Mouthpiece and Bell Effects on Trombone Resonance Michael C. LoPresto, Henry Ford Community College, Dearborn, MI
T
The effects of the mouthpiece and bell on the frequencies of the vibrating air column in a trombone can be demonstrated quite readily by first calculating the expected resonant frequencies of a piece of PVC pipe that is the same length as a trombone, then replacing portions of the PVC pipe of the same length with first a cup-shaped mouthpiece and then a flaring-bell section of a trombone. The trombone components used were from a new pBone plastic trombone1 that is not only much less expensive than a regular instrument, which is made mostly of brass, but also features components that are much lighter than those of a conventional brass trombone and is therefore easier to work with. The purported effect on the resonant frequencies of a cylindrical tube closed at one end of replacing a portion with an equal-length cup-shaped mouthpiece section is that the presence of the mouthpiece will add an effective length to the tube that increases with frequency and will therefore have more effect on the frequencies of the higher modes, bringing them down.2-6 The effect of replacing a section of the cylindrical tube with a flaring-bell section of equal length on the open end of the tube opposite the mouthpiece is that the lower frequency modes will not penetrate as far into the bell as the higher modes do, so the air-column length will be shortened more for the lower modes, raising their frequencies.2,7-10 The combined effects of the mouthpiece lowering the frequencies of the higher modes and the bell raising the frequencies of the lower ones is to compact the set of odd harmonics, n=1, 3, 5 …, sounded by a cylinder closed at one end into a more musically useful set that includes both the even and odd members, n=2, 3, 4…, of a harmonic series. See Fig. 1.2,11
Background and procedure
It is well known that a cylinder of length L closed on one end sounds a set of harmonic frequencies fn that are odd-integer multiples of a lowest, n=1, fundamental frequency, fn = (2n – 1)c/4L, (1) 594
Fig. 1. The effect of adding a mouthpiece to a cylinder is to lower the higher modes and the effect of adding a bell is to raise the lower modes, transforming a set of odd harmonics into a complete harmonic series.2,11
where c is the speed of sound. Brass musical instruments, such as the trombone and trumpet, become tubes that are closed on one end when the player blows air into the horn while vibrating her lips at the resonant frequencies of the air column. It is also, however, well known that brass instruments resonate at not only the odd harmonics, but also at the even harmonics, making musical pitches of the frequencies of an entire harmonic series, with the exception of the fundamental2 available (this will be explained below). This is what makes brass instruments musically useful. The presence of the even harmonics, similar to a tube open at both ends, often propagates a myth that brass instruments are in fact open at both ends, but they are not, as again they are clearly closed at the mouthpiece end by the player’s lips. First, the resonant frequencies of a length of cylindrical PVC pipe that is the same length, in this case 274 cm, as a pBone plastic trombone1 (Fig. 2) are calculated with Eq. (1), and then the playing frequencies are measured after sections of the cylindrical pipe are replaced by equal lengths of first a 13-cm trombone mouthpiece section (Fig. 3) and then a 59-cm flaring-bell section (Fig. 4). The PVC pipe into which the trombone components fit best is called ½-in PVC pipe, although its inner diameter is really about 0.6 in (about 1.5 cm). 12
Results
Figure 5 is a plot comparing the calculated resonant frequencies of the cylinder to the measured playing frequencies13 of the PVC pipe with a 13-cm section replaced by the mouthpiece section of a pBone and then with an additional 59-cm section replaced by a flaring-bell section. The measured playing frequencies of an actual 274-cm length pBone from which the components used came are also plotted. The length of the trombone and its playing frequenFig. 2. A pBone plastic trombone.1 cies were measured with the trombone’s slide section completely closed. This means that the trombone’s tubing is
THE PHYSICS TEACHER ◆ Vol. 52, J ANUARY 2014
cylinder (Hz) mouthpiece (Hz) bell & mouthpiece (Hz) trombone (Hz)
the comparison of only three sets of frequencies rather than five sets makes the demonstration more simple. The plots in Fig. 5 show the effects of adding a mouthpiece and bell on the frequencies of the cylindrical air column (red) quite well. The addition of the mouthpiece (blue) raised the frequencies of the lower modes and lowered the frequencies of the higher modes, showing that its effective length does indeed Fig. 4. The flaring-bell section of a pBone increase with frequency. Adding the bell plastic trombone attached to ½-in PVC pipe. (orange) raised the frequencies of all the modes but affected the lower modes more, raising them by a higher percentFig. 