Math and Music

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Math and Music

• Or how we get from this.to this

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What connections did early mathematicians make
between math and music?
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Lots! So, my new question becameHow did math
affect the development of the musical scale?
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In the beginning.

• There was Pythagorus
• 569 - 479 BCE

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The monochord
6

• 12/6 2/1 the octave
• Use the arithmetic mean to find
• 12/9 3/2 the fifth

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The Harmonic Mean

• Starting with 12 and 6, find the number between
  the two, such that
• The first decreased by a certain part results in
  the same number as the second increased by that
  same part.
• Huh?

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(2 12 6) / 12 6 8

• Since 12 - 4 8, and 4 is 1/3 of 12
• and
• 6 2 8 and 2 is 1/3 of 6
• 8 is the harmonic mean of 12 and 6

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• 12 8 9 6

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The only problem is.

• ,

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Because 2 and 3 are prime,

• The fifths will never form a closed system with
  the octaves

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Who cares?

• OK for Gregorian chants
• Small instruments, certain music
• Trouble if you want to
• Transpose
• Use a larger instrument, like piano or organ

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Ptolemy steps in with a solution

• Ramos and Zarlinos (1500s) propose
• Just intonation
• syntonic diatonic
• Natural scale

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Its all about compromise

• between the ear and the math.

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The perfection of 6

• Well, it is perfect 1 2 3 1 2 3
• God created the earth in 6 days
• Hymns should only be sung for 6 generations
• ???

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4 5 6

• C E G
• F A C
• G B D

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Dang, forgot the whole steps!

• Still doesnt work!

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Mean Intonation

• Cracks me up every time!

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C to G is 5/4

• Since intervals between CEG, etc. were 5/4, the
  mean is v5/4.

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Irrational string lenghts?

• How blasphemous can you get?

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So what do we use now?

• Equal temperament
• A compromise between beautiful sound and
  functionality.

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Equal temperament, invented in China?

• In1596
• Proposed for Western music by Simon Stevin
  (1548-1620)
• Became widely known through Mersenne in 1635

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Based on powers of 2

• We want (T5)12 2 7
• When you solve, you get 1.498
• Pretty close to 3/2, but still creates a closed
  system