Physics in the ArtsRevised Edition

WHAT IS THE COMPLEMENTARY SCIENCE SERIES?

We hope you enjoy this book. If you would like to read other quality sciencebooks with a similar orientation see the order form and reproductions of thefront and back covers of other books in the series at the end of this book.

The Complementary Science Series is an introductory, interdisciplinary, andrelatively inexpensive series of paperbacks for science enthusiasts. The seriescovers core subjects in chemistry, physics, and biological sciences but oftenfrom an interdisciplinary perspective. They are deliberately unburdened byexcessive pedagogy, which is distracting to many readers, and avoid the oftenplodding treatment in many textbooks.

These titles cover topics that are particularly appropriate for self-studyalthough they are often used as complementary texts to supplement stan-dard discussion in textbooks. Many are available as examination copies toprofessors teaching appropriate courses.

The series was conceived to fill the gaps in the literature between conventionaltextbooks and monographs by providing real science at an accessible level,with minimal prerequisites so that students at all stages can have expert insightinto important and foundational aspects of current scientific thinking.

Many of these titles have strong interdisciplinary appeal and all have a placeon the bookshelves of literate laypersons.

Physics in the ArtsRevised Edition

P.U.P.A. GilbertUniversity of Wisconsin-Madison

and

W. HaeberliUniversity of Wisconsin-Madison

AMSTERDAM • BOSTON • HEIDELBERG • LONDONNEW YORK • OXFORD • PARIS • SAN DIEGO

SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Academic Press is an imprint of Elsevier

The cover image, by Pupa Gilbert, captures all the main themes of this book: light, color, color generatingmechanisms, photography, music, sound and waves. White light illuminating the surface of a compact disc isseparated into the colors of the spectrum, and photographed. Music is represented by the CD itself, a commonmusic medium, and a sound wave. This oscillation is computer-generated, and similar to a sound wave detected by amicrophone and displayed on the screen of an oscilloscope.

Academic Press is an imprint of Elsevier225 Wyman Street, Waltham, MA 02451, USARadarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands

c© 2012 Elsevier Inc. All rights reserved.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical,including photocopying, recording, or any information storage and retrieval system, without permission in writingfrom the publisher. Details on how to seek permission, further information about the Publisher’s permissionspolicies and our arrangements with organizations such as the Copyright Clearance Center and the CopyrightLicensing Agency, can be found at our website: www.elsevier.com/permissions

This book and the individual contributions contained in it are protected under copyright by the Publisher (other thanas may be noted herein).

NoticesKnowledge and best practice in this field are constantly changing. As new research and experience broaden ourunderstanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using anyinformation, methods, compounds, or experiments described herein. In using such information or methods theyshould be mindful of their own safety and the safety of others, including parties for whom they have a professionalresponsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liabilityfor any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, orfrom any use or operation of any methods, products, instructions, or ideas contained in the material herein.

Library of Congress Cataloging-in-Publication DataGilbert, Pupa.Physics in the arts / P.U.P.A. Gilbert and W. Haeberli.

p. cm.Includes bibliographical references and index.ISBN 978-0-12-391878-9 (alk. paper)1. Light. 2. Music–Acoustics and physics. I. Haeberli, W. (Willy) II. Title.QC355.3.G55 2011700.1'05–dc23

2011015964

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

For information on all Academic Press publicationsvisit our Web site at: www.elsevierdirect.com

Printed in China

11 12 13 14 15 9 8 7 6 5 4 3 2 1

�

Introduction

Light and Sound

Through light and sound we perceive the world. We have perfected severalways of communicating with each other, and most of these use either light orsound, or both. The highest form of communication is art, and most of the artsuse light and/or sound as their expression medium. The visual arts use light,so we can see the works of art; music uses sound so we can hear it; and ballet,movies, and music videos with computer graphics use both.

A deep yet accessible analysis of the physics of light and sound, and howour eyes and ears detect them, is not only intellectually enjoyable, but alsouseful to understand and interpret the world in which we live, all the pheno-mena that take place around us, and how we perceive them—in short, how weinterface with our planet, its inhabitants, and their creations. Understandingthe physics of light and sound may also increase the appreciation for worksof art and stimulate the artists among the readers to deepen their knowledgeof their media, of how people interface with them, and perhaps inspire newideas.