3. The mouthpiece of a pBone plastic age. Note the frequencies measured for trombone attached to a ½-in PVC pipe. an actual trombone (purple) are very Frequencies vs Harmonics similar to those measured for the cylinder with the mouthpiece and bell attached. The (n=1) fundamental frequency is not included in the 400 measurements because the lowest actual resonance produced by a trombone (and a trumpet as well) is not part of, or intune with, the rest of the harmonic series produced by the 300 instrument. The resonance has a frequency that is actually about 20%8 to 30%14 below a “fictitious fundamental,”2 a resonance that is half the frequency of the n=2 mode, but is 200 not actually present. The harmonic series produced by a brass instrument is based on the lowest musically useful resonance being the n=2 mode. The lowest resonance, also known as the pedal tone (see Fig. 1), is of a sound quality inferior to that of 100 the other modes and is rarely used as it is generally not musi7 5 1 3 Harmonic cally useful.2,15,16 Fig. 5. Plot comparing the calculated resonant frequencies of a Linear fits (not shown in Fig. 5 to allow the points to be 274-cm long cylinder of PVC pipe (red) and with the measured more readily seen) to the frequencies plotted in Fig. 5 for the frequencies,13 first when a trombone mouthpiece section is cylinder with the mouthpiece and bell sections attached and attached (blue), then a bell section (orange), and finally to those the actual trombone showed very high correlation (100% and of an actual trombone (purple). 99.9%), very low errors (1.33 and 1.67), and slopes of 58.3 Hz at its shortest possible length, known to players as “first posiand 58.0 Hz, respectively. The equations for the fits were of tion.” The trombone’s tuning slide section was also closed for the form f(n)=nf1, where the slope f1 represents the aboveall measurements. mentioned “fictitious fundamental”1 of the harmonic series. It was not possible to produce sustainable tones with the The measured n=2 frequencies for both the cylinder with the cylinder from which the frequencies could be measured withmouthpiece and bell attached and the actual trombone were out some sort of mouthpiece attached to the cylinder. Using f2 = 117 Hz, half of which (58.5 Hz) is very close to slope of a connecting joint for PVC pipe did allow measurable tones both curve fits. to be produced, but they had frequencies much different than The effects of the mouthpiece and bell may be shown even the expected frequencies of the cylinder calculated with more clearly when an effective cylindrical length, Eq. (1) Eq. (1). They were in fact very similar to frequencies measolved for L, is calculated for each measured playing frequensured when the pBone mouthpiece was attached to the cylcy. These values are plotted in Fig. 6. inder. This suggests that the connecting joint was having an It can be seen in Fig. 6 that the effective length of the cyleffect on the air column similar to that of the mouthpiece. inder with the mouthpiece (blue) is less than the actual cyThese are also the reasons that frequencies could not be lindrical length (red) for the lower modes and longer for the measured with only a flaring bell attached to the PVC pipe higher modes. This again shows that the effective length addwithout a mouthpiece present. Fortunately, the effect of the ed to the air column by the mouthpiece does indeed increase bell on the air column can still be observed by adding the bell with frequency. The apparent “spike” in the effective length of section when the mouthpiece section is already present, and the cylinder with a mouthpiece (blue) at n=7 is because that THE PHYSICS TEACHER ◆ Vol. 52, J ANUARY 2014
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Harmonic Relationships bell & mouthpiece trombone
15 2.8
mouthpiece
2.6
2.4
2.2
1
3
Harmonic
5
5
1
7
Fig. 6. Plot of the effective lengths calculated with the measured frequencies13 of a PVC pipe with a trombone mouthpiece section attached (blue), then a bell section added (orange), and an actual trombone (purple) compared to the 274-cm length of cylindrical PVC pipe (red).
harmonic is actually slightly out of tune with the rest of the series. This results in the musical note being of too low a frequency, or flat, and not musically useful. This also means that the effective length should be longer Fig. 7. The lower modes of vibration do not than expected. penetrate as far into the flaring-bell section When the bell is as the higher modes.17 This is verified by added, the effec- the plot (Fig. 6, blue) of air column effective tive lengths (or- lengths when the bell is added being shorter for the lower harmonics than for the higher ange) of the lower ones. modes are clearly shown to drop much more than those of the higher modes. This is expected since, as mentioned above, the lower modes do not penetrate as far into the bell section. This is shown in Fig. 7. 17 Also, again note that as with the frequencies, the similarity between the values for the cylinder with both the bell and mouthpiece attached (orange) to those of the actual pBone (purple). Finally, the effects on the harmonics themselves of adding the components can be seen in Fig. 8. The cylinder sounds only odd harmonics (red), and the addition of the mouthpiece (blue) brings down all the harmonics but, as with the frequencies and effective lengths, has more effect on the higher harmonics. The addition of the bell (orange) finishes the transition, bringing all the harmonics very close to those of a complete set similar to those of an actual pBone (purple). 596
10
cylinder
cylinder (m) mouthpiece (m) bell & mouthpiece (m) trombone (m)
Effective Lengths vs Harmonic
3
5
7
Harmonic
Fig. 8. Plot of the harmonic relationships between the measured frequencies plotted in Fig. 5.
