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Math for Liberal ArtsMAT 110: Chapter 11 NotesMath and ArtDavid J. Gisch

Mathematics and Music

Sound 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.

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.

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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.

Strings and Frequency• Each String has its own fundamental frequency which

depends on characteristics including:▫ length, ▫ density, and ▫ tension of the string

Music Scales 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 Scales and Mathematics

260 ∗ 1.05946 275275 ∗ 1.05946 292292 ∗ 1.05946 309

Cycles Per Second--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 Step

• 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 Scales as Exponential Growth

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FrequencyExample 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?

FrequencyExample 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?

FrequencyExample 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?

FrequencyExample 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 sample 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 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

• Reflection symmetry: An object remains unchanged when reflected across a straight line.

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

Rotation symmetry: 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 FrequencyExample 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 Polygon180 2

Tessellation

6 180 6 26

7206 120°

120°

120°120°

120° 120° 120° 360°

360120 3

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

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