Deriving its name from phusis, the Greek for “nature,” physics is thescience that studies naturally occurring phenomena concerning energy ormatter. If we can understand a natural phenomenon, then we have contributedto mankind’s general knowledge, but often we can also harness it and use itto develop a better life for ourselves and all others on this planet. We thinkthis exciting concept can be exported to the arts and stimulate art productionsthrough the understanding of the physical phenomena underlying them. Thisvery thought motivates our desire to teach physics to artists and humanists

xi

xii Introduction

and to communicate the physics concepts in simple terms to a more generalaudience.

Both sound and light are wave phenonema. These are different kinds ofwaves—mechanical waves in sound and electromagnetic waves in light—butthey both oscillate like the surface of a water pond after a stone has beenthrown into it. Newton first coined the word spectrum of visible light, andthen subdivided it into seven colors. Why seven? This choice was arbitrary,but he chose seven by analogy with the seven notes of the musical scale. Wenow know that the normal human eye can distinguish almost a million colors!For reasons that we will explain later, we will, however, subdivide the visiblespectrum into three colors only: red, green, and blue.

Have you ever wondered why the human eye is sensitive to the specificradiation called “light”? Although the sun emits radiation, which, after filte-ring through the atmosphere, comprises the visible range, there is also a lot ofinfrared and ultraviolet light that reaches Earth’s surface. Why don’t we seein those ranges as well? Radiation in the visible range has the most effectiveenergy to interact with the objects of the world. It therefore best informs us onthe structure and behavior of objects around us.

The measurable electromagnetic spectrum extends from high-energygamma rays to low-frequency radio waves. This covers about 30 orders ofmagnitude in frequency (or energy, or wavelength). From the beginning to theend of what we call visible range (violet to red), the frequency varies by a merefactor of 2. In music, the frequency doubles in one octave. In the measurableelectromagnetic spectrum the frequency doubles 100 times (2100 � 1030).Poetically put, we can only “see one octave” on an idealized electromagneticpiano keyboard of 100 octaves! Yet so much happens in this narrow regionbecause radiation of these energies strongly interacting with electrons gene-rates nonflat, interesting absorption spectra, which in turn generate the millioncolors to which we are sensitive.

Compared to light, we hear as sound a much larger region of the usablefrequencies. The frequency range for audible sound is 20−20,000 oscilla-tions per second or 20−20,000 Hz. Higher frequencies, or ultrasounds, areused for medical imaging (10 MHz) and for communication by dolphins (upto 170 kHz), whales (up to 200 kHz), and bats (up to 120 kHz). Lower fre-quencies, or infrasounds, are produced by earthquakes, avalanches, volcanoes,nuclear tests, and even elephants (14 Hz)—for communicating with otherelephants within a 10 km distance.

The color and sound we perceive do not depend only on the physical,measurable stimuli, but also on the physiological and psychological responseof our eyes, ears, and brain to the stimuli. Color and sound, therefore, are bestdescribed by psycho-physical parameters. These are, as we will describe indetail later, hue, saturation, and brightness for color, and pitch, loudness, and

Introduction xiii

The bedroom, 1888 (oil on canvas) by Gogh, Vincent van (1853–90) c© Van Gogh Museum,Amsterdam, The Netherlands/The Bridgeman Art Library Nationality/copyright status:Dutch/out of copyright. Ursus Wehrli, Tidying up Van Gogh’s Bedroom at Arles. Copyrightc© Ursus Wehrli. From: Ursus Wehrli, Kunst aufraumen. Copyright c© 2002 KEIN & ABERAG, Zurich.

timbre for sound. A trombone and a viola can immediately be distinguished bythe listener even if they play the exact same tone. The attribute distinguishingthem is called the timbre or, by another analogy with light, “tone color.”