The sounding harmonics were calculated by dividing each measured frequency by half of the n=2 member of its series, the “fictitious fundamental”1 of the series discussed above.
Effects on tone quality The presence of a mouthpiece and bell also have an effect on the sound or tone quality also known, in musical terms, as the timbre of the instrument. When the PVC pipe is played with the mouthpiece only or even with the bell added, it does not sound like or have the tone quality of an actual trombone. The quality or timbre of a sound is caused by the waveform that is a result of the harmonic or Fourier spectrum of the sound wave.18 Figure 9 shows the resulting waveforms when the PVC pipe was played with only the mouthpiece, then with the mouthpiece and the bell attached compared to the waveform of an actual pBone. Figure 10 shows Fourier spectra of each waveform. Note that the waveforms and spectra are simpler with only the mouthpiece or even the mouthpiece and bell present than for the actual instrument. Sounds with more complex waveforms and spectra have a fuller or richer sound quality than those with simpler waveforms and spectra that tend to sound more hollow or dull. This was the case when the quality of the actual pBone’s sound was compared to that of the PVC pipe played with its components.
Conclusion
The ready availability and low price of PVC pipe as well as the relatively low cost and ease of working with the lighter components of a pBone could make this exercise useful, not only as a demonstration that clearly shows the effects of the mouthpiece and bell on the playing frequencies, but possibly as a laboratory in a Science of Sound or Musical Acoustics course.19 Each laboratory group would need only two lengths of ½-in PVC pipe, one 13 cm shorter (261 cm) than the 274 cm length as the pBone for attaching the mouthpiece
THE PHYSICS TEACHER ◆ Vol. 52, J ANAURY 2014
References
1. 2. 3. 4. 5. 6. 7. 8.
9. 10. Fig. 9. The waveforms13 of the tones produced at the n=4 harmonic of the PVC pipe with only the pBone mouthpiece section attached, the pBone mouthpiece and bell sections attached, and the actual pBone.
11.
12. 13.
14. 15. 16. 17. 18. 19. Fig. 10. The Fourier spectra13 of the tones produced at the n=4 harmonic of the PVC pipe with only the pBone mouthpiece section attached, the pBone mouthpiece and bell sections attached, and the actual pBone.
section and one 13 cm + 59 cm= 72 cm shorter (202 cm) for attaching the mouthpiece and bell sections, a pBone, and computer hardware and software for measuring the playing frequencies. The mouthpiece section of a pBone fits snugly into ½-in PVC pipe and the flaring-bell sections can easily be attached with duct, electrical, or even just masking tape. To measure the playing frequencies, a laboratory group would also need a member capable of playing a brass instrument. This is usually not a problem in a general education science course designed for music students.
http://www.webwiki.com/jiggspbone.com. J. Backus, The Acoustical Foundations of Music, 2nd ed. (Norton, New York, 1977), pp. 260-268. R. W. Pyle, “Effective length of horns,” J. Acoust. Soc. Am. 57, 1313–1314 (1975). A. H. Benade, Fundamentals of Musical Acoustics (Dover, New York, 1990), pp. 414–416. http://www.phys.unsw.edu.au/jw/brassacoustics. html#mouthpiece. See http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html, under Sound and Hearing; Musical Instruments; Brass Instruments; Brass Concepts; Mouthpiece Effect. Ref. 4, pp. 407-409. N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments, 2nd ed. (Springer, New York, 1998), pp. 431–433. http://www.phys.unsw.edu.au/jw/brassacoustics.html#bells. See http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html, under Sound and Hearing; Musical Instruments; Brass Instruments; Brass Concepts; Bell Effect. Figure from http://hyperphysics.phy-astr.gsu.edu/hbase/ hframe.html, under Sound and Hearing; Musical Instruments; Brass Instruments; Brass Concepts; Forced Harmonic. Sequence used with permission of webmaster
[email protected]. http://flexpvc.com/PVCPipeSize.shtml. Data taken with Logger Pro, Vernier Software & Technologies, http://www.vernier.com/. Data could be taken with any data collection hardware and software able to capture sound waves, display their form, and calculate frequencies. T. D. Rossing, F. R. Moore, and P. A. Wheeler, The Science of Sound, 3rd ed. (Addison-Wesley, San Francisco, 2002), p. 232. http://www.phys.unsw.edu.au/jw/brassacoustics.html#pedal. See http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html, under Sound and Hearing; Musical Instruments; Brass Instruments; Brass Concepts; Pedal Tone. From Ref. 14. Used with permission. Ref. 14, p. 135. An online appendix is available at [EPAPS] for teachers interested in a more mathematically advanced calculation.
