HHDP: 16 – 05
Air modes of the Bacon internal resonator banjo
David Politzer
California Institute of Technology, 452-48 Caltech, Pasadena CA 91125 (Dated: August 8, 2016)Sound measurements on a sequence of related, similar constructions with slightlydifferent dimensions confirm a simple picture of the air modes of the internal res-onator banjo’s body. For the purpose of this study, the air modes are decoupled fromthe soundboard (i.e., [drum] head) modes by replacing the head with 3 / (cid:48)(cid:48) plywood.The resulting characteristic features survive the strong coupling of the air modesto the head and are in accord with the qualitative distinctions recognized by banjoplayers.************************** contact info: [email protected], (626) 395-4252, FAX: (626) 568-84731 a r X i v : . [ phy s i c s . pop - ph ] O c t I. INTRODUCTION
Virtually every banjo design ever made continues to have enthusiasts and remains inproduction to this day. Almost all fall into one of two categories: open-back or resonator.Of the alternatives and variations introduced over the past century and a half that areneither, the “internal resonator,” first patented[1] and put in production by stage performerFred Bacon in 1916, is something of a cult favorite. The originals are coveted, and the basicdesign is still produced by independent luthiers.Banjos are stringed instruments whose body is a drum and whose string-head interfaceis a floating bridge. The most common drum is a cylinder (called the “rim”), with the headstretched taut over its top edge. If that top edge is made out of a separate piece, that edgepiece is called the “tone ring.” Tone rings can be hardwood, but more often they are metal.They can be solid, hollow, or a combination. They range from simple lb rings made of (cid:48)(cid:48) diameter brass rod to elaborate cast and machined combinations, weighing up to 3 lbs.Open-back banjos are just that: the drum has no back or bottom. However, the player’sbody effectively forms a back, and the sound hole is the space between rim and body, whereair can enter from and escape to the ambient surroundings. Resonator banjos have a woodenback. However, it does not seal against the bottom edge of the rim. Rather, there is a smallspace all around which serves as a sound hole.[2]The novel feature of Bacon’s design is the internal structure of the drum. The purposeof this study is to gain a qualitative understanding of the acoustics of that structure. Likeall banjos, almost all the sound is produced by the vibration of the head. Subtle variationsarise from the interaction of the head with the internal air pressure variations. The couplingis strong because the head is light and flexible and makes contact with the internal air overits whole area. Bacon’s and subsequent internal resonator banjos used the standard headdesigns of their times. And the head/internal air coupling is not particularly amenable todetailed study or analysis. The focus here is on the internal air itself.The drum (or “pot”) of the internal resonator banjo has a partial back attached to therim and an additional internal wall that divides the pot interior into a central cylinder thathas an open back and an enclosed annular region, as diagramed in FIG. 1.There is a small space between the top edge of the inner wall and the head that serves asan air connection between the two regions. Bacon’s original models also had metal tone FIG. 1. Schematic cross section of the internal resonator potFIG. 2. Back view of an original Bacon Professional ff and the modified Deering Goodtime rings, weighing about 1 lb, of a design that has come to be known by his name. Those tonerings have a solid (or almost solid) core, with a sheet metal wrapping that forms horizontaland vertical flanges. The vertical, outer flange is fitted snug to the outer side surface of thewood rim, while the inner, horizontal flange is free to vibrate. That inner flange also furtherdefines the air passage between the central cylinder and the outer annulus. When played,the central cylinder is partially closed by the player’s body, and the sound hole is the wholepassage between the partially enclosed interior cylinder and the outside air.Even Fred Bacon himself was not particular about the exact dimensions or even designdetails. In the early years of his company, he sold banjos made in part or entirety by leadingmanufacturers of the time. Many of those instruments exist to this day, and they varyconsiderably in their details. However, they all fit the description given above. What wasstudied for the analysis that follows is a new banjo with interchangeable internal resonatorsof different dimensions and identical banjos with and without a Bacon-style tone ring. FIG. 2is a photo of an original “Professional ff” (the Bacon Banjo Company’s name for this design)and a modified Deering Goodtime banjo, equipped with an internal resonator and replicaBacon tone ring. On both, the edge of the metal tone ring inner, horizontal flange is justbarely visible.
