an acoustic analysis of single-reed woodwind instruments with an emphasis on design and ...

244 Pages·1998·3.48 MB·English

AN ACOUSTIC ANALYSIS OF SINGLE-REED WOODWIND
INSTRUMENTS WITH AN EMPHASIS ON DESIGN AND
PERFORMANCE ISSUES AND DIGITAL WAVEGUIDE
MODELING TECHNIQUES
a dissertation
submitted to the department of music
and the committee on graduate studies
of stanford university
in partial fulfillment of the requirements
for the degree of
doctor of philosophy
Gary Paul Scavone
March 1997
(cid:13)c Copyright 1997 by Gary Paul Scavone
All Rights Reserved
AN ACOUSTIC ANALYSIS OF SINGLE-REED WOODWIND INSTRUMENTS
WITH AN EMPHASIS ON DESIGN AND PERFORMANCE ISSUES AND
DIGITAL WAVEGUIDE MODELING TECHNIQUES
Gary Paul Scavone
Stanford University, 1997
Currentacoustictheoryregardingsingle-reedwoodwindinstrumentsisreviewedandsummarized,
with special attention given to a complete analysis of conical air column issues. This theoretical
acoustic foundation is combined with an empirical perspective gained through professional perfor-
mance experience in a discussion of woodwind instrument design and performance issues. Early
saxophone design speci(cid:12)cations, as given by Adolphe Sax, are investigated to determine possible
influencesoninstrumentresponseandintonation. Issuesregardingsaxophonemouthpiecegeometry
are analyzed. Piecewise cylindrical and conical section approximations to narrow and wide mouth-
piece chamber designs o(cid:11)er an acoustic basis to the largely subjective examinations of mouthpiece
e(cid:11)ects conductedin the past. The influence of vocal tract manipulations in the controland perfor-
mance of woodwind instruments is investigated and compared with available theoretical analyses.
Several extendedperformancetechniques are discussed in terms of acousticprinciples.
Discrete-time methods are presented for accurate time-domain implementation of single-reed
woodwindinstrumentacoustictheoryusingdigitalwaveguidetechniques. Twomethodsforavoiding
unstable digital waveguide scattering junction implementations, associated with taper rate discon-
tinuities in conical air columns, are introduced. A digital waveguide woodwind tonehole model is
presented which incorporates both shunt and series impedance parameters. Two-port and three-
port scattering junction tonehole implementations are investigated and the results are compared
withtheacousticliterature. Severalmethodsformodelingthesingle-reedexcitationmechanismare
discussed.
Expressive controls within the context of digital waveguide woodwind models are presented,
as well as model extensions for the implementation of register holes and mouthpiece variations.
Issues regarding the control and performance of real-time models are discussed. Techniques for
verifying and calibrating the time-domain behavior of these models are investigated and a study
is presented which seeks to identify an instrument’s linear and nonlinear characteristics based on
periodicprediction.
iii
Acknowledgements
IamgreatlyindebtedtotheDepartmentofMusicandtheCenterforComputerResearchinMusic
and Acoustics (CCRMA) for their (cid:12)nancial support throughoutmy studies at Stanford University.
CCRMAprovidedawonderfulinterdisciplinaryenvironmentinwhichtopursuemyinterests. When
I (cid:12)rst sought out a means for combining my musical and technical interests, I had no clear vision
of where this search would lead ...I’d like to thankJohn Chowning for encouragingme to apply to
the Ph.D. program in Computer-Based Music Theory and Acoustics without a speci(cid:12)c project in
mind and the trust in my abilities he demonstratedby admitting me.
