HHDP: 19 – 01 one of several prototypes
Banjo Ring from Stretching String: A Zero Break Angle Demo
David Politzer ∗ (Dated: March 7, 2019)A novel bridge and tailpiece design allows direct comparison of the sound of zerobreak angle with same banjo (and all its parts) configured to have an angle of 13 o .This lends additional support to the 2014 proposal that a key element in banjo soundis the frequency modulation produced by string stretching due to a floating bridge,break angle, and head with substantial motion. When playing a banjo tune in the0 o configuration, there are enough audio clues that it still sounds like a banjo. Thecomparison allows you to judge for yourself to what extent it’s lost its ring or sparkle. ∗ a r X i v : . [ phy s i c s . pop - ph ] M a r Banjo Ring from Stretching String: A Zero Break Angle Demo
Curiosity about tailpieces and string break angles over the bridge led to an investigationof some of the physics involved.[1] A string with fixed ends must stretch when it undergoestransverse vibrations. However, that stretch is proportional to the square of the vibrationamplitude. Assuming the stretch to be negligible for small amplitudes gives rise to a verysatisfactory physics description of the vibrating string. In contrast, the motion of a floatingbridge followed by a non-zero break angle produces stretching that is linear in the amplitude.And that produces tension and frequency modulation which are not necessarily negligible.The sound of a tone whose modulation frequency is also in the audio range was discoveredby John Chowning in the early 1970’s.[2] He described it as sounding “metalilic.”The presence of first order (i.e., linear) string stretching in the banjo is an unequivocalmatter of geometry. The subtle issue is a question of psychoacoutics. What is it that allowspeople to identify banjo sound, irrespective of whether the instrument is a bluegrass banjowith lots of metal parts or a 150 year old banjo with not a single scrap of metal? Banjoswere said to ring well before Stephen Foster wrote about them in the 1850’s. The suggestionthat banjo “ring” is an aspect of Chowning’s metalilc sound was supported by a variety ofexamples, demonstrations, and players’ observations.[1]The details are not repeated here. Rather, what is presented is the starkest possiblecomparison: a normal banjo with and without a break angle over the bridge.
ZERO DEGREE BREAK ANGLE DESIGN
On a banjo with a normal break angle, the stretching of the strings is first order in thevertical motion of the bridge, i.e., the former is linearly proportional to the latter. Theconstant of proportionality goes to zero as the break angle gets smaller. At zero breakangle, the stretching is proportional to the square of the bridge motion. However, a normalbridge and tailpiece will not work at zero break angle. The strings would buzz at the bridgeas they vibrate upwards; likewise, the bridge would buzz on the head.The design used here obviates those problems. It also allows a sound comparison withthe same banjo, bridge, and tailpiece but with a break angle of 13 o — simply by flippingthe tailpiece over and reinstalling the strings.The strings pass through holes in the bridge. The hole diameters are all about 10% larger b r i dg e t a il p i ece head s t r i ng FIG. 1: Bridge-tailpiece schematic – not to scalethan the strings. The holes angle down towards the tailpiece at 5 o relative to the head. Thestrings enter the top of the tailpiece through ferrules. They are secured by their ball ends insteel-washer-lined holes in the bottom. The 5 o means that the centerlines of the holes drop0 . (cid:48)(cid:48) as they traverse the 0 . (cid:48)(cid:48) thickness of the bridge. The string gauges are 10-12-14-22w-10. So the oversize holes and the 5 o drop ensure that strings sit snugly on the bottomsof the holes (much like they normally sit in slots on the top of a bridge) and do not bindin some awkward way in the hole. Even when the string vibration motion is upward, the 5 o break is sufficient to keep it in contact with the bottom of its hole. (That’s clear because 5 o is perfectly adequate on a normal Old-Time banjo to keep it from buzzing.) The hole heightwas chosen to provide the same action as a normal 5 / (cid:48)(cid:48) bridge and a Presto-style tailpiece.(Note that, with zero break angle, the head is perfectly flat, even with the strings at fulltension.)The bridge is glued to the smooth side of a bottom-frosted head and further secured withthree brass / (cid:48)(cid:48) screws. (I didn’t find any glue that adhered to the mylar, even aftersanding; so I used contact cement.) The final weight of the bridge, including screws, is 2.6gm (i.e., pretty standard for a steel string and mylar head set-up).The tailpiece is designed so that only very minor adjustment is needed, when under fullstring tension, to bring the bridge-to-tail strings into parallel with the bridge-to-nut strings.The slight off-set due to the 5 o channels has no impact on string stretching. In particular,the stretch due to bridge motion is precisely the same as would occur for upward bridgemotion with strings going straight over a normal bridge with no break angle. The wood ofthe tailpiece is 35 gm, which is relatively hefty — chosen to minimize its motion. Includingaluminum spacer and steel screw, washer, and three nuts, the tailpiece is 63 gm.FIG. 2: The bridge and tailpiece described above and used in the sound samples with zeroand 13 o break angle; the woods are maple and purpleheart, respectively. A COUPLE OF TUNES FOR COMPARISON
The banjo is a year 2000 Deering Goodtime, played with a solid disk back spaced 1 / (cid:48)(cid:48) ~ politzer/zero-break/A.mp3 and then switch A to B,C, and D. If you can’t hear which is which, this exercise is pretty bootless. DISCUSSION
