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The Mathematics of Music

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  1. The Mathematics of Music By: Phil Boyer, Mark Forman, Brian Glenn, and Arvind Kanna

  2. What is a harmonic? • Any of a series of musical tones whose frequencies are integral multiples of the frequency of a fundamental tone • A wave whose frequency is a whole number multiple of that of another

  3. frequencies

  4. Harmonic Series • This series Diverges. • For each whole number a harmonic exists. • In music, only the odd harmonics sound pleasing to the ear.

  5. EAR

  6. Structure of Human Ear • Sounds travel through the ear by way of the cochlea and basilar membrane

  7. Why Sine Waves? • Sine waves most accurately model how a sound particle travels through the ear. • The equation that best represents how musical sound waves travel through the ear is the equation for simple harmonic motion

  8. Simple Harmonic Motion

  9. Vibrating strings

  10. Vibrating Strings Equation • n is the value of the harmonic being played.

  11. Musical Scale • The Western Musical Scale consists of twelve notes forming an octave. • There are 7 natural notes and 5 sharp/flat notes. • # means “sharp”, b means “flat” • A A# B C C# D D# E F F# G G# • A Bb B C Cb D Eb E F Gb G Ab

  12. Musical Harmonies • There are many musical harmonies that can be played on guitar or any instrument for that matter. • Harmonies can be played between different types of instruments. • Most popular artists use harmonies in their songs. Even samples and electronic beats use them.

  13. Why Do Mathematical PeopleExcel in Music? • 1. generally this is so because of the mind set and thought, which is similar between both subjects. • 2. math as a subject is generally about thinking, problem solving, seeing patters. • 3. math is also about working with what you know to find a method of exploring and understanding new concepts. • 4. Music is very similar to math in this sense.

  14. 5.       Arranging a melody on an instrument and finding fingerings that correspond to certain sequences of notes(like piano for example) is definitely a type of math problem. • 6. Playing the same melody on different instruments is math, as is playing a stringed instrument and changing the tuning.

  15. 7. Playing harmonies, playing the same song in different keys--these things all involve recognizing the structure of a piece of music. • 8. Good musicians can often listen to a song, observe the musical structure, and play along with it, without really knowing it or rehearsing it, because they recognize familiar patterns. • 9. this type of pattern recognition and mind set of math oriented individuals helps them excel in music.

  16. Bibliography • Dave, Benson. "Music : A Mathematical Offering." Mathematics and Music. 14 May 2006. Cambridge University. 25 May 2006 <http://www.maths.abdn.ac.uk/~bensondj/html/maths-music.html>. • Harrison, David M. "Standing Waves." 24 Mar. 2003. University of Toronto. 25 May 2006 <http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Vibrations/Vibrations.html>. • Reid, Harvey. "On Mathematics and Music." Woodpecker Records. Nov. 1995. Woodpecker Records. 25 May 2006 <http://www.woodpecker.com/writing/essays/math+music.html>.