Vibration

Figure 1 shows some of the ways in which a string can vibrate: A produces a certain note; B produces the first harmonic, an octave higher; and C fifth higher

      y = sin x cos t for vibration A
y = sin 2x cos 2t for vibration B
y = sin 3x cos 3t for vibration C

Graph for each musical notes

Please click on each "empty box" for more information :)

C, C#	C, D
C, D#	C, E
C, F	C, F#
C, G	C, G#
C, A	C, A#
C, B	C, C

Trigonometry and sound

Sound of Music and Trigonometric Function

research findings on sin, cos graphs! :)

A wave might be representable as a combination such as  1.4 cos x + 2.8 sin x - 1.103 cos 2x + 3.32 cos 3x + 5.6 sin 3x. Normally, a wave selected at random can only be represented by an infinite number of such terms; in other words, the combination doesn't stop, but goes on forever, though the numbers in front of the sines and cosines get smaller and smaller.

Musically, when an instrument plays a note, the basic note can be represented by A.cos Lt + B.sin Lt. The number L has to do with how high the note is (pitch), and A and B have to do with how loud it is; t is time.

Trigonometry is the mathematics of sound and music.

A microphone measures air pressure. Sound is just the variation of air pressure. Sound can be represented mathematically by a function f(t) giving the air pressure at time t . More precisely, f(t) represents the difference between the air pressure at time t and the average air pressure.

The simplest sounds, called pure tones are represented by functions of the form

f(t) = A sin(2 pi w t)

This family of functions has two parameters that we can hear -- w, the frequency, and A, the amplitude.

The b in both of the graph types

• y = a sin bx
• y = a cos bx

affects the period (or wavelength) of the graph. The period is the distance (or time) that it takes for the sine or cosine curve to begin repeating again.

 

Sound of Music and Trigonometric Function

1.       http://www.youtube.com/watch?v=5k_Fzmn3CJo&feature=related

How Is Trigonometry Used in Music?

Harmonics

• Every note (pitch) in music is determined by the size of its sine wave---that is, it is determined by its frequency. Notes with wide sine waves are lower in pitch and have fewer cycles per second, while notes that have narrow sine waves are higher in pitch and have more cycles per second. Musicians can manipulate their timbre by manipulating the sine waves produced. For instance, if a player plays a note with a frequency of 512 hertz, then a harmonic or partial is produced above it at 1,024 hertz, and you may hear a base note with the same note an octave higher. Violin players use knowledge of harmonics frequently, and tuning is related to how the base frequencies and harmonics interact.

Read more: How Is Trigonometry Used in Music? | eHow.com http://www.ehow.com/how-does_4969029_how-trigonometry-used-music.html#ixzz1TN7OYFap

How Is Trigonometry Used in Music?

Production

• Studio producers have the job of making a musical recording sound balanced. They use many different computer programs in order to do this. The computer programs can allow a producer to see the sound waves that have been recorded as different types of graphs. These graphs are produced as the program uses trigonometric equations to quickly calculate how the graph should appear based on individual points---the sine wave of a singer's voice, for instance, can be viewed visually using this process. The producer then can tweak things like pan and volume based on the visual cues in the graphs.

Read more: How Is Trigonometry Used in Music? | eHow.com http://www.ehow.com/how-does_4969029_how-trigonometry-used-music.html#ixzz1TN7720Zj

How Is Trigonometry Used in Music?

Engineering

• The sounds that we hear every day, including music, reach our ear as sound waves. These sound waves travel through the air at different angles from the original sound source. The sound then bounces off whatever is nearby, such as people or the walls of a concert hall. If a building is designed in such a way that the sound does not bounce back to the listener's ear well, then the music can be hard to hear or it can sound unbalanced. Engineers use trigonometry to figure out the angles of the sound waves and how to design a room or hall so that the waves bounce to the listener in a balanced and direct manner. Studio producers or hall managers sometimes install panels that hang from the ceiling of the room---these panels can be adjusted at specific angles to get the sound waves to bounce correctly.

Read more: How Is Trigonometry Used in Music? | eHow.com http://www.ehow.com/how-does_4969029_how-trigonometry-used-music.html#ixzz1TN6qwWSl

Links:)

http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=5735961

Abstract of Analysis

This gave us a deeper insight to the world of trigonometry in music:
Harmonic detection is the basis of harmonic issues and harmonic detection method is the key of harmonic measurement. Based on Fourier Transform and Wavelet Transform, ways and means of detecting and analyzing are developed, but both of them have the problem of spectrum leakage. In this paper, a novel power harmonic algorithm based on the family of trigonometric functions is developed for the characterization of harmonics. The algorithm has designed a set of discrete reference signals and only estimated the amplitudes and the phases which are needed. Besides it hasn't been involved in the influence of the leakage, this algorithm is also easy to carry out and has much high resolution precision than the traditional Fourier algorithm and Wavelet algorithm. The algorithm also has the characteristic of being more sensitive to sampling frequency and less sensitive to sampling points. Using this characteristic, the algorithm can adjust sampling parameters in order to give consideration to both precision and real-time performance. The simulation results have proved that the outstanding advantage of the algorithm is that when chosen a proper sampling frequency the algorithm can pick out harmonics components with high precision in very few fundamental periods. Comparing with Fast Fourier Transform (FFT), Windowed-FFT and Wavelet Package Transform, the algorithm has its certain advantage.

Power harmonic analysis based on Orthogonal Trigonometric Functions Family

Where the Analysis took place: 

Ruoyu Wang;   Yuefeng Zhang;   Jinghong Guo;  
Huzhou Electr. Power Bur., Huzhou, China 

This paper appears in: Electricity Distribution (CICED), 2010 China International Conference on
Issue Date: 13-16 Sept. 2010
On page(s): 1 - 7
Location: Nanjing
Print ISBN: 978-1-4577-0066-8
INSPEC Accession Number: 11876277
Date of Current Version: 22 March 2011 

Links:)

http://www.cliffsnotes.com/study_guide/Simple-Harmonic-Motion.topicArticleId-11658,articleId-11649.html

Math Research

How is trigonometry used in music?

1.     Sound consists of waves, and that sin(x) and cos(x) are waves.

2.     A wave might be representable as a combination such as  1.4 cos x + 2.8 sin x - 1.103 cos 2x + 3.32 cos 3x + 5.6 sin 3x.

3.     Musically, when an instrument plays a note, the basic note can be represented by A.cos Lt + B.sin Lt. The number L has to do with how high the note is (pitch), and A and B have to do with how loud it is; t is time. But that function is a "pure tone" that seems to sound like a flute.

4.     The unique sounds of oboes and clarinets and so forth are due to almost inaudible notes at the octave, octave+fifth, etc.

5.     Every note (pitch) in music is determined by the size of its sine wave---that is, it is determined by its frequency.

6.     Notes with wide sine waves are lower in pitch and have fewer cycles per second, while notes that have narrow sine waves are higher in pitch and have more cycles per second. Musicians can manipulate their timbre by manipulating the sine waves produced.

7.     For instance, if a player plays a note with a frequency of 512 hertz, then a harmonic or partial is produced above it at 1,024 hertz, and you may hear a base note with the same note an octave higher. Violin players use knowledge of harmonics frequently, and tuning is related to how the base frequencies and harmonics interact.

8.     The simplest sounds, called pure tones are represented by functions of the form

f(t) = A sin(2 pi w t) 

Math Project

Phenomenon: Setting and rising of the Sun via trigonometry properties.
Applied: Music and sound waves via trigonometry