Along the way, Julius Smith and Perry Cook listened patiently to my questions and always
hadenlightenedsuggestionstomake. ChrisChafe’senthusiasm,of course,wasconstantandgreatly
appreciated. DougKeefekindlyallowedmetovisithislaboratoryinSeattleandgraciouslyagreedto
beareaderforthisthesis,longbeforeIevenbeganwritingit! TheDSPgroupatCCRMAingeneral,
and Tim Stilson, Bill Putnam, Scott Van Duyne, and Dave Berners in particular, provided much
helpduringthecourseofthiswork. FernandoLopez-Lezcanosu(cid:11)eredthroughmyseeminglyendless
computerglitchesand troubles. My brother,John, did his parttohelp maintain my sanitythrough
timely o(cid:11)ers to visit his Lake Arrowhead cabin, as well as enticing ski trip proposals. Meanwhile,
my family in Western New York just kept sending those packages of candy and encouragingcards.
Despite the appearance that I would never (cid:12)nish this dissertation, their support never faltered
...many thanks to Jackie, Jerry, Sheri, Peter,Paul, Gail and Noni. My grandmother,Eleanor, has
been and will always continue to serve as a source for encouragementin my life ...her picture sits
prominently on top of my computer monitor. Finally, Melissa provided companionship, listened to
my troubles, and helped take my mind o(cid:11) the work at hand, all of which played a large part in
allowing me to enjoy this undertaking through to its completion. To all these people, I express my
sincerestgratitude.
iv
Preface
Thestudyofmusicalinstrumentacousticbehavioris arelativelyyoungscience. Earlyworkwith
directapplicationtowoodwindinstrumentswasconductedbyWeber(1830),Helmholtz(1954),and
BouasseandFouch(cid:19)ee(1930). Itisclear,however,thattheinstrumentsthemselves\came(cid:12)rst." The
art of instrument building was acquired principally through empirical means. Over the course of
severalcenturies,knowledgewas handeddownfrom generationtogenerationandincrementalgains
wereachieved. Mostmodernacousticinstrumentsreachedtheirpresentform by thelatenineteenth
centuryand have undergonerelatively little improvement since.
Inthepast,therewaslittleassociationbetweentheinstrumentbuilderandthephysiciststudying
musical instrument acoustics. Several large instrument manufacturers, such as C. G. Conn Ltd.,
employedscientistsforthepurposeofimprovingtheirinstrumentdesigns,thoughtheirresearchwas
largelybasedonexperimentalmeasurements. Onlyinrecentyearshasmusicalinstrumenttheoretical
acousticsreachedanadvancedenoughleveltoo(cid:11)erpotentialrewardstothedesignprocess. Oneaim
of this study is to demonstrate that these theoreticaladvances have necessitatedclose cooperation
between scientists and professionalinstrument performers. That is, the theory has progressed to a
pointwheresubtlecontrolissuesincorporatedbyprofessionalperformers,butotherwiseunknownto
thephysicist,canhavegreatimportanceincon(cid:12)rmingorrefutingtheoreticalmodels. Theinsightof
expertperformerswithreasonablegroundingin acousticfundamentalswill play a signi(cid:12)cantrole in
re(cid:12)ningcurrenttheory,aswellasin furtherexplorationsintodelicateissuesofinstrumentbehavior.
Theprimaryobjectiveofthisstudyistodetailtheimplementationofthebestavailable acoustic
theoryregardingsingle-reedwoodwindinstrumentsusingdigitalwaveguidetechniques. Digitalwave-
guide modeling is an e(cid:14)cient discrete-time method for time-domain simulation of one-dimensional
wave propagation,which has gained increasing popularity in the (cid:12)eld of musical acoustics research
sincethepublicationofMcIntyreet al.(1983). Severalstudieshave previouslybeenconductedwith
regard to waveguide modeling of woodwind instruments (Hirschman,1991; Va¨lim¨aki, 1995). What
distinguishesthis workis its comprehensivescope,togetherwith itsfoundationon themostcurrent
literature from the musical acoustics research community. Issues regarding conical air columns in
general, and saxophones in particular, are analyzed in great detail. Instabilities associated with
v
PREFACE vi
discrete-time models of conical-section discontinuities are discussed and stable solutions are pre-
sented. A variety of extensions regarding toneholes, mouthpieces, and performance expression are
introduced.
