Measured Tones: The Interplay of Physics and Music,2nd Edition

385 Pages·2002·12.749 MB·English

by  Ian Johnston

MEASURED TONES
The interplay of physics and music
Second edition
Ian Johnston
Institute of Physics Publishing
Bristol and Philadelphia
Copyright © 2002 IOP Publishing Ltd.
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Firstpublished1989
Reprintedwithcorrections1993
Reprinted1994,1997,2001
Secondedition2002
Reprintedwithcorrections2003
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Acknowledgments
Theproductionofabookofthischaracterwouldbeimpossiblewithouta
largenumberofillustrations,particularlyofmusicalinstruments. Wehave
tried to obtain permission for all illustrations which are still in copyright,
butthereareoneortwowherewehavebeenunsuccessful. Mythanksare
duetothefollowingorganizationsandindividualswhogavetheirpermis-
siontousematerial:
H.Bagot,p.276l;BostonSymphonyOrchestra,p.178;CordonArt,Baarn,
Holland(forM.C.Escherheirs)p.111u,p.112u;DiagramVisualInforma-
tion Ltd, p.224u, p.270u, p.303u; Dover Publications Inc, p.57u, p.59ul,
p.60ul, p.118ul, p.137, p.167; and (from Music, A Pictorial Archive of
WoodcutsandEngravingsbyJimHarter1980)p.41,p.43ul,p.54u,p.118m,
p.120u, p.123l, p.125u, p.211l, p.213, p.224l, p.226l, p.227ul, p.230ul,
p.231, p.232, p.269ul, p.273, p.274; Harper and Row Inc (from Harper’s
Dictionary of Music by Christine Ammer, drawings by C.M. Ciampa and
K.L. Donlon, 1972) p.87l, p.89u, p.218l, p.270l, p.271m, p.272u, p.275u;
S. Levarie, p.2, p.275l; Lord Rennell of Rodd, p.338; Macmillan Pub-
lishers Ltd (fromThe New Grove Dictionary Of Music and Musiciansed.
Stanley Sadie, 1980) p.303l; McGraw-Hill Publishing Co. (from R.D.
Chellis,PileFoundations,1961),p.63;OxfordUniversityPress(fromThe
New Oxford Companion to Music, ed Denis Arnold, 1983) p.128, p.307;
The Ann Ronan Picture Library, p.21, p.22, p.24, p.27, p.72, p.93, p.154,
p.185,p.236,p.237;W.J.Strong,p.276u,p.277l;UnwinHymanLtd(from
C.A. Ronan The Astronomers, 1964) p.11, p.23; John Wiley and Sons
(from J.W. Kane and M.M. Sternheim, Life Science Physics 1978) p.241;
MichaelNicholson/CORBIS,p.107;Bettmann/CORBIS,p.107.
Copyright © 2002 IOP Publishing Ltd.
Contents
List of tables x
Prologue xii
1. Why these and not others? 1
Lengthofstrings.Harmoniousintervals.Pythagoras.Pentatonic
andPythagoreanscales. Ptolemy. Boethius.
2. Music and scientific method 17
Medieval music. Josquin. Copernicus. Kepler. Galileo. Vi-
brations. Pendulumsandstrings. Consonance. Measurementof
frequency. Standardsofpitch.
Interlude 1. Brass instruments 37
3. The harmonies of a mechanical universe 56
Mechanical forces. Energy. Oscillation theory. Mersenne’s
Laws. Baroquemusic. Instruments. Equaltemperament.
Interlude 2. The piano 74
4. Overtones of enlightenment 86
Newton and the Age of Reason. Visualization of oscillations.
Overtones. Harmonicanalysisandsynthesis. Rameauandhar-
mony. EndoftheEnlightenment.
Interlude 3. The violin 109
Copyright © 2002 IOP Publishing Ltd.
