Math for Liberal Arts MAT 110: Chapter 11 Notes Mathematics and - - PowerPoint PPT Presentation

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Math for Liberal Arts MAT 110: Chapter 11 Notes

Math and Art David J. Gisch

Mathematics and Music

S

• und and Music
• Any vibrating object produces sound. The vibrations

produce a wave.

• 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.

S

• und 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.

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Frequency

• The lowest possible frequency for a particular string,

called its fundam ental 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 harm onics.

S trings and Frequency

• Each String has its own fundamental frequency which

depends on characteristics including:

▫ length, ▫ density, and ▫ tension of the string

Music S cales and Mathematics

• 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.

Musical Notes

Each half step increases the frequency by approximately 1.05946 cycles per second.

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Music S cales and Mathematics

260 ∗ 1.05946 275 275 ∗ 1.05946 292 292 ∗ 1.05946 309

Cycles Per S econd--Octave

• Recall that increasing an octave doubles the frequency.

Middle C at 260 CPS Next C at 520 CPS Next C at 1040 CPS Lower C at 130 CPS 260 ∗ 2 520 520 ∗ 2 1040 260 2 130 Up an Octave Up an Octave Down an Octave

Calculation Of Frequency For Each Half S tep

• Each half step the frequency increases by same multiply

factor 1.05946.

• C to C#

▫ C= 260 CPS, so C#=260*1.05946=275 CPS

• C# to D

▫ C#= 275 CPS, so D=275*1.05946= 292 CPS

• What if I want to jump 10 half-steps or 30 half-steps?

If Q0 is the initial frequency, then the frequency of the note n half-steps higher is given by 1.05946 Note that this is an exponential growth equation, which we studied in chapter 9.

Musical S cales as Exponential Growth

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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.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.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.4: To make a sound with a higher pitch, what needs to be done with the frequency?

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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.

• Today, most of us listen to digital recordings of music.
• 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 Digital Age

• 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.

• CD audio has a sam ple rate of 44.1 kHz (44,100

samples per second) and 16-bit resolution per channel.

• 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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• Perspective
• Symmetry
• Proportion

At least three aspects of the visual arts relate directly to mathematics:

Connection Between Visual Arts and Mathematics

Perspective

Side view of a hallway, showing perspectives.

Perspective and Vanishing Point

• 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.

▫ 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.

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.

• Lines that are parallel in the actual scene but not

perpendicular to the canvas intersect at their own vanishing point, called the horizon line.

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Example Example Example Example

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Example S ymmetry

• Sy m m etry refers to a kind of balance, or a repetition of

patterns.

• In mathematics, sy m m etry is a property of an object

that remains unchanged under certain operations.

S ymmetry

• Reflection sym m etry: An object

remains unchanged when reflected across a straight line.

 Translation sym m etry:

A pattern remains the same when shifted to the left or to the right.

 Rotation sym m etry: An object

remains unchanged when rotated through some angle about a point.

Example

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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.

A tiling is an arrangement of polygons that interlock perfectly without overlapping.

Regular Polygon Tessellations

Tilings (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

• A form of art called tiling or tessellation of a region

involves:

▫ Must fill ▫ No overlapping ▫ No gaps

• Mush go into 360° evenly—it tessellates

Interior Angle of a Regular Polygon

180 2

• Tessellation

6 1806 2 6 720 6 120° 120° 120° 120° 120° 120° 120° 360° 360 120 3

Tiling?

Example 10.B.3: Can a square, regular triangle and regular hexagon tessellate the plane?

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