Music and Mathematics

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Music and Mathematics

• are they related?

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What is Sound?

• Sound consists of vibrations of the air.
• In the air there are a large number of molecules
  moving around at about 1000 mph colliding into
  each other. The collision of the air molecules
  is perceived as air pressure.
• When an object vibrates it causes waves of
  increased and decreased pressure in the air,
  which are perceived by the human ear as sound.
• Sound travels through the air at about 760 mph.
  (That is, the local disturbance of the pressure
  propagates at this speed).

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Four Attributes of Sound

• Amplitudethe size of the vibration and the
  perceived loudness.
• Pitchcorresponds to the frequency of vibration
  (measured in Hertz (Hz) or cycles per second).
• Durationthe length of time for which the note
  sounds.
• Timbrethe quality of the musical note.

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Timbre

• Imagine a note played by banging a stick on an
  aluminum can, then imagine the same note being
  played on a guitar string.
• Although the amplitude, pitch and duration may
  all be the same, there is a discernible
  difference to the ear of the quality of the note
  each instrument produced this is timbre.

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Visual Picture of Sound

• Mathematically these attributes can be pictured
  by a sine wave as illustrated.
• This picture illustrates one cycle of the sine
  wave.
• The Amplitude (or height) of the wave is the
  maximum y value (in this case one) the higher
  the amplitude, the louder the sound.

• If the x axis represents time, this wave has a
  frequency of 1/6 cycle per second. This sound
  would not be audible to the human ear.

• The length of the wave, in this case just over 6
  seconds, gives the duration of the sound.

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A 20 Hz Sound

• The picture below shows a 20 Hz sound wave
  lasting for 1 second.

If you count the cycles you will see that there
are 20 cycles of the sine wave in this one second
interval.
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Example

• Example 1 Look at the pictures representing
  sound waves below.
• Which sound would be louder?
• Which has the highest pitch?
• Which would sound the longest?

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The 440 Hz sound (A note)
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Fundamental Frequency and Overtones

• Although we talk about a frequency of an
  individual sound wave, most vibrations consist of
  more than one frequency.
• If, for example, an A is played on a guitar
  string, a frequency of 440 Hz, then the string is
  muted other sounds can still be heard, these are
  the other frequencies that play simultaneously
  with the 440 Hz frequency.
• The 440 Hz frequency in this example would be
  called the fundamental frequency, the other
  frequencies heard are called overtones.

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Sum of Sine waves (Fundamental Overtones)
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Harmonics

• An integer multiple of the fundamental frequency
  is called a harmonic.
• The first harmonic is the fundamental frequency,
  the second harmonic is twice the fundamental
  frequency, the third harmonic is 3 times the
  fundamental frequency and so on.
• For example, if the fundamental frequency is 100
  Hz, then the second harmonic is 200 Hz, the third
  is 300 Hz, etc.

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Harmonics as Overtones

• Recall that on most instruments, like a guitar,
  there are overtones that sound out with the
  fundamental frequency.
• These overtones are higher pitched, which would
  mean they have shorter wave lengths, since there
  are more cycles per second (Hz).
• The overtones are actually the different
  harmonics. The wave lengths are 1/2, 1/3, 1/4,
  1/5, 1/6, etc. the wavelength of the fundamental
  frequency. (see illustration on next slide)

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Illustration


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The Harmonic Series

• Notice that one cycle of the sine wave is 1/2,
  1/3, and 1/4 the fundamental frequency for the
  2nd, 3rd, and 4th harmonics respectively.
• In mathematics the sum 1 1/2 1/3 1/4 1/5
  (denoted by ) is called the harmonic
  series.
• In mathematics the harmonic series diverges so
  what does this mean musically?

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Example

• Example 4 If Sound A is the fundamental
  frequency, then which harmonic is Sound B? What
  is the frequency of each sound?

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The 12-tone (chromatic) Scale

• On a 12-tone scale the frequency separating each
  tone is called a half-step. These half steps
  correspond to keys on the piano keyboard as
  illustrated below

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Ratio of Frequencies to the Fundamental Frequency.

• Each half step is separated by a common
  multiplicative factor, say f that is C?f C,
  C?f D, etc.
• So, from C to C weve increased the frequency by
  a factor of f 12 times or by f12.
• Since we know that the second C is an octave
  above the first, that means its frequency has
  doubled, hence f12 2.
• Consequently f .

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Table of Frequencies
Note Frequency (in Hz) (rounded) Ratio to Frequency of Middle C
C 262 1
C 277 ? 1.05946
D (second) 294 ?1.12246
D 311 ? 1.18921
E (third) 330 ?1.25992 ? 5/4
F (fourth) 349 ? 1.33483 ? 4/3
F 370 ? 1.414214
G (fifth) 392 ? 1.498307 ? 3/2
G 415 ?1.58740
A (sixth) 440 ?1.68179 ? 5/3
A 466 ? 1.78180
B (seventh) 494 ?1.88775
C (octave) 524 2

• If we accept that middle C has a frequency of
  261.6 Hz, then we can find the frequencies of all
  the notes in a 12-tone scale by successively
  multiplying by see table to the right.

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Major Scales

• A major scale consists of 8 notes.
• The major C scale is C-D-E-F-G-A-B-C.
• Notice that between C and D are two half-steps,
  or a whole-step, and between D and E is a
  whole-step, but between E and F its only a
  half-step (refer to keyboard picture).
• The next step from F to G is a whole-step, G to A
  is a whole-step, A to B is a whole-step, and then
  from B to C is another half-step.
• So we see the pattern for a major scale starting
  at any note we will take a whole-step,
  whole-step, half-step, whole-step, whole-step,
  whole-step, half-step.

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Example

• Example Find the notes in the major A scale.

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Minor Scales

• The pattern for minor Scales starting from the
  fundamental note Whole step, half step, whole
  step whole step, half step, whole step whole step
• Example Find the notes in an A-minor (Am) Scale.

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Relative Minors

• The major C scale is C-D-E-F-G-A-B-C
• The A minor Scale is A-B-C-D-E-F-G-A
• Notice the same notes in both scales but in a
  different order
• Thus, Am is called the relative minor of C.
• Relative minor chords have a similar sound

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Contact Information

• Angie Schirck-Matthews
• Broward College Mathematics Central Campus
• 3501 SW Davie Road, Davie FL 33314
• Office 954.201.4918
• Cell 954.249.5331
• Email amatthew_at_broward.edu