Math for Liberal Arts MAT 110: Chapter 11 Notes

Transcription

1 Math for Liberal Arts MAT 110: Chapter 11 Notes Math and Art David 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. 1

2 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 cycles per second. 2

3 Music Scales and Mathematics Cycles Per Second--Octave Recall that increasing an octave doubles the frequency. Next C at 1040 CPS Up an Octave Next C at 520 CPS Up an Octave Middle C at 260 CPS Down an Octave Lower C at 130 CPS Calculation Of Frequency For Each Half Step Each half step the frequency increases by same multiply factor C to C# C= 260 CPS, so C#=260* =275 CPS C# to D C#= 275 CPS, so D=275* = 292 CPS Musical Scales as Exponential Growth If Q 0 is the initial frequency, then the frequency of the note n half-steps higher is given by 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? 3

4 Frequency 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 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 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 10.A.4: To make a sound with a higher pitch, what needs to be done with the frequency? 4

5 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 5

6 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 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. 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. 6

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8 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. 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. 8

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10 Frequency 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) 10

11 Tiling A form of art called tiling or tessellation of a region involves: Must fill No overlapping No gaps Interior Angle of a Regular Polygon Mush go into 360 evenly it tessellates Tessellation Tiling? B.3: Can a square, regular triangle and regular hexagon tessellate the plane?

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