11/9/2012 Math for Liberal Arts MAT 110: Chapter 11 Notes Mathematics and Music Math and Art David J. Gisch S ound and Music S ound and Music • Any vibrating object produces sound. The vibrations • The frequency of a vibrating string is the rate at which it produce a wave . 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. 1
11/9/2012 Frequency S trings and Frequency • The lowest possible frequency for a particular string, • Each String has its own fundamental frequency which called its fundam ental frequency , occurs when it depends on characteristics including: vibrates up and down along its full length. ▫ length, ▫ density, and • Waves that have frequencies that are integer multiples of ▫ tension of the string the fundamental frequency are called harm onics. Music S cales and Mathematics 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. 2
11/9/2012 Music S cales and Mathematics Cycles Per S econd--Octave • Recall that increasing an octave doubles the frequency. Next C at 1040 CPS 260 ∗ 1.05946 � 275 Up an Octave 520 ∗ 2 � 1040 275 ∗ 1.05946 � 292 292 ∗ 1.05946 � 309 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 Musical S cales as Exponential Growth Calculation Of Frequency For Each Half S tep • Each half step the frequency increases by same multiply If Q 0 is the initial frequency, then the frequency of the note factor � � 1.05946 . n half-steps higher is given by � � � � �1.05946� � • C to C# ▫ C= 260 CPS, so C#=260*1.05946=275 CPS • C# to D Note that this is an exponential growth equation, which we ▫ C#= 275 CPS, so D=275*1.05946= 292 CPS studied in chapter 9. • What if I want to jump 10 half-steps or 30 half-steps? 3
11/9/2012 Frequency Frequency Example 10.A.1: If a note of F has a frequency of 347 CPS, Example 10.A.2: If a note of A has a frequency of 437 CPS, what is the frequency of the note 8 half-steps higher? what is the frequency of the note 20 half-steps higher? Frequency Frequency Example 10.A.3: One note has a frequency of 292 CPS and Example 10.A.4: To make a sound with a higher pitch, what another note has a frequency of 365 CPS, will they sound needs to be done with the frequency? “pleasing” together? 4
11/9/2012 The Digital Age The Digital Age • Until the early 1980s, nearly all music recordings were • To digitize this the computer cannot continually etch based on the analog picture of music. For example, into a vinyl record so it has to take samples. The more records etched the sound wave into the vinyl. samples per second the better quality. • Today, most of us listen to digital recordings of music. • CD audio has a sam ple 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 • The higher the bit rate the more “levels” of sound you analog electrical signal, which is then digitized by a can measure at one instant. computer. 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 Perspective and Symmetry should be used to encode it. This makes sense as quiet portions would not need as much sampling as other portions. 5
11/9/2012 Perspective Connection Between Visual Arts and Mathematics 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. • Lines that are parallel in the actual scene but not • Notice that the lines on the floor, which are parallel in real life perpendicular to the canvas intersect at their own (perpendicular to the canvas), are not parallel in the painting. vanishing point, called the horizon line . 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. 6
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11/9/2012 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 Example • Reflection sym m etry : An object remains unchanged when reflected across a straight line. Rotation sym m etry : An object remains unchanged when rotated through some angle about a point. Translation sym m etry : A pattern remains the same when shifted to the left or to the right. 8
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11/9/2012 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) Tilings (Tessellations) • Some tilings use irregular polygons. A tiling is an arrangement of polygons that interlock perfectly without overlapping. • Tilings that are periodic have a pattern that is repeated throughout the tiling.(PURE) Regular Polygon Tessellations • Tilings that are aperiodic do not have a pattern that is repeated throughout the entire tiling.(SEMI-PURE) 10
11/9/2012 Tiling Interior Angle of a Regular Polygon • A form of art called tiling or tessellation of a region ����� � 180�� � 2� involves: � ▫ Must fill ▫ No overlapping ▫ No gaps • Mush go into 360° evenly—it tessellates Tessellation Tiling? Example 10.B.3: Can a square, regular triangle and regular hexagon tessellate the plane? � � 6 ����� � 180�6 � 2� 6 � 720 � 120° 6 120° 120° 120° 120° � 120° � 120° � 360° �� 360 120 � 3 11
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