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Music and Mathematics

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Title: Music and Mathematics


1
Music and Mathematics
  • are they related?

2
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).

3
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.

4
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.

5
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.

6
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.
7
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?


8
The 440 Hz sound (A note)
9
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.

10
Sum of Sine waves (Fundamental Overtones)
11
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.

12
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)

13
Illustration


14
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?

15
Example
  • Example 4 If Sound A is the fundamental
    frequency, then which harmonic is Sound B? What
    is the frequency of each sound?


16
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

17
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 .

18
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.

19
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.

20
Example
  • Example Find the notes in the major A scale.

21
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.

22
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

23
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
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