Transcript Music and Mathematics

Music and Mathematics
are they related?
What is Sound?
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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).
Four Attributes of Sound
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Amplitude—the size of the vibration and the
perceived loudness.
 Pitch—corresponds to the frequency of
vibration (measured in Hertz (Hz) or cycles per
second).
 Duration—the length of time for which the note
sounds.
 Timbre—the quality of the musical note.
Timbre
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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.
Visual Picture of Sound
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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.
A 20 Hz Sound
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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.
Example
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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?
The 440 Hz sound (A note)
Fundamental Frequency and
Overtones
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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.
Sum of Sine waves
(Fundamental + Overtones)
Harmonics
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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.
Harmonics as Overtones
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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)
Illustration
The Harmonic Series
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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
1/5 + … (denoted by  n ) is called the
harmonic series.
 In mathematics the harmonic series diverges;
so what does this mean musically?

n 1
Example
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Example 4: If Sound A is the
fundamental frequency, then which
harmonic is Sound B? What is the
frequency of each sound?
The 12-tone (chromatic) Scale
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On a 12-tone scale the frequency
separating each tone is called a halfstep. These half steps correspond to
keys on the piano keyboard as illustrated
below:
Ratio of Frequencies to the
Fundamental Frequency.
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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 we’ve 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.
12
 Consequently f = 2 .
Table of Frequencies
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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 12 2 ; see
table to the right.
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
Major Scales
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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 it’s 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.
Example
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Example: Find the notes in the major A
scale.
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.
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: [email protected]
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