The Mathematics of Music By: Phil Boyer, Mark Forman, Brian Glenn, and Arvind Kanna
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
Harmonic Series • This series Diverges. • For each whole number a harmonic exists. • In music, only the odd harmonics sound pleasing to the ear.
Structure of Human Ear • Sounds travel through the ear by way of the cochlea and basilar membrane
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
Vibrating Strings Equation • n is the value of the harmonic being played.
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
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.
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.
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.
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.
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>.