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Acoustics of the xiao : a case study of modern methods for the design of woodwind instruments Lan, Yang 2014

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Acoustics of the xiao:A case study of modern methods for the design ofwoodwind instrumentsbyYang LanB.Sc., Huazhong University of Science and Technology, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2014c© Yang Lan 2014AbstractThe xiao is a Chinese end-blown flute with a history over a thousand yearsand well known in China for its elegant sound. The xiao is little known out-side China, but its close relative, the Japanese shakuhachi, is better knowninternationally. The xiao has not been well developed or standardized (be-cause of the bamboo’s varying geometries) for the contemporary musicalrequirements and is imperfect in tuning, tone range, tonal stability andplayability of high notes. In acoustics, all these imperfections can be char-acterized by the acoustical impedance. As an air-reed instrument, the xiaoplays at its input impedance minima. In the work reported in this thesis,the xiao was modelled by a modified transmission-matrix method, and animpedance tube was built for measuring the xiao’s acoustical impedance tovalidate the model (accurate to a few cents). Then player effects were takeninto account by an empirical formula, and the model was able to predict theplaying frequencies of a xiao with any tone hole positions, sizes, and arbi-trary bore shape along the symmetry axis. Based on this model, numericaloptimizations were applied to improve the xiao, and a set of optimal fin-gerings for the xiao were obtained systematically. Several xiaos made fromPVC pipes with optimized tone holes show good tuning over three octaves.A xiao with additionally optimized bore shape was machined out of acrylic,showing improved tonal stability and rich harmonics.iiPrefaceThis thesis focuses on an acoustical study of the xiao, which belongs tothe broad area of woodwind research. The use of acoustical impedance wasadvised by my supervisor Chris Waltham, who built the first version of ourimpedance tube. The experimental apparatus described in this thesis(theimpedance tube and its accessories, in section 2.2 and 2.2.3) were built bythe author. The measurements were done by the author, who also wrote,from scratch, the Python scripts for data analysis.Figure 1.1, 1.3, 1.10, 1.11, 1.12, 3.2 and 3.3 were taken from publishedliterature. In Figure 3.4 the CAD model used for illustrating the xiao headwas originally plotted by Martin O’Keane of our departmental machine shop.The TMM and optimization codes were originally written by the author(based on the formalism of TMM code written in C by Dickens (2007),and the application of the L-BFGS-B algorithm was derived from Lefebvre(2010)).The xiao models pipe-xiao-1 to pipe-xiao-8 and xiao heads were madeby the author. The acrylic xiaos were made by Dan Skjaveland of ourdepartmental machine shop.The main ideas of this thesis (and results before section 5.1.1) have beenpresented at the International Symposium of Musical Acoustics (2014) andpublished in the proceedings (Acoustical impedance of the xiao; Lan andWaltham (2014)). The paper was written by this author and edited by mysupervisor Chris Waltham. Some text and figures of the paper were used inthis thesis.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction to the xiao . . . . . . . . . . . . . . . . . . . . . 11.1.1 History and classifications . . . . . . . . . . . . . . . 11.1.2 Playing style, fingerings and tone range . . . . . . . . 41.1.3 Making and tuning of xiao and shakuhachi . . . . . . 41.2 Sound waves in pipes and woodwind instruments . . . . . . . 61.2.1 Sound wave equations . . . . . . . . . . . . . . . . . . 71.2.2 Transmission matrices . . . . . . . . . . . . . . . . . . 91.3 Acoustical impedance . . . . . . . . . . . . . . . . . . . . . . 131.3.1 Acoustical impedance of a cylindrical pipe . . . . . . 141.3.2 Acoustical impedance of the xiao . . . . . . . . . . . 151.4 Resonances characterized by the acoustical impedance . . . . 161.4.1 Resonance of a cylindrical pipe . . . . . . . . . . . . . 161.4.2 Resonance and playing frequency of the xiao . . . . . 171.5 Goals of designing a good xiao . . . . . . . . . . . . . . . . . 192 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1 Impedance tube . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.2 Singularities and the singular factor . . . . . . . . . . 23ivTable of Contents2.1.3 Calibration of the impedance tube . . . . . . . . . . . 252.2 Building an acoustical impedance measurement system . . . 262.2.1 General and refined excitation signal . . . . . . . . . 292.2.2 Noise estimation and microphone requirements . . . . 322.2.3 Two new calibration methods . . . . . . . . . . . . . 352.2.4 Effects of reference plane mismatch . . . . . . . . . . 392.2.5 Determine the microphone locations . . . . . . . . . . 412.2.6 Cautions and tips in measurement . . . . . . . . . . . 432.2.7 A complete measurement procedure . . . . . . . . . . 462.3 Acoustical Impedance Measurements of the Xiao . . . . . . . 473 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1 TMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1.1 Radiation impedance and radiation loss . . . . . . . . 513.1.2 Tone holes . . . . . . . . . . . . . . . . . . . . . . . . 523.1.3 Tone hole undercuts . . . . . . . . . . . . . . . . . . . 563.2 A TMM model for the xiao . . . . . . . . . . . . . . . . . . . 563.2.1 Modelling the xiao embouchure . . . . . . . . . . . . 563.2.2 Modelling the full xiao . . . . . . . . . . . . . . . . . 613.2.3 Sound pressure and flow inside the xiao . . . . . . . . 653.3 Cross fingering, impedance irregularities and the woodwindcut-off frequencies . . . . . . . . . . . . . . . . . . . . . . . . 654 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.1 Optimization objectives . . . . . . . . . . . . . . . . . . . . . 704.1.1 Objective function . . . . . . . . . . . . . . . . . . . . 714.2 Optimization algorithm . . . . . . . . . . . . . . . . . . . . . 714.3 Optimization of the xiao . . . . . . . . . . . . . . . . . . . . 724.3.1 Phase one: optimizing the tone holes . . . . . . . . . 724.3.2 Phase two: optimizing the bore shape . . . . . . . . . 724.4 Speed and efficiency of the code . . . . . . . . . . . . . . . . 764.5 Optimizing the xiao fingerings . . . . . . . . . . . . . . . . . 774.5.1 A systematic investigation . . . . . . . . . . . . . . . 775 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.1 Phase one: PVC xiaos with optimized tone holes . . . . . . . 825.1.1 Based on optimized fingerings . . . . . . . . . . . . . 845.2 An unfinished bamboo xiao . . . . . . . . . . . . . . . . . . . 865.2.1 Measurement of the Bore Shape . . . . . . . . . . . . 865.3 Phase two: an acrylic xiao with optimized bore shape . . . . 88vTable of Contents6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94AppendicesA Fingering Chart of the Xiao . . . . . . . . . . . . . . . . . . . 98A.1 Traditional fingerings . . . . . . . . . . . . . . . . . . . . . . 98A.2 Optimized fingerings . . . . . . . . . . . . . . . . . . . . . . . 99B Measurement and Modelling Results of the Xiaos . . . . . 100B.1 Pipe-xiao-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100B.2 Pipe-xiao-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113B.3 Acrylic xiao . . . . . . . . . . . . . . . . . . . . . . . . . . . 123viList of Tables2.1 Geometries and usage of several accessories for the impedancetube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1 Geometries and embouchure end-corrections of the xiao heads 615.1 Geometry of pipe-xiao-3 . . . . . . . . . . . . . . . . . . . . . 835.2 Geometries of pipe-xiao-7 . . . . . . . . . . . . . . . . . . . . 855.3 Geometries of pipe-xiao-8 . . . . . . . . . . . . . . . . . . . . 86A.1 Traditional xiao fingering chart . . . . . . . . . . . . . . . . . 98A.2 Optimized xiao fingering chart . . . . . . . . . . . . . . . . . 99viiList of Figures1.1 One of the eight chibas preserved in the Sho¯so¯in treasury, Japan 21.2 Two northern style xiaos . . . . . . . . . . . . . . . . . . . . . 41.3 Typical shakuhachi bore shapes . . . . . . . . . . . . . . . . . 61.4 Plane waves inside a cylindrical pipe . . . . . . . . . . . . . . 81.5 Spherical waves in a conical pipe . . . . . . . . . . . . . . . . 91.6 Input and output of a cylindrical pipe . . . . . . . . . . . . . 101.7 Input and output of a conical pipe . . . . . . . . . . . . . . . 111.8 A cylindrical pipe of a typical xiao’s size . . . . . . . . . . . . 141.9 Acoustical impedance and standing waves of a cylindrical pipe 171.10 Behavior of air-jet . . . . . . . . . . . . . . . . . . . . . . . . 181.11 Dependency of the jet length on notes . . . . . . . . . . . . . 201.12 Dependency of playing frequencies on the jet velocity . . . . . 212.1 Singular factor of sample microphone setups . . . . . . . . . . 252.2 Setup of the acoustical impedance measurement system . . . 272.3 Accessories for the impedance tube . . . . . . . . . . . . . . . 282.4 A general excitation signal . . . . . . . . . . . . . . . . . . . . 312.5 Microphone signals of a sample measurement . . . . . . . . . 322.6 Microphone spectra and the background noise . . . . . . . . . 332.7 A set of resonance-free short pipes for calibration . . . . . . . 352.8 Ripples in the measured impedance curve . . . . . . . . . . . 372.9 Measured impedance curve after optimization of sound speed 392.10 Measured impedance of the open pipe-2 . . . . . . . . . . . . 402.11 Impedance measurement calibrated by a 8mm-cavity . . . . . 412.12 Impedance measurement with reference plane mismatch . . . 422.13 Impedance measurement using old microphone locations . . . 432.14 Singular factor of 4 microphones . . . . . . . . . . . . . . . . 442.15 A failed connection mechanism for the impedance tube . . . . 452.16 Brass pipe connected to the impedance tube by transparenttape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.17 Use modelling clay to seal the connection of a short cavity . . 47viiiList of Figures2.18 Connecting a xiao to the impedance tube . . . . . . . . . . . 493.1 Demonstration of modelling woodwind bore by TMM . . . . 503.2 Geometry of a woodwind tone hole . . . . . . . . . . . . . . . 523.3 Equivalent circuit of a woodwind tone hole . . . . . . . . . . 533.4 Geometry of a xiao embouchure . . . . . . . . . . . . . . . . . 573.5 The xiao head SXH1 . . . . . . . . . . . . . . . . . . . . . . . 583.6 Impedance of the xiao head SXH1. . . . . . . . . . . . . . . . 593.7 Xiao heads with different geometries . . . . . . . . . . . . . . 603.8 A full xiao pipe-xiao-3 . . . . . . . . . . . . . . . . . . . . . . 623.9 Acoustical impedance of pipe-xiao-3 with all finger holes closed 633.10 Acoustical impedance of pipe-xiao-3 with fingering ooxo ooxo 643.11 Pressure and flow inside pipe-xiao-3 with fingering xxxx xxxx 663.12 Pressure and flow inside pipe-xiao-3 with fingering oxxx ooxo 673.13 Pressure and impedance of pipe-xiao-2 with fingering xxxx xxxx 683.14 Pressure and impedance of pipe-xiao-2 with fingering oxxx ooxo 694.1 Optimized bore shape of xiao-v1 . . . . . . . . . . . . . . . . 744.2 Optimized bore shape of xiao-v2 . . . . . . . . . . . . . . . . 754.3 Optimized bore shape of xiao-v3 . . . . . . . . . . . . . . . . 764.4 Impedance curves of sorted fingerings for B6 . . . . . . . . . . 804.5 Impedance curves of sorted fingerings for C7 . . . . . . . . . . 814.6 Impedance curves of sorted fingerings for D7 . . . . . . . . . 815.1 Pipe-xiao-1 and pipe-xiao-2 . . . . . . . . . . . . . . . . . . . 835.2 Pipe-xiao-4, pipe-xiao-5 and pipe-xiao-6 . . . . . . . . . . . . 845.3 Pipe-xiao-7 and pipe-xiao-8 . . . . . . . . . . . . . . . . . . . 855.4 A probe mounted on lathe for measuring bamboo bore . . . . 875.5 Measured bore shape of a bamboo pipe. . . . . . . . . . . . . 885.6 Measured and fitted impedance of the bamboo pipe . . . . . 895.7 Bore shape of the acrylic xiao . . . . . . . . . . . . . . . . . . 905.8 An acrylic xiao with optimized bore shape . . . . . . . . . . . 915.9 Impedance before and after the tone hole undercuts . . . . . 92ixAcknowledgementsI would like to thank my supervisor Chris Waltham for providing the pre-cious research opportunity of studying the Chinese xiao at UBC and hisadvice and comments on this thesis. Thanks to Evert Koster for usefuldiscussions and being the second reader of my thesis. Thanks to MurrayHodgson of UBC Mechanical Engineering for sharing the anechoic chamber.Thanks to Alan Thrasher of UBC School of Music for discussions and helpwith playing the xiao. Thanks to Martin O’Keane and Dan Skjaveland formaking the acrylic xiao possible. Thanks to the WestGrid teams for pro-viding the computing facilities. Finally, I would like to thank my fianceeMeiling for her persistent spiritual support and her common passion for thexiao.xChapter 1Introduction1.1 Introduction to the xiao1.1.1 History and classificationsThe xiao (Thrasher, 2014) is a Chinese end-blown flute with a history of onethousand years or longer, traditionally made of bamboo. The terminologyand form of Chinese end-blown flutes have evolved throughout its history.The most notable period of xiao’s evaluation is that of the Tang dynasty(618 - 907 AD)Common Origin with the Japanese ShakuhachiDuring the Tang, the most widely used Chinese end-blown flute was thechiba. The chiba’s name means its length is 1.8 chi (chi is the Chinese footunit, and ba is eight in Chinese). The chiba has six finger holes including onelocated at the back as a thumb hole. With all finger holes closed, the chibaplays one octave lower than the yellow bell (Kuttner, 1965) pitch standard(corresponding to a tube of 0.9 chi). The chiba embouchure’s blowing edgehas an “arc” shape formed by cutting off an oblique plane from outside ofthe bamboo pipe.The chiba was imported to Japan in the 8th century and is believed tobe the origin of the shakuhachi (Kishibe et al., 2014; Yoshikawa, 2012). Theshakuhachi’s name means the instrument’s length is 1.8 shaku (Japanese footunit). Nowadays, there are still eight originally imported chiba preserved inthe Sho¯so¯in treasury (Jimusho, 1967) in Japan. Figure 1.1 shows one of thepreserved chiba and its measured geometry. The length of the chiba shownis only 38.25 cm, because at that time the foot unit was shorter.11.1. Introduction to the xiaoFigure 1.1: One of the eight chibas preserved in the Sho¯so¯in treasury, Japan.The figure is taken from Goodman (2009), originally from Jimusho (1967,pp. 133-134).21.1. Introduction to the xiaoAfter the Tang dynasty, for some reasons the chiba become rarely usedin China, and the term xiao (or donxiao) started to be used for the end-blown flute. Some changes in the end-blown flute’s form also took place.The significant changes were:• New types of embouchure were used by notching from inside of thebamboo pipe, and the blowing edge could be in “U” or “V” shape.• Pairs of holes were added beyond the finger holes. The instrumentbecome longer, and there is no longer pitch-length correspondence.• Two semi-tone finger holes were added, increasing the finger hole num-ber to eight.The last change was made quite recently (in the 1930s), and the eight-hole xiao become popular in the recent decades. However there are still aconsiderable number of six-hole xiaos in use and being made today.It is worthwhile to mention that the shakuhachi preserves much of thechiba’s form that the xiao doesn’t. e.g. the “arc” shape embouchure andthe pitch-length relationship. In the current Japanese foot unit, 1.8 shaku ≈54.5 cm is the standard length of a shakuhachi in key of D (the lowestnote is D4, 293.7 Hz). There are also changes in the shakuhachi comparedto the ancient chiba. A noticeable change is the finger hole number beingdecreased to five for the Japanese musical scale. A hard-to-notice but signif-icant change is the shakuhachi’s bore. The shakuhachi bore gets delicatelyshaped for producing rich harmonics (Ando, 1986; Tukitani et al., 1994).Good shakuhachis are highly esteemed by musicians around the world.The traditional xiao has no specific refinement in the bore. However, inthe last two or three decades, the international impact of the shakuhachi hasreached back to China. As an amateur xiao player and maker, the authorlearnt that some xiao makers in Taiwan and mainland China are starting topay attention to the bore of xiao.ClassificationsNowadays the xiao’s forms depends on region of China where they are used,classified as the northern and southern style. In the northern style, thebamboo’s node is kept with a rectangular or an oval cutting, and the blowingedge is usually in a “U” shape, as shown in Figure 1.2. The northern styletends to be long as two to three pairs of additional holes (labelled in thefigure as a1 1 to a3 3) are usually located beyond the finger holes.31.1. Introduction to the xiaoFigure 1.2: Two northern style xiaos. f1 to f8 are eight finger holes. a1 1 toa3 3 are three pairs of additional holes.In the southern style, the node is broken through and the blowing edgecan be “U”, “V” or “arc” shape . The top end of the southern xiao is blockedby the jaw of the player when playing, leaving only the blowing side of theembouchure open (similar as playing the northern style). The southern styletend to be short, with at most one pair of additional hole.1.1.2 Playing style, fingerings and tone rangeThe xiao is played vertically by holding it with an angle of about 45◦ withthe body; each hand controls half of the six or eight finger holes, with thetop finger hole (thumb hole) located at the back. For convenience of playing,the finger pad is used instead of finger tips for closing the finger holes. Forthe first octave, finger holes are opened in sequence to raise a half tone ora whole tone (for all upper finger holes, lower holes are kept open, two ofthem are mostly closed for holding the xiao). The second octave comes fromover-blowing with the same fingering except for the cross fingering one. Forall notes in the third octave, cross fingerings are used. Traditional fingeringsof all the notes were shown in appendix A.1. The xiao’s lowest note (e.g.D4, 293.7 Hz) and the note played by opening all the lower hand finger holes(e.g. G4) define the key of the instrument. All the xiaos under study in thisthesis are in key of DG and have a U-shape embouchure edge with eightfinger holes. The lengths of the xiaos range from 540 to 980 mm, the longerones have up to three pairs of adjustment holes (tassel holes) starting ataround 540 mm from the embouchure end.1.1.3 Making and tuning of xiao and shakuhachiThe making of xiao is not well standardized because the naturally grownbamboo pieces have varying geometries; also additional holes enable thexiao to have very different lengths. As a result, the required finger hole41.1. Introduction to the xiaolocations/diameters may be different for each xiao. With knowledge fromexperienced xiao makers and practice of making several xiaos, the authorlearnt that the current xiao makers locate the finger holes (and the loweradjustment holes) using parameters from existing instruments or given byexperienced makers. The xiao is played to test its tunings and the fingerholes’ positions/diameters are slightly adjusted accordingly. However eachfinger hole controls pitches of at least two notes and sometimes the adjust-ments cannot correct tuning of every note.In addition, the bore of a xiao is also known to have effects on tuning(especially for balancing the different octaves). A conical shape tapering toa smaller diameter at the pipe’s end is preferred (similar to the Baroqueflute). A bamboo pipe near the root part of the plant is usually chosen tomake the xiao because the tapering will be naturally formed after removingthe bamboo nodes and smoothing the inner wall.ShakuhachiShakuhachis of the same key (pitch standards) have roughly the same lengthbecause they do not have additional holes. The shakuhachi makers payspecial attention to the instrument’s bore, and the bore diameter at differentpositions (the bore shape) is adjusted to control the tuning and timber (thebore shape tuning mechanism will be discussed in detail in section 4.3.2).After centuries of evolution, the shakuhachi is mostly standardized in itsbore and finger holes. Figure 1.3 shows the typical bore shape of shakuhachismade in the modern era and at the beginning of 19th century. Adjustingthe bore shape allows the shakuhachi tuning and timbers to be delicatelycontrolled. The specially formed bore shape is known to be the reason ofshakuhachi’s unique timber – very rich harmonics.Current imperfections and limitationsCurrently the xiao has imperfections in the following aspects:1. The cross fingering of C5 provides a pitch change far less than 100cents from C5], so the two notes cannot be both in tune.2. Most xiaos have a tone range limited to 2.5 octaves or less.3. The tuning is not always good, especially the high notes.4. The high notes are usually hard to play, and the high note fingeringsare inconsistent for different xiaos.51.2. Sound waves in pipes and woodwind instrumentsFigure 