The goal of this book is not to tidy up art, rationalize it, and explain itin scientific terms. Other excellent authors did that, as shown in the figuresabove. Our goal is to add another component—physics—to the enjoyment ofart. Understanding the form and function of musical instruments adds to themusic. Similarly, understanding color, color vision, and color mixing can onlyexpand the palette of visual artists and the intellectual enjoyment of all peoplelooking at their art.

Both authors collaborated on all parts of the book, however, Pupa Gilberttakes primary responsibility for the part on Light, and Willy Haeberli for thaton Sound. We wish you an enjoyable read!

�Physics in the Arts

P.U.P.A. Gilbert andWilly Haeberli

University ofWisconsin-Madison

Contents

Introduction xi

Light and Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Light and Light Waves 01

1.1 Speed of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 051.2 Electromagnetic Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 061.3 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 07

2 Reflection and Refraction 10

2.1 Specular Reflection of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Refraction of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Total Internal Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Reflection and Refraction in Diamonds . . . . . . . . . . . . . . . . . . . 212.5 The Rainbow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

v

vi Contents

3 Lenses 30

3.1 The Prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Converging and Diverging Lenses . . . . . . . . . . . . . . . . . . . . . . . . 313.3 Focal Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 Images—Real and Virtual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5 Three Easy Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.6 The Lens Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6.1 Note on Magnification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.7 Lens Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.7.1 Chromatic Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.7.2 Spherical Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.8 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 The Eye 56

4.1 Accommodation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Eyeglasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3 Nearsighted Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4 Farsighted Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.5 Astigmatic Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 Photography 63

5.1 The Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Focusing the Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3 Choosing the Exposure Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.4 Choosing the Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.5 Depth of Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.5.1 Why the f Number? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.6 The Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.7 Digital Photography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.8 Putting it All Together: Taking a Photograph. . . . . . . . . . . . . . . 765.9 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Color and Color Vision 82

6.1 Color . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.2 Color Sensitivity of the Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.3 Physical and Psychological Color . . . . . . . . . . . . . . . . . . . . . . . . 89

Contents vii

6.4 Color: Hue, Saturation, and Brightness . . . . . . . . . . . . . . . . . . . . 906.5 Light Interaction with other Objects . . . . . . . . . . . . . . . . . . . . . . 926.6 Scattering or Diffuse Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . 926.7 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7 Additive Color Mixing 99

7.1 Primary Colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.2 Adding Primary Colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.3 The Color Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.4 Low-Brightness Colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.5 Spectral Colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.6 Non-Spectral Colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.8 Additive Color Mixing in Painting . . . . . . . . . . . . . . . . . . . . . . . . 1147.9 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8 Subtractive Color Mixing 118

8.1 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1188.2 Subtractive Primary Colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.2.1 Subtractive primaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.3 Color Photography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.4 Pigments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.5 Change in Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1288.6 Why Do Blue and Yellow Make Green? . . . . . . . . . . . . . . . . . . . 1308.7 Change in Hue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318.8 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

9 Color-Generating Mechanisms 136

9.1 Illuminating Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369.2 Pigments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369.3 Structural Color: Iridescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1379.4 More Color-Generating Mechanisms Due to Iridescence . . . . 1399.5 Color in Gemstones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1429.6 Mineral Color Due to Charge Transfer . . . . . . . . . . . . . . . . . . . . 1449.7 Mineral Color Due to Color Centers . . . . . . . . . . . . . . . . . . . . . . 1449.8 Color in Gems Due to Band Gap Absorption of Light . . . . . . . 145

viii Contents

10 Periodic Oscillations 148

10.1 Displacement Graph: Position x Changes with Time t . . . . . . 15110.2 The Period T and the Frequency f . . . . . . . . . . . . . . . . . . . . . . . 15310.3 Large and Small Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15410.4 Speed of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15410.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

11 Simple Harmonic Motion 158

11.1 The Spring Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16011.2 Oscillation Frequency for Simple Harmonic

Motion (SHM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16111.3 Wave Shape of Simple Harmonic Motion. . . . . . . . . . . . . . . . . . 16311.4 Phase Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16511.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