II. SUMMARY OF RESULTS
As with most stringed instruments, the lowest body resonance is a Helmholtz resonance,whose frequency is determined by the enclosed volume and the geometry of the soundhole. The partial back provided by the internal resonator design decreases the interfacearea between the enclosed volume and the Helmholtz bottle “neck.” The partial back alsoincreases the volume of that “neck.” Both of those features lower the resonant frequency.(If it were a bottle of ideal geometry with neck-to-interior-volume interface area A and neckvolume V neck , the frequency would be proportional to A/ √ V neck .) In spite of there being twointernal volumes separated by a constriction, for geometries relevant to banjo construction,these volumes act as one — at least with respect to Helmholtz resonance physics. There isa single Helmholtz resonance whose frequency depends only on the size of the hole in thepartial back and not on the height of the internal wall that separates the inner cylinderfrom the outer annulus. Apparently, the area of the interface between the two regions is toolarge and the volume of the interface is too small for the system to behave anything likethe coupled Helmholtz oscillators envisaged by Rayleigh as a logical possibility.[3] An idealHelmholtz resonator has a frequency that is independent of the shape of the main volume.So it is consistent that the one observed Helmholtz resonance frequency is independent ofthe internal wall height.The closed-body air resonances are identified as coupled versions of the separate cen-tral cylinder and outer annulus. The cylinder modes are calculable from the dimensions.The annulus, while not exactly soluble, has modes that are well-approximated by periodicboundary conditions applied to a straight pipe with the annulus’ cross section and effectivecircumference as its length. Hence, the consequence of the wall is two-fold. The lowestresonant frequency of the annulus is significantly lower than that of a cylinder without thewall. (The longest annulus wavelength is essentially π times the diameter, while the longestwavelength of the no-wall pot is shorter by a factor of ∼ / .
84, coming from the zero ofthe appropriate Bessel function derivative). And once the frequency spectra of the annulusand inner cylinder overlap, the combined system is has a greater density of resonances infrequency, which gives a more even response to driving frequencies.For most banjos, the lowest half-octave or so is an example of a “missing fundamental”in comparing the radiated sound to the plucked string frequencies. The internal resonatorgives more support to these lower frequencies than present in the standard design.The final element of Bacon’s design is the tone ring itself. In general, a metal tone ringprovides a harder and stiffer edge to the vibrating area of the head than is presented by thesame pure wood rim without it. This means less dissipation. Usually, this is most apparentat high frequencies (i.e., the “hardness” aspect). However, the net effect of the Bacon tonering is mostly the opposite because of the horizontal, vibrating flange. The flange dissipatesa noticeable amount of sonic energy that, in its absence, would have gone into radiatedsound. This is most apparent at the high ringing resonances of the flange itself. Frequency-dependent damping means that the timbre of the transient sound of a pluck changes withtime. Discerning players refer to the “finish” of a note, and some people particularly likethe finish of the Bacon ring in which the highest frequency overtones die off more rapidly.The flanges provide extra stiffness to the rim that reduces flexing in their respective planes.That reduces energy loss at low frequencies.[4]In summary, Bacon’s modifications offer a banjo with response stronger at low frequen-cies, smoother across all frequencies, and more subdued at high frequencies, especially as asweeter finish to pluck sounds. Those were his goals[1] and are consistent with the verbalcharacterizations offered by modern players.
III. INTERNAL RESONATOR HELMHOLTZ RESONANCE
Investigation of the Helmholtz resonance(s) begins with comparing partial backs withdifferent size holes. The bottom of a Bacon-tone-ring-equipped Goodtime rim was cut flatand fitted with six threaded inserts. The rings shown in FIG. 3 were cut from 3.0 mm,7-ply birch and could be attached with a narrow retaining ring of the same plywood and sixscrews.