1.3: Typical shakuhachi bore shapes. No 7 and No 47: bore shape ofshakuhachis made by contemporary famous makers. No 33 and No 34: boreshape of exquisite shakuhachis made by famous makers at the beginning ofthe 19th century. Figure from Ando (1986).The imperfections can by partly solved (or have already been solved bysome experienced makers) by the bore shape tuning method as describedabove. However there are limitations on the bore shape tuning. The boreshape cannot be easily measured, so the tuning is done by adding/removingmaterial at/from the inside wall of the instrument. This process is usuallyslow, the makers may need days or more to tune one instrument. Moreover,changes of the bore at one position for some notes may ruin tuning of someoriginally correct notes. The bore shape tuning technique is an empiricalapproach, and sometime cannot reach a good tuning result.1.2 Sound waves in pipes and woodwindinstrumentsSome acoustics basics are given in this section as the necessary material forintroducing the concept of acoustical impedance in section 1.3. Below is alist of symbols commonly used in acoustics:61.2. Sound waves in pipes and woodwind instrumentsp sound pressure (abbreviated to “pressure”)u acoustical velocity (abbreviated to “velocity”)S cross-sectional areaU = uS volumetric flow (abbreviated to “flow”)Z acoustical impedancec speed of sound in free airv wave speed inside a pipeα attenuation coefficient from wall lossesk = ωv − iα wave number including wall losses1.2.1 Sound wave equationsDerivation of sound wave equations are shown in this section for two typesof geometry: cylindrical pipe and conical pipe. For most woodwind instru-ments, including the xiao, these are the only two geometries that need to beconsidered.Plane waves in a cylindrical pipeIn a cylindrical pipe with a rigid wall, sound waves are confined to propa-gate inside a cylindrical symmetrical space. The wave equation has a seriesof solutions, but below a cutoff frequency fc = 1.84c2pia (Fletcher and Rossing,1998), only plane waves propagate along the pipe and form steady reso-nances. Here a is radius of the pipe. For xiaos of different keys (pitchstandards), the maximum radius is about 15 mm, corresponding to a cutofffrequency of about 6.7 kHz. The frequency of interest for studying the xiaois 150 to 4000 Hz, which is below the cutoff frequency. So only plane wavesare considered for the cylindrical pipe.To illustrate the derivation of several sound wave equations, wall losses(described later in section 1.2.2) are neglected for now and the wave numberis: k = ωc . Sound pressure inside the pipe can be written as:p(x, t) = (Ae−ikx +Beikx)eiωt, (1.1)where A and B are the amplitude of waves propagating in the positive andnegative directions, and x is the position along the pipe. For the planewaves, the sound pressure is uniform at a cross section of the pipe, as shownin Figure 1.4.71.2. Sound waves in pipes and woodwind instruments2axpU ABFigure 1.4: Plane waves inside a cylindrical pipe. The sound pressure isuniform in this cross sectional plane. A and B shows amplitude of wavesthe positive and negative direction.Acoustical velocity is related to the sound pressure by the 1D equationof motion:∂p∂x= −ρ∂u∂t. (1.2)For steady resonaces, acoustical velocity in the pipe can be derived fromEq. (1.2) by inserting Eq. (1.1) and integrate over t:u(x, t) =1ρc(Ae−ikx −Beikx)eiωt. (1.3)Since U = Su, the volumetric flow is:U(x, t) =1Z0(Ae−ikx −Beikx)eiωt, (1.4)where Z0 =ρcS is defined as the pipe’s characteristic impedance neglectingthe wall losses.Spherical waves in a conical pipeSound waves propagating in a conical pipe are actually a fraction of sphericalwaves. As shown in Figure (1.5), the center of the spherical waves is thecone’s apex.81.2. Sound waves in pipes and woodwind instrumentsxABx xFigure 1.5: Cross-sectional view of spherical waves in a conical pipe. Theblue circles show the spherical waves; the blue solid line shows the actualfraction of waves inside the conical pipe.For derivation of the spherical wave equations, its helpful to use thevelocity potential φ, which has a simple form of solution for a sphericalwave:φ(x, t) =1x[Ae−ikx +Beikx]eiωt, (1.5)where x is the distance to the cone’s apex. The velocity potential is relatedto the acoustical velocity by u = ∂φ∂x in 1D. So,u(x, t) = −1x2[A(1 + ikx)e−ikx +B(1− ikx)eikx]eiωt. (1.6)p(x, t) can be derived from u using Eq. (1.2) and rewrite ω = ck:p(x, t) = −iρck1x[Ae−ikx +Beikx]eiωt. (1.7)The volumetric flow U is:U(x, t) = −Sx2[A(1 + ikx)e−ikx +B(1− ikx)eikx]eiωt, (1.8)where S is the cross-sectional area of the cone at position x.1.2.2 Transmission matricesFor a segment of cylindrical or conical pipe, its pressure and flow at inputx1 and output x2 can be related by a 2× 2 matrix:[p1U1]= Tx1,x2[p2U2]. (1.9)91.2. Sound waves in pipes and woodwind instrumentsThe transmission matrix can be used for calculating acoustical impedanceas shown in section 1.3, and it is also a key idea for modelling the xiao asshown in Chapter 3.Elements of the transmission matrix are calculated using the same man-ner as electrical two-port networks:T11 =p1p2∣∣U2=0T12 =p1U2∣∣p2=0T21 =U1p2∣∣U2=0T22 =U1U2∣∣p2=0.(1.10)Transmission matrices of cylindrical and conical pipes are derived from theirsound wave equations in section 1.2.1 as below.Transmission matrix of a cylindrical pipeThe cylindrical pipe under study is shown in Figure 1.6. According tox2x1L2ap1U1 AB p2U2Figure 1.6: Input and output of a cylindrical pipe.Eq. (1.1) and Eq. (1.4), the pressure and flow at the output x2 can bewritten as:p(x2) = (Ae−ikx2 +Beikx2) = A′ +B′ (1.11)U(x2) =1Z0(Ae−ikx2 −Beikx2) =1Z0(A′ −B′). (1.12)Now set B′ = A′ (U2 = 0) to calculate T11 and T21:p(x1) = (Ae−ikx1 +Beikx1) = A′(eikL + e−ikL) (1.13)U(x1) =1Z0(Ae−ikx1 −Beikx1) =A′Z0(eikL − e−ikL) (1.14)p(x2) = 2A′. (1.15)101.2. Sound waves in pipes and woodwind instrumentsHere L = x2 − x1 is the length of the pipe. Then T11 and T21 can becalculated from Eq. (1.13), Eq. (1.14) and Eq. (1.15):T11 =eikL + e−ikL2= cos(kL) (1.16)T21 =eikL − e−ikL2= i1Z0sin(kL). (1.17)Similarly, T12 and T22 can be calculated by setting B′ = −A′ (p2 = 0):T12 =eikL + e−ikL2= iZ0 sin(kL) (1.18)T22 =eikL − e−ikL2= cos(kL). (1.19)Writing the results together, the transmission matrix of a cylindrical pipeis:Tcylin =[cos(kL) iZ0 sin(kL)i 1Z0 sin(kL) cos(kL)]. (1.20)Transmission matrix of a conical pipeThe conical pipe under study is shown in Figure 1.7. According to Eq. (1.7)xABx xp1U1 p2U2x1x2LFigure 1.7: Input and output of a conical pipe.and Eq. (1.8), pressure and flow at the output x2 of a conical pipe can bewritten as:p(x2) = −iρck1x2[Ae−ikx2 +Beikx2](1.21)U(x2) = −S2x22[A(1 + ikx2)e−ikx2 +B(1− ikx2)eikx2]. (1.22)111.2. Sound waves in pipes and woodwind instrumentsAgain use A′ = Ae−ikx2 and B′ = Beikx2 . Now set B′ = −1+ikx21−ikx2A′ (U2 = 0)to calculate T11 and T21:p(x1) = −iρckA′ 1x1(eikL −1 + ikx21− ikx2e−ikL)(1.23)U(x1) = −S1x22A′((1 + ikx1)eikL − (1− ikx1)1 + ikx21− ikx2e−ikL)(1.24)p(x2) = −iρck1x2A′(1−1 + ikx21− ikx2). (1.25)Expressions of T11 and T21 seem to be complicated from the above equations,but they can be efficiently simplified after introducing several trigonometricnotations (Kulik, 2007; Olson, 1957):T11 = −rt2 sin(kL− θ2) (1.26)T21 = irS1ρct1t2 sin(kL+ θ1 − θ2), (1.27)where r = x2x1 , S1 is approximated to be the cross-sectional area of the coneat x1, t1 =√( 1kx1 )2 + 1 = arcsin(kx1), t2 = arcsin(kx2), θ1 = arctan(kx1)and θ2 = arctan(kx2). Similarly, T12 and T22 can be calculated by settingB′ = −A′ (p2 = 0):T12 = irρcS2sin(kL) (1.28)T22 = rS1S2t1 sin(kL+ θ1), (1.29)where S2 is approximated to be cross-sectional area of the cone at x2. Writ-ing the results in matrix form:Tcon = r[−t2 sin(kL− θ2) iρcS2sin(kL)iS1ρc t1t2 sin(kL+ θ1 − θ2)S1S2t1 sin(kL+ θ1)]. (1.30)The transmission matrix Eq. (1.30) is only valid for a converging conicalpipe (S1 < S2). For a diverging conical pipe (S1 > S2), the transmissionmatrix is (Lefebvre, 2010, p. 20):Tdiv =1T11T22 − T12T21[T22 T12T21 T11], (1.31)where T11, T12, T12 and T22 are elements of converging transmission matrixTcon. It is worthwhile to mention that Tdiv is not the inverse of Tcon becausethe flow has opposite directions in these two situations.121.3. Acoustical impedanceAt this point, it is also necessary to consider the wall loss effects. Afterincluding these, the wave number is varies with the radius a of the pipe, sofor the conical transmission matrix Eq. (1.30) the mean of the wave numberalong the pipe needs to be used:k =1L∫ x2x1k(x)dx. (1.32)The wall losses are described below.Wall losses in pipesIn real pipes and woodwind instrument bores, the air has viscous and ther-mal effects near the wall. The effects can be accounted for as wall losses andby writing the wave number k in complex form:k =ωv− iα, (1.33)where v is the wave speed in he pipe and α is the attenuation coefficient.Both v and α are frequency dependent. The following formulae have ac-ceptable accuracy within the xiao’s frequency range of interest (Fletcherand Rossing, 1998):v ≈ c[1−1.65× 10−3af1/2], (1.34)α ≈3× 10−5f1/2a. (1.35)Some more recent and detailed expressions of wall losses can be found inKeefe (1984) and Lefebvre (2010, section 1.3.2).After taking the wall losses into account, the characteristic impedanceZc =Skρω(1.36)is a complex number. The characteristic impedance is only a scaling fac-tor throughout the acoustical impedance calculations, so the approximationZc ≈ Z0 = Sρc is used in this thesis.1.3 Acoustical impedanceMost woodwind instruments, including the xiao, can be characterized bytheir acoustical impedance. The acoustical impedance Z is defined as the131.3. Acoustical impedanceratio of sound pressure p over volumetric flow U at a reference plane:Z =pU. (1.37)The acoustical impedance is usually studied in frequency domain, whereboth Z, p and U are complex numbers.1.3.1 Acoustical impedance of a cylindrical pipeA cylindrical pipe with similar size to a typical xiao (key of DG) is studiedfirst, as shown in Figure 1.8. The pipe has radius a = 9 mm and lengthL=54 cmL''=55.1 cm2a=1.8 cmx1 x2x1' x2'Figure 1.8: A cylindrical pipe of a typical xiao’s size. x1 and x2 are positionof the pipe’s left and right end, L is the pipe’s length. x′1 , x′2 and L′′ arepositions and length after taking the end-corrections into account.L = 0.54 m (this is also the typical size of a Japanese Shakuhachi in key ofD). To obtain the unknown acoustical impedance at the pipe’s head x1 =0 m, the analysis needs to start from the pipe’s end x2 = 0.54 m, where thepipe has known radiation impedance Zr =p2U2. The radiation impedancecan be approximated by an short extension of the pipe, widely known as theend-correction:Zr ≈ iδkZ0, (1.38)where δ is length of the end-correction and its value depends on the pipeend’s flange condition. For the most usual cases when the pipe’s wall is nottoo thick (unflanged), δ = 0.61a (Fletcher and Rossing, 1998).With the end-correction approximation, the pipe is ideally open at its“extended” end x′2 and has zero impedance. Note that here the approx-imated radiation impedance Eq. (1.38) is only used for illustration of thesimple pipe and it keeps only the imaginary part. A complete descriptionof the radiation impedance is made in section 3.1.1.Next, the cylindrical pipe’s transmission matrix Eq. (1.20) relates thepipe’s pressure and flow at x1 and x2:[p1U1]= Tcylin[p2U2]=[cos(kL) iZ0 sin(kL)i 1Z0 sin(kL) cos(kL)] [Zr1]U2. (1.39)141.3. Acoustical impedanceFrom Eq. (1.39), the pipe’s impedance at x1 is:Z1 =p1U1=cos(kL)Zr + iZ0 sin(kL)iZrZ0 sin(kL) + cos(kL). (1.40)The above equation shows how the impedance at different positions of thepipe can be calculated by the transmission matrices. In fact the pipe’simpedance at its head can be further simplified using the end-correctionapproximation:Z1 = iZ0 tan(kL′). (1.41)Where the pipe is treated as having effective length L′ = L+δ, and radiationimpedance of this “extended” pipe is Z ′r = 0. It can be shown that, whenthe end-correction is small compared to the wavelength (δ  λ, or kδ  1),Z1 = iZ0sin(kL) cos(kδ) + sin(kδ) cos(kL)cos(kL) cos(kδ) + sin(kL) sin(kδ)≈ iZ0sin(kL) + kδ cos(kL)cos(kL)= iZ0 tan(kL) + Zr.(1.42)So, within the frequency limit (which is true in most cases), the effectof the short end-correction can be treated as a direct addition ofan acoustical impedance. This is similar to two electrical elements in aseries circuit.1.3.2 Acoustical impedance of the xiaoCompared to a cylindrical pipe, the xiao is complicated in the followingaspects:• The xiao’s embouchure connects to the free atmosphere through asmall and special opening.• The xiao’s inside bore radius varies along its length.• There could be open finger/additional holes that connect to the outsideair.• Closed finger holes form small cavities at the inside wall.The above geometrical complications makes it necessary to build a modelin order to calculate the acoustical impedance of the xiao. The xiao modelis described in chapter 3. The xiao’s impedance can also be measured fromits embouchure as described in chapter 2.151.4. Resonances characterized by the acoustical impedance1.4 Resonances characterized by the acousticalimpedanceThe acoustical impedance determines the resonant behaviour of air in thepipe; this the reason why the acoustical impedance is studied.1.4.1 Resonance of a cylindrical pipeFor a pipe open at both ends, its resonances occur at the acoustical impedanceminima. Considering the actual resonance phenomena, the end correctionat the x1 side also needs to be taken into account. The overall impedanceof the open pipe at the x1 side is (according to Eq. (1.41) and Eq. (1.42)):Zoverall = iZ0 tan(kL′′), (1.43)where L′′ = L+ 2δ.Resonances occur at the impedance minima because the pipe’s endsare connected to the free atmosphere. Maximum sound power occurs atimpedance minima:P = p2/Z, p =√PZ (1.44)The pipe of the typical xiao’s size in section 1.3.1 is again used as anexample here. In the upper part of Figure 1.9, Zoverall is plotted with a solidline according to Eq. (1.43), Z1 is plotted with a dashed line according toEq. (1.41). In the lower part of the figure, the first four standing waves forthe open pipe are plotted with solid lines. Wavelengths of the four standingwaves are 2L′′, L′′, 2L′′/3 and L′′/2. The dashed lines shows the standingwaves of the pipe if it is closed at x1; their wavelengths are 4L′, 4L′/3,4L′/5 and 4L′/7. Resonance frequencies of the closed pipe correspond tothe impedance maxima of Z1. It can be seen that frequencies of the standingwaves are not in perfectly integer multiples, because the wave speed insidethe pipe is frequency dependent (see Eq (1.34)).161.4. Resonances characterized by the acoustical impedance0 500 1000 1500 2000Frequency (Hz)10-310-210-1100101102Z overall0309.8 621.5 933.5 1245.8 1558.3 1870.5155.8 470.3 785.5 1100.8 1416.3 1731.8ZoverallZ10.0 0.1 0.2 0.3 0.4 0.5Position (m)2.01.51.00.50.00.51.01.52.0Normalized Sound pressure309.8 Hz621.5 Hz 933.5 Hz1245.8 Hz 155.8 Hz470.3 Hz 785.5 Hz1100.8 HzLL'L''end correctionend correctionx1 x2Figure 1.9: Acoustical impedance and standing waves of a cylindrical pipe.This cylindrical pipe also demonstrates that the resonance frequency ofthe pipe is almost a semi-tone higher than a pipe of the same length with axiao/shakuhachi embouchure. (A xiao or shakuhachi of the same size playsat D4, 293.7 Hz.)1.4.2 Resonance and playing frequency of the xiaoBased on the resonance phenomena of the cylindrical pipe, it can be assumedthat resonances of the xiao will also occur at its overall impedance minima:Zoverall = Z + Zemb. (1.45)171.4. Resonances characterized by the acoustical impedanceHere Z is the impedance of the xiao introduced in section 1.3.2, and Zembis the radiation impedance at the embouchure. The assumption can be vali-dated by exciting the xiao’s resonances with a speaker near it, and measuringthe resonance frequencies near or inside the xiao with a small microphone.The validation is not done in this thesis because Zemb is difficult to be ob-tained accurately (because of the complicated flange condition formed by theplayer’s lips and face), also because the playing frequency is of our ultimateinterest.Excitation mechanism and playing frequencyThe sound generation of the xiao is excited by an air-jet impinging theblowing edge of the embouchure. The air-jet couples to the oscillation of theresonator and flows into/out of the embouchure at the resonance frequency.Oscillation of the air-jet is nonlinear, and harmonics (oscillations of integermultiples of the fundamental frequency) are usually generated (Fletcher andDouglas, 1980). This excitation mechanism is the same as the flute and organpipes. As shown in Figure 1.10.Figure 1.10: Behavior of the air-jet shown by Sketches of smoke-laden jetviewed stroboscopically, the numbers are phase of the air-jet during an os-cillation cycle. Figure from Coltman (1968).The playing frequencies of a xiao are usually close to its resonance fre-181.5. Goals of designing a good xiaoquencies, but the resonance/playing frequencies also depends on the playerat the following aspects:• As part of the overall impedance, Zemb depends on the flange geometryaround the embouchure (the player’s lips and face) and opening areaof the embouchure.• Impedance Z of the resonator (xiao) depends on the wave speed, whichcan be affected by the temperature and air content inside the instru-ment.• The playing frequencies depend on the velocity (controlled by the blow-ing pressure) of the air-jet.The first aspect is well known by the xiao/flute players as pitch bending,and is a technique for adjusting the instrument’s tuning. In order to playthe high notes easily, the players usually decrease the gap between their lipsand the blowing edge to play at a smaller jet velocity (Coltman, 1966). Inthis way, the opening area of the embouchure tends to be smaller at highernotes. The dependency of the gap on the different notes was measured byColtman (1966), as shown in Figure 1.11.On the xiao, the gap’s dependency on different notes may not be soobvious because the xiao’s embouchure structure is different from the flute,and the player’s lips has less adjustable gap with the blowing edge. Thismay be the reason why xiao is hard to play on the high notes (the gap islimited by the depth of the embouchure, and cannot be decreased to be assmall as the flute).The last aspect is also widely observed by xiao/flute players, and can beexplained by the jet drive mechanism described by Coltman (1976). Figure1.12 shows an organ pipe’s playing frequencies depends on the jet velocity.Effects of the air-jets have been intensively studied during the 1960s to1990s. However, to the author’s knowledge, a generally accepted expressionfor the jet-resonator interaction has not been obtained. So, an empiricalexpression for the Zemb was used in this thesis, as described in section 3.2.2.1.5 Goals of designing a good xiaoRegarding the xiao’s imperfections as mentioned in section 1.1.3, a goodxiao has the following goals of improvements:1. A correct cross fingering for C5.191.5. Goals of designing a good xiaoFigure 1.11: Dependency of the jet length (gap) on the notes been played.o: gap calculated from the measured resonance frequencies. ∆: length ofthe gap measured from photographs. Figure from Coltman (1966).2. A wide tone range (3 octaves expected) and being always in tune.3. Consistent fingerings for the high notes, easy to play and convenientto change between notes.4. Rich in harmonics, as a requirement of the above goal.All the goals can be characterized by the xiaos acoustical impedance(except the fingerings needs to be adjusted according to the players). Toreach these goals, measurement of the acoustical impedance will be describedin chapter 2; a model for calculating the impedance of the xiao will bedescribed in chapter 3; the model was then used for the optimization asdiscussed in chapter 4; finally, several optimized xiaos will be shown inchapter 5.201.5. Goals of designing a good xiaoFigure 1.12: An organ pipe’s dependency of playing frequencies on jet ve-locity. The playing frequency is compared to the frequencies of edge toneunder the same jet velocity.21Chapter 2Acoustical ImpedanceMeasurementMeasurement of the acoustical impedance of woodwind instruments wasonce a challenge because it requires a large dynamic range (sometimes over80 dB) and high frequency resolution (0.5%, or a few cents). Measurementtechniques have been under development since 1940s, but the precision hasnot been good enough for quantitatively characterizing woodwind instru-ments until the very recent decades.2.1 Impedance tubeMeasuring the acoustical impedance needs knowledge of the sound pressurep and flow U at a reference plane (for woodwinds, the reference plane isopening of the embouchure or mouth piece). Sound pressure can be easilymeasured with a microphone, while the flow is relatively difficult to mea-sure. An impedance tube with several microphones is used to eliminate thenecessity of flow measurement.2.1.1 MethodMost impedance tubes have a cylindrical shape. In ideal cases, if one ignoresthe microphone’s acoustical compliance, pressure and flow at any positioninside the tube can be related by a transmission matrix as Eq. (1.20). ForM microphones with the mth located at xm, a matrix A related the pressureand flow at the reference plan with all the microphone signals (Dickens et al.,2007; Jang and Ih, 1998):A11 A12......Am1 Am2......AM1 AM2[pZ0U]=b1...bm...bM. (2.1)222.1. Impedance tubeHere U is multiplied by Z0 so that all elements in A are dimensionless.For simplicity of notation, Eq. (2.1) can be written as: Ax = b, wherex =[pZ0U]. Certainly, a matrix A is required for each frequency undermeasurement.Eq. (2.1) has two unknowns (p and U). So M = 2 will be just enoughfor solving for this linear equation:[pZ0U]=[A11 A12A21 A22]−1 [b1b2]. (2.2)If two microphones are used, and if the two microphones have the samegain (ratio of electrical voltage output to sound pressure), the matrix A canbe written asAideal = g[cos(kL1) i sin(kL1)cos(kL2) i sin(kL2)]. (2.3)Here g is the microphone gain, Lm = x0 − xm (m = 1, 2) is displacement ofthe mth microphone to the reference plane x0. In practice, more than twomicrophones are usually used to avoid the singularities, and reduce errors inthe measurement. For more than two microphones, Eq. (2.1) can be solvedby the least squares method.2.1.2 Singularities and the singular factorThe singularities occur when kL1 − kL2 = Npi (N = 0, 1, 2, · · · ). Thecorresponding singular frequencies are fSG = Nfsg, where fsg is the firstnon-zero singular frequency:fsg =v2L12. (2.4)Here v ≈ c is the wave speed inside the impedance tube, and L12 is thedistance between the two microphones. Note that N = 0 means f = 0 Hzis also a singular frequency.At the singular frequencies, the two rows of Eq. (2.3) become the same,the matrix become singular and has no inverse. So p and U can not besolved. Namely, the two microphones provide the same data, because thesound waves are periodic in the impedance tube.Around the singular frequencies, the measurement error will be large.This can be shown by continuing the calculation of Eq. (2.2):[pZ0U]=1A11A22 −A12A21[A22b1 −A12b2−A21b1 +A11b2], (2.5)232.1. Impedance tubeZ =pU= Z0A22 − b2b1A12−A21 + b2b1A11. (2.6)When the frequency is close to the singular frequency (f ≈ fSG and k(L1−L2) ≈ Npi), there will be A21 ≈ A11, and b2 ≈ b1 according to the periodicsound waves. The denominator in Eq. (2.6) will be close to zero. So, anysmall noise in b (described later in section 2.2.2) and calibration error in A(described later in section 2.1.3) will result in large measurement error in Z.Singular factorThe measurement errors caused by the singularities can be convenientlycharacterized by the singular factor (Jang and Ih, 1998):SF =√∑jΛ−2j . (2.7)Here Λj is the jth element in Λ, and Λ comes from the singular decomposi-tion of A:A = UΛVH. (2.8)The value of SF is the factor of noise in b being propagated to x (and Z), sosmall value of SF is expected in the frequency of interest. For measurementof the xiao, the frequency of interest is 150 to 4000 Hz. The value of SF inseveral microphone setups is shown in Figure 2.1. As the figure shows, atmicrophone spacing of 50 mm, fsg ≈ 3430 Hz. Reducing the microphonespacing to 40 mm pulls the the singular frequencies out of the measurementrange (fsg ≈ 4288 Hz).Considering the singularities, two general rules can be obtained for mi-crophone locations setup:• Use larger microphone spacing for lower frequencies and smaller mi-crophone spacing for higher frequencies.• Increasing the microphone number makes improvements over the entirefrequency range.These two rules and the singular factor were used for determining the mi-crophone locations in section 2.2.5.242.1. Impedance tube0 500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)12345678910SF2_mics_50mm2_mics_40mm3_mics_40mmFigure 2.1: Singular factor of several microphone setups. 2 mics 50mm:2 microphones spaced 50 mm (e.g. x1 = 50 mm, x2 = 100 mm);2 mics 40mm: 2 microphones spaced 40 mm (e.g. x1 = 40 mm, x2 = 80mm); 3 mics 40mm: 3 microphones spaced 40 mm (e.g. x1 = 40 mm,x2 = 80 mm, x3 = 120 mm).2.1.3 Calibration of the impedance tubeIn the previous section, the matrix A is determined from theory based ontwo ideal assumptions:1. The microphone’s acoustical compliance can be ignored.2. Gain of all the microphones is the sameIn practice, none of the assumptions can be safely made, and the matrix Aneeds to be determined from experimental calibration.The calibration uses an impedance tube to measure some loads withknown acoustical impedance, then determines A reversely from Eq. (2.1).Gibiat and Laloe¨ (1990) proposed a widely used calibration routine calledTwo-Microphone-Three-Calibration (TMTC) method, which is a completecalibration (no assumptions of the microphones’ locations and gain rationeed to be made). Dickens et al. (2007) improved the TMTM calibrationby taking measurements on three resonance-free loads:252.2. Building an acoustical impedance measurement system1. An infinite impedance (by blocking the reference plane).2. An infinite flange (made out of a large plate with a center hole fittedto the impedance tube).3. A pure resistance (a 91 m pipe of the same radius with the impedancetube).This improved calibration method is adopted in this thesis with some changesand improvements, as described in section 2.2.3.2.2 Building an acoustical impedancemeasurement systemTo measure the acoustical impedance of the xiao, an impedance tube needsto be built with a frequency range from 150 Hz to 4 kHz to cover all thenotes and some important harmonics.The basic setups were made mostly following Dickens et al. (2007). Animpedance tube was made out of brass with inner diameter 7.94 mm, outerdiameter 9.53 mm and total length 400 mm. One end of the tube was coupledto a speaker (a Selenium D250-X compression driver) through a small cone.The other end, defined as the reference plane, was to be connected to thexiao’s embouchure or the calibration loads. Four microphones were used inthe impedance tube.Figure 2.2 shows the measurement system setup in the anechoic chamberat UBC Department of Mechanical Engineering. The chamber is fully ane-choic including its floor, which is covered by a steel mesh. It has a geometryof 4× 4× 2.5 m3 (Waltham et al., 2013) and background noise of about 23dB (A) in Sound Pressure Level (SPL). The impedance tube was suspendedfrom the ceiling to isolate it from vibrations, and to be easily connectedto the loads. The microphones were mounted by four homemade micro-phone mounters. The mounters were originally made for an impedance tubeof bigger diameter (19.05 mm, as shown in Figure 2.15), so a narrow alu-minum block was shaped to fit the 9.5 mm impedance tube to the aluminummicrophone mounters.An Audiobox 44VSL 24 bit external sound card was used to collect themicrophone signals and output an excitation signal to the speaker, usingPresonus Studio One software. The external sound card was connected toa computer outside the anechoic chamber through a USB cable, so that theoperations were done outside the chamber to reduce environmental noise.262.2. Building an acoustical impedance measurement systemFigure 2.2: Setup of the acoustical impedance measurement system in ananechoic chamber. The microphones are inside the aluminum mounter andcannot be seen.The four channels of microphone signals were recorded and saved as 32-bitWAV files. The WAV files were read through the scikits.audiolab packageand processed in a Python script.Each impedance measurement takes 10.92 s, corresponding to 220 datapoints at sampling rate of 96 kHz. The microphone signals were trans-formed to the frequency domain through the Fast Fourier Transform (FFT)and averaged for every 8 frequencies. The final frequency resolution is∆f =0.73 Hz.Several accessories for the impedance tube were used as shown in Figure2.3. Geometries and usages of these accessories are described in Table 2.1.The length of the several brass pipes were measured accurately on a latheand milling machine with digital read-out.272.2. Building an acoustical impedance measurement systemFigure 2.3: Accessories for the impedance tube.Name Geometry How it works UsageLid The central hole:same diameter asthe impedance tube’soutside.Block the referenceplane to provide aninfinite impedance.CalibrationInfiniteFlangeThe square: 500 ×500 mm2; the cen-tral hole: same diam-eter as the impedancetube’s outside.Act as an effectivelyinfinite flange.CalibrationPipe-1 Length: 3.2995 m; di-ameters: same as theimpedance tube.As an extension of theimpedance tube.Calibrationand testmeasure-ment.Pipe-2 Length: 0.8798 m; di-ameters: same as theimpedance tube.Same as pipe 1. Test mea-surement.pipe-3 Length: 0.9990 m;inner diameter:6.35 mm; outerdiameter: 7.94 mm.Create a diametermismatch at thereference plane.Test mea-surement.Table 2.1: Geometries and usage of several accessories for the impedancetube in Figure 2.3.282.2. Building an acoustical impedance measurement systemThe other setups were determined by experiment: the excitation signalwill be discussed in section 2.2.1; the microphone requirements will be es-timated in section 2.2.2; two new calibration methods will be proposed insection 2.2.3 and the optimal microphone locations will be determined insection 2.2.5.2.2.1 General and refined excitation signalThe excitation signal is sent to the speaker to generate sound waves insidethe impedance tube. In this thesis, a log-swept signal was used as theexcitation signal. The log-swept signal consists of sinusoidal waves withlogarithm increments of frequency over time, so that all frequencies haveequal fraction of waveform. The range of the swept signal was set to be100 to 4200 Hz, a little wider than the measurement range. Amplitudes ofdifferent frequencies were adjusted according to the microphone signals toobtain large Signal to Noise Ratio (SNR). The adjustments were based onthe following rules:• At frequencies of large microphone signals, decrease the excitationsignal’s amplitudes to decrease the maximum of the microphone signal(which is limited by the Analog to Digital Converter (ADC) or themicrophones’ measurement range).• At frequencies of small microphone signal, increase the excitation sig-nals’ amplitudes to salvage the microphone signal from the environ-mental/electrical background noise.General excitation signalA general excitation signal is intended to be used for all kinds of loads undermeasurement. Considering the SNR of microphones will be small at lowsound pressure, and response functions of sound pressure at the microphonepositions depend on the load under measurement. In order not to have toosmall signals at some frequencies, the general excitation signal should havealmost equal amount of sound power distributed for each frequency.According to the speaker’s specifications, the Selenium D250-X compres-sion driver is of mid-range and has -10 dB frequency response in 400 to 9000Hz. So the swept signal was amplified at the low frequency region (100to 600 Hz). The sound power at different frequencies also depends on theimpedance of the impedance tube system, as shown by Eq. (1.44). Theoverall impedance of the impedance tube system depends on what load is292.2. Building an acoustical impedance measurement systemconnected, but it will generally show a logarithmically decreasing envelope(see the impedance of a simple cylindrical pipe in Figure 1.9). So the am-plitude of the excitation signal was adjusted to have a logarithmic increasewith the frequency, as shown by Figure 2.4 (a) (the first half is the amplifiedlow frequency region). As the combined effects of log-sweep and increasedamplitude along the frequency, the signal’s spectrum is flat except for theamplified low frequency region (see Figure 2.4 (b)).The general excitation signal for our system was produced by a PythonScript through the following steps:1. Generate a log-scaled swept signal s1(t) with equal amplitude.2. Apply the Fast Fourier Transform (FFT) on s1(t) to obtain the fre-quency domain signal s2(f).3. Apply the amplitude modulation as described in the previous para-graph on s2(f) and obtain s3(f).4. Apply the inverse Fast Fourier Transform (iFFT) on s3(f) to obtainthe final excitation signal s4(t) (use the real part).5. Write s4(t) to a WAV file through the scikits.audiolab package.Here FFT and iFFT were used to process the swept signal in order to preventthe spectrum leakage at the beginning and end of the swept (Mu¨ller andMassarani, 2001). By preserving the phase of different frequencies, s4(t)mostly maintains its sweeping characteristics, although a small fraction ofits beginning and end is not purely sinusoidal.In fact, effects of the spectral leakage are not significant, and can bedealt with by extending the swept frequency range to be a little wider thanthe measured frequency range. So the FFT and iFFT processes are notcompletely necessary, and the steps 2, 3 and 4 can be simplified by directlymodulating amplitudes of the swept signal s1(t).The waveform and spectrum of the general excitation signal are shownin Figure 2.4. Microphone signals of a sample measurement are shown inFigure 2.5. The measurements were taken with the reference plane blocked(equivalent to infinite impedance). It is shown that the maxima of the mi-crophone signal at different frequency ranges are of similar amplitude. Themaxima/minima of the microphone signal are caused by the resonace/anti-resonance of the measurement system (and the load under measurement),and they can further balanced by a refined excitation signal, as describednext.302.2. Building an acoustical impedance measurement systemFigure 2.4: A general excitation signal. (a): waveform; (b): spectrum. Thelight blue region indicates the waveform is sparse at low frequencies.Refined excitation signalAs shown by Figure 2.5, under the general excitation signal, the microphonesignals only make good use of the ADC near the resonance frequencies.There are still frequencies where the signal and SNR are small (near theanti-resonance). The general excitation signal may not be good enough forsome impedance measurements systems if they have some other limits suchas small microphone dynamic range and large environmental/electrical noise(as described in the next section). For these reasons, the excitation signalcan be further refined specifically for each measurement, as reported byDickens et al. (2007) by modulating the amplitudes of the excitation signalat expected frequencies, and by Lefebvre and Scavone (2008) by pre-filteringthe excitation signal with the inverse of the system’s frequency response.However, in our measurements, attempts at refining the excitation didn’tresult in better measurement results. This may be explained by the con-tamination from the speaker, as described in the next section. Meanwhile,our general excitation signal produced satisfactory results.312.2. Building an acoustical impedance measurement systemFigure 2.5: Microphone signals of a sample measurement (with the referenceplane blocked). The microphone locations in this measurement are 25 mm,50 mm,100 mm and 200 mm.2.2.2 Noise estimation and microphone requirementsThe measurements require clear signals over the frequency range, and thesignal should be proportional to the sound pressure. Regarding these, thereare four aspects to be considered in the measurement system:1. Background noise of the environment and electrical noise of the mi-crophone.2. Dynamic range of the microphone (maintaining a linear relationshipof output signal and sound pressure).3. Dynamic range of the ADC and pre-amplifier.4. Contamination from the speaker (caused by vibration of the impedancetube or propagation through paths other than the impedance tube aircolumn).322.2. Building an acoustical impedance measurement systemNoise estimationIn order to estimate requirements in the above aspects, the spectra of themicrophone signals shown in Figure 2.5 are plotted in Figure 2.6. Back-ground noise (when no excitation signal was sent to the speaker) is plottedat the lower part of the Figure.Figure 2.6: Microphone spectra and the background noise. The spectra areobtained by FFT from waveform shown in Figure 2.5.Figure 2.6 shows that the microphone signal spectra are slightly abovethe noise floor. The noise floor is about 30 dB at 1 kHz, caused by theelectrical noise of the microphone. The noise floor levels are different for thefour microphones, indicating either the microphones or the pre-amplifiershave different gains. The measurement was taken inside the UBC anechoicchamber, with a background noise of about 23 dB (A) in SPL. Variationof sound pressure as shown by the microphone signals is over 64 dB (3.2orders of magnitude). This variation of sound pressure requires that theADC, pre-amplifier and the microphones should all have dynamic range ofat least 64 dB in the frequency range. The Audiobox 44VSL has built in332.2. Building an acoustical impedance measurement systempre-amplifier with 97 dB SNR, and its 24-bit ADC has a higher dynamicrange. So the main consideration need to be given to the microphones, asdescribed later. The figure also shows that even though some microphonespectra are high above the background noise floor, they are still noisy. Theonly possible reason is contamination of the speaker, as described in theaspect 4 earlier. The speaker contamination may also be the reason whyour refined excitation signal failed to produce better measurement results asdescribed in the previous section. The reason is, by refining the excitationsignal, the speaker’s sound power (and its contamination to the microphonesignals) of frequencies near the anti-resonances were increased proportionallyto the enhancement of microphone signal. So the SNR of the microphonesignal cannot be improved any more.Microphone requirementsThe noise estimation above sets a relatively high requirement on the mi-crophones, considering normal microphone has electrical noise of about 30dB and SPL over 94 dB may be out of general microphones’ range to keepa linear gain. So usually high quality measurement microphones are usedwith impedance tubes.We started our experiment with low cost microphones and found thePanasonic WM61A microphones with the Linkwitz modification (Linkwitz,2005) give satisfactory results.The original WM61A microphone has a signal to noise ratio (SNR) of62 dB at 1 Pa (or 94 dB in Sound Pressure Level (SPL)) at 1 kHz. So themicrophone’s electrical background noise is 32 dB. Linkwitz reports thatthe WM61A microphone in its original setup starts to show distortions ata moderate SPL, because the built-in Field-Effect Transistor (FET) is notconfigured properly. By changing the FET from common source to commondrain (source-follower) mode, the microphone remains linear at large soundpressures. Linkwitz’s microphone circuit was adopted to our system with aslight modification to fit the Audiobox 44VSL microphone channel, whichsupplies a 48 V phantom power. Note that the phantom power drops downto about 25 V when the circuit is connected, since the Audiobox 44VSL has1.7 kΩ input impedance at its microphone channels.As long as the modified WM61A microphones worked well with ourimpedance tube, no further investigations were made on evaluating the mi-crophone properties or comparing with the original WM61A microphone.342.2. Building an acoustical impedance measurement system2.2.3 Two new calibration methodsThe improved calibration method by Dickens et al. (2007) worked well ontheir system, but their purely resistive load is 91 m in length. Consideringthe practical difficulties in building such a long pipe, the resistive load isnot used in our measurement system. Two new calibration methods weretested.In both methods, a short cavity (closed pipe) was needed. A set of suchshort cavities were made using the milling machine flat-end mill cutters,as shown in Figure 2.7. The short cavities consist of two concentric holes.The first hole was about 5 mm in depth, and the same diameter as theimpedance tube’s outer diameter. The first hole was used to fit the cavity tothe impedance tube. The second hole is the actual depth of the short cavity,with inner diameter of 7.94 mm (5/16 inch), the same as the impedancetube’s inner diameter. Depths of the cavities were calibrated with a digitalcaliper with a resolution of 0.01 mm to be:• 2mm-cavity: 1.94± 0.04 mm.• 5mm-cavity: 4.94± 0.01 mm.• 8mm-cavity: 7.97± 0.04 mm.• 12mm-cavity: 11.97± 0.03 mm.• 15mm-cavity: 14.95± 0.02 mm.Figure 2.7: A set of resonance-free short pipes made from aluminum. Fromleft to right: 2mm-cavity, 5mm-cavity, 8mm-cavity, 12mm-cavity and 15mm-cavity.Method 1: an accurate resonance loadDickens et al. (2007) chooses resonance-free loads because the calibrationerror is large at the resonance frequencies. The calibration error is caused by352.2. Building an acoustical impedance measurement systeminaccurate value of the loads’ impedance. The inaccuracies mostly show nearthe resonance frequencies, since the overall impedance amplitude variationscan be accurately characterized by the wall losses as described in section1.2.2.The new method 1 proposed in this thesis needs four loads:• The lid (infinite impedance).• The infinite flange.• The closed pipe-1.• A short cavity (a 8mm-cavity was used here for example).The first three loads were described in Table 2.1.The closed pipe-1 is a resonance load, and its impedance (especially nearresonance frequencies) needs to be accurately determined. That is to say, thepipe-1’s length and the environmental temperature needs to be accuratelydetermined.The pipe-1’s length was measured in the UBC departmental studentmachine shop by making several circular marks around the pipe using alathe with a tiny cutter at intervals of about 0.5 m. The distances betweenthese markers were then measured accurately on a milling machine with adigital read-out and added up to be 3.2995 m. The uncertainly of the lengthmeasurement is about 2×10−4 m ( 0.2 mm), mostly caused by human errorsin judging the markers’ locations.In order to measure the environmental temperature accurately, the tem-perature in the anechoic chamber was measured by a thermometer (withresolution of 0.1 ◦C) and a MCC USB-502 digital temperature logger (withresolution and accuracy of 0.5 ◦C). Results measured by the two devicesagree with each other. However, impedance of the closed pipe-1 calcu-lated using the measured temperature Tmea is not accurate. Inaccuracyof this closed pipe-1’s impedance cause ripples in the measured impedanceof the 8mm-cavity, as shown in Figure 2.8. The measurement was takenat Tmea = 14.6 ◦C. The ripples has a frequency interval of about 53 Hz,correctly corresponds to the resonance interval of the closed brass pipe.Inaccuracy of the closed brass tube has several possible reasons:1. Inaccurate measurement of the environmental temperature Tmea.2. Inaccurate calculation of the sound speed.362.2. Building an acoustical impedance measurement system0 500 1000 1500 2000 2500 3000 3500 4000100101102Z/Z 0MeasurementTheory0 500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)1.701.651.601.551.501.451.40Phase (rad) MeasurementTheoryFigure 2.8: Impedance measurement of the 8mm-cavity. Ripples in themeasured impedance curve were caused by inaccurate calculation of theclosed brass pipe’s impedance.3. Inaccurate measurement of the pipe-1’s length (systematical error ofthe milling machine’s digital read-out).The formula used in this thesis to calculate the speed of sound is givenby Keefe (1984):c = 347.23× (1 + 0.00166(T − 26.85)). (2.9)The formula does not take the environmental humidity into account, sothe reason 2 can be checked by using more rigorous calculation (Cramer,1993). However, this was not attempted. Instead, we turn to directly findthe actual speed of sound (which will let the ripples disappear). This processwas done by setting the speed of sound c as an optimization parameter, anduse the optimization process as described in chapter 4. The optimization372.2. Building an acoustical impedance measurement systemobjective function is written as:O(c) =fend∑fstart(φ(f)− φ). (2.10)Here φ(f) is phase of the measured impedance at frequency f , φ is the meanof the phases in the from fstart to fend. The theoretical value of the phasewas not used in Eq. (2.10) in order to make the experiment independentfrom the theory. The optimization process was done by iteratively adjustingc to minimize O(c).The optimized speed of sound was 340.5 m/s. Based on Eq. (2.9) the re-derived temperature was Tder = 15.18 ◦C (0.58◦C higher than the measuredtemperature). Tder might not be the actual environmental temperature,but the speed of sound obtained from optimization would be the correctphysical quantity. The measured impedance of the 8mm-cavity based atthis temperature (or speed of sound) is shown in Figure 2.9.To demonstrate the capability of method 1 in calibration, impedance ofthe pipe-2 (with its far end open) was measured and compared with theory,as shown in Figure 2.10.Method 2: a resonance-free short cavityImpedance measurement results on the 8mm-cavity (as shown in Figure 2.9)shows that the acoustical impedance of such short cavities (closed pipes) canbe accurately known. This indicates the possibility of using a short cavityas one of the calibration loads. Resonance frequencies of the 8mm-cavity fallfar out of our frequency range, so it can also be classified as a resonance-freecalibration load. Compared with the resonance load and the resistive load,the short pipes were easier to manufacture and be installed in the impedancetube.An impedance measurement of the open brass pipe calibrated by the8mm-cavity (together with an infinite impedance and an infinite flange) isshown in Figure 2.11.Compared with a calibration made by an accurate resonance load asshown in Figure 2.10, impedance measurement of calibration made by the8mm-cavity has some amplitude error at the low frequency range (high-lighted by the circles). Calibrations were also made using the other shortcavities (except the 2mm-cavity), the resultant impedance measurements aremostly similar to Figure 2.11 except for slightly different amplitude errorsat the low frequency range.382.2. Building an acoustical impedance measurement system0 500 1000 1500 2000 2500 3000 3500 4000100101102Z/Z 0MeasurementTheory0 500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)1.651.601.551.501.451.40Phase (rad) MeasurementTheoryFigure 2.9: Impedance measurement of the 8mm-cavity. The ripples disap-pear after using the right temperature (speed of sound)Based on the above results, it can be concluded that calibration made bymethod 2 is acceptable if the resonance frequencies are of the main interest,and if the impedance amplitude error is not very important (which is truefor studying the resonance frequencies of woodwind instruments).2.2.4 Effects of reference plane mismatchAfter the above calibrations, what is measured is the impedance at the refer-ence plane with a circular area of diameter 7.94 mm, which is poorly matchedto our xiaos’ smaller U-shaped openings. To get the actual input impedanceat the embouchure, the mismatch (discontinuity) needs to be modelled bymultimodal theory (Pagneux et al., 1996). In this thesis, complicated multi-modal calculations were not attempted. Instead, end-corrections were usedto account for the reference plane mismatch. Test measurements were madeon pipe-3, which has inner diameter 6.35 mm (similar to the opening areaof our xiaos). In order to check the validity of using the end-correction,392.2. Building an acoustical impedance measurement system0 500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)10-1100101Z/Z 0MeasurementTheoryFigure 2.10: Measured impedance of pipe-2 with its far end open. TheCalibration was done by method 1.pipe-2 was measured with its far end both open and closed. End-correctionswere obtained by the optimization process described in chapter 4. Then theend-correction δ was set as an optimization parameter, and the optimizationobjective function is written as:O(δ) =fend∑fstart[Zmea(f)− Ztheory(f)]. (2.11)The optimization (fitting) process was done by iteratively adjusting δ tominimize O(δ). The measured and fitted impedance of pipe-3 are shown inFigure 2.12, and the obtained end-corrections for the closed and open pipe-3are: {δ(o) = −3.78 mm (Open end),δ(c) = −3.22 mm (Closed end).(2.12)402.2. Building an acoustical impedance measurement system0 500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)10-1100101Z/Z0MeasurementTheoryFigure 2.11: Impedance measurement of the open pipe-2. The calibrationwas done by method 2 with an infinite impedance, an infinite flange and a 8-mm cavity. The amplitude errors in the low frequency range are highlightedby the circles.There is a noticeable difference between δ(o) and δ(c) (about 15%), however,both of them are small compared to the length of pipe-3. In both conditions,the calculated (fitted) impedance matches well with measurement, so end-corrections were accepted as the way to deal with reference plane mismatch.2.2.5 Determine the microphone locationsSection 2.1.2 shows that the microphone locations are important in choosingthe measurement range and to avoid measurement errors.Initially, we learned from Dickens et al. (2007), who used three micro-phones located at 10, 50, and 250 mm from the reference plane. We used4 microphones, so we simply placed them at 10, 50, 100 and 250 mm fromthe reference plane (this microphone locations will be referred to as the oldmicrophone locations in this thesis). The measurement results were mostly412.2. Building an acoustical impedance measurement system250 500 1000 2000 400010-1100101Z/Z 0 165 334 504674164.9 334.1 504.0673.2FittingMeasurement250 500 1000 2000 4000Frequency (Hz) (log-scaled)10-1100101Z/Z 0 250 420 590 760249.9 419.1589.0FittingMeasurementFigure 2.12: Impedance measurement with reference plane mismatch. Pipe-3 under measurement has inner diameter of 6.4 mm, smaller than innerdiameter of the impedance tube, so an end-correction is applied in theory.Upper: pipe-3 with its far end open; lower: pipe-2 with its far end closed.acceptable, and actually Figure 2.10, 2.11 and 2.12 were measured earlierusing the old microphone locations. However, the measurement noise anderror can be obviously seen by plotting measurement results of the shortcavities, as shown in Figure 2.13.Measurement results in Figure 2.13 are generally good. But in compari-son with impedance measurement of the 8mm-cavity by our final microphonelocations as shown in Figure 2.9, measurement made by the old microphonelocations have obvious noise and error at frequency range of 3000 Hz to 4000Hz (as circled in the Figure).The singular factor as described in section 2.1.2 can be used to explainthe observed noise and error. The SF of the old and new microphone lo-cations were calculated according to Eq. (2.7) and plotted in Figure 2.14.The large SF in the old microphone locations is circled in the Figure.422.2. Building an acoustical impedance measurement system0 500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)100101102Z/Z02mm-cavity Measurement5mm-cavity Measurement8mm-cavity Measurement12mm-cavity Measurement15mm-cavity MeasurementTheoryFigure 2.13: Impedance measurement of the short cavities made using theold microphone locations. The region highlighted by the oval has large errorand noise, corresponding to the large SF highlighted in Figure 2.14The SF for an impedance tube with equal spacing microphone locationsis also plotted in Figure 2.14. Jang and Ih (1998) shows that equal-spacedmicrophones have a smaller SF than random-spaced ones. It is true that theequal spaced microphones have the smallest SF among the three setups atfrequency range of 750 Hz to 3500 Hz. However the lower frequency limitis not good enough. Our final microphone locations used in our impedancetube were chosen to to let them have equal increase in the ratio of spacing:25, 50, 100, 200 mm, so that the short-spaced microphone pairs can do theirbest to remedy singularities of the long-spaced ones. The first unavoidablesingularity occurs at about 6.9 kHz.2.2.6 Cautions and tips in measurementMany impedance measurements have been made in our experiments, butnot all of the measurement results can be used. The most frequent reason432.2. Building an acoustical impedance measurement system0 500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)246810SFNew_2.5_5_10_20Old_1_5_10_20Equal_4_8_12_16Figure 2.14: Singular factor of 4 microphones at three location setups.New 2.5 5 10 20: The new microphone locations (2.5, 5, 10 and 20 cm).Old 1 5 10 25: The old microphone locations (1, 5, 10 and 25 cm).Equal 4 8 12 16: The equal microphone locations (4, 8, 12 and 16 cm).causing the failed measurements is leaking in the connections. So makingsure the connections be well sealed is the first caution.Connections at the reference planeThere are several different types of loads to be connected to the referenceplane. The lid and the infinite flange can be easily installed, while theextension pipes are relatively hard. The various woodwind instruments areeven more difficult.Initially, in order to tightly connect all kinds of loads to the referenceplane, a connection mechanism was designed and built, as shown in Figure2.15. But it doesn’t work well. Although the mechanism holds the loadstightly to the reference plane, the connection was hard to reach, and thesealing condition was hard to ensure. So the connection mechanism was442.2. Building an acoustical impedance measurement systemreplaced by the transparent tape and modelling clay.Figure 2.15: A failed connection mechanism for the impedance tube. Theimpedance tube shown was an earlier one with inner diameter 19.05 mmThe transparent tape was used for connecting and sealing pipe-1 andpipe-2, as shown in Figure 2.16, where the connected pipe was hung at thesame height as the impedance tube.Modelling clay was used for all the other cases that need to be sealed:the lid, short cavities, pipe-3, and so on. Figure 2.17 shows the process ofusing modelling clay to make sure the sealing condition of a 12mm-cavity.Microphone installationIn our impedance tube, the microphones detect sound pressure inside thetube through a 1 mm diameter side-hole. The microphone needs to be sealedto the outside of the impedance tube to prevent leaking of the impedancetube (because any leaking will cause non-plane waves to be measured bythe microphone). If no further adjustment is required, the microphones canbe glued to the impedance tube. However in our initial experiment, themicrophone locations were not quite certain. So an rubber o-ring was gluedto each microphone, then the microphones were sealed to the impedancetube by tightening screws in the microphone mounters. A set of 1-mm holeswere drilled on the impedance tube, the holes not in use were simply blockedby a small piece of modelling clay.452.2. Building an acoustical impedance measurement systemFigure 2.16: Brass pipe connected to the impedance tube by transparenttape. The connection was carefully made to ensure sealing.2.2.7 A complete measurement procedureAs a summary of the built measurement system, a complete measurementprocedure is as below:1. Calibrate the impedance tube1 using method 1 as described in sec-tion 2.2.3 by doing measurements on four loads: the lid (infiniteimpedance), the infinite flange, the closed pipe-1 and a short cavity.2. Use the first three measurements to calculate the matrix A, and run anoptimization code (using the measured impedance of the short cavity)to obtain the actual environmental sound speed and the optimizedmatrix A.3. Measure the actual target loads and obtain results using the optimizedmatrix A.The measurements in step 1 can be done in 10 minutes, while the optimiza-tion process in step 2 takes more than half an hour (calculating the matrixA for each iteration). The steps 1 and 2 have to be done at the beginningeach time, because a temperature change of even 0.1 ◦C can cause noticeableripples in the measured impedance curve if an older matrix A was used.1Our measured temperature disagree with the actual sound speed (about 0.5 ◦C higherthan measured). So the new calibration method 1 was always used in our measurements .462.3. Acoustical Impedance Measurements of the XiaoFigure 2.17: A short cavity sealed to the impedance tube by modelling clay.If the short cavity was made of transparent acrylic, the sealing conditioncan be better checked.If one has a reliable and precise enough thermometer, the method 2in section 2.2.3 can be used. Then the closed pipe-1 will not need to bemeasured, and the optimization process in step 2 can be skipped. Thecalibration process will take less time.2.3 Acoustical Impedance Measurements of theXiaoThe xiao was measured as described in section 2.2.7. The extra consider-ations are to attach the xiao to the impedance tube and to deal with themismatch (discontinuity) at the reference plane.Different fingerings of the xiao were measured by blocking some fingerholes with pads made of modelling clay.Connection to the impedance tubeA simple mounter was made for the xiao, as shown in Figure 2.18. A piece ofmodelling clay was shaped to fit the xiao. The advantage of this mounter isthat, it can be removed from the reference plane, and the xiao embouchure’slocation can be accurately determined by matching to the mounter’s hole (asshown by the figure (C)). The sealing condition can be checked through thetransparent acrylic plates. After fixing the xiao to the mounter by modelling472.3. Acoustical Impedance Measurements of the Xiaoclay and a tape, the mounter and the xiao were connected to the impedancetube. The two screws shown in figure (C) were then tightened. Similar toFigure 2.17, a piece of modelling clay was used near the reference plane toseal the connection.Reference plane mismatchFigure 2.18 shows the embouchure opening is approximately a semi-circularshape. It does not connect to the reference plane smoothly, and its equivalentcircular area has a smaller radius than the impedance tube’s inner diameter.Similar as the pipe-3 discussed in section 2.2.4, the reference plane mis-match at the xiao embouchure is included by an end-correction as describedby section 3.2.1 on the xiao embouchure model.482.3. Acoustical Impedance Measurements of the XiaoFigure 2.18: Connecting a xiao to the impedance tube. The xiao undermeasurement is made of transparent acrylic.49Chapter 3ModellingA model is built for the xiao to calculate its acoustical impedance. Thexiao model is largely based on the Transmission-Matrix Method (TMM).Which was chosen because it is easy to implement in programming andhas a relatively light computational load compared to the Finite ElementMethod (FEM).3.1 TMMThe TMM has been intensively developed during the last two decades, it hasbeen used for mutual validation of experiments (Dickens, 2007) and FEM(Lefebvre, 2010). This improves the accuracy of TMM and makes it veryreliable.The TMM approximates the woodwind instrument bores as short sec-tions of cylindrical and conical pipes, as shown in Figure 3.1. For a wood-xnxn-1x2x1pnUnp1U1ZrZ1Figure 3.1: Demonstration of modelling woodwind bore by TMM.wind instrument with any axial symmetrical bore (without side holes), theacoustical impedance at the mouse-piece position x1 can be calculated by aseries of transmission matrix:U1[Z11]= Tx1,x2Tx2,x3 · · ·Txn−2,xn−1Txn−1,xn[Zr1]U2, (3.1)503.1. TMMwhere xn is position of the bore end, Zr is the radiation impedance, and willbe described in section 3.1.1. Transmission matrix of the ith section Txi−1,xiwill either be Tcylin as Eq. (1.20), Tcon as Eq. (1.30) or Tdiv as Eq. (1.31).Effects of the mouth piece has not taken into account in Eq. (3.1) yet.Most woodwind instruments have tone holes. In ideal cases, the toneholes can be taken into account as impedances in parallel with the impedanceof the bore at the tone hole’s central position. In practice, the holes con-tribute both series and parallel impedance, as described later in section3.1.2.3.1.1 Radiation impedance and radiation lossThe radiation impedance is caused by the radiation of sound wave to thefree atmosphere and can be expressed byZr = β(ka)2Z0 + iδkZ0. (3.2)Here β and δ are dependent on the flange condition. The real part of Zrdetermined by β is related to the radiation loss, and is neglected in mostcases because it has no obvious effect on the resonance frequency. However,the value of β affects the amplitude of the impedance curves and the depthof the impedance minima (and the playability of the notes), so it is non-negligible in this case. The imaginary part acts as the end-correction andaffects the resonance frequency.For a frequency range that satisfies ka  1, theoretical values exist forthe two typical flange conditions (Buick et al., 2011; Fletcher and Rossing,1998):• unflanged (negligible wall thickness): β0 = 14 , δ0 = 0.6133a;• infinitely flanged: β∞ = 12 , δ∞ = 0.8216a.In real woodwind instruments like the xiao, the wall thickness is not neg-ligible. This type of radiation impedance is classified as a result of circularflange by Dalmont et al. (2001), and a pipe end with outer diameter b hasend-correction termδcirc = δ∞ +ab(δ0 − δinfty) + 0.057ab[1− (ab)5)]a. (3.3)This formula is originally Eq. (41) in Dalmont et al. (2001), and the radi-ation loss is described by their Eq. (42). However, the radiation loss termβ in this thesis is determined by experiment because β is also dependent onsharpness of the tone hole (Buick et al., 2011). The value of β is assumedto be within the two extreme flange conditions: 0.25 < β < 0.5513.1. TMM3.1.2 Tone holesThe geometry of a sample woodwind tone hole is shown in Figure 3.2, wherethe bore has radius a, the tone hole has radius b and thickness t.Figure 3.2: Geometry of a woodwind tone hole (Dalmont et al., 2002). Top:side view; bottom: cross-sectional view at the center of the tone hole.An equivalent circuit has been well developed for such a tone hole (Ned-erveen et al., 1998), as shown in Figure 3.3. Since the tone hole’s thickness ismuch smaller than the wavelength, the tone hole is treated as a combinationof several lumped elements:• Za – impedance in series with the bore caused by disturbance of thetone hole.• Zi – impedance in parallel with the bore caused by interactions of flowin the tone hole and bore.• Zm – impedance in parallel with the bore caused by the matchingvolume, as shown in Figure 3.2.523.1. TMMcorrectionsFigure 3.3: Equivalent circuit of a woodwind tone hole (Nederveen et al.,1998).• Zh – impedance in parallel with the bore caused by the small segmentof the tone hole.• Zr – radiation impedance of the tone hole.Tone hole series impedanceThe above impedance elements are usually written in terms of equivalentlength. For example,Za = i tan(kta)Z0 ≈ iktaZ0. (3.4)Here ta is the length correction (equivalent length) of Za, and its expressiondepends on whether the tone hole is open or closed. Formulae for ta used inthis thesis are given by Lefebvre and Scavone (2012):{t(o)a =[− 0.35 + 0.06 tanh(2.7t/b)]bδ2 (open),t(c)a =[− 0.12− 0.17 tanh(2.4t/b)]bδ2 (closed).(3.5)Here δ = ba is the ratio of tone hole radius to the bore radius.533.1. TMMOpen tone hole shunt impedanceThe other four elements Zi, Zm, Zh and Zr in Figure 3.3 can be expressedby an overall shunt impedance. For an open tone hole,Z(o)s = i[kti + tan(k(tm + t+ tr)]Z0(h). (3.6)Here Z0(h) =pib2ρc is characteristic impedance of the tone hole.In the above equation, the inner correction ti is given by Lefebvre andScavone (2012)ti = 0.822− 0.095δ − 1.566δ2 + 2.138δ3 − 1.640δ4 + 0.502δ5; (3.7)the matching volume length correction is calculated by Nederveen et al.(1998) from the match volume shown in Figure 3.2tm =bδ1 + 0.207δ3; (3.8)t is thickness of the tone hole; tr is the length correction of the radiationimpedance at the outside of the tone hole, determined bytr =1karctan[ ZriZ0(h)]. (3.9)As mentioned in the previous section 3.1.1, the radiation impedance has anon-negligible real part. The real part of tr is determined by Dalmont et al.(2001) and classified as a cylindrical flange length correctionδcylin = 0.822− 0.47[ ba+ t]0.8. (3.10)The imaginary part of tr, corresponding to the real part of Zr, is determinedby experiment based on the specific geometry of the tone hole in this thesis.This is actually to determine the tone hole radiation loss term βr(h) in theradiation impedance:<(Zr(h)) = βr(h)(kb)2Z0(h). (3.11)In practice, a loss term βi(h) was also roughly applied to our inner end-correction to account for the vortex losses at the inside of the tone hole.The loss terms βr(h) and βi(h) were determined by experiments.543.1. TMMClosed tone hole shunt impedanceA closed tone hole differs from an open one only by having an infiniteimpedance at the outside terminal of the hole, its shunt impedance is givenby Eq. (7) of Nederveen et al. (1998)Z(c)c = iZ0h[kti − cot[k(t+ tm)]]. (3.12)For the frequency range of most woodwind instruments, both ti and t+ tmare small compared to the wavelength, so the first term kti can usually beneglected.In actual woodwinds, the effective tone hole thickness is affected by theway it is closed. The study of Dickens (2007) on flutes found that an tonehole closed by finger have an correction termtfigner = −0.76δb. (3.13)The final expression for closed tone hole shunt impedance used in the xiaoisZ(c)c = iZ0h[kti − cot[k(t+ tm+ tfinger)]]. (3.14)Here the tfinger was experimentally determined, as described in section 3.2.2.Transmission matrices of the tone holesThe series and shunt impedance of tone holes described in the previoussection are taken into account in the TMM by writing in matrix form. Forthe series impedance Za/2 at both side of the tone hole,Ta =[1 Za/20 1]. (3.15)For the shunt impedance Zs,Ts =[1 01/Zs 1]. (3.16)Then the complete matrix of a tone hole is obtained by multiplying the threematrices (Lefebvre, 2010, p.27):Th = Ta ·Ts ·Ta =[1 + Za2Zs Za(1 + Za4Zs)1Zs1 + Za2Zs]. (3.17)The tone hole matrix is then inserted into the bore matrices at the tone holecenter.553.2. A TMM model for the xiao3.1.3 Tone hole undercutsThe undercut means wall of a tone hole being cut at its inside and forma slope, either for tuning or timber controlling purposes. The undercutscan be modelled by using a conical transmission matrix in calculating thetone hole shunt impedance, and length corrections tfinger need to be usedto account any inaccuracies. The length corrections were determined byexperiments.The tone hole undercuts can reduce the tone hole losses (Giordano, 2014),and were expected to have effects on the value of βr(h) and βi(h) (and am-plitude of the impedance curve).3.2 A TMM model for the xiaoThe TMM provides a very good characterization of the woodwind bore andtone holes. The xiao’s body part can be mostly modelled with the existingTMM parameters from literature and a few practical correction parameters.To build a TMM model for the xiao, the xiao’s embouchure is the onlyspecial component to be considered.3.2.1 Modelling the xiao embouchureThe geometry of a xiao is shown in Figure 3.4. The embouchure outside haswidth w and farthest distance to the blowing edge d. In the lower right ofthe figure shows equivalent geometry of the embouchure as a truncated cone.The embouchure’s outside is equivalent to a circle of the same area (radiusr1); similarly, the embouchure’s inside is equivalent to a circular of radiusr2. The truncated cone’s height is the same as the embouchure’s thicknesst.To model the xiao embouchure, the woodwind tone hole model as de-scribed in section 3.1.2 was used with modifications:• A short conical pipe as shown in Figure 3.4 was used to represent thegeometry of the embouchure, and an end-correction te was applied atthe bottom of the cone (inner of the embouchure).• The embouchure (equivalent as the truncated cone) is in series withthe xiao body.• A short closed pipe from the embouchure’s inner center position to thepipe end is in parallel with the xiao body.563.2. A TMM model for the xiaor1r2tteEmbouchure outsideEmbouchure insideBlocked by the player's lip and jawwdFigure 3.4: Geometry of a xiao embouchure. Original Solidworks modelplotted by Martin O’Keane.The other impedance components of a tone hole (ti, tm, and ta) were conse-quently used for the embouchure. The radiation end-correction tr was nottaken into account because its effect will not show in the impedance mea-surement (tr was taken into account in Zemb in the playing condition of thexiao).The end-correction te also takes into account the reference plane mis-match in the impedance measurement as described in section 2.3 and wasthe only unknown parameter for the xiao embouchure model. So te wasdetermined experimentally from the impedance measurement.Determine end-correction te from experimentA xiao head (SXH1) of inner radius a = 7.6 mm, wall thickness t = 2.8 mmand pipe length L = 100 mm was made of a PVC pipe as shown in Figure3.5. The U-shaped embouchure hole was formed by cutting from one end ofthe pipe with a 7.9 mm drill, the pipe was mounted with a 45◦ angle to the573.2. A TMM model for the xiaodrill to form an oblique edge.Figure 3.5: The xiao head SXH1.According to the geometrical annotation routine of Figure 3.4, the xiaohead’s embouchure has d = 4.5 mm, w = 7.9 mm, r1 = 3.03 mm and r2 =4.03 mm. To determine te, acoustical impedance of SXH1 was measured withits far end open and closed. (The embouchure end x1 was also blocked whendoing the impedance measurement, leaving only the U-shape embouchureopen.)The impedance of SXH1 can be calculated using the TMM model. Thecalculation starts from the far end x2 = 100 mm. For the far end open,the impedance at x2 is the radiation impedance (Eq. (3.2) and Eq. (3.3)).The radiation loss term β will be also be determined from the measuredimpedance. For the far end closed, the impedance at x2 is infinity.The measured acoustical impedance of SXH1 is shown as blue curves inFigure 3.6. The impedance was normalized to the characteristic impedanceof the xiao embouchure’s outside openingZ0(e) =ρcpir21. (3.18)The unknown parameters te(o), β and te(c) were obtained by fitting thecalculated impedance to the measured ones.In a similar manner as obtaining the end-correction of a mismatchingpipe in section 2.2.4, the unknown parameters were set as an optimizationparameters. The optimization objective function was written as Eq. (2.11).The fitting (optimization) results were{t(o)e = 3.72 mm, β = 0.212 (Open end),t(c)e = 3.52 mm (Closed end).(3.19)583.2. A TMM model for the xiao10-210-1100101102Z/Z0(e)(open) FittingMeasurement500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)10-310-210-1100101102Z/Z0(e)(Closed)FittingMeasurementFigure 3.6: Impedance of the xiao head SXH1. Upper: far end open; lower:far end closed.The difference between t(o)e and t(c)e is not significant, so the end-correctionte was assumed to be suitable for the xiao embouchure. Later, t(o)e was usedfor the xiao embouchure model.Dependency of te on the embouchure geometriesThe embouchure end-correction te is dependent on the xiao head geometry( r1, r2, t and a). An ideal way to find the dependency was to calculate theembouchure model by FEM and to check effects of the individual variables.However, for the lack of time and computing resources, the parameters werefound experimentally.A set of xiao heads with different geometries were made as shown in Fig-ure 3.7. Their acoustical impedances were measured, and the end-correctionste were obtained as described above. Table 3.1 shows geometries and fittedte (far ends open) for a selection of the xiao heads. The dimensionless values593.2. A TMM model for the xiaot′e, defined by te = t′eγ2r2, were also shown in the table. Here γ = r2/a isthe radius radio of the embouchure and the bore. In comparison, Dickens(2007, p. 81) got t′e = 0.5 for the flute embouchure.Figure 3.7: Xiao heads with different geometries.603.2. A TMM model for the xiaoTable 3.1: Geometries and end-corrections of the xiao heads. Sym-bols used for the geometries ( w, d, t, r1, r2) were indicated inFigure 3.4, in unit of mm. And a is the radius of the xiao head atthe embouchure (in mm).Name a w d t r1 r2 te(mm) t′eSXH11 7.6 8.12 4.7 2.8 3.12 4.12 2.56 3.09SXHA 7.6 7.9 3.86 2.85 2.76 3.84 2.98 2.93SXHB 7.6 8 6.22 2.81 3.69 4.56 1.31 2.15SXHC 7.6 7.67 4.54 2.86 3.05 4.03 2.36 2.68SXHD 7.6 8.21 7.06 2.84 3.94 4.77 1.05 1.96SXHH 7.6 6.28 4.54 3.12 2.79 3.74 3.01 2.73SXHI 7.6 6.42 6.51 2.85 3.44 4.21 1.3 1.68SHXJ2 7.6 9.45 3.72 1.7 2.81 3.61 1.43 1.16SXHL 7.6 9.47 3.58 1.8 3 3.8 1.87 1.77BXHA 10.2 7.83 4.04 3.01 2.84 3.94 6.24 3.68BXHB 10.2 8.05 6.4 3.11 3.76 4.7 2.22 2.22BXHC 10.2 7.82 5.48 3.08 3.41 4.39 2.68 2.19BBXH1 8.22 8.02 5.04 4.43 3.26 4.68 4.58 6.95BBXH2 9.64 8.09 4.56 3.44 3.07 4.27 5.22 4.39BBXH33 8.38 7.9 4.53 4.4 3.05 4.51 2.37 3.1BBXH4 8.33 7.93 4.9 4.17 3.2 4.56 3.1 4.231 This xiao head was used for pipe-xiao-5, pipe-xiao-6 and pipe-xiao-7 (discussed later in section 5.1).2 This xiao head was used for pipe-xiao-8 (discussed later in sec-tion 5.1).3 Geometry and te of this xiao head were used for the optimizationof the acrylic xiao (discussed later in section 5.3).3.2.2 Modelling the full xiaoThe full xiao model is a combination of the xiao embouchure model and thexiao body TMM model. The xiao body also has several unknown parametersto be determined from experiments:1. The tone hole losses term βr(h) and βi(h).2. The xiao end radiation loss term β.3. The closed finger hole length correction t(c)f caused by fingers (mod-elling clay pads) protruding into the finger hole.613.2. A TMM model for the xiao4. An open finger hole length correction t(o)f caused by any inaccuracy ofthe calculated tone hole shunt impedance. (Dickens, 2007, p. 92).The embouchure end-correction te of a full xiao model can be determinedeither from a xiao head as described above, or by fitting with measuredimpedance of the full xiao.The full xiao used as an example for here is pipe-xiao-3 (as shown inFigure 3.8). The pipe-xiao-3 has similar embouchure geometries with SXH1Figure 3.8: A full xiao pipe-xiao-3. This xiao has eight finger holes, labelledfrom f1 to f8, and two pairs of additional holes, labelled from a1 1 to a2 2.as described in section 3.2.1. So the embouchure length correction te =3.72 mm was used; the other parameters as listed above were determined byfitting to the measured acoustical impedance.The closed finger hole length correction t(c)f was determined by fittingto the measured impedance of all the finger holes closed. There were stillholes open (a1 1 to a2 2), so the tone hole losses and the xiao end radiationloss term cannot be reliably distinguished. As a simplification, one value βcwas used for all the three loss terms. The measured and fitted acousticalimpedance was shown in Figure 3.9. The fitting results were:t(c)f = −1.65, βc = 0.41 (3.20)The open hole length correction t(o)f and the tone hole loss term βr(h),βi(h) were determined by fitting to the measured impedance of most fingerholes open. The fingering can be written as ooxo ooxo, where the eightsymbols represent the eight finger holes from the embouchure to the end; xmeans closed and o means open. The is put between the finger holes ofleft hand and right hand to make the notation easy to read. The tone holelosses term and the xiao end radiation loss term still cannot be distinguishedfrom each other, so one value βo was used for the three terms (β, βr(h) andβi(h)).623.2. A TMM model for the xiao500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)10-210-1100101Z/Z 0TMM fittingMeasurementFigure 3.9: Acoustical impedance of pipe-xiao-3 with all finger holes closed.The measured and fitted acoustical impedance was shown in Figure 3.10.The fitting results were:t(o)f = −0.08, βo = 0.33 (3.21)For all other fingerings, the obtained value t(c)f = −1.65, t(o)f = −0.08and βo = 0.33 were used, where βo were used for all the three loss terms (β,βr(h) and βi(h)). The measured and fitted acoustical impedance cannot bedirectly used for predicting the playing frequency of the xiao, because theradiation impedance of the embouchure has not been taken into account.Embouchure radiation impedance and the playing frequenciesThe xiao’s embouchure has a small but non-negligible radiation impedanceZemb, which is difficult to obtaine accurately, as described in section 1.4.2,so an empirical formula was taken from Dickens (2007, Eq. 7.2):Zemb(f) = Z∞(e)[2.9370 log(m)− 11.6284), (3.22)633.2. A TMM model for the xiao500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)10-210-1100101102Z/Z 0TMM fittingMeasurementFigure 3.10: Acoustical impedance of pipe-xiao-3 with fingering ooxo ooxo.where Z∞(e) is the radiation impedance of an infinitely flanged pipe of thesame opening area as the embouchure, and m = 69 + 12 log2(f/440) is thecorresponding MIDI number of the note at frequency f . The empiricalexpression of Zemb was determined by comparing the measured acousticalimpedance minima and playing frequencies of the flute. In this way, effectsof the temperature and air content changes inside the instrument are alsotaken account.For practical consideration, another length correction thead = −1.6 mmwas applied to the blowing end of the xiao considering the player’s jawprotrudes into this side a little (applies to all the xiaos measured in thisthesis). So the xiao’s impedance Z ′ at the playing state is a little differentfrom the measured impedance. Finally, the minima of the overall impedance(Z ′ + Zemb) will be the xiao’s playing frequencies. For convenience, theoverall admittance Y was introduced:Y = Z0(e)/(Z′ + Zemb), (3.23)and peaks of the admittance Y will be the final calculated resonance/playing643.3. Cross fingering, impedance irregularities and the woodwind cut-off frequenciesfrequencies of the xiao.3.2.3 Sound pressure and flow inside the xiaoThe TMM model can also be used to calculate the pressure and flow dis-tribution inside the xiao. Each frequency will have its own pressure andflow distribution, and usually only the resonance frequencies are of inter-est. The pressure and flow inside the xiao at the resonance frequencies werecalculated by the following steps:1. Measure/calculate the acoustical impedance, determine the resonancefrequencies f and value of the overall acoustical impedance Zoverall(f).2. Set the reference pressure at the embouchure to be pemb = 1 Pa;the flow at the embouchure should be pemb/Zoverall(f) to meet theresonance condition.3. Calculate pressure and flow at other positions along the xiao by thetransmission matrices.The pressure and flow of pipe-xiao-3 were calculated by the above steps.Results of two fingerings (xxxx xxxx and oxxx ooxo) are shown in Figure3.11 and Figure 3.12. The figures show that the pressure and flow hasalmost no noticeable change at the closed holes. At the open holes, the borewas connected to the outside air; but the pressure inside the bore does notdecrease much. This is because the open hole’s shunt impedance has someequivalent length, and acts as a buffer to the bore. The flow has a significantchange as the result of a shunt flow. The pressure has an abrupt change atthe embouchure because the embouchure effects were accounted at positionof its geometrical center. Results of all fingerings were normalized to theirmaxima and plotted in appendix B.1.3.3 Cross fingering, impedance irregularities andthe woodwind cut-off frequenciesIt is known that an open or closed cylindrical or conical pipe has mostlyharmonic impedance curves (see the typical open pipe impedance in Figure1.9, and pipe-xiao-7 with all its finger holes closed in appendix B.2). How-ever impedance of pipes with open side holes is usually not harmonic. Whilestudying the impedance and cavity modes of a guqinWaltham et al. (2014),653.3. Cross fingering, impedance irregularities and the woodwind cut-off frequencies0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70510152025Sound pressure(Pa)xxxx_xxxx: 293.8Hz0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Distance to the embochure side (m)0.0000000.0000020.0000040.0000060.0000080.000010Volumetric flow (m3/s)Figure 3.11: Absolute values of pressure and flow inside pipe-xiao-3 withfingering xxxx xxxx. The vertical lines indicates locations of the tone holes.Solid lines mean closed holes and dashed lines mean open holes.it was recognized that the irregularities in impedance curves come from res-onances happening at shorter segments of the cavity/pipe. The resonanceswere validated as below.The sound pressure of different frequencies of pipe-xiao-2 (later shownin Figure 5.1) was calculated using method described in the previous sec-tion and shown in Figure 3.13. The contour shows the real part of pressure,red for positive and blue for negative. The pressure is normalized to theembouchure pressure, and input impedance at the embouchure is superim-posed. At about 1.2 kHz, a pattern can be seen between the pipe end andthe adjustment hole a1 1. Input impedance at a1 1 was calculated by TMMand plotted under the contour, it correctly shows minima at around 1.2 kHz.This may explains why the pipe-xiao-2 can be played at the first three andthe fifth impedance minima, but cannot be played at the fourth (becausearound this frequency, a resonance occurs at the short segment between a1 1663.3. Cross fingering, impedance irregularities and the woodwind cut-off frequencies0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70510152025Sound pressure(Pa)oxxx_ooxo: 524Hz0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Distance to the embochure side (m)0.0000000.0000010.0000020.0000030.0000040.0000050.0000060.0000070.0000080.000009Volumetric flow (m3/s)Figure 3.12: Absolute values of pressure and flow inside pipe-xiao-3 withfingering oxxx ooxo.and the xiao end, instead of the embouchure). In contrast, pipe-xiao-3 has ashorter segment formed by the xiao end to a1 1. As its impedance in Figure3.9 indicates, pipe-xiao-3 with all its finger holes closed can be played at thefirst five and the seventh impedance minima, but not at the sixth one (be-cause its segment between the xiao end and a1 1 and has higher resonancefrequency, around the sixth impedance minima).The impedance irregularities can be better shown by the cross fingeringoxxx ooxo of pipe-xiao-2. A pattern shows at about 900 Hz and position ofthe three closed holes. Input impedance at f8 is plotted below the contour,it has minima at about 500 Hz and 900 Hz. Blowing at f8 can exactly soundthe two resonances.Based on the observed relationship of impedance irregularities and lo-cations of resonances along the pipe, we can state that any irregularities inthe impedance curve have a corresponding resonance at a shorter segmentS of the pipe, formed by the pipe end with an open hole, or formed by two673.3. Cross fingering, impedance irregularities and the woodwind cut-off frequenciesFrequency (Hz)0100200300400500600700Distance to the embouchure (mm)10−1100101Z/Z 0(e)ZZ0(e)  at embouchure500 1000 1500 2000Frequency (Hz)10−1100101Z/Z 0(e)ZZ0 (e)  at hole a1_1Figure 3.13: Pressure distributions and impedance curves of pipe-xiao-2with all finger holes closed. The horizontal lines indicates hole locations,solid line for closed holes and dashed lines for open holes.open holes. The resonances of segment S can only be excited by blowing ateither of its end (for example, at Figure 3.14, the resonance at about 900Hz cannot be played at the emboucure).The above statement can be related to the woodwind tone hole lat-tice cutoff frequency fc(lattice)(Benade, 1959, 1990). Above fc(lattice), theimpedance curve’s amplitude reduces and its number of minima increases,interval of the minima correspond to standing waves of the whole instru-ment as if the holes were closed (Wolfe and Smith, 2003). For example,impedance of pipe-xiao-3 with fingering oxxx ooxo (as shown in appendixB.1) has fc(lattice) ≈ 1.5 kHz. However, above fc(lattice), pipe-xiao-3 with thisfingering can still be played at about 1.6 and 2.5 kHz, which is the third andfifth impedance minima of the embouchure segment. All other impedanceimpedance minima cannot be played at the embouchure, but some of themcan be played at the open holes of their segment, traced by their resonancelocations in the pressure contour (not shown). So for frequencies abovefc(lattice) the xiao is not equivalent to a whole pipe with all holes closed,683.3. Cross fingering, impedance irregularities and the woodwind cut-off frequenciesFrequency (Hz)0100200300400500600700Distance to the embouchure (mm)10−1100101Z/Z 0(e)ZZ0(e)  at embouchure500 1000 1500 2000Frequency (Hz)10−1100101Z/Z 0(e)ZZ0(e)  at hole f8Figure 3.14: Pipe-xiao-2 with fingering oxxx ooxo, see the caption of Figure3.13 for details.although the standing waves were observed to extend to the whole pipe inthe contour.69Chapter 4OptimizationThe numerical optimization’s principle is to minimize an objective functionbased on iteratively adjusting the optimization parameters.The TMM model for xiao as described in the previous chapter makes thenumerical optimizations possible for the design of xiao. For the xiao, theobjective function is described in section 4.1.1; the optimization parametersare the tone hole locations/diameters and the bore shape.The optimization process failed sometimes; then some changes neededto be made manually. The changes were mostly the xiao’s fingerings. Theoptimal fingerings were found systematically using the some optimizationresults, as described in section 4.5.4.1 Optimization objectivesBased on the goals of a good xiao as described in section 1.5. The optimiza-tion objectives are as follows:1. Adjust the impedance minimum of each note to the correct frequency,to make the notes in tune.2. Adjust the impedance minima of integer multiples of notes in the firstoctave (except for the cross fingering note), to make the notes haverich harmonics (in this way they will also be able to be used for higheroctaves).3. Increase the depth of high notes’ impedance minima, to make the highnotes easy to play.4. Find a set of correct fingerings (especially for the high notes).704.2. Optimization algorithm4.1.1 Objective functionThe objective function for optimizing the xiao is written as below:O(geometry) =∑nt=oc1∑n=har[1200× logfn(nt)n fstd(nt)]+∑nt=cf[1200× logf(nt)fstd(nt)]+∑nt=ht[abs(Z(nt))/Zref (nt)].(4.1)In the first term, oc1 = [D4, E4, F4, F4], G4, A4, B4, C4]] is the xiao’normal fingerings in the first octave. fn(nt) is the nth mode of the fingeringand fstd(nt) is the note’s standard frequency in equal temperament scale(A4=440 Hz). Under the second summation symbol, har = [1,· · · , N ] isthe number of harmonics. In the second term, cf contains all the crossfingering notes. The first two terms were the total tuning and harmonicsvariation of the xiao. Minimizing the first two terms can achieve the first twoobjectives as described above. In the third term, abs(Z(nt)) is amplitudeof the impedance minima of the high notes. Minimizing the third term canmake the impedance minima deeper and achieve the third objective. Thefourth objective was not taken account into the objective function and thefingerings were adjusted manually as described in section 4.5.4.2 Optimization algorithmTwo optimization algorithms were used in this thesis, both of which canadd constraints to the optimization parameters. The first one is a gradi-ent/hessian based algorithm L-BFGS-B (Byrd et al., 1995), it was proposedfor optimizing the design of flute by Lefebvre (2010). The L-BFGS-B algo-rithm has been included in the SciPy module of Python, and was primarilyused in this thesis. The L-BFGS-B algorithm needs to know the parameters’gradient of the objective function to decide the next optimization step. Bydoing so, the TMM code needs to run opN times (change each parameter bya small value ) to obtain the gradient. The L-BFGS-B was used when thenumber of optimization parameters is small to medium (up to about 100).The second algorithm in this thesis is the Rosenbrock’s method (Rosen-brock, 1960), proposed by Kausel (2001) for optimization on the bore shapeof brass instruments. The Rosenbrock’s method does not need to calculatethe gradient, so it is advantageous when the optimization parameters is large(opN > 100).714.3. Optimization of the xiao4.3 Optimization of the xiaoDepending on the goals of the optimization and the manufacture of the xiao,the optimization of the xiao was as two phases. In phase one the bore shapeis fixed, either a cylindrical bore or a measured bamboo bore shape is used.The tone hole locations/diameters (and possibly thickness) were used as theoptimization parameters. In phase two the bore shape is also allowed tovary.4.3.1 Phase one: optimizing the tone holesThe xiao has 8 finger holes and up to 3 pairs of additional holes. So thenumber of optimization parameters opN in phase one will be within 16 to24 (locations and diameters of two additional holes in a pair were set to bethe same). Based on such limited number of optimization parameters, ansimplified objective function was used by taking only the two terms of Eq.(4.1), and number of harmonics N was set to be 2 ≤ N ≤ 4.The finger hole diameters were initially constrained within 6 to 10 mm,later changed to 8 to 10 mm. The lower limit was set higher because smallfinger holes have poor sound radiation, so their notes will have too smallvolumes. The finger hole diameter’s upper limit was set as 10 mm to fitthe finger pad of normal players. Constraints on the finger hole locationswere manually set up, only to avoid getting finger hole locations that werenot convenient for the players to reach. The additional hole had diameterconstraint within 4 to 12 mm.It was found that the tone hole thickness and diameter were inverselyrelated, so adding the tone hole thickness as optimization parameters won’tincrease the effective number of optimization parameters. In practice, asparameters not for the optimization, the tone hole’s thickness was adjustedmanually if its diameter reached the constraints and limits the optimizationresult.The optimization starts from an initial tone hole geometry determinedfrom the required wave length of the first octave and a rough estimation ofthe tone hole’s equivalent length correction.4.3.2 Phase two: optimizing the bore shapeAfter including the bore shape as optimization parameters, the high notes’impedance minima were considered by including the third term of Eq. (4.1).More number of impedance minima were also expected to be harmonically724.3. Optimization of the xiaoaligned (up to 8 for the lowest note D4).Bore shape tuning mechanismThe bore shape can be used for tuning the woodwind instrument by thedifferent effects of bore shape changes made at different positions:• If the diameter is increased near a pressure node, the resonance fre-quency will be raised.• If the diameter is increased near a flow node, the resonance frequencywill be lowered.• Vice versa for decreasing the diameters.The above bore shape tuning mechanism was proposed by Lord Rayleighin the 19th century, and developed by Benade (1959) to be the woodwindperturbation method (Benade, 1990; Fletcher and Rossing, 1998).Nowadays, the bore shape tuning method is widely used in the shakuhachi,and is starting to be used on the xiao. The tuning is done by removing somematerial from the inner wall of the instrument at the pressure/flow nodes,if the pitch needs to be raised/lowered. In shakuhachi tuning, material mayalso be added to some positions of the bore.Optimization validationThe optimization was validated in reverse, by letting the code optimize abore with a normal xiao embouchure but without tone holes. A specialobjective function is set asOv1(geometry) = 1200× logf1fstd(D4)+ 1200× logf2fstd(C5), (4.2)where f1 and f2 are the first two resonance frequencies. The second reso-nance was intentionally set to be a whole tone (200 cents) lower than twiceof f1 to check the bore shape’s tuning effect.The optimization was run by both algorithms, and the results (namedxiao-v1) were shown in Figure 4.1. The results show that the optimizedxiao-v1 had increased diameter around 1/4 and 3/4 distance to the em-bouchure end. The two locations were flow nodes of the second resonance –exactly as the bore shape tuning mechanism predicted. However, the boreshape changes abruptly, and the bore shape from Rosenbrock’s method has734.3. Optimization of the xiao0 100 200 300 400 500 6006.57.07.58.08.59.09.5Radius (mm)l-bfgs-bRSB0 100 200 300 400 500 600Distance to embouchure end(mm)201001020Figure 4.1: Optimized bore shape of xiao-v1 by both algorithms.zig-zags around the 1/4 position. These two features result from the opti-mization algorithms, because no special requirement was applied on the boreshape, and the bore shapes shown in the figure do meet the optimization re-quirements. However, these two features were not favored in real woodwindinstrument bores.To remove the abrupt changes and zig-zags in the bore shape optimiza-tion, a modified objective function were used:Ov2(geometry) = 1200× logf1fstd(D4)+1200× logf2fstd(C5)+xn∑x1[g∂2r(x)∂2x].(4.3)Here the third term was the total variation of the bore radius, with a co-efficient g was used to set the required smoothness. In order to maintainthe tuning requirements, g was set to let the contribution of the third termhave almost equal contribution to the Ov2 when the optimization resultsconverge. Optimization results (named xiao-v2) by the modified objectivefunction were shown in Figure 4.2. It is shown that the bore shapes of744.3. Optimization of the xiaoxiao-v2 were considerably smoother.0 100 200 300 400 500 6006.57.07.58.08.5Radius (mm)l-bfgs-bRSB0 100 200 300 400 500 600Distance to embouchure end(mm)201001020Figure 4.2: Optimized bore shape of xiao-v2 by the modified objective func-tion.Test optimization on harmonic alignmentHarmonic alignment of the impedance minima is one of the main purposesof the bore shape optimization. The optimization was tested by trying tooptimize a xiao without tone holes, and let the first 10 impedance minimabe harmonically aligned. The objective function was written as:Ov2(geometry) =10∑n=1[1200× logfnn fstd(D4)]+xn∑x1[g∂2r(x)∂2x]. (4.4)Here the bore smoothness constraint was also applied as the second term.Result of this test optimization (named xiao-v3) is shown in Figure 4.3.Bore shape of xiao-v3 shows 10 significant variations in radius, corre-sponding to the 10 pressure and flow nodes of the 10th harmonic.754.4. Speed and efficiency of the code0 100 200 300 400 500 6007.27.47.67.88.08.28.48.68.8Radius (mm)RSB0 100 200 300 400 500 600Distance to embouchure end(mm)201001020Figure 4.3: Optimized bore shape of xiao-v3 by the Rosenbrock’s method.4.4 Speed and efficiency of the codeThe TMM and optimization codes were written from scratch in Python.Speed of the TMM code was crucial when it needs to be run iteratively inthe optimization.As the first attempt to enhance the TMM code’s speed, in each TMMcalculation, the transmission matrices and tone hole matrices (both open andclosed) were first calculated for all frequencies and stored, then impedancesof different fingerings were calculated by using the corresponding open orclosed tone hole matrices.The time each TMM calculation took was proportional to the numberof frequencies nf to be calculated, and the number of points nb to representthe bore shape. So attempts were made to minimize both Nf and NT toincrease the code speed.Minimizing nf : In order to cover the frequency range of 280 to 2600 Hz(3.2 octaves, or 3840 cents) in the optimization of the xiao, and maintain764.5. Optimizing the xiao fingeringsfrequency resolution of a few cents (e.g. 2 cents), the frequencies shouldbe log-scaled and nf = 1920. Since the impedance had a quite linear de-pendence on frequency around its minima, nf can be significantly loweredusing interpolations. The impedance minima can be determined by the zerocrossing points of its first order derivative (or by the zero crossing pointsof the impedance’s phase). Using the interpolations, nf = 200 gave similaraccuracy (2 cents) in the impedance minima.Minimizing nb: This attempt was applied only to xiaos with bore shapeas a part of the optimization parameters. Since the small differences in thebore shape had limited effects on the impedance minima frequencies anddepth, nb was set to be small (e.g. 80) at first for reasons of speed. Aftera period of optimization, nb was set to the required number of points torepresent the bore shape (e.g. 320), and the optimization was continuedusing the result of the previous optimization as initial parameters.After the above attempts, for a xiao with cylindrical bore, the TMM codetakes less than one second to calculate the impedance of all its fingerings;for a xiao with nb = 320, it takes about 30 seconds.The xiao optimizations were run on the WestGrid clusters (www.westgrid.ca).Generally it took about 10 hours to optimize a xiao with a cylindrical boreshape, and several days to optimize a xiao with varying bore shape.4.5 Optimizing the xiao fingeringsThe traditional xiao fingerings as shown in appendix A.1 come from themakers/players’ experience, and may not be the best choice. Especially onthe cross fingering notes and high notes. In this thesis, all the fingeringsfor the xiao were investigated systematically and optimized. The optimizedfingerings were shown in appendix A.2.4.5.1 A systematic investigationIn this systematic investigation, the xiao fingerings were determined basedon the following guidelines:1. The fingerings should be as normal as possible (open in sequence).2. Fingerings for higher octaves should have as a small number of changesas possible compared to the lower octave fingering.3. Fingerings for the high notes should have deep impedance minima torender the high notes easy to play.774.5. Optimizing the xiao fingerings4. The fingerings should be convenient to the player (e.g. avoid the fin-gering oooo oooo because the xiao will be even hard to hold).Normal fingeringsBased on guideline 1, fingerings for the first octave were to open the fingerholes in sequence. f6 is closed in xoxo oooo and ooxo oooo to help holdthe xiao. The fingerings for the first octave were similar to the traditionalfingerings.Based on guideline 2, fingerings in the first octaves should have theirimpedance minima harmonically aligned, so that the same fingering or mi-nor modified fingering can be used for the second octaves and high notes.Regarding this requirement, f2 were kept open (other than xxxx xxxo andxxxx xxxx), because the traditional fingerings with closed f2 were found tobe obviously inharmonic in their 3rd and 4th impedance minima.The properties of the musical scale makes the fingering with harmonicimpedance minima useable for the higher octaves. For example, a twelfthabove (three times in frequency) F4 is C6. So the fingering of E4 xxxx xxoocan also be used for C6. In practice, f7 was opened as an resister hole todamp the first two resonance frequencies (xoxx xxoo is one of fingerings forC6 in the traditional fingerings). In this way, by using more of the twelfths,fingering of the following high notes were determined as:• C6]: xxxxvxooo; no suitable register hole.• D6: oxxx oooo; f8 as the register hole.• E6: xxxo oooo; F6: xxoo oooo; F6]: xoxo oooo: no suitable registerhole.By using more of the 3rd octave (4 times in frequency), the following highnotes fingering were determined:• G6: xoxx oooo; f7 as the register hole.• A6: oxxo oooo; f8 as the register hole.A correct cross fingerings for C5Notes not covered by the normal fingerings have to use cross fingerings. Thefirst task is to find a correct cross fingering for C5.784.5. Optimizing the xiao fingeringsThe traditional fingering for C5 is oxxx ooxo, similar to the westernBaroque flute. The problem of the traditional C5 fingering is, compar-ing with the fingering of C5], closing the two holes doesn’t cause sufficientchange of the xiao’s effective length. Since the note C5] is less used in thexiao, the xiaos are usually made to have C5 in tune, but C5 ] is too flat (usu-ally more than 30 cents). The problem can be solved by covering more holesto enhance the cross fingering effect, such as using oxxx xoxo or oxxx xxoofor C5. However they are not used in practice, because of the two fingerings’inconvenience in playing.In this thesis, oxxx xxxo was chosen as the new fingering for C5. Thisfingering is not considered by the current xiao makers/players because itis an alternative fingering for C6 (by its second resonance) if the fingeringxoxx xxoo is not in tune. When C6 is in tune with this fingering, its firstresonance about 50 cents lower than C5. In this thesis, the fingering for C6is chosen to be xoxx xxoo as described above, and it is made to be in tuneby the optimization process.High notes fingeringsNow only fingerings for B6 or higher have not been systematically deter-mined. They need to be determined following the guideline 3 and 4, and annumerical approach was used in this thesis.Fingerings for the high notes need to have deep impedance minima lo-cated at the correct frequencies. The TMM model makes it possible tocalculate impedance curves of all the 256 fingerings of the xiao. For eachhigh note, absolute values of each fingering were compared at the note’sfrequency and sorted from small to large. In this way, the good fingeringswill be listed in front of the bad fingerings.Calculated impedances of a PVC xiao (pipe-xiao-7) were used to demon-strate how the high note fingerings were found.Figure 4.4 shows the first 14 of the sorted fingerings’ impedance curvesfor the note B6 (1975.5 Hz). The second listed fingering xxox oooo plottedby the solid green line was chosen for this fingering because its impedanceminima depth is similar as the blue solid line. Also because xxox oooo isclose to fingering of the note before it (A6: oxxo oooo).Figure 4.5 shows impedance curves for the note C7 (2093 Hz). The 12thlisted fingering xoox oooo had once been considered, because its impedanceminima around C7 is very deep (in practice this fingering is very easy toplay), also because it is close to the fingering of B6 (xxox oooo). However,use of the fingering xoox oooo in several optimizations does not produce794.5. Optimizing the xiao fingeringsFigure 4.4: The first 14 of the sorted fingerings’ impedance curves for noteB6. The black line is the standard frequency of B6. A zoom-in view ofimpedance minima near the standard frequency is shown.a good tuning result (also shown in Figure 4.5 as the tuning deviation ).Finally the third listed fingering xoxx oxxo was chosen; this fingering alsoproduces good optimization results for other xiaos.Figure 4.6 shows impedance curves for the note D7 (2349.3 Hz). Thefirst listed fingering oxxo xooo was simply chosen.804.5. Optimizing the xiao fingeringsFigure 4.5: The first 14 of the sorted fingerings’ impedance curves for noteC7.Figure 4.6: The first 14 of the sorted fingerings’ impedance curves for noteD7.81Chapter 5ResultsThe results will be presented in two phases in the same sequences as theprevious chapter on optimization. A failed optimization on an unfinishedbamboo xiao with measured bamboo bore shape will be discussed betweenthese two phases.The optimizations were made by assuming the environmental tempera-ture to be 23 ◦C. However the measurements were done on different days,so the impedance minima and playing frequencies in the results had somedeviations from the standard equal temperament pitches. The xiaos wereplayed and recorded by the author (an amateur xiao player with six yearsof playing experience) right after the impedance measurements, so that theenvironmental temperature was the same as when the impedance measure-ments were made. As described in section 1.4.2, the playing frequencies canbe largely dependent on the player effects. So the measurements were madeat a moderate playing volume and a normal playing condition; intentionalpitch adjustments were avoided. The playing spectra were obtained from therecorded notes. Each note was played for 5 seconds, recorded and Fouriertransformed. The spectra were then averaged with 1 Hz resolution.5.1 Phase one: PVC xiaos with optimized toneholesAll the PVC xiaos were made from optimization results. The first two ofthem were named pipe-xiao-1 and pipe-xiao-2, as shown in Figure 5.1. Bothof the two xiaos were 700 mm in length, 7.6 mm in inner radius, and 2.8 mmin wall thickness. As a result of balancing the problematic cross fingeringC5 with C5], their eighth finger hole f8 has a small diameter (the undersizedfinger hole for cross fingering notes on flute was also pointed out by Benade(1990, p. 452)). However, these two xiaos were made before the TMMmodel was fully optimized (especially the embouchure length correctionste), so they were not well tuned.Acoustical impedance and playing spectra of the two PVC xiaos were825.1. Phase one: PVC xiaos with optimized tone holesFigure 5.1: Pipe-xiao-1 and pipe-xiao-2measured. Their embouchure length correction te were obtained by fittingto the measured impedance, and the xiao model’s accuracy consequentlyimproved.Then the first successful PVC xiao was made and named pipe-xiao-3(used as the example of a full xiao model in section 3.2.2), its geometrieswere shown in Table 5.1.Table 5.1: Geometry of the pipe-xiao-31.Name Position2(mm) Thickness (mm) Radius (mm)a1 1 574.98 2.8 4.08a1 2 575.18 2.8 4.13a2 1 529.21 2.8 3.14a2 2 529.43 2.8 3.12f1 458.76 2.8 3.99f2 434.43 2.8 4.07f3 400.91 2.8 3.94f4 380.81 2.8 3.94f5 328.29 2.8 3.95f6 312.41 2.8 4.18f7 283.86 2.8 3.94f8 248.37 2.8 2.99Emb3 3.33 2.8 r1, r241 Total length: 650 mm. Bore inner radius: 7.6 mm, out-side radius: 10.4 mm.2 Distance to the embouchure end.3 Geometrical center of the embouchure’s inside.4 r1=3.12, r2=4.09 are the embouchure’s equivalent radiusof outside and inside.The pipe-xiao-3 was optimized in tone range from D4 to G6 with thetraditional fingerings (see appendix A.1). The optimization objective on835.1. Phase one: PVC xiaos with optimized tone holespipe-xiao-3 was to achieve best tuning, so its harmonics were only optimizedto N = 2 (in fact they were notes of the second octave). The maximumtuning deviation of pipe-xiao-3 was 23 cents, occurring on the problematiccross fingering note C5. The pipe-xiao-3 was measured and played for notesfrom D4 to G6 (except F5 missing), the results were shown in appendix B.1.The measurements were made at environmental temperature of 18.43 ◦C.The pipe-xiao-3 results were presented in the International Symposiumof Musical Acoustics (2014) at Le Mans, France (Lan and Waltham, 2014)(named xiao P in that paper).The next PVC xiao made was pipe-xiao-4 as shown in Figure 5.2. Ithad eight finger holes and no additional holes. As advised by Kongxincai(Kongxincai, 2014) in the xiaoyaji forum, the thickness of its f8 was increasedto enable a larger diameter. However, the spacing of f1 and f2 (indicated inthe figure as d12) was too large and inconvenient for playing.Figure 5.2: Pipe-xiao-4, pipe-xiao-5 and pipe-xiao-6.Pipe-xiao-5 and pipe-xiao-6 were made with only their body parts (forthe xiao head SXH1). The two xiaos were built for testing and optimizingthe xiao fingerings as described in section 4.5; however these xiaos were notsuccessful.5.1.1 Based on optimized fingeringsThe successful PVC xiaos based on the optimized fingerings (as shown inappendix A.2) were pipe-xiao-7 (with xiao head SXH1) and pipe-xiao-8 (withxiao head SXHJ), as shown in Figure 5.3. Their geometries are shown inTable 5.2 and 5.3.The two xiaos have very similar finger hole locations and diameters, re-sulting from the same fingerings and optimization objective function. Har-monics of the two xiaos were optimized to N = 4 for the normal fingeringnotes in the first octave (except N = 3 was set for the fingering xoxo oooo845.1. Phase one: PVC xiaos with optimized tone holesFigure 5.3: Pipe-xiao-7 and pipe-xiao-8.Table 5.2: Geometries of the pipe-xiao-71.Name Position2(mm) Thickness (mm) Radius (mm)f1 465.57 2.8 4.8f2 439.75 2.8 4.5f3 413.98 2.8 4.5f4 387.72 2.8 4.5f5 337.91 2.8 4.99f6 317.84 2.8 4.5f7 295.32 2.8 4.98f8 252.99 5.7 4.66Emb3 3.28 2.8 r1, r241 Total length: 543.09 mm. Bore inner radius: 7.6 mm,outside radius: 10.4 mm.2 Distance to the embouchure end.3 Geometrical center of the embouchure’s inside.4 r1=3.12, r2=4.12 are the embouchure’s equivalent radiusof outside and inside.because its closed f6 caused some inharmonicity). The optimized tone rangeof these two xiaos was from D4 to D7, and they had maximum tuning devi-ation within 10 cents within the tone range according to the optimization.A selection of fingerings were measured on the two xiaos, and results ofpipe-xiao-7 are shown in appendix B.2. The measurements were taken at anenvironmental temperature of 21.82 ◦C. The calculated pressure and flowinside the xiao as described in section 3.2.3 are also shown. Because of thesimilarities, results of pipe-xiao-8 are not be shown.855.2. An unfinished bamboo xiaoTable 5.3: Geometries of the pipe-xiao-81.Name Position2(mm) Thickness (mm) Radius (mm)a1 1, a1 2 608.5 2.8 3.48a2 1, a2 2 561.66 2.8 4.49a3 1, a3 2 531.58 2.8 3.14f1 462.49 2.8 4.5f2 438.75 2.8 4.5f3 411.94 2.8 4.5f4 386.64 2.8 4.5f5 336.43 2.8 4.9f6 316.64 2.8 4.5f7 291.69 2.8 4.62f8 251.04 5.7 4.64Emb3 2.17 2.8 r1, r241 Total length: 730.58 mm. Bore inner radius: 7.6 mm, outsideradius: 10.4 mm.2 Distance to the embouchure end.3 Geometrical center of the embouchure’s inside.4 r1=2.81, r2=3.61 are the embouchure’s equivalent radius ofoutside and inside.5.2 An unfinished bamboo xiaoPVC xiaos with cylindrical bore may not be favored by most xiao players.Using the same phase one optimization method as the PVC xiaos, fingerholes for a bamboo pipe could also be optimized. But to build a bambooxiao with optimized finger holes, the first task is to have its bore measured.5.2.1 Measurement of the Bore ShapeA long probe was made for measuring the bore shape of bamboo pipes, asshown in Figure 5.4 (a). The probe was mounted on a lathe to measure theinside of the bamboo bore, as shown in the figure (b) and (c). The probeworked by being electrically connected (judged by resistance measured by amultimeter) when it attached the inner wall of the xiao, then the coordinateswere obtained from the lathe’s digital read-out.The bamboo pipe was obtained by removing its nodes, then its innerwall was smoothed manually by long rods covered with sandpaper. Thebamboo bore may not be axis symmetrical, so four sides along the bore865.2. An unfinished bamboo xiaoFigure 5.4: A probe mounted on a lathe for measuring the bamboo pipe’sbore shape.were measured (marked as 1, 2, 3 and 4 in the Figure 5.4 (b)). Then crosssectional areas along the position were approximated as ellipses, the majorand minor radii were determined by half of the spacing measured along thefour sides. Then the bore shape was obtained as a sequence of equivalentradius along the position. The bore shape is shown in Figure 5.5 with asolid blue line. The dashed red line shows a conical shape determined fromtwo ends of the bamboo.A xiao embouchure was made on the bamboo pipe; the pipe’s acousticalimpedance was measured and fitted by the modified TMM model to verifythe bore shape measurement, and to determine its embouchure length cor-rection. The fitting was done using both the measured bamboo bore shapeand the conical pipe shape as shown in Figure 5.5. The measured and fittedimpedance results are shown in Figure 5.6. Both of the fittings matched withthe measured impedance very well, but the measured bamboo bore shapehad slightly better fitting results, indicating the bore shape measurement isreliable.875.3. Phase two: an acrylic xiao with optimized bore shape0 100 200 300 400 500 600 700 800Distance to the embouchure side (mm)8.68.89.09.29.49.69.810.010.2Radius (mm)Bore shape measurementConical pipeFigure 5.5: Measured bore shape of the bamboo pipe. The dashed line is aconical shape determined from the bamboo pipe’s ends.Then the tone holes optimization was run with the pipe’s measured boreshape. Locations/diameters of eight finger holes and three pairs of additionalholes were to be determined. However the optimization results were notgood (even correct tuning in the first two octaves cannot be reached). Sothe bamboo xiao was left unfinished.5.3 Phase two: an acrylic xiao with optimizedbore shapeAs a final confirmation of the optimization technique, a xiao with optimizedtone holes and bore shape was made. Being able to vary the bore shapeenabled the optimization to follow more strict requirements as below:• Tone range of the optimization was set to be from D4 to D7.• N = 8 was set as harmonic number of fingering xxxx xxxx, N = 4 forother normal fingerings except for N = 3 for xoxo oooo and N = 2for ooxo oooo.• The top five high notes’ impedance minima depths were taken into885.3. Phase two: an acrylic xiao with optimized bore shape0 500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)10-210-1100Z/Z 0(e)MeasurementBamboo fittingCone fittingFigure 5.6: Measured and fitted impedance of the bamboo pipe. Bamboofitting: fitting done by the measured bamboo bore shape. Cone fitting:fitting done by the conical shape shown in Figure 5.5.account.The bore shape was represented by 320 points (initially 80 for fasteroptimization speed, as described in section 4.4) as shown in Figure 5.7 (a).In the figure (b), it was shown that the finger holes’ thicknesses vary, theywere to help limit the finger hole’s radius within 4 to 5 mm. The optimizationresults had a maximum tuning deviation of 6.7 cents within the tone range,and increased depth of the high notes impedance minima. For now, thereis no a simple explanation for the optimized bore shape. However, it has8 noticeable variations along the positions, which should corresponds tothe pressure and flow nodes of the eighth harmonic of fingering xxxx xxxx(similar as xiao-v3 discussed in section 4.3.2).The optimized bore shape was machined out of transparent acrylic bya CNC milling machine in the UBC Physics and Astronomy departmentalmachine shop, as shown in Figure 5.8 (a), then the outside was machined895.3. Phase two: an acrylic xiao with optimized bore shape0 100 200 300 400 500 600Position (mm)7.07.58.08.59.09.5Radius (mm)(a)0 100 200 300 400 500 600Position (mm)3020100102030Radius (mm) (b)Figure 5.7: Bore shape of the acrylic xiao represented by 320 points alongthe position (distance to the embouchure end). (a): radius of the bore atdifferent positions. (b): the bore shape plotted in real scale; outer diame-ters around the finger holes were machined to meet the required tone holethickness.on a lathe as shown in the figure (b).In addition, undercuts were done for the acrylic xiao’s eight finger holes.The undercuts were done manually from the inside of the two halves, with1 mm cut depth and 45◦ angle with walls of the finger holes. The under-cuts were accounted for in the xiao model as described in section 3.1.3.Impedance measurements were done before and after the undercuts, asshown in Figure 5.9. As expected, the results showed increased variationof the impedance amplitudes after the undercuts.All of the measurement and modelling results of the acrylic xiao areshown in appendix B.3. The measurements were taken at an environmentaltemperature of 16.53 ◦C (while the xiao was designed for room temperatureat 23 ◦C), so the resonance frequencies were considerably lower than theA4=440 Hz standard pitches.The measured and modelled acoustical impedance of the acrylic xiaomostly agreed with each other within several cents, except for frequency905.3. Phase two: an acrylic xiao with optimized bore shapeFigure 5.8: An acrylic xiao with optimized bore shape and tone holes. (a):the bore shape made by the CNC milling machine as two halves. (b): Out-side of the xiao machined by a lathe. Tapes were used to temporarily fixthe two halves and Vaseline was applied to ensure sealing.range of about 1300 to 1800 Hz, probably caused by inaccurate character-ization of the tone hole undercuts. For most of the notes, the admittancepeaks correctly predicted the playing frequencies; however for F6 and higher,deviations of the playing frequencies and calculated admittance peaks wereover 20 cents. These may be caused either by inaccuracies in calculatingthe embouchure radiation impedance (Eq. (3.22)), or by the player effectsas described in section 1.4.2. The xiao’s playing frequencies depends on theopening area of the embouchure and the jet velocity; the variations can beover 50 cents. The playing frequencies of the acrylic xiao were tested onlyby the author and only for one set of recording, so the player effects may bethe reason of the measured deviations.Finally, the additionally optimized bore shape showed very rich harmon-ics in playing the note D4. Its high notes were easy to play. Noticeably, thehighest note D7 can be also played with fingering xxxx xxxx, because of thedeep 8th impedance minima (as a result of increasing impedance minimumdepth of the actual D7 fingering oxxo xooo).915.3. Phase two: an acrylic xiao with optimized bore shape0 500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)10-210-1100101Z/Z 0(e)After undercutsBefore undercutsFigure 5.9: Measured acoustical impedance of the acrylic xiao before andafter its tone hole undercuts (with fingering xoxo oooo).92Chapter 6ConclusionAn acoustical impedance measurement system was built (as described inchapter 2) and calibrated by two new calibration methods. The accuracyof the measurement system was validated by measuring simple cylindricaltubes.A modified TMM model was built for the xiao (as described in chapter3); the acoustical impedance of the xiao calculated by this model agrees withmeasurements mostly within several cents. The modified TMM model wasalso used to calculate the pressure and flow inside the xiao; relationshipsbetween the cross fingerings, impedance irregularities and the woodwindcutoff frequencies were examined.Based on the modified TMM model, numerical optimizations were madefor the xiao’s finger holes and bore shape. A set of optimized fingerings forthe xiao was obtained by systematically analyzing the xiao’s fingering rules,and a method was proposed for finding the best fingerings for the high notes.A PVC xiao with optimized tone holes using traditional fingerings at-tained good tuning over 2.5 octaves. Two PVC xiaos made with optimizedtone holes using optimized fingerings showed good tuning over three octaves.Finally, an acrylic xiao with additionally optimized bore shape showed richharmonics and easily playing high notes. Playing frequencies of the acrylicxiao had some obvious deviations from the model on the high notes, proba-bly caused by the player effects.93BibliographyAndo, Y. 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The Journal of the Acoustical Society of America,131(4):3297–3297.97Appendix AFingering Chart of the XiaoA.1 Traditional fingeringsAs described in section 1.1.3, the traditional fingerings are not well stan-dardized because the fingering depends on the instruments, especially forthe cross fingering note C5 and the high notes. Below is a selection of fin-gerings applicable to the bamboo xiaos we have. The fingerings were takenfrom the Chinese online xiao forum (Xiaoyaji, 2010).Note D4 to C5] D5 to C6] D6 and aboveD xxxx xxxx xxxx xxxx oxxxvooxoAlternative oxxx xxxx oxxx xxxxE xxxx xxxo xxxx xxxo xxxo ooxxF xxxx xxoo xxxx xxooF] xxxx xoxo xxxx xoxo xxox xxxxG xxxx ooxo xxxx ooxo xoxx xoxxA xxxo ooxo xxxo ooxoA] xxoo ooxo xxoo ooxoB xoxo ooxo xoxo ooxoC oxxx ooxo xoxx xxooAlternative oxxx oxxx oxxx xxxoC] ooxo ooxo ooxo ooxoTable A.1: Traditional fingering chart of the xiao; x means closed and omeans open. Sequence of the fingering notation corresponds to finger holesfrom left to right as shown in the figure above (f8 to f1).98A.2. Optimized fingeringsA.2 Optimized fingeringsNote D4 to C5] D5 to C6] D6 to C7 D7D xxxx xxxx xxxx xxxx oxxxvoooo oxxo xoooE xxxx xxxo xxxx xxxo xxxo ooooF xxxx xxoo xxxx xxoo xxoo ooooF] xxxx xooo xxxx xooo xoxo ooooG xxxx oooo xxxx oooo xoxx ooooA xxxo oooo xxxo oooo oxxo ooooA] xxoo oooo xxoo ooooB xoxo oooo xoxo oooo xxox ooooC oxxx xxxo xoxx xxoo xoxx oxxoC] ooxo oooo xxxx xoooTable A.2: Optimized fingering as described in section 4.5. Rules of thefingering notation are the same as the traditional fingerings in AppendixA.1.99Appendix BMeasurement and ModellingResults of the XiaosMeasurement and modelling results of three xiaos (pipe-xiao-3, pipe-xiao-7 and the acrylic xiao) are shown. Zemb is the embouchure’s radiationimpedance and Z ′ is the xiao’s playing impedance (caused by the player’sjaw protrudes into the xiao for about 1.6 mm as shown in the figures),as described in section 3.2.2. The pressure and flow are calculated by theTMM model, as described in section 3.2.3. The xiaos are designed to playin tune at 23 ◦C. The measurements were taken at different days, so the en-vironmental temperatures are different. The playing spectra were measuredright after the impedance measurement to maintain the same environmentaltemperature.B.1 Pipe-xiao-3The measurements on pipe-xiao-3 were taken at an environmental temper-ature of 18.43 ◦C.100B.1. Pipe-xiao-3xxxx_xxxx: D40 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 294 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)293 586 879 1172 1467Playing spectrum 10−1100101102Normalized impedance and admitance296 596 895 1196 1492297 595 891 11931488294 586 880 1176 1472 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_xxxo: E40 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 331 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)330 661 992 13221553Playing spectrum 10−1100101102Normalized impedance and admitance335 671 1004 13181553335 668 1001 13141558331 658 985 1298 1542Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)101B.1. Pipe-xiao-3xxxx_xxoo: F40 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 350 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)350 700 1049 13971560Playing spectrum 10−1100101102Normalized impedance and admitance355 712 107014111573354 710 1067 14041579350 697 1048 13901569Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_xoxo: F4]0 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 372 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)373 745 11191398Playing spectrum 10−1100101102Normalized impedance and admitance378 757 112813981566379 752 112313921568372 740 110513901548Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)102B.1. Pipe-xiao-3xxxx_ooxo: G40 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 393 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)393 786 118114101572Playing spectrum 10−1100101102Normalized impedance and admitance400 802 120214281619400 799 119814231618393 783 117714271594 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxo_ooxo: A40 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 442 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)444 888 13321412Playing spectrum 10−1100101102Normalized impedance and admitance452 905 13401423 1725151455 902 13301420442 881 13141420 1710Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)103B.1. Pipe-xiao-3xxoo_ooxo: A4]0 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 468 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)469 939 1409 1878Playing spectrum 10−1100101102Normalized impedance and admitance479 962 145017691867151481 960 1447468 935 142117681858Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xoxo_ooxo: B40 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 500 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)5021006 1506Playing spectrum 10−1100101102Normalized impedance and admitance513 10261510 17361836151515 10241497500 996 1483 17291833Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)104B.1. Pipe-xiao-3oxxx_ooxo: C50 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 524 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)523898 1170 1423 1692Playing spectrum 10−1100101102Normalized impedance and admitance539 916 120814301694151538 902 120714291692524 905 118014291666 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)ooxo_ooxo: C5]0 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 548 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)551 1101Playing spectrum 10−1100101102Normalized impedance and admitance565 1127158917521834151569 11241573548 1092157317381830Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)105B.1. Pipe-xiao-3xxxx_xxxx: D50 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 586 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)29058787511731469Playing spectrum 10−1100101102Normalized impedance and admitance296 596 895 1196 1492297 595 891 11931488294 586 880 1176 1472 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_xxxo: E50 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 658 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)29565998513181546Playing spectrum 10−1100101102Normalized impedance and admitance335 671 1004 13181553335 668 1001 13141558331 658 985 1298 1542Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)106B.1. Pipe-xiao-3xxxx_xoxo: F5]0 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 740 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 3707401102 13711556Playing spectrum 10−1100101102Normalized impedance and admitance378 757 112813981566379 752 112313921568372 740 110513901548Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_ooxo: G50 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 783 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 391781117214021563Playing spectrum 10−1100101102Normalized impedance and admitance400 802 120214281619400 799 119814231618393 783 117714271594 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)107B.1. Pipe-xiao-3xxxo_ooxo: A50 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 881 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 439872130614061757Playing spectrum 10−1100101102Normalized impedance and admitance452 905 13401423 1725151455 902 13301420442 881 13141420 1710Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxoo_ooxo: A5]0 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 935 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB) 47092913981867Playing spectrum 10−1100101102Normalized impedance and admitance479 962 145017691867151481 960 1447468 935 142117681858Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)108B.1. Pipe-xiao-3xoxo_ooxo: B50 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 996 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB) 496992 148716971813Playing spectrum 10−1100101102Normalized impedance and admitance513 10261510 17361836151515 10241497500 996 1483 17291833Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xoxx_xxoo: C60 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1051 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 492 83910471462Playing spectrum 10−1100101102Normalized impedance and admitance502865 107615011584151501853 107214931589490858 1051 14861570Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)109B.1. Pipe-xiao-3ooxo_ooxo: C6]0 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1093 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 54510881714Playing spectrum 10−1100101102Normalized impedance and admitance565 1127158917521834151569 11241573548 1092157317381830Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)oxxx_ooxo: D60 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1180 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 531 87311711411 1658Playing spectrum 10−1100101102Normalized impedance and admitance539 916 120814301694151538 902 120714291692524 905 118014291666 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)110B.1. Pipe-xiao-3oxxx_xxxx: D60 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1179 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)475589 97811771526Playing spectrum 10−1100101102Normalized impedance and admitance469598 997 12021551151 464596 990 12011551464587 984 1179 1538 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxo_ooxx: E60 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1323 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 88213231696Playing spectrum 10−1100101102Normalized impedance and admitance452 90410531352 1725151454 900 1346442 88010531323 1709Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)111B.1. Pipe-xiao-3xxox_xxxx: F6]0 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1473 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)454 937 12791479Playing spectrum 10−1100101102Normalized impedance and admitance46367495513111496151461667950 1303 149345367393012991473 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xoxx_xoxx: G60 100 200 300 400 500 600 700Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1556 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)506 1004 12131560Playing spectrum 10−1100101102Normalized impedance and admitance5058901031 1237 1588151504878 1028 1233 15794928841012 1228 1556 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)112B.2. Pipe-xiao-7B.2 Pipe-xiao-7The measurements on pipe-xiao-7 were taken at an environmental temper-ature of 21.82 ◦C. As shown in the figures, the finger hole f8 has increasedthickness. xxxx_xxxx: D40 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 292 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)292 585 876 1167 1459Playing spectrum 10−1100101102Normalized impedance and admitance295 595 892 1197 1500292 593 889 11921494292 584 877 1177 1479 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)113B.2. Pipe-xiao-7xxxx_xxxo: E40 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 330 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)331 659 994 1325Playing spectrum 10−1100101102Normalized impedance and admitance334 671 1007 13331623333 670 1007 13301624330 658 988 1312 1607 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_xxoo: F40 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 349 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)348 699 1047 1396Playing spectrum 10−1100101102Normalized impedance and admitance354 710 1069 14211705352 708 1067 1415349 695 048 13971698Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)114B.2. Pipe-xiao-7xxxx_xooo: F4]0 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 368 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)369 739 1109 14791712Playing spectrum 10−1100101102Normalized impedance and admitance374 750 1131 15061731374 749 1132 1499368 733 1108 14811731Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_oooo: G40 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 390 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)389 778 1167 1560 1950Playing spectrum 10−1100101102Normalized impedance and admitance397 795 1200 15992005151396 795 1196 1602390 776 1174 1571 1979 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)115B.2. Pipe-xiao-7xxxo_oooo: A40 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 441 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)438 874 1311 17492084Playing spectrum 10−1100101102Normalized impedance and admitance450 900 13501782 2084151449 899 1343440 876 1320 1760 2072Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xoxo_oooo: B40 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 494 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)494 988 14821897 2094Playing spectrum 10−1100101102Normalized impedance and admitance507 1016 15171899 2102153508 1015 1508494 986 148318922079Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)116B.2. Pipe-xiao-7ooxo_oooo: C5]0 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 552 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)548 1097 1648 18842194Playing spectrum 10−1100101102Normalized impedance and admitance570 11401669 1901 2199153570 11431682552 1105 1638 1895 2180Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_xxxx: D50 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 585 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 29858587311681488Playing spectrum 10−1100101102Normalized impedance and admitance295 595 892 1197 1500292 593 889 11921494292 584 877 1177 1479 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)117B.2. Pipe-xiao-7xxxx_xxxo: E50 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 658 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 32965998313171604Playing spectrum 10−1100101102Normalized impedance and admitance334 671 1007 13331623333 670 1007 13301624330 658 988 1312 1607 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_xxoo: F50 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 695 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 348694104213891735Playing spectrum 10−1100101102Normalized impedance and admitance354 710 1069 14211705352 708 1067 1415349 695 1048 13971698Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)118B.2. Pipe-xiao-7xxxx_xooo: F5]0 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 733 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 374735110614691714Playing spectrum 10−1100101102Normalized impedance and admitance374 750 1131 15061731374 749 1132 1499368 733 1108 14811731Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_oooo: G50 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 776 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 389778116715591945Playing spectrum 10−1100101102Normalized impedance and admitance397 795 1200 15992005151396 795 1196 1602390 776 1174 1571 1979 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)119B.2. Pipe-xiao-7xxxo_oooo: A50 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 876 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)43587213151745Playing spectrum 10−1100101102Normalized impedance and admitance450 900 13501782 2084151449 899 1343440 876 1320 1760 2072Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xoxo_oooo: B50 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 986 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB) 49298414772076Playing spectrum 10−1100101102Normalized impedance and admitance507 1016 15171899 2102153508 1015 1508494 986 148318922079Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)120B.2. Pipe-xiao-7xxxx_xooo: C6]0 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1108 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)370 73511121480 1741Playing spectrum 10−1100101102Normalized impedance and admitance374 750 1131 15061731374 749 1132 1499368 733 1108 14811731Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxo_oooo: E60 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1320 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB) 454 869132817722077Playing spectrum 10−1100101102Normalized impedance and admitance450 900 13501782 2084151449 899 1343440 876 1320 1760 2072Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)121B.2. Pipe-xiao-7xoxx_oooo: G60 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1572 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)482988 136815822058Playing spectrum 10−1100101102Normalized impedance and admitance504 9981349 1600 2079151502 9981344 1593491 9701338 1571 2046Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)oxxo_oooo: A60 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1771 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)5141078 139417812128Playing spectrum 10−1100101102Normalized impedance and admitance561 10921415 18012132153560 10951411544 1063 1400 1770 2127Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)122B.3. Acrylic xiaoB.3 Acrylic xiaoThe measurements on the acrylic xiao were made at an environmental tem-perature of 16.53 ◦C. As shown in the figures, the finger holes have varyingthicknesses. xxxx_xxxx: D40 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 289 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)292 584 877 1169 1461 1751 2044 2336Playing spectrum 10−1100101102Normalized impedance and admitance292 591 889 1174 14561755 2027 2348292 591 891 1176 1464 1764 2028 2351289 580 873 1156 1435 1738 2008 2319 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)123B.3. Acrylic xiaoxxxx_xxxo: E40 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 326 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)329 657 986 13131514 1747 2049Playing spectrum 10−1100101102Normalized impedance and admitance330 663 9951314 1518 1755 2045 2368330 664 998 13131527 1764 2044 2370326 650 975 1294 1508 1738 2024 2339Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_xxoo: F40 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 345 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)347 694 1039 1386173818022425Playing spectrum 10−1100101102Normalized impedance and admitance350 704 105414031534 1808 2146 2426350 703 1059 14041543 1815 2142 24 5345 688 1032 13801531 1786 2125 2403Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)124B.3. Acrylic xiaoxxxx_xooo: F4]0 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 366 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)368 736 1103 1469 1839 2205 2573Playing spectrum 10−1100101102Normalized impedance and admitance372 748 11181479 1890 2275 2533 2786151373 747 1122 1479 18 62268 2523366 730 1095 1456 186 2247 2524 2781Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_oooo: G40 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 386 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)387 773 1157 1546 1933 23192560 2785Playing spectrum 10−1100101102Normalized impedance and admitance393 792 1185 15701963 2334 2539 2803151393 793 1186 1592 19662333 2535386 772 1159 1547 1938 2308 2533 2797Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)125B.3. Acrylic xiaoxxxo_oooo: A40 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 434 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)436 871 1307 17442012 234326052804Playing spectrum 10−1100101102Normalized impedance and admitance444 892 13341730 1998 2339 2591 2801151447 897 13371751 2003 23342592434 867 1304 1711 1984 2309 2586 2795Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxoo_oooo: A4]0 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 460 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)461 923 1385 18422008 2393 2859 3458Playing spectrum 10−1100101102Normalized impedance and admitance472 947 141618161999 2388 28583484151474 952 142018202003 2383460 919 138318061982 2356 2852 3483Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)126B.3. Acrylic xiaoxoxo_oooo: B40 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 489 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 49298514801791 20722467Playing spectrum 10−1100101102Normalized impedance and admitance502 100714951789 2069 24442585 2907151503 1014 151118022071 24412589489 976 14621786 2039 24242575 2904Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)oxxx_xxxo: C50 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 517 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)523104613231569178320862378Playing spectrum 10−1100101102Normalized impedance and admitance530 682 11091334 1574 1777 2088 2385152538682 11191330 1584 1794 2084 2390517 677 1079 1326 15661751 2076 2351Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)127B.3. Acrylic xiaoooxo_oooo: C5]0 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 548 HzPressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)551 1103 16541812 2450 2953Playing spectrum 10−1100101102Normalized impedance and admitance566 113116321791 2153 2439 2954 3339152566 113616551804 2159 2434548 109416091782 2133 2420 2951 3337Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_xxxx: D50 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 580 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)291581 871 1161 14511742 2032 2322Playing spectrum 10−1100101102Normalized impedance and admitance292 591 889 1174 14561755 2027 2348292 591 891 1176 1464 1764 2028 2351289 580 873 1156 1 35 1738 2008 2319 Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)128B.3. Acrylic xiaoxxxx_xxxo: E50 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 650 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 327651975 1304 1501 1746 2038Playing spectrum 10−1100101102Normalized impedance and admitance330 663 9951314 1518 1755 2045 2368330 664 998 13131527 1764 2044 2370326 650 975 1294 1508 1738 2024 2339Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_xxoo: F50 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 688 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 344688 1032 13761525 1791 21372411Playing spectrum 10−1100101102Normalized impedance and admitance350 704 105414031534 1808 2146 2426350 703 1059 14041543 1815 2142 24 5345 688 1032 13801531 1786 2125 2403Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)129B.3. Acrylic xiaoxxxx_xooo: F5]0 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 730 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 3667331099 14662564Playing spectrum 10−1100101102Normalized impedance and admitance372 748 11181479 1890 2275 2533 2786151373 747 1122 1479 18 62268 2523366 730 1095 1456 1865 2247 2524 2781Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxxx_oooo: G50 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 772 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB) 3857701155 1540 1930 23102564 2795Playing spectrum 10−1100101102Normalized impedance and admitance393 792 1185 15701963 2334 2539 2803151393 793 1186 1592 19662333 2535386 772 1159 1547 1938 2308 2533 2797Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)130B.3. Acrylic xiaoxxxo_oooo: A50 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 867 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB) 433863130217261986 231525912786Playing spectrum 10−1100101102Normalized impedance and admitance444 892 13341730 1998 2339 2591 2801151447 897 13371751 2003 23342592434 867 1304 1711 1984 2309 2586 2795Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxoo_oooo: A5]0 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 919 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 459919137818381977 2859 3465Playing spectrum 10−1100101102Normalized impedance and admitance472 947 141618161999 2388 28583484151474 952 142018202003 2383460 919 138318061982 2356 2852 3483Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)131B.3. Acrylic xiaoxoxo_oooo: B50 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 976 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 48997614672029 244325702928Playing spectrum 10−1100101102Normalized impedance and admitance502 100714951789 2069 24442585 2907151503 1014 151118022071 24412589489 976 14621786 2039 24242575 2904Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xoxx_xxoo: C60 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1033 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 490 88810311522 1929 2487Playing spectrum 10−1100101102Normalized impedance and admitance495890 1055 15401930 2149 2506 2834152492892 1053 15651925 2145 2507482881 1033 14721523 1917 2125 2483Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)132B.3. Acrylic xiaoxxxx_xooo: C6]0 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1095 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 365 7311097146224842559Playing spectrum 10−1100101102Normalized impedance and admitance372 748 11181479 1890 2275 2533 2786151373 747 1122 1479 18 62268 2523366 730 1095 1456 1865 2247 2524 2781Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)oxxx_oooo: D60 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1163 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)538 92011611667 197323242871 3208Playing spectrum 10−1100101102Normalized impedance and admitance549934 1193 16771987 2365 28863241152555938 1198 1705 1988 23632523533925 1163 1649 1971 2331 2880 3239Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)133B.3. Acrylic xiaoxxxo_oooo: E60 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1304 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB) 450 86313181970 23132626Playing spectrum 10−1100101102Normalized impedance and admitance444 892 13341730 1998 2339 2591 2801151447 897 13371751 2003 23342592434 867 1304 1711 1984 2309 2586 2795Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxoo_oooo: F60 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1383 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)447 96214092000 2376 3503Playing spectrum 10−1100101102Normalized impedance and admitance472 947 141618161999 2388 28583484151474 952 142018202003 2383460 919 138318061982 2356 2852 3483Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)134B.3. Acrylic xiaoxoxo_oooo: F6]0 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1462 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)48010201487179320482426Playing spectrum 10−1100101102Normalized impedance and admitance502 100714951789 2069 24442585 2907151503 1014 151118022071 24412589489 976 14621786 2039 24242575 2904Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xoxx_oooo: G60 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1548 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB) 989 130515682056 24252558Playing spectrum 10−1100101102Normalized impedance and admitance498 9851298 1575 206424202554 2896151497 98912991606 206924032552484 9571288 1548 2032 24112542 2893Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)135B.3. Acrylic xiaooxxo_oooo: A60 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1736 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)01020304050607080SPL (dB) 585 14021774 237229003337Playing spectrum 10−1100101102Normalized impedance and admitance557 10781385 1763 2025 2369 28883337152560 10891387 1786 2028 2372540 1049 1370 1736 2018 2333 2882 3336Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)xxox_oooo: B60 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 1949 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080100SPL (dB)445937 1347 160819762373 2861Playing spectrum 10−1100101102Normalized impedance and admitance463 922132315151595 1978 2383 2858151466 927 1329 1610 198023782517452 896 1301151615801949 2354 2852Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)136B.3. Acrylic xiaoxoxx_oxxo: C70 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 2063 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB) 524 1166 14261582184720952497Playing spectrum 10−1100101102Normalized impedance and admitance497 9741152 1417156518382094 2490152501 9811154 1420158618302099 2493484 9491146 14081543 1835 2063 2464Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)oxxo_xooo: D70 100 200 300 400 500 600Distance to the embouchure end (mm)0.00.20.40.60.81.0Normalized pressure and flow 2318 Hz PressureFlow500 1000 1500 2000 2500 3000 3500 4000Frequency (Hz)020406080SPL (dB)561 133215201677191023572882Playing spectrum 10−1100101102Normalized impedance and admitance557 1070132315051650 1893 2356 2872151558 107913241516671 1895 2360539 1043 1310 150516351870 2318 2867Z/Z0(e), TMMZ/Z0(e), MeasurementY=Z0(e)/(Z′ +Zemb)137

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