Download Math for Liberal Arts MAT 110: Chapter 11 Notes

11/9/2012
Math for Liberal Arts
MAT 110: Chapter 11 Notes
Mathematics and Music
Math and Art
David J. Gisch
Sound and Music
• Any vibrating object produces sound. The vibrations
produce a wave.
Sound and Music
• The frequency of a vibrating string is the rate at which it
moves up and down. The higher the frequency (more
vibrations per second), the higher the pitch.
• Most musical sounds are made by vibrating strings
(guitar), vibrating reeds (saxophone), or vibrating
columns of air (trumpet).
• One basic quality of sound is pitch. The shorter the
string, the higher the pitch.
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Frequency
• The lowest possible frequency for a particular string,
called its fundamental frequency, occurs when it
vibrates up and down along its full length.
• Waves that have frequencies that are integer multiples of
the fundamental frequency are called harmonics.
Music Scales and Mathematics
Strings and Frequency
• Each String has its own fundamental frequency which
depends on characteristics including:
▫ length,
▫ density, and
▫ tension of the string
Musical Notes
• Raising the pitch by an octave corresponds to a
doubling of the frequency.
• Pairs of notes sound particularly pleasing when one note
is an octave higher than the other note.
▫ Because they integer multiples in terms of frequency.
• The musical tones that span an octave comprise a scale.
Each half step increases the frequency by approximately 1.05946 cycles
per second.
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Music Scales and Mathematics
Cycles Per Second--Octave
• Recall that increasing an octave doubles the frequency.
Next C at 1040 CPS
260 ∗ 1.05946
275
275 ∗ 1.05946
292 ∗ 1.05946
292
309
Up an Octave
520 ∗ 2
1040
Next C at 520 CPS
Up an Octave
260 ∗ 2
520
Middle C at 260 CPS
Down an Octave
260
2
130
Lower C at 130 CPS
Calculation Of Frequency For Each Half Step
• Each half step the frequency increases by same multiply
factor
1.05946.
• C to C#
Musical Scales as Exponential Growth
If Q0 is the initial frequency, then the frequency of the note
n half-steps higher is given by
1.05946
▫ C= 260 CPS, so C#=260*1.05946=275 CPS
• C# to D
▫ C#= 275 CPS, so D=275*1.05946= 292 CPS
Note that this is an exponential growth equation, which we
studied in chapter 9.
• What if I want to jump 10 half-steps or 30 half-steps?
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Frequency
Example 10.A.1: If a note of F has a frequency of 347 CPS,
what is the frequency of the note 8 half-steps higher?
Frequency
Example 10.A.3: One note has a frequency of 292 CPS and
another note has a frequency of 365 CPS, will they sound
“pleasing” together?
Frequency
Example 10.A.2: If a note of A has a frequency of 437 CPS,
what is the frequency of the note 20 half-steps higher?
Frequency
Example 10.A.4: To make a sound with a higher pitch, what
needs to be done with the frequency?
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The Digital Age
The Digital Age
• Until the early 1980s, nearly all music recordings were
based on the analog picture of music. For example,
records etched the sound wave into the vinyl.
• To digitize this the computer cannot continually etch
into a vinyl record so it has to take samples. The more
samples per second the better quality.
• Today, most of us listen to digital recordings of music.
• CD audio has a sample rate of 44.1 kHz (44,100
samples per second) and 16-bit resolution per channel.
• When a recording is made, the music passes through an
electronic device that converts sound waves into an
analog electrical signal, which is then digitized by a
computer.
• The higher the bit rate the more “levels” of sound you
can measure at one instant.
MP3 & MP4
• MP3 and MP4 files are now common for iPods and
iPhones. However, these files sample at much lower
rates, which reduces the file size but also reduces quality.
• VBR: iTunes now uses a variable bit rate. What VBR
encoders do is that they analyze each frame of audio to
be encoded and decide what is the minimum bitrate that
should be used to encode it. This makes sense as quiet
portions would not need as much sampling as other
portions.
Perspective and Symmetry
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Connection Between Visual Arts and Mathematics
Perspective
At least three aspects of the visual arts relate
directly to mathematics:
• Perspective
• Symmetry
• Proportion
Side view of a hallway, showing perspectives.
Perspective and Vanishing Point
Perspective
• Lines that are parallel in the actual scene, but not parallel
in the painting, meet at a single point, P, called the
principle vanishing point.
• All lines that are parallel in the real scene and
perpendicular to the canvas must intersect at the principal
vanishing point of the painting.
• Notice that the lines on the floor, which are parallel in real life
(perpendicular to the canvas), are not parallel in the painting.
Similarly with the edges of the ceiling.
• Lines that are parallel in the actual scene but not
perpendicular to the canvas intersect at their own
vanishing point, called the horizon line.
▫ All of these lines meet at a point, called the vanishing point.
• The parallel lines on the floor not perpendicular to the canvas do not
cross.
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Example
Example
Example
Example
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Example
Symmetry
• Symmetry refers to a kind of balance, or a repetition of
patterns.
• In mathematics, symmetry is a property of an object
that remains unchanged under certain operations.
Symmetry
Example
• Reflection symmetry: An object
remains unchanged when reflected
across a straight line.


Rotation symmetry: An object
remains unchanged when rotated
through some angle about a point.
Translation symmetry:
A pattern remains the
same when shifted to the
left or to the right.
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Example
Example
Example
Example
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Example
Frequency
Example 10.B.1: Identify the type of symmetry for each
letter.
(a) Identify the types of symmetry in the letter M.
(b) Identify the types of symmetry in the letter X.
Tilings (Tessellations)
A tiling is an arrangement of polygons that
interlock perfectly without overlapping.
Regular Polygon Tessellations
Tilings (Tessellations)
• Some tilings use irregular polygons.
• Tilings that are periodic have a pattern that is repeated
throughout the tiling.(PURE)
• Tilings that are aperiodic do not have a pattern that is
repeated throughout the entire tiling.(SEMI-PURE)
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Tiling
Interior Angle of a Regular Polygon
• A form of art called tiling or tessellation of a region
involves:
180
2
▫ Must fill
▫ No overlapping
▫ No gaps
• Mush go into 360° evenly—it tessellates
Tessellation
Tiling?
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180 6 2
6
720
120°
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120°
Example 10.B.3: Can a square, regular triangle and regular
hexagon tessellate the plane?
120°
120°
120°
120°
120°
360
120
360°
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