FIG. 3. Partial backs with various hole diameters and the adjustable-height internal resonator
With strings, neck, and tailpiece removed (but coordinator rod in place), the soundsof head taps with a piano hammer were recorded for various bottom hole diameters. Thelargest was the stock Goodtime, whose inner diameter is 9 (cid:48)(cid:48) . The smallest was 7 (cid:48)(cid:48) , whichis the diameter of the internal resonator insert that appears later.I mounted a synthetic belly, made of closed cell foam, cork, and Hawaiian shirt on theback to serve as an approximation of the how the banjo is normally played. Those materialswere chosen to mimic the absorption and reflection of the player’s body. The opening tothe outside air was chosen to approximate typical playing and is far more reproducible thanholding the instrument up to one’s body. The genesis and details of this back are discussedin ref. [5]. Since a Helmholtz resonance is characterized by motion of air in and out of thesound hole, I placed the microphone right at the largest portion of the opening.FIG. 4 shows the spectra for long series of those head taps, plotted for 100 to 500 Hz. FIG. 4. Head taps with foam belly-back; curves labeled by back hole diameter (no internal wall)
The two lowest peaks show a systematic decrease in frequency with decreasing back holediameter. All higher frequency features show no appreciable frequency dependence on theback hole dimension.This is the qualitative behavior expected from the Helmholtz resonator formula. Smallerhole diameter implies smaller A and larger V neck . There are two peaks for each back thatreflect this behavior because the internal pot Helmholtz resonance couples strongly to thelowest drum mode of the head. Not only do they both push the same plug of air in and out,but they also push on each other over the whole head surface. The higher frequency modesare due to other physics. It is typical that the lowest two modes of the body of a stringedinstrument are the coupled versions of the Helmholtz and lowest sound board modes.[6] Notethat on typical banjos the fundamental frequencies of all strings but the short 5 th string arebelow 300 Hz.To separate Helmholtz from sound board physics on violins and guitars, experimentershave occasionally buried the instrument in sand — to immobilize the sound board motion.It is easier on a banjo. I replaced the regular mylar head with (cid:48)(cid:48) plywood. To drive theHelmholtz resonance, I mounted a 3 (cid:48)(cid:48) speaker in the middle of that plywood head. Thathead is the one not attached to the rim in FIG. 5 and installed on the rim in FIG. 6. (Theattached solid head and rim-mounted 1 (cid:48)(cid:48) speaker and microphone in FIG. 5 are described insection IV.) The speaker is driven with a signal generator and audio amplifier with a slow FIG. 5. Plywood heads, speakers, & mic sweep, logarithmic in frequency, over the desired ranges.So, the frequencies of the lowest two pot resonances are substantially lowered by thepartial back in a way whose physics is qualitatively understood. The next question is theimpact of the cylindrical wall of the internal resonator that divides the interior into a smaller,central cylinder and an outer annular volume. I fabricated a variety of internal resonators,all with the same cylinder diameter and back hole size but with various wall heights. Thecylinders were cut from 3.0 mm, 5-ply maple drum shell stock to produce wall heightsranging from (cid:48)(cid:48) to 2 (cid:48)(cid:48) . However, the key to understanding the Helmholtz resonance andthe cavity resonances (section IV) turned out to be internal walls of the roughly standardheight, 2 (cid:48)(cid:48) , and higher. To this end, I fabricated an adjustable height insert. It had a splitring that could be inserted into a 2 (cid:48)(cid:48) high cylinder. The top edge of the split ring couldbe placed carefully at any particular distance from the head when assembled and tightenedsnugly with a shim in the gap in its circumference. That is the upper right construction inFIG. 