The Mathematics of Music

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The Mathematics of Music

• By Phil Boyer, Mark Forman, Brian Glenn, and
  Arvind Kanna

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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

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frequencies
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Harmonic Series

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

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EAR
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Structure of Human Ear

• Sounds travel through the ear by way of the
  cochlea and basilar membrane

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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

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Simple Harmonic Motion
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Vibrating strings
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Vibrating Strings Equation

• n is the value of the harmonic being played.

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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

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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.

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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.

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• 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.

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• 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.

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Bibliography

• Dave, Benson. "Music A Mathematical Offering."
  Mathematics and Music. 14 May 2006. Cambridge
  University. 25 May 2006 lthttp//www.maths.abdn.ac.
  uk/bensondj/html/maths-music.htmlgt.
• Harrison, David M. "Standing Waves." 24 Mar.
  2003. University of Toronto. 25 May 2006
  lthttp//www.upscale.utoronto.ca/GeneralInterest/Ha
  rrison/Vibrations/Vibrations.htmlgt.
• Reid, Harvey. "On Mathematics and Music."
  Woodpecker Records. Nov. 1995. Woodpecker
  Records. 25 May 2006 lthttp//www.woodpecker.com/wr
  iting/essays/mathmusic.htmlgt.