5. Over the waves 131
Wave theory. Speed of sound. Wave properties: reflection, ab-
sorption,diffraction. HuygensandYoung. Standingwaves.
Interlude 4. Acoustics in architecture 152
6. The romance of electricity 171
TheRomanticmovement. Studyofelectricity,Faraday. Acous-
tics. Energy coupling and acoustic impedance. Mismatch the-
orem. Standingwavesinaircolumns.
Interlude 5. Woodwind instruments 195
7. Summertime in Heidelberg 221
Wagner and Helmholtz. Theory of resonance. Working of the
ear. Pitch recognition and discrimination. Psycho-acoustics.
Theoryofconsonance.
Interlude 6. Percussion instruments 247
8. O brave new world 265
Communication devices: microphones, gramophones, valves,
transistors. Effect on music. Modern sound engineering: fre-
quencyresponse,transients. Technology.
Interlude 7. Electronic instruments 287
9. I think, therefore I am 304
Theoryofcommunication. Information. Computers. Computer
composition.
Interlude 8. Sublimest of instruments, the voice 319
Epilogue 339
Copyright © 2002 IOP Publishing Ltd.
Appendix 1. Musical notation 341
Appendix 2. Logarithms 346
Appendix 3. Measurement of pitch intervals 350
Appendix 4. Measurement of loudness 353
Appendix 5. Acoustic impedance 357
Appendix 6. Pentatonic scales 363
Bibliography 369
Accompanying web site 375
Copyright © 2002 IOP Publishing Ltd.
List of Tables
Just scale frequencies 36
Vibration frequencies, cylinders and cones 43
The harmonic series 46
Equal tempered and just scales 72
Physical properties of strings 75
Fourier harmonics 98
Representative wavelengths in air 141
Standing waves on a string 151
Absorption coefficients of materials 160
Standing waves, closed pipe 186
Standing waves, open pipe 186
Vibration frequencies, closed/open pipes 187
Loudness levels 238
Vibration frequencies of a membrane 253
Vibration frequencies of a metal bar 259
Copyright © 2002 IOP Publishing Ltd.
Representative formant frequencies 332
Musical notation, duration 342
Musical notation, pitch 345
Powers of ten 346
Logarithms (integer) 347
Logarithms (fractional) 348
Pythagorean tuning 352
Just tuning 352
Equal tempered tuning 352
Acoustic power, representative instruments 353
Intensity, representative instruments 354
Loudness, decibel scale 355
Representative impedance mismatches 359
Copyright © 2002 IOP Publishing Ltd.
Prologue
The purpose of this book is easy to explain. I am a physicist and not a
musician. Yet I am interested in music and am aware of just how much
thetwofieldshavetooffertooneanother. Iwouldliketomakethismore
widely known. I believe that your enjoyment of music can be increased
therebyandsocanyourunderstandingofphysics. Whateveryourinterest,
alittlebitofextraknowledgecanonlyhelp. Itisassimpleasthat.
Thereareofcoursemanybooksonthissubject. Theyareusuallywrit-
tenbyphysicists(Ihaveneverseenonewrittenbyamusiciantellingscien-
tistswhattheyshouldknowaboutmusic). Mostofthemhavewhatseems
tomeagravefault: theypresupposeacertainframeofmindintheirread-
ers. Theirearlychaptersaredevotedtosimplephysics—vibrations,waves,
Fourier analysis, etc. The second half deals with musical instruments, ar-
chitectural acoustics, scales, electrical recording and reproduction. It is a
classicalscientificapproach—theoryfirstandapplicationslater.