3.The resulting spectra for driving with the 3 (cid:48)(cid:48) head-mounted speaker and listening with amicrophone at the rim-belly-back opening (as shown in FIG. 6) are plotted in FIG. 7. Now,for each pot geometry, there is only one, low, broad peak between 200 and 300 Hz. With FIG. 6. Wood “head” with 3 (cid:48)(cid:48) speaker and cork & foam belly-back this set-up, the higher resonances are all considerably weaker. All versions are with the samerim with its Bacon tone ring. The curve labeled “stock” refers to the standard, open-backGoodtime rim. The curve labeled “7 (cid:48)(cid:48) ” ring, is the partial back with no cylindrical wall.(That 7 (cid:48)(cid:48) is the same partial back hole size as the internal resonator.) The dashed anddotted curves refer to internal resonators that have a (cid:48)(cid:48) and (cid:48)(cid:48) space, respectively, betweenthe top of the internal cylinder and the inner surface of the head. The “no gap” curve refersto an inner cylinder that touches the head and seals off the outer annulus from the innercylinder.The relations between the stock, 7 (cid:48)(cid:48) and no-gap curves are standard Helmholtz resonatorphysics. (As before, referring to the ideal Helmholtz bottle, A is the main volume/neckinterface area, and V neck is the volume of the neck; let V be the main volume. Then theideal Helmholtz frequency is f H = v s π A √ V V neck .) The stock and 7 (cid:48)(cid:48) ring have the same V , butthe ring has a smaller A and a larger V neck . The ring and no gap have the same A and V neck ,0 FIG. 7. Spectra with plywood head with 3 (cid:48)(cid:48) speaker and foam belly-back but the no gap has a smaller V . Stock and no gap differ in all three parameters: the no gaphas a smaller V , a smaller A , and a larger V neck . So the sign of the difference depends ondetails of the actual values. However, stock and no gap have approximately equal values of A/ √ V , which accounts for the sign of the observed difference in frequencies.A very important lesson from these measurements, which was not altogether obviousbeforehand, is that the Helmholtz resonances of the 7 (cid:48)(cid:48) ring, (cid:48)(cid:48) gap, and (cid:48)(cid:48) gap curves,i.e., all of the pots with the same size partial back, are essentially indistinguishable. Thatmeans that the height of the internal cylindrical wall, going from zero up to the height inthe standard, finished banjo (i.e., reaching to (cid:48)(cid:48) from the inner surface of the head) andeven beyond by another (cid:48)(cid:48) , does not effect the Helmholtz resonance. They are all thesame — as if there were no inner wall at all. The simple Helmholtz resonance picture saysthat the resonant frequency is independent of the shape of the cavity. So, apparently, thewall is simply an alteration in the shape. And these three configurations have the same V , V neck , and A . However, one might ask whether there could be a wall sufficiently highthat it divides the original cavity into two Helmholtz resonators in series — just as Rayleighsuggested could arise,[3] at least for some design. Apparently, in practice, the answer isno, not for the internal resonator geometry. There are two obstacles. Friction becomes animportant force with yet smaller gaps. And the volume of the purported neck between theannulus and the central cylinder is too small relative to the interface area.1 IV. CAVITY MODES
The internal wall certainly does something, and that is revealed by a study of the higherfrequency cavity modes. Again, the coupling to the head modes is removed by using a solid (cid:48)(cid:48) plywood head. Since these air modes are essentially internal to the pot, the sound holegap can be eliminated — allowing for cleaner and clearer resonances. The sound hole wasonly crucial to the Helmholtz mode. So I chose to seal the back with solid plywood. Andthat required putting a driving speaker and a recording microphone inside the pot. Thathead, speaker, and mic assembly is shown in FIG. 5.Again, the rim is the Goodtime fit with a Bacon tone ring. Logarithmic frequency sweeps,with the small, internally mounted speaker and microphone, yielded the spectra shown inFIG. 