Michael C. LoPresto is in his 24th year of teaching physics and astronomy at HFCC; he has been teaching Sound & Light in Fine-Arts for the last 10 years, and is a regular contributer to TPT. Henry Ford Community College, Dearborn, MI 48128;
[email protected]
Acknowledgments I thank the students of HFCC’s Sound & Light in Fine Arts class for whom this demonstration was developed, and the members of the AAPT Michigan Section who attended the meeting where it was presented. THE PHYSICS TEACHER ◆ Vol. 52, J ANUARY 2014
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Appendix: Comparing Measured and Calculated Frequencies
Table I. Comparison of the measured and calculated frequencies of the cylinder with the mouthpiece section and a cylinder with both the mouthpiece and bell section.
for “Mouthpiece and Bell Effects on Trombone Resonance,” by Michael C. LoPresto, Henry Ford Community College
Harmonic
This portion is likely beyond the scope of a non-mathematical Science of Sound or Musical Acoustics course, but could possibly be of interest as an independent study for a more mathematically advanced student. The playing frequencies of a trombone can be calculated with the use of an expression for the frequencies of what is known as a Bessel horn.1,2 The frequencies depend on the parameter γ that defines the rate of flare of the bell in the equation a = b(x+x0)γ, where a is the radius of the bell a distance x from the large open end and x0 is an end correction.1,2 For trombones, usually γ = 0.7 and x0 = 0.1 cm,3 and β = 0.0.639 is another parameter.
fn =
c ⎡(2n − 1) + β[γ ( γ + 1)]1/ 2 ⎤ ⎥⎦ 4( L + x0 ) ⎢⎣
(3)
The effective length that a mouthpiece adds to the air column to which it is attached can be calculated with the expression 2,3
L=
⎡ (2π f / c ) L0 ⎤ c ⎥. tan−1 ⎢ ⎢1 − (4 fL 1 / c ) 2 ⎥ 2π f ⎣ ⎦
(4)
This expression depends on frequency and L0, which is the length of the tubing to which the mouthpiece is attached that has the same volume of the mouthpiece and L1 that is equal to the length of cylindrical tubing that has the same resonant frequency as the mouthpiece.4 A mouthpiece can be considered a Helmholtz resonator with its own resonant frequency.5,6 For a trombone mouthpiece the resonant frequency is about f = 550 Hz.4 Both of the lengths are easily calculated. L0=V/πr2 = 6.5 cm, where V is the volume of the mouthpiece (about 10 cm3 for a pBone mouthpiece, measured by filling the mouthpiece with water and then pouring the water into a graduated cylinder) and r is the inner radius, about 0.75 cm for the ½-in PVC pipe used. L1 = c/4f = 15.6 cm, where c is the speed of sound and f = 550Hz, the above mentioned mouthpiece resonant frequency. Table I shows the measured frequencies of the cylinder with a mouthpiece section and the cylinder with both a mouthpiece and bell section. Note the close agreement with the calculated values. The frequencies for the cylinder with a mouthpiece section were calculated by subtracting the actual 8-cm length of the mouthpiece from the 274-cm length and replacing it with the effective mouthpiece length calculated with Eq. (4). The frequencies for the cylinder with a mouthpiece and bell were calculated with Eq. (3), using a length L of 274 cm –8 cm = 266 cm plus the effective mouthpiece lengths calculated with Eq. (4).
598
Cylinder with mouthpiece (measured)
Cylinder with mouthpiece (measured)
Cylinder with mouthpiece & bell (measured)
Cylinder with mouthpiece & bell (calculated)
f (Hz)
f (Hz)
f (Hz)
f (Hz)
n 2
99
94
117
113
3
158
156
176
174
4
219
218
234
235
5
275
280
294
295
6
336
341
350
355
7
389
401
411
413
8
458
460
466
470
References 1. 2. 3. 4. 5. 6.
THE PHYSICS TEACHER ◆ Vol. 52, J ANAURY 2014
http://www.phys.unsw.edu.au/jw/brassacoustics. html#mouthpiece. See http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html, under Sound and Hearing; Musical Instruments; Brass Instruments; Brass Concepts; Mouthpiece Effect. F. J. Young, “The natural frequencies of musical horns,” Acustica 10, 91–97 (1960). A. H. Benade, Fundamentals of Musical Acoustics (Dover, New York, 1990), pp. 414–416. M. Morse, Vibration and Sound, 4th ed. (Acoustical Society of America, Melville, NY, 1991), pp 234–235. N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments, 2nd. ed. (Springer, New York, NY, 1998), pp. 433–437. Michael C. LoPresto, Henry Ford Community College, Dearborn, MI 48128 ;
[email protected]