Audacity R (cid:13) ’s frequency analyses of the finger-picked tune are displayed in FIG.s 3, 4, & 5.These serve as a guide to what you’re hearing. All graphs show that there is little differencebetween the tailpiece set-ups below 1500 Hz. That range includes all the fundamentalfrequencies of the notes played and many of their first few harmonics. (The fundamentalfrequencies of those notes range from 131 Hz [open 4 th ] to 1047 Hz [12 th fret 1 st string].)And it accounts for virtually all of the sound energy. So the two set-ups are equally loud.But the 0 o version is generally much weaker above 1500 Hz. (The next section, SINGLESTRING PLUCKS , takes a closer look at the time evolution of individual notes.)FIG. 3: low resolution spectra for the full 34 seconds of the picked tune — 13 o on the left,& zero on the right(Note on spectrograms: the choice of frequency resolution, ∆ ν , produces a time resolu-tion, ∆ t , subject to the relation ∆ ν ∆ t ≈
1, i.e., the “uncertainty relation.” Hence, highFIG. 4: high resolution version of FIG. 3FIG. 5: spectrograms of the first few seconds of the picked selection, 13 o (left) and 0 o (right), Hz. vs. secondsresolution in frequency necessarily smears out the results in time.)Nevertheless, both versions sound like a banjo. (What did you expect? A French hornor an oboe?)The mental identification of “banjoness” (performed effortlessly and unconsciously, atleast by someone who has listened to a significant amount of banjo music) is, presumably,based on many clues. Facial and voice recognition of a particular person certainly workthat way. It’s likely that no one clue is absolutely essential. Here are a couple of examplesfrom the far corners of banjo playing that illustrate how perceptions can be confused. TheReverend Gary Davis occasionally picked up a 6-string banjo and even recorded some on one.He played the same repertoire and in exactly the same style as his guitar playing. It’s easy tomiss any difference. Adrian Legg’s compositions include a few tunes and arrangements thatare immediately recognizable as banjo music. But it’s all played on guitar (admittedly witha healthy dose of real-time electronic effects). Harvey Reid sometimes plays a 6-string banjo.Even straight acoustically, he can make it sound like almost any plucked string instrument,e.g., from his “faux frailing” banjo sound to a whining, driving solid body guitar.The other obvious physical feature common to all banjos is the head. A mylar or skinhead sounds nothing like a wooden head. Instruments with the latter sound like dulcimers,irrespective of their appearance or how they’re played. In ref. [1], I argued that, under theaction of similar strings, a proper drum head would move through much greater distancethan a wood sound board. That obviously implies that the string energy is turned intosound faster with the drum head. So a normal banjo is louder but for a shorter period. Butthe timbre changes, too. Part of that could be more high frequency dissipation with thewood head while the vibrational energy is still in the string, i.e., before being converted intosound. But there’s also the fact that the string is stretching much more while it is vibratingwith the mylar or skin head. On the other hand, a proper acoustician would point out that adrum head is reasonably approximated as a “membrane,” whose restoring force is principallydue to tension, while a sound board is properly described as a thin plate, whose restoringforce is stiffness. The spectra of resonant frequencies of the two are rather different.Ref. [1], contains explicit examples of the sound of frequency modulation, both generatedanalog, i.e., by manually moving the bridge up and down and then speeding up the wholerecording, and digitally, i.e., computer generated files of sinusoidally modulated sinusoidalfunctions. In all cases, there is a discernible ring. But the actual sound of a played banjo isquite complex. Many things are going on at once. The bridge rocks and goes up and downunder the influence of all the strings. And that, in turn, effects the motion of all the strings.So the strings certainly influence the motion of each other.Chowning’s insight was that frequency modulation could be the basis of a very richrange of synthesis possibilities. In fact, that’s how and why it became the basis of StanfordUniversity’s second biggest money-making patent. It was licensed to Yamaha. Jointly withStanford, they developed the DX7. Released in 1983, it was the first commercially successfulelectronic keyboard synthesizer. The point is that FM synthesis can easily get you almostanywhere in timbre space. So it is not clear that successfully synthesizing banjo sound usingfrequency modulation would add any additional support to the notion that break angle andstretching are the essential elements. Chowning himself began with the hitherto vexingbrass instrument sound. With an appropriately chosen amplitude envelope function andfrequency modulation of the note, he found he could make plausible imitations of brasses,from tubas to cornets. (Note added: A colleague pointed out that frequency modulation canbe added to an otherwise generic plucked-string-sound synthesizer to get plausible banjo-likesounds.[3])The logic of the identification of frequency modulation as the source of “ring” is, therefore,necessarily a bit indirect. As is well-accepted among players, steeper break angle makes thesound “ringier.” In the comparisons presented here, the only physical change between banjosis the dramatic change in break angle. From a physics perspective, one must ask, “Whatis the relevant change in the sound producing system as a result of that angle?” Steeperangles do not improve the motion transfer from the strings to the bridge (discussed in § TAILPIECE DOWN PRESSURE below). The only consequence I can identify in theequations of motion for the system is the increased stretching of the strings as the floatingbridge moves up and down. This effect is a continuous function of angle. In words (ratherthan in math concepts), the break angle determines the amount of stretching, and thestretching is smallest at zero angle.
SINGLE STRING PLUCKS
Listening to and analyzing the pluck of a single string offers further insight into the timeevolution of the sound. Among the many possibilities, I found the plucks of the open 4 th string the most revealing. Here, the other strings are left open (undamped). I chose onetypical pluck for each tailpiece configuration. In the following example, