The Acoustics of the Clarinet: An Observation of Harmonics, Frequencies, Phases, Complex Specific

Acoustic Impedance, and Resonance

Kristin Holz Engineering and Physics Department, Abilene Christian University

ABSTRACT

Acoustical measurements were conducted on a wooden Buffet R-13 Bb clarinet to determine its characteristic properties and the effects of its structure. Sound recordings of concert F3, F4, F5, and two separate fingerings of F6 on the tempered scale were made and analyzed to observe the amplitudes, frequencies, and phases of the harmonics. Com-plex specific acoustic impedance analyses were conducted over a range of frequencies for the same notes as well as along the bore of the instrument for four harmonics of concert F4. Data from these experiments as well as longitudinal mechanical resonance measure-ments identified two formants of the clarinet. Each of these experiments aided in produc-ing a larger picture describing the acoustics of the clarinet.

I. Background and Introduction A. The Clarinet The clarinet is a woodwind instrument that uses a reed to induce vibrations in the air column and make sound. The reed is made out of an elastic material called Arundo Do-nax, a variety of cane. This reed is attached over the window of the mouthpiece where the clarinetist blows air into the system. As the clarinet is played, a pressure antinode forms in the mouthpiece, which makes the clarinet similar to a pipe closed at one end and open at the other. Different notes containing several har-monics are produced by opening and closing various tone holes along the bore of the in-strument. The register key, located one third of the way down the clarinet from the mouth-piece end, when opened allows the clarinet to play notes a twelfth (i.e. an octave plus a mus-ical fifth) higher than using the same fingering without the register key. Other effective reg-ister keys are created by opening other tone holes near the mouthpiece to raise the pitch.1

B. Harmonics Only certain modes of pressure and longi-tudinal displacement waves occur in various air column configurations. Since the clarinet is similar to a cylindrical pipe with one end open and one end closed, an open-closed pipe, the focus for this section will be on this air column configuration. Figure 1 depicts the first three modes of displacement and pressure waves within an open-closed pipe. The closed end allows for a displacement node and a pressure antinode while the open end allows the opposite. The distance between a node and antinode is a quarter wavelength, so that

  1 4Lλ = (1.1) where is the wavelength for the first wave mode and is the length of the pipe. The fundamental frequency then be-comes,

4

(1.2)

where c  is the speed of sound. Using wavelengths and pipe lengths to calculate subsequent mode frequencies in

terms of the fundamental frequency, a pattern develops. The wavelengths become

4

(1.3)

and the corresponding frequencies are

(1.4)

where is an odd integer. Therefore, only frequencies that are odd integer multiples of the fundamental are produced in an open-closed pipe. The first mode is referred to as the fundamental or first harmonic, which is followed by the third harmonic, fifth harmon-ic, and so on.2

Figure 1: The first three modes of the displace-ment and pressure waves for an open-closed pipe are pictured including the frequency of each in terms of the fundamental frequency.3

C. Inharmonicity Musical instruments such as the clarinet do not exhibit ideal harmonics because they have a more complex geometry than a cylin-drical pipe for example. Inharmonicity is a measure of the deviation of each harmonic from its theoretical frequency. The percent deviation ∆ is calculated as follows:

∆ 100 (1.5)

where is the experimental ratio between two consecutive harmonic frequencies and

is the theoretical ratio between the same consecutive harmonic frequencies.

There are 100 cents between semitones on the tempered musical scale and the inharmo-nicity can be converted from percent devia-tion to deviation in cents. If the value of the percent deviation is negative, the conversion uses the equation

1781.715 1 (1.6)

where is the number of cents high from the theoretical pitch. If the value of the per-cent deviation is positive, the conversion uses a similar equation

1681.718 1 (1.7)

where is the number of cents low from the theoretical pitch.4 D. Acoustic Impedance A unique characteristic to every instru-ment is its specific acoustic impedance. Complex specific acoustic impedance is the ratio of complex pressure to complex particle velocity and is a quality independent of the player of the instrument. Equation 1.8 shows the longitudinal complex specific acoustic impedance as a function of complex pressure and complex particle velocity

 at point where 0 and is a positive distance.

(1.8)

This impedance varies for different frequen-cies. The clarinet specifically operates at maxima in the impedance, only allowing cer-tain frequencies to be played for each finger-ing.5 II. Method Four types of experimentation were done on a wooden Buffet R-13 Bb clarinet. These included a harmonic analysis, a frequency spectrum impedance analysis, an analysis of the acoustic impedance along the bore of the clarinet for concert F4 on the tempered scale,

and a spectral analysis of longitudinal me-chanical resonance. A. Harmonic Analysis Initially, a harmonic analysis was con-ducted for four octaves of concert F on the tempered scale. These included F3, F4, F5, and two separate fingerings of F6 as shown in Fig-ure 2.

Figure 2: This chart shows the fingerings used for each note used in experimentation where black keys and holes represent finger placement. The key and hole at the top left of each fingering dia-gram are the thumb hole and register key at the back of the instrument. The notes on the staff are the notes in treble clef that the clarinetist reads for each fingering shown, and the concert pitch is written below each note.

Each note was tuned and played into a Behringer ECM8000 reference microphone for a duration of about 5 seconds, and five digital recordings were taken. These 24-bit recordings were clipped at the beginning and end to attain a steady sample of each recorded note without the transient effects of articula-tion or amplitude variations. The harmonics were filtered from each sound clip using a MATLAB-based analysis program created by Joseph Yasi.6 The data from this analysis in-cluded information about frequency, ampli-tude, and phase for each of the harmonics. B. Frequency Spectrum Impedance

Analysis The next experiment performed measured the input and output acoustic impedance over a range of frequencies for each concert F. The setup involved simulating playing the clarinet by using a modified mouthpiece and covering the correct tone holes for each fin-gering. Microphones measuring complex

pressure and particle velocity were placed at each end of the instrument to calculate acous-tic impedance as a function of frequency. The mouthpiece used, seen in Figure 3, was a Selmer Bundy student Bb clarinet mouthpiece. A piezoelectric disc transducer was glued to the top of the window on the mouthpiece and a copper plate was glued be-neath it to cover the rest of the opening in or-der to simulate a vibrating reed. The voltage output from a sine wave function generator was applied across the piezoelectric transduc-er to allow the disc to vibrate at specific fre-quencies. Additionally, holes were drilled into the reed table and back of the mouthpiece where complex pressure and particle velocity microphones were inserted. These were se-cured with Apiezon Sealing Compound, which also sealed the remainder of the holes.

Figure 3: A Selmer Bundy student Bb clarinet mouthpiece was modified to simulate playing the clarinet. A piezoelectric disc transducer was at-tached to the window of the mouthpiece in addi-tion to a copper plate to simulate a vibrating reed while microphones measuring complex pressure and particle velocity were inserted into the cavity of the mouthpiece.

The tone holes were covered by plugging them with rubber stoppers secured with pieces of PVC-coated wire as depicted in Figure 4 for concert F3. Any keys that needed to be held down were closed by propping various levers with rubber stoppers or wire. Once the proper holes were covered, the clarinet was played to make sure there were no air leaks in the system.

Figure 4: Rubber stoppers and PVC-coated wire was used to close the tone holes necessary for spe-cific notes tested. This picture shows the wooden Buffet R-13 Bb clarinet configured for concert F3 before doing a frequency spectrum impedance analysis.

The Buffet R-13 Bb clarinet was placed in a wooden box lined with foam in the interior to avoid picking up noise from the room. Two microphones, one measuring complex pressure and one measuring complex particle velocity, were placed at the bell end of the instrument centered at the edge of the bell. SRS 830 dual-channel lock-in amplifiers were used to measure the pressure and particle velocity at the input and output of the clarinet for specific frequencies of piezoelectric trans-ducer vibration. Measurements were taken from 29.5 Hz to as high as 3030.5 Hz in 1 Hz steps for each of the five fingerings. C. Acoustic Impedance Along Bore An input and output acoustic impedance analysis was also done along the bore of the clarinet for concert F4 on the tempered scale. The modified Selmer Bundy mouthpiece in-cluding complex pressure and particle veloci-ty microphones was attached and no tone holes needed to be covered for this note. A pine wood frame was constructed for the cla-rinet so that the joint below the barrel and the joint above the bell rested on wooden slots coated in foam. This frame was clamped along the y-axis of a near-field acoustic holo-graphy apparatus with large C-clamps. A long steel rod held a complex pressure and a particle velocity microphone to be inserted

into the bell end of the clarinet along the bore. This rod was clamped onto a ring stand so that it was positioned horizontally. The wooden frame was set to move in 1 mm steps along the y-axis while the steel rod with the micro-phones remained stationary. This apparatus is shown in Figure 5. Measurements of complex pressure and particle velocity were taken at the mouthpiece and the location of the steel rod microphones for positions of the end of the rod ranging from just between the mouth-piece and barrel to just outside the bell.

Figure 5: This diagram depicts the apparatus used for experimentation measuring the acoustic im-pedance along the bore of the clarinet.

D. Spectral Analysis of Longitudinal Mechanical Resonance

The final experiment done on the Buffet R-13 Bb clarinet was a measurement of longi-tudinal mechanical vibrations due to induced longitudinal vibrations on the bell over a range of frequencies. This experiment used a Pyne Signature Symphonic Bb clarinet mouthpiece, a Rovner Eddie Daniels II Bb clarinet leather ligature, and a size 4 Vando-ren V-12 reed. The apparatus, pictured in Figure 6, involved wrapping the clarinet’s thumb rest with electrical tape and clamping the tape with a C-clamp to protect the silver from scratches. This clamp was held by a three-pronged clamp attached to a ring stand so that the clarinet was vertical and upside-down. The screw of the ligature rested gently on some foam to keep the clarinet steady, but limit contact to avoid perturbation. Two piezoelectric transducer discs were attached on the bell of the instrument 180° from each other. A 20 gram brass weight and some electrical tape were placed on top of each transducer to keep them steady. Fre-

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Results and

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Discussion

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B. Frequency Spectrum Acoustic Imped-ance Analysis

A frequency spectrum analysis was done across a range of frequencies using the appa-ratus described in section II. B. Complex pressure and particle velocity microphones took measurements at the mouthpiece and bell of the clarinet and the data was used to calcu-late the input and output impedance using Eq-uation 1.8 for concert F3, F4, F5, and two fin-gerings of F6 on the tempered scale. The complex input impedance graphs are docu-mented in Figures 23-27 as they represent the best quality and most information of the data gathered. The impedance maxima represent places of maximum pressure and minimum particle velocity while the impedance minima represent maximum particle velocity and min-imum pressure at the bell end of the instru-ment. The impedance maxima are located at frequencies that produce playable sounds.

1. F3 Analysis Figure 23 shows the complex input im-pedance measured in acoustic Ohms from 29.5 Hz to 2613.5 Hz induced on the piezoe-lectric transducer. The plot shows a rather ideal pattern of minima and maxima except around 1500 Hz because of mechanical reson-ances in the clarinet. The first maximum is the fundamental at 185.50 Hz, which is 11.08 Hz higher than concert F3 on the tem-pered scale at 174.42 Hz. This could be due to 80°F high and 41% humidity in Urbana which could cause the wood to expand and

therefore reduce the diameter of the bore cavi-ty. However, the acoustic physics lab was kept at a much cooler temperature. The high-er frequency could also be due to the nature of the apparatus using the modified mouthpiece and rubber stoppers to close necessary tone holes.

Figure 23: This data was obtained from a fre-quency spectrum acoustic impedance analysis for concert F3 on the clarinet. The magnitude input complex acoustic impedance measured in acoustic Ohms and logarithmically scaled is on the y-axis while the frequency of the piezoelectric transducer attached to the modified clarinet mouthpiece is located on the x-axis.

Table 1 lists the frequencies at several of the complex input impedance extrema, the theoretical harmonic number for each, and the type of extrema for the concert F3 acoustic impedance frequency spectrum data. All the odd harmonics with the exception of the 9th are at impedance maxima, meaning that the playable sounds for this fingering only occur at odd harmonics in accordance with the geo-metry of the clarinet similar to an open-closed cylindrical pipe. The fundamental is the only frequency intentionally played with this fin-gering, but when the reed vibrates at other input impedance maxima frequencies a squeak is produced at that frequency.

Concert F3 Impedance Extrema Measured Frequency

(Hz)

Theoretical Harmonic Number

Type of Local

Extrema 185.50 1 Maximum 361.50 2 Minimum 549.5 3 Maximum 715.5 4 Minimum 912.5 5 Maximum 1070.5 6 Minimum 1286.5 7 Maximum 1428.5 8 Minimum 1654.5 9 Minimum 1871.5 10 Minimum 2019.5 11 Maximum 2192.5 12 Minimum 2452.5 13 Maximum Table 1: This table represents several data points from Figure 23 describing the complex input im-pedance extrema of the clarinet for concert F3.

2. F4 Analysis A frequency spectrum impedance analysis was done for concert F4 from 29.5 Hz to 3030.5 Hz. A plot of the complex input im-pedance data can be seen in Figure 24. Simi-lar to the data taken for concert F3, the imped-ance minima and maxima look uniform until about 1500 Hz due to a formant of the clari-net. The first maximum is at the frequency for the note tested at 382.5 Hz, 33.27 Hz above the theoretical frequency for concert F4 at 349.23 Hz. This could be due expansion in the clarinet’s wood from the 63% humidity in Urbana on the day of experimentation as well as imperfections in the apparatus.

Figure 24: This semi-log plot shows the magni-tude of the complex input acoustic impedance data for concert F4 on the clarinet across a range of fre-quencies.

The values of the frequencies at several of the complex input impedance extrema are listed in Table 2 along with the theoretical harmonic number for each. The 5th and 6th harmonics are at the opposite extrema than expected, but the frequencies of these har-monics fall within a formant of the clarinet.

Concert F4 Impedance Extrema Measured Frequency

(Hz)

Theoretical Harmonic Number

Type of Local

Extrema 382.5 1 Maximum 733.5 2 Minimum 1123.5 3 Maximum 1474.5 4 Minimum 1875.5 5 Minimum 2247.5 6 Maximum 2652.5 7 Maximum Table 2: The concert F4 complex input acoustic impedance extrema, measured frequency, and theoretical harmonic number are listed from the data gathered in Figure 24.

3. F5 Analysis A frequency spectrum acoustic impedance analysis was also run for concert F5 on the tempered scale for the wooden Buffet R-13 Bb clarinet from 29.5 Hz to 3030.5 Hz. The complex input impedance data as a function of frequency is displayed in Figure 25, and the second maximum is at the frequency of the note being tested. This maximum occurs at 745.5 Hz, a whole 47.04 Hz higher than the

frequency specified for concert F5 on the tem-pered scale at 698.46 Hz. The Urbana tem-perature of 78°F and humidity level of 56% played a large role in this discrepancy along with the apparatus’ errors in perfectly simulat-ing a person playing the clarinet. Apart from the frequency values, the maxima and minima look regular until about 1500 Hz where the clarinet experiences mechanical resonance.

Figure 25: An acoustic impedance frequency spectrum analysis was conducted for concert F5 on the clarinet where the magnitude of the complex input acoustic impedance was plotted as a function of frequency with a semi-log scale.

Table 3 contains measured frequency val-ues and theoretical harmonic numbers for several complex input acoustic impedance extrema. It is easy to see the regular pattern of maxima and minima before reaching the 9th harmonic. The frequency at each impedance maximum is a playable pitch and the third harmonic frequency is the intended playable note for this fingering.

Concert F5 Impedance Extrema Measured Frequency

(Hz)

Theoretical Harmonic Number

Type of Local

Extrema 306.5 1 Maximum 518.5 2 Minimum 745.5 3 Maximum 972.5 4 Minimum 1233.5 5 Maximum 1434.5 6 Minimum 1663.5 7 Maximum 1929.5 8 Minimum 2220.5 9 Minimum 2416.5 10 Minimum 2651.5 11 Minimum Table 3: This table lists the measured frequen-cies, theoretical harmonic numbers, and type of extrema for several data points in Figure 25 in-volving a frequency spectrum acoustic impedance analysis done for concert F5.

4. F6 Analysis: First Fingering The same impedance analysis was per-formed on the clarinet with the first fingering of concert F6 shown in Figure 2. This finger-ing is also intended for concert A5 on the tem-pered scale. The complex input acoustic im-pedance was calculated from the measured complex pressure and particle velocity and plotted across a range of frequencies from 29.5 Hz to 3030.5 Hz as seen in Figure 26. The second impedance maximum is for con-cert A5 with a theoretical frequency of 880.00 Hz, and the third impedance maximum is for concert F6 with a theoretical frequency of 1396.90 Hz. The measured frequency val-ues for these notes were 946.5 Hz and 1523.5 Hz respectively, each higher than the theoretical values by 66.5 Hz and 126.6 Hz. This can be explained by a high humidity of 82% in Urbana on the day of experimentation, non-ideal apparatus, and the register of the clarinet as these notes tend to be sharp on the instrument.

Figure 26: This is a semi-log plot of the magni-tude of the complex input acoustic impedance data across a range of vibration frequencies of the pie-zoelectric transducer on the modified clarinet mouthpiece for testing done with the first fingering of concert F6 which can also be used for playing concert A5.

Values for the measured frequencies at various impedance extrema are listed in Table 4 along with the theoretical harmonic number and type of local extrema for each one. The 8th harmonic shows strange behavior as it falls at a local maximum instead of a local mini-mum.

Concert F6: First Fingering Impedance Extrema

Measured Frequency

(Hz)

Theoretical Harmonic Number

Type of Local

Extrema 362.5 1 Maximum 647.5 2 Minimum 946.5 3 Maximum 1195.5 4 Minimum 1523.5 5 Maximum 1846.5 6 Minimum 2192.5 7 Maximum 2598.5 8 Maximum 2911.5 9 Maximum Table 4: The frequency spectrum acoustic imped-ance analysis done for the first fingering of concert F6 produced the complex input impedance data found in Figure 26. The measured frequencies, theoretical harmonic numbers, and type of local extrema are listed in the table for several data points in this plot.

5. F6 Analysis: Second Fingering The second fingering for concert F6 was used to conduct a final frequency spectrum acoustic impedance analysis. The complex input acoustic impedance data from this expe-riment are shown in Figure 27 over a frequen-cy range from 29.5 Hz to 3030.5 Hz. The 4th local impedance maximum is at the 7th har-monic, the playable note for this fingering. Instead of the ideal frequency of 1396.90 Hz, the measured frequency was 85.6 Hz higher at 1482.5 Hz. This was due in part to the 73°F temperature and 52% humidity in Urbana causing the wood of the clarinet to expand as well as the non-ideal apparatus used and the register containing concert F6.

Figure 27: This semi-log plot displays the magni-tude of the complex input acoustic impedance measured in acoustic Ohms as a function of fre-quency in Hertz for the second fingering for con-cert F6.

The measured frequencies, theoretical harmonic numbers, and type of local extrema for several extrema in Figure 27 are listed in Table 5. The maxima and minima follow a regular pattern besides the 8th harmonic which falls within a range of mechanical resonance for the clarinet.

Concert F6: Second Fingering Impedance Extrema

Measured Frequency

(Hz)

Theoretical Harmonic Number

Type of Local

Extrema 340.5 1 Maximum 550.5 2 Minimum 691.5 3 Maximum 835.5 4 Minimum 1056.5 5 Maximum 1251.5 6 Minimum 1482.5 7 Maximum 1636.5 8 Maximum 1859.5 9 Maximum 2061.5 10 Minimum 2194.5 11 Maximum 2387.5 12 Minimum 2564.5 13 Maximum 2709.5 14 Minimum Table 5: This table displays the measured fre-quencies, theoretical harmonic numbers, and type of local extrema for several complex input imped-ance extrema from the frequency spectrum acous-tic impedance analysis with the second fingering of concert F6 on the clarinet.

6. Acoustic Impedance Inharmonicity Analysis

An inharmonicity analysis was repeated except instead of using the filtered harmonic frequencies from the harmonic analysis, cer-tain frequencies from the complex input acoustic impedance extrema were used. The reference frequencies were set to the meas-ured frequency of the first playable note, and the subsequent theoretical frequencies were determined based on these reference values. The experimental harmonic numbers were then calculated with the reference frequency as an integer harmonic, and equations 1.5-1.7 were used to determine the inharmonicities. Figure 28 shows the deviation in cents from the ideal harmonics for each harmonic number of each note tested. The data for the first fingering of F6 is labeled as A5 on the graph, and the measured frequency for A5 was also used as the reference frequency for this note’s inharmonicity data. Clearly this plot varies from Figure 22 based on the harmonic analysis in that the harmonics deviate even beyond 500 cents. The deviations are less

dramatic close to the reference harmonic, and the worst inharmonicities occur at the lowest harmonics for the highest notes. This is rela-tively insignificant in the playing of the clari-net, however, because the notes at those har-monics are not intended to be played. This input impedance-based inharmonici-ty plot differs significantly from the harmonic analysis-based inharmonicity plot, which is disconcerting. One cause for this discrepancy is the inability to simulate a person’s lips in interaction with the mouthpiece and reed as well as the effects of blowing into the instru-ment. The data from the three highest notes that use the register key have the greatest in-harmonicities at the lowest harmonics which could be due to the open hole one third of the way down the clarinet.

Figure 28: The frequency values at complex input acoustic impedance extrema were used to deter-mine inharmonicities in the concert pitches of F3, F4, F5, A5, and F6.

C. Acoustic Impedance Along Bore for Concert F4

The complex specific acoustic impedance was calculated from measurements of com-plex pressure and particle velocity in the mouthpiece and along the bore of the clarinet for concert F4 using the apparatus pictured in Figure 5. Figures 29-32 show the complex specific acoustic impedance as a function of distance from the mouthpiece. Just past 10 cm, the vent hole that is opened by the reg-ister key has a short metal tube protruding into the bore to avoid clogging the hole with condensed moisture from the player’s breath.7

1. 382.5 Hz Analysis The piezoelectric transducer on the mod-ified mouthpiece was set to vibrate at 382.5 Hz. This frequency was chosen by ma-nually scanning frequencies near the funda-mental for concert F4 until a complex specific acoustic impedance maximum was reached in the mouthpiece. Figure 29 shows the acoustic impedance data as a function of distance from the mouthpiece. The spike near 10 cm is the impedance near the protrusion in the vent hole showing a pressure maximum and particle velocity minimum for this frequency. The data becomes rather noisy starting around 45 cm from the mouthpiece of the clarinet as the microphones become more exposed to room and ventilation noise.

Figure 29: This semi-log plot displays the magni-tude of the complex specific acoustic impedance in acoustic Ohms as a function of distance in centi-meters from the mouthpiece of the clarinet for the fundamental frequency of concert F4.

2. 738.5 Hz Analysis A second data set, seen in Figure 30, was obtained for the second harmonic frequency of concert F4 located at a maximum in input acoustic impedance. This data is similar to that for 382.5 Hz in that a maximum in the impedance occurs at the vent hole and the da-ta becomes noisy further along the bore, in this case starting around 30 cm. This would indicate that there is a pressure antinode and particle velocity node at the vent hole.

Figure 30: The magnitude of the complex specif-ic acoustic impedance measured along the bore of the clarinet in 1 mm steps is plotted for 738.5 Hz, the second harmonic of concert F4.

3. 1123.5 Hz Analysis The third harmonic frequency was found manually to be at 1123.5 Hz, again based on a maximum in the input impedance. The com-plex specific acoustic impedance was deter-mined along the bore and displayed in the semi-log plot in Figure 31. This time the vent hole seems to be the location of an inflection point in the acoustic impedance.

Figure 31: This semi-log plot shows the magni-tude of the complex specific acoustic impedance in as a function of distance from the mouthpiece of the clarinet for the third harmonic frequency of concert F4.

4. 1413.5 Hz Analysis The final impedance analysis done for F4 was at the fourth harmonic frequency found manually to be 1413.5 Hz. Figure 32 shows the plot for the magnitude of the complex specific acoustic impedance along the bore of the clarinet for this frequency. There are two

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opportunity to utilize his sound analysis equipment and computer programs. The REU program is supported by NSF Grant PHY-1062690. VI. Footnotes, Endnotes and References 1J. Backus, The Acoustical Foundations of

Music, 2nd ed. (W. W. Norton & Compa-ny, Inc., New York, 1977), pp. 225-228.

2J. Backus, The Acoustical Foundations of

Music, 2nd ed. (W. W. Norton & Compa-ny, Inc., New York, 1977), pp. 66-67.

3D. E. Hall, Musical Acoustics, 2nd ed.

(Brooks/Cole Publishing Company, Pacif-ic Grove, 1991).

4S. Errede, “Calculation of Inharmonicities in

Musical Instruments,” 5-7 (2000-2010) (unpublished). <http://courses.physics.

illinois.edu/phys406/Lab_Handouts/ Musical_Instrument_Inharmonicities.pdf>

5J. Wolfe, “What is Acoustic Impedance and Why is it Important?” (1997). <http:

//www.phys.unsw.edu.au/jw/z.html> 6J. Yasi, “An Algorithm for Extracting the

Relative Phase of Harmonics from a Peri-odic Digital Signal,” (2004) (unpub-lished). <http://courses.physics.illinois.edu/phys

406/NSF_REU_Reports/2004_reu/Joe_ Yasi_Final_Paper.pdf> 7J. Backus, The Acoustical Foundations of

Music, 2nd ed. (W. W. Norton & Compa-ny, Inc., New York, 1977), pp. 228.

8The Vintage Clarinet Doctor, “Clarinet Fin-

gering Systems,” <http://www.thevintageclarinetdoctor.

com/-fingering-systems.html>