Chapter1 of this studypresentsa detailed and comprehensivediscussion of the acoustic behav-
ior of single-reed woodwind instruments, as determined from available musical acoustics research
literature. No new theoryis presentedin this chapter,thoughthe discussion with regard to conical
air columns is particularly comprehensive in comparison to most sources. Chapter 2 seeks to build
on the material of the previous chapter and combine it with insights gained from professional per-
formance expertise. Several issues relevant to instrument design and the performance community
areanalyzedin termsof acousticprinciples. Chapter3 shadowsChapter1, detailingtheimplemen-
tation of the theory presented in the (cid:12)rst chapter using digital waveguide techniques. A variety of
improvements and/or extensions to current waveguide modeling theory are discussed. Chapter 4
o(cid:11)ers further extensions to digital waveguide models of woodwind instruments which are inspired,
in part, from the discussion of Chapter 2. Further, techniques for calibrating the models based on
time-domain measurements are reviewed and an investigation of linear and nonlinear instrument
behavior is presented.
Contents
Acknowledgements iv
Preface v
List of Figures ix
List of Tables xv
List of Symbols xvi
1 Single-Reed Woodwind Acoustic Principles 1
1.1 Sound Propagationin Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Lumped AcousticSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Woodwind InstrumentBores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Cylindrical Bores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.2 Conical Bores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.3 Boundary Layer E(cid:11)ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.4 Time-Domain Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4 Sound Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.4.1 Non-Flaring Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.4.2 Horns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.4.3 Toneholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.5 The Nonlinear Excitation Mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2 Acoustical Aspects of Woodwind Design & Performance 51
2.1 Woodwind Air Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.1.1 Cylinders and Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.1.2 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.1.3 The Saxophone’s \Parabolic"Cone . . . . . . . . . . . . . . . . . . . . . . . . 66
2.2 Woodwind Toneholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.2.1 The Tonehole Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.2.2 The Single Tonehole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.2.3 Register Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.3 The Single-Reed Excitation Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3.1 Mouthpieces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3.2 Reeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.4 The Oral Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.5 ContemporaryPerformanceTechniques . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.5.1 Multiphonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
vii
CONTENTS viii
2.5.2 The Saxophone’s Altissimo Register . . . . . . . . . . . . . . . . . . . . . . . 89
2.5.3 Additional ContemporaryTechniques . . . . . . . . . . . . . . . . . . . . . . 91
3 Digital Waveguide Modeling of Single-Reed Woodwinds 92
3.1 Modeling Sound Propagationin One Dimension . . . . . . . . . . . . . . . . . . . . . 93
3.2 Modeling Lumped AcousticSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3 Modeling Woodwind InstrumentBores . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.3.1 Cylindrical Bores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.3.2 Conical Bores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.3.3 Diameter and Taper Discontinuities . . . . . . . . . . . . . . . . . . . . . . . 108
3.3.4 Fractional Delay Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.3.5 Boundary-Layer E(cid:11)ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.4 Modeling Sound Radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.4.1 Non-Flaring Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.4.2 Horns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.4.3 Toneholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.5 Modeling the NonlinearExcitation Mechanism . . . . . . . . . . . . . . . . . . . . . 143
3.5.1 The Pressure-DependentReflectionCoe(cid:14)cient . . . . . . . . . . . . . . . . . 145