8. The horizontal frequency scale is linear. “Stock” refers to the standard rim. “ (cid:48)(cid:48) gap” is the standard internal resonator, whose cylindrical wall is 2 (cid:48)(cid:48) high, which bringsit to (cid:48)(cid:48) from the inner surface of the head. The task at hand is to understand how thestandard internal resonator converts the stock spectrum into the one labeled (cid:48)(cid:48) . The are noHelmholtz resonances in this configuration because there is no in-and-out air motion. Thelowest closed cavity resonances are the ones shown.The internal diameter of the pot was 9 . (cid:48)(cid:48) . In terms of the obvious cylindrical coordinates r , θ , and z , the standard r - θ resonance frequency values are indicated by the solid verticallines at the bottom of FIG. 8. (That calculation won’t be perfect because it ignores thepresence of the speaker, microphone, tone ring, and other hardware inside.) The dottedlines, first appearing around 2300 Hz, are the calculated frequencies of the additional modesthat involve wave components in the z direction for an internal height of 2 . (cid:48)(cid:48) .The key to understanding what is going on is to consider a wall that leaves no gap betweenitself and the head. In that case the smaller inner cylinder and the outer annular regionare distinct. A small acoustical coupling between the two was introduced in the form of a (cid:48)(cid:48) × (cid:48)(cid:48) hole in the internal wall. The corresponding spectrum is the red “no gap” curve inFIG. 8.The calculated resonant frequencies for the no-gap system, under the assumption that thecoupling of its two parts is weak enough to ignore, are also indicated by lines at the bottomof FIG. 8. The long dash lines are the standard cylinder mode frequencies, higher than thesolid lines simply by the ratio of the stock diameter to the internal resonator diameter, at2 FIG. 8. Comparison of the stock pot & the internal resonator with various gaps; lines at thebottom denote frequencies calculated from the actual physical dimensions least for the z -independent, lower frequency range. The two cylinders have the same height,and the higher frequency contributions from waves in the z direction are added in also.The calculated mode frequencies of the annular volume are indicated with short dashesand use the approximation described in section II. They begin around 500 Hz, which issubstantially lower than the lowest mode of the stock cylinder. Note that the smallestdimension of the annulus is 1 . (cid:48)(cid:48) in the r direction. That is only first excited around 6700Hz.FIG. 8 also displays the measured spectra for intermediate values of the rim-wall-headgap, illustrating how the spectrum evolves continuously from no gap to its final (cid:48)(cid:48) value. V. THE BACON TONE RING
Bacon included a metal tone ring in his design. It sits on the top edge of the wood rim,and the drum head is stretched over it. A detailed study of its vibrations and their effect onthe banjo’s sound is presented in ref. [4]. In summary: the (cid:48)(cid:48) solid diameter core hardens3the rim edge to reduce high frequency absorption relative to a pure wood rim. The two thinflanges stiffen the rim (at the cost of little extra weight) to reduce large rim motions thatotherwise absorb low frequencies. And vibrations of the free horizontal flange absorb someof the vibrational energy that without the flange would have gone into sound.For the study of air modes, all of the comparisons are made with a Bacon tone ringinstalled on a single 11 (cid:48)(cid:48) Goodtime rim. Whatever stiffening and vibrating the tone ring does,it does so similarly for the different back and internal geometries. Only the comparisonsin section VI involve an all-wood, standard Goodtime rim, where it is contrasted witha Bacon-modified Goodtime, i.e., with tone ring and full-size internal resonator installed.This comparison involves all the effects at once.