12 Damped Oscillations and Resonance 168

12.1 Damped Oscillations—The Concept of “Damping Time” . . . 16812.2 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17012.3 Build-up and Decay of Musical Tones . . . . . . . . . . . . . . . . . . . . 17512.4 Applications in Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

12.4.1 Resonators in Musical Instruments . . . . . . . . . . . . . . . . . . 175

12.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

13 Adding Sound Sources: Beats and Harmony 179

13.1 Principle of Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17913.2 Two Pure Tones of the Same Frequency . . . . . . . . . . . . . . . . . . . 18013.3 Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18213.4 Harmony. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18413.5 For the Fun of It: Lissajous Figures . . . . . . . . . . . . . . . . . . . . . . . 18513.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

14 Sound Waves 190

14.1 Propagation of a Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19014.2 Longitudinal and Transverse Waves . . . . . . . . . . . . . . . . . . . . . . . 19214.3 Sound Waves in Air Are Longitudinal Waves . . . . . . . . . . . . . . 19314.4 Speed of Sound in Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Contents ix

14.5 Wavelength and Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19614.5.1 Relevance to Size of Instruments or Loudspeakers . . . . . . 197

14.6 Sound Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19814.7 Interference of Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19914.8 Concert Hall Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20114.9 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

15 Sound Perception: Pitch, Loudness, and Timbre 206

15.1 Loudness and Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20715.2 Loudness and Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21015.3 Pitch Discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

16 The Ear 214

16.1 The Parts of the Ear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21416.2 Place Theory of Pitch Perception . . . . . . . . . . . . . . . . . . . . . . . . . 21616.3 What Do the Auditory Nerves Tell the Brain? . . . . . . . . . . . . . . 217

17 Vibration of Strings 220

17.1 Single Modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22017.2 Higher Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22217.3 Traveling Versus Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . 22317.4 The Voicing Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22517.5 How Do Modes Relate to Music? . . . . . . . . . . . . . . . . . . . . . . . . . 22617.6 Damping of Higher Partials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22717.7 Plucked Strings: Missing Partials . . . . . . . . . . . . . . . . . . . . . . . . . 22717.8 Playing Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22817.9 Real Strings Have Some Stiffness . . . . . . . . . . . . . . . . . . . . . . . . 22817.10 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

18 Pipes 231

18.1 Pressure Pulse in a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23118.2 Reflections in Open and Closed Pipes . . . . . . . . . . . . . . . . . . . . . 232

18.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

18.3 Standing Waves in Open Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . 23318.4 Fundamental Frequency of Open Pipe. . . . . . . . . . . . . . . . . . . . . 23418.5 Higher Modes of Open Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23518.6 Fundamental Frequency of Closed Pipe . . . . . . . . . . . . . . . . . . . 237

x Contents

18.7 Higher Modes of Closed Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23818.8 Playing Tunes on Wind Instruments: Fingerholes

and Overblowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24018.9 Other Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24018.10 Acoustic Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24118.11 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

19 Fourier Analysis 243

19.1 The Fourier Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24319.2 Sound Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24419.3 Fourier Analyzer (Sound Analyzer) . . . . . . . . . . . . . . . . . . . . . . . 24919.4 Fourier Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25119.5 Why Can’t We Synthesize a Stradivari? . . . . . . . . . . . . . . . . . . . 25219.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

20 Musical Scales 256

20.1 Musical Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25720.2 Consonance (Harmony): Simple Number Ratios . . . . . . . . . . . 25820.3 The Major Triad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25920.4 Constructing a Scale: The Just Scale . . . . . . . . . . . . . . . . . . . . . . 26020.5 Whole and Half Tone Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 26320.6 Names of Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26420.7 Transposing: Why Black Keys? . . . . . . . . . . . . . . . . . . . . . . . . . . 26620.8 Perfection Sacrificed: The Tempered Scale . . . . . . . . . . . . . . . . 26720.9 Major and Minor Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27320.10 The Natural Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27320.11 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

21 Musical Instruments 275

21.1 Structure of Musical Instruments . . . . . . . . . . . . . . . . . . . . . . . . . 27521.2 Excitation Mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27621.3 Playing a Tune . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27821.4 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

22 Solutions to Problems 284

Index 307