3.5.2 The Reed-ReflectionPolynomial . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.5.3 The Dynamic WoodwindReed Model . . . . . . . . . . . . . . . . . . . . . . 150
3.5.4 Excitation Mechanisms Attached to Truncated Cones . . . . . . . . . . . . . 153
4 Digital Waveguide Model Extensions and Calibration 156
4.1 Extensions to Digital Waveguide Models of Woodwinds . . . . . . . . . . . . . . . . 157
4.1.1 Register Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.1.2 MouthpieceModels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.2 PerformanceExpression in Digital Waveguide WoodwindModels . . . . . . . . . . . 163
4.2.1 Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.2.2 Control Issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.3 Calibrating Digital Waveguide WoodwindModels . . . . . . . . . . . . . . . . . . . . 169
4.3.1 Measuring Woodwind InstrumentResponsesin the Time Domain. . . . . . . 169
4.3.2 PeriodicPredictionfortheDeterminationofLinearandNonlinearInstrument
Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5 Conclusions and Future Research 193
5.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.2 Suggestionsfor Future Research. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
A The Surface Area of a Spherical Wave Front in a Cone 196
B Saxophone Air Column Measurements 198
C Experimental Equipment 199
D MATLAB Code 200
D.1 openpipe.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
D.2 boundary.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
D.3 openhole.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
D.4 closhole.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
D.5 branch.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
D.6 clarinet.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
List of Figures
1.1 Wave-induced pressure variations on a cubic section of air. . . . . . . . . . . . . . . . 3
1.2 Linear acoustic system block diagrams.. . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 The Helmholtz resonatorand its mechanical analog. . . . . . . . . . . . . . . . . . . 8
1.4 A cylindrical pipe in cylindrical polar coordinates. . . . . . . . . . . . . . . . . . . . 9
1.5 A non-uniform bore(a) and its approximation in terms of cylindrical sections (b). . 14
1.6 Theoretical input impedance magnitude, relative to Z at x = 0; of the cylindrical
0
section structureshown in Fig. 1.5(b). . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 A conical section in sphericalcoordinates. . . . . . . . . . . . . . . . . . . . . . . . . 16
1.8 A divergent conicalsection and its associateddimensional parameters. . . . . . . . . 20
1.9 A convergent conical sectionand its associateddimensional parameters. . . . . . . . 20
1.10 Partial frequencyratios, relative to f ; for a closed-openconic frustum. . . . . . . . 22
0
1.11 A non-uniform bore (a) and its approximation in terms of cylindrical and conical
sections (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.12 Theoretical input impedance magnitude, relative to Z = (cid:26)c=S at x = 0; of the
0
cylindrical and conic sectionstructureshown in Fig. 1.11(b). . . . . . . . . . . . . . 25
1.13 Phase velocity (v ) normalized by c vs. frequency (top); Attenuation coe(cid:14)cient ((cid:11))
p
vs. frequency(bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.14 Input impedance magnitude of an ideally terminated cylindrical bore with viscous
and thermallosses (borelength = 0.3 meters,bore radius = 0.008 meters). . . . . . 28
1.15 Ageneralwindinstrumentrepresentedbylinearandnonlinearelementsinafeedback
loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.16 Theoretical input impedance magnitude (top) and impulse response (bottom) of a
cylindrical bore,normalized by the bore characteristicimpedance. . . . . . . . . . . 31
1.17 Theoretical input impedance magnitude (top) and impulse response (bottom) of a
conical bore,relative to R =(cid:26)c=S(0) at the small end of the bore. . . . . . . . . . . 33
0
1.18 A conical frustum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.19 Reflectancemagnitudeandlengthcorrection(l=a)versusthefrequencyparameterka
for a flanged and unflanged circular pipe. . . . . . . . . . . . . . . . . . . . . . . . . 37
1.20 Cross-section of a horn and the displacement of a volume element within it. . . . . . 38
1.21 Basic tonehole geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.22 T section transmission-line element representingthe tonehole. . . . . . . . . . . . . . 43
1.23 L section transmission-line element representingthe tonehole. . . . . . . . . . . . . . 44