VI. FULLY ASSEMBLED BANJO PLUCKED AND PLAYED
How does a Bacon internal resonator banjo sound and how does it differ from an otherwiseidentical instrument? People familiar with 5-string banjo music can hear the difference.However, that does not mean that the differences can be easily discerned from numbers andgraphs. Shown below are spectrographs of individual plucks and frequency analysis of twoentire 35 second played samples. The only obvious differences in FIG.s 9 and 10 are in theenhanced response at the lowest frequencies, as to be expected from the discussion of theair modes of the pots. There really is no substitute for listening.[7].The instruments compared are two fully assembled banjos: a totally normal Goodtimeand the Bacon-inspired modified Goodtime, i.e., with tone ring and internal 2 (cid:48)(cid:48) high in-ternal resonator. The strings, heads, and head tensions (as measured by a DrumDial) werethe same. (Deering Goodtime banjos were chosen because they are mass-produced, hand-finished, high-quality instruments that are about as identical as complex constructions madeof maple can be.)FIG. 9 is a spectrograph of four typical single string plucks, with the other stings left freeto vibrate. The microphone was at 20 (cid:48)(cid:48) in front of the head. All plucks were at the secondfret. The first one is the 4 th string of the normal Goodtime; the second is the 4 th string ofthe Bacon-modified Goodtime; the third is the 1 st string of the normal Goodtime; and thefourth is the 1 st string of the Bacon-modified Goodtime.FIG. 10 presents the frequency spectra for an entire 35 second selection, played and4recorded on the two banjos as identically as possible. FIG. 9. Spectrograph of four typical plucks (left to right): 4 th string stock, 4 th string Bacon-modified, 1 st string stock, 1 st string Bacon-modifiedFIG. 10. Spectra of 35 seconds of frailing on a Goodtime stock vs internal-resonator-fitted banjo VII. CONCLUSION
The cylindrical symmetry typical of banjo design simplifies the analysis of internal airmodes. The “internal resonator,” a modification of the inside geometry of an open-back5banjo, defines a smaller central cylinder and an outer annular region. A simple calculationof air modes of the ideal, separated, two-volume system and a sequence of measurementsof increasingly coupled volumes demonstrates that this is a useful picture. The whole con-struction produces a richer spectrum, starting at a lower frequency than a simple cylinderof the same outer dimensions. The lowest frequency corresponds to a Helmholtz resonancethat turns out to be insensitive to the division of the internal volume into annular and cen-tral cylindrical regions — at least for practical geometries. That frequency is lower thanthe one for a simple open-back banjo of the same outer dimensions because of the internalresonator’s impact on the effective sound hole.On the other hand, the pressure-sensitive drum head/sound board makes intimate contactover its entire area with that internal air. The head is not particularly close to an idealmembrane nor a plate. Being very light and flexible, the head has interactions with the airthat are neither ignorable nor trivial. So analysis of its motion presents a serious challenge.But that motion is what makes almost all of the sound. A more modest question is acomparison. How do Bacon’s design modifications alter the sound? A big part of that isclearly the altered dynamics of the air in the pot as studied in this note. However, that isnot the complete story because it is still missing the details of the interaction with the headthat drives the internal air motion and also responds to it.There is a further caveat that goes with any discussion of plucked strings that is based onnormal modes and spectral analysis. The relevant motions are all transients. Steady statebehavior is an approximation — and not always a good one, especially if one is interested insubtle distinctions. Sonically interesting features of the transients can be sensitive to smalldifferences in details.[8] This is certainly true of banjos.
REFERENCES
The Resonator Banjo Resonator, part 1 and part 2 , ~ politzer, also available from the permanent archiveshttp://arxiv.org/ and http://authors.library.caltech.edu/. The other D. Politzer papers citedbelow are available likewise.[3] J. W. Strutt, 3 rd Baron Rayleigh,
The Theory of Sound , v. 2, 1878, 1896,... 1945, DoverPublications (New York), § A Bacon Tone Ring on an Open-Back Banjo , HDP: 16 – 01; the clearest demonstra-tion of the Bacon tone ring’s effects can be heard in the head tap sound samples accompanyingthat paper.[5] D. Politzer,
The Open Back of the Open-Back Banjo , HDP: 13 – 02[6] The two coupled oscillator model of the guitar lowest modes is reviewed in
The Physics ofMusical Instruments , N. H. Fletcher and T. D. Rossing, Springer (New York) 1998, 2010, § The Science of String Instruments , T. D. Rossing ed., Springer (2010) § The Physics of the Bacon Internal Resonator Banjo , HDP: 16 – 02[8] E.g., in general, the modes of definite complex frequency of coupled, damped systems are notorthogonal; see G. Weinreich,
Coupled piano strings , J. Acoust. Soc. Am. (6), 1474–1484(1977); D. Politzer, The plucked string: an example of non-normal dynamics , Am. J. Phys.83