Thetroubleisthatthisapproachreallyonlysuitsscientificallyminded
readers. MostmusicalpeoplethatIknow,findthephysicalconceptsquite
obscure,andhavegreatdifficultyrecallingthemforlaterapplications. So
Iwanttotryadifferentapproach,todevelopthephysicalconceptsandthe
musical applications together. This approach will make for rather an un-
usualordering. Forexample, itispossibletounderstandalotaboutbrass
instruments with relatively little physical knowledge, so I will talk about
them quite early. But woodwinds are much more complicated theoreti-
cally,andmustwaittilltowardstheend. Theveryfirsttopicwillbemusi-
calscalesandconsonance,whichisanextremelysubtlemusicalquestion,
butcanbediscussedwithonlyafewverysimplescientificobservations.
So my plan has been firstly to explain the physics that is involved in
music, trying all the time to simplify the details as much as possible. To
thisendIhavescrupulouslyavoidedalgebraicmathematicalequations,in
the belief that they tend to frighten non-scientists. It seems to me that if
they are written out in words, they look more user-friendly. Secondly, I
have tried at each stage to make it clear where the physics is important in
a musical context. However I am not a expert on music or musicology—
although I think that everything I have to say is perfectly well known and
accepted, and can be checked in one of the standard musical encyclope-
dias. So please keep in mind that, when I talk about musical matters, I
Copyright © 2002 IOP Publishing Ltd.
amtryingtosaythatthephysicalconceptshaveapplicationhere, without
beingdefinitive.
One word of warning. I have arranged the material historically, be-
causethatseemsanaturalandlogicalorderingfortheapproachIwantto
take. Physicalideashavetobeunderstoodinsequence,theeasyonesfirst
and later ones building on the earlier. And by and large, that is how they
developed. So this book will, in parts, read like a history of the subject.
Butitisimportant,Ithink,tomakeitclearthatthisdoesnotclaimtobea
serioushistoricalwork.
Scientists, when trying to explain science, have always used an ‘his-
torical’ approach—thumbnail sketches of great figures, the dates and cir-
cumstances of various discoveries, and so on. A lot of what they say is
probably apocryphal. Did Archimedes really jump out his bath shouting
“Eureka!”? DidRo¨ntgenreallydiscoverX-raysbyleavingalumpofpitch-
blendeonsomeunexposedfilm? Inaway,literalhistoricalaccuracyisnot
important. The message usually is: “This is a simple and easy to observe
fact, but it is not trivial. There was certainly a time when people did not
knowitordidnotrealizeitssignificance”.
Itmaybethatmyapproachinthisbookshouldbeputintothis‘apoc-
ryphal history of science’ category. To give you one example, many his-
torians and philosophers of science (though not working scientists) seem
to dislike the idea that science ‘progresses’. The story, as I tell it, very
definitely progresses from the time of Pythagoras to the present day. I
havechosenpeopleandideastotalkaboutinordertobuildupacomplete
picture,eachconceptrestingonthefoundationofthepreviousones. Ide-
liberately chose not to talk about the dead ends, the ideas that didn’t lead
anywhere,regardlessofwhetherornotthispresentsanaccuratepictureof
howscienceisactuallydone. NeverthelessIhavetriedtobeasaccurateas
Icaninmyuseofhistoricalinformation,andIhavebeencarefultoinclude
areferencebibliographyfromwhichyoucancheckonwhatIhavesaidor
canfollowitfurtherifyouwish.
But most importantly, I have never attempted to explain a physical
idea in any but modern terms. To present a correct explanation of some
effectbytalkingaboutanincorrectexplanationfirst,usuallyjustincreases
confusion in the reader’s mind, no matter how valid that approach might
be in historical terms. In the end, the history is there as a context for the
physicsandthemusic. ItisalsotherebecauseIfinditinteresting. Ihope
youdotoo.
Lastly, my treatment of musical instruments, for reasons that I have
hintedatearlier, willnotfitintothissimplehistoricalordering. SoIhave
called the chapters dealing with them interludes, and you can skip over
them if you are looking for a connected story. Within these interludes I
have talked about the historical development of each class of instrument
becauseitisusuallyeasiertoappreciatethephysicsinthesimplerinstru-
Copyright © 2002 IOP Publishing Ltd.