1.24 Inputimpedancemagnitude(top),relativetothemainborewave impedanceZ ;and
0
reflectance (bottom)for the written note G on a simple six-hole flute, as described
4
in Keefe (1990). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.25 A single-reed woodwind mouthpiece. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.26 The single-reed as a mechanical oscillator blown closed. . . . . . . . . . . . . . . . . 46
ix
LIST OF FIGURES x
1.27 The volume flow and pressure relationships for a single-reed oral cavity/mouthpiece
geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.28 Steady flow (u ) througha pressure controlled valve blown closed. . . . . . . . . . . 49
ro
1.29 Dynamic flow througha pressure controlled valve blown closed for p =p =0:4: . . 50
oc C
2.1 Open-end radiation characteristics for an unflanged cylindrical pipe: (top) Load
impedance magnitude relative to the wave impedance Z ; (bottom) Tuning devia-
0
tion in cents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2 Thermoviscous e(cid:11)ects for wave travel over two lengths of a cylindrical pipe of length
0.5metersandradius0.008meters: (top)Attenuationcharacteristic;(bottom)Tuning
deviation in cents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3 Partial frequency ratios for a closed-open conic section versus (cid:12); the ratio of closed-
to open-endradii. The frequencies are normalized by f ; the fundamental resonance
0
for an open-openpipe of the same length as the frustum. . . . . . . . . . . . . . . . 57
2.4 Partial frequency ratios for a closed-open conic section versus (cid:12); the ratio of closed-
to open-end radii. The frequencies are normalized by f ; the fundamental resonance
c
for the same conic section complete to its apex. . . . . . . . . . . . . . . . . . . . . . 58
2.5 Partial frequency ratios for a closed-open conic section versus frustum length, for a
cone of half angle 2(cid:14) and small-end radius of 0.005 meters. The partial frequencies
are normalized by f ; the fundamental resonancefor an open-open pipe of the same
0
length as the frustum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.6 The input impedancemagnitude,relative toR =(cid:26)c=S at its input end, of a conical
0
frustum of length 0.5 meters,small-end radius 0.005 meters,and half angle of 2(cid:14): . . 60
2.7 Exponentialhorn pro(cid:12)les for flare parametersof m=2 ({) and m=3 (- -). . . . . . 63
2.8 Partialfrequencyratios,relative tothefundamentalresonance,forexponentialhorns
de(cid:12)ned by flare parametersof m=2 ({) and m=3 (- -). . . . . . . . . . . . . . . . 63
2.9 Partialfrequencyratios,relativetothefundamentalresonanceofanopen-openconic
frustum of the same length, for exponential horns de(cid:12)ned by the flare parameters
m=2 ({) and m=3 (- -).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.10 The pro(cid:12)le of an alto saxophone built by Adolphe Sax, in exaggerated proportions
[after (Kool,1987)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.11 Quadraticsurfaces: (a)Circularcone;(b)Ellipticcone;(c)Hyperboloidofonesheet;
(d) Elliptic paraboloid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.12 Various views of a \Parabolic cone." . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.13 Pro(cid:12)le of an alto saxophonebore,up to its lower bow. The boresectionbetweenthe
neckpipe and lower bow is represented by a conical frustum ({) and an exponential
horn (- -). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.14 Partialfrequencyratios,relativetothefundamentalresonanceofanopen-openconic
section of the same length, of a combined cone/exponential horn structure ({) and
a closed-open conical frustum (- -). The dotted lines indicate exact integer multiple
relationships to the (cid:12)rst closed-openconical frustum resonance. . . . . . . . . . . . . 71
2.15 An early saxophonemouthpiece design. . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.16 Approximate saxophone mouthpiece structures: (top) Wide chamber design; (bot-
tom) Narrow chamber design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.17 Theoreticalmouthpieceandair column structureinput impedances,relative to R =
0
(cid:26)c=S at the input of the structure: (top) Wide chamber design; (bottom) Narrow
chamber design.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.18 Normalized partial frequency ratios for the theoretical mouthpiece and air column
structures vs. air column length: ({) Wide chamber design; (- -) Narrow chamber
design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.19 A B[ clarinet multiphonic (cid:12)ngering and the correspondingwritten notes it produces. 89