D: 587.33 5C#: 554.37
4B: 493.88 4A: 440.00
4G: 392.00 4F# 369.99
4E: 329.63 4D: 293.66 Step 2: Write down frequencies of the distinct notes: In real life, we never actually hear one tone, however, every sound is a harmonic (composed of multiple tones).
For example, on a violin, the A note will be a combination of A harmonics, (A220, A440, A880). In order to play a chord (a combination of tones), the resultant tone is the sum of all the individual sine waves. Note: Sound cancellation uses the same concept
By broadcasting a wave with a phase shift of half the original period, with the same amplitude and pitch, all noise is canceled out. The sound from the violin resonates from the f-holes which are in the middle of the body. The sound comes from the vibrations made from the strings. Rosin helps the bow stick better to the string allowing for a more uniform and level vibration as the player draws the bow across the string. This is supposed to be the chin rest. Generating sounds: Using mathematica, function "play".

Inside the function "play", define a waveform sin(2*pi*freq*time) (Angular frequency)

As from the previous slide, the tones of the notes are the frequency, and time is the length of the individual note (half, quarter, etc.) Note: Each measure is 2.7 seconds long at 82 beats per minute.

Input all variables, and add the sines to cascade each notes in series.
For example: F5, E5
sin(2pi*739.99*(0 to 2.7/2)), sin(2pi*659.26*(2.7/2 to 2.7)) The Results: The resulting sound of the first 30 notes of the song using pure notes: Remember, we never hear pure tones, all tones are many harmonics added together. The resulting sound of the first 30 notes of the song using harmonics: Sample of the program: Waveform of pure F5 Waveform of harmonic F5 Individual tones: Cascade program: Example of the part of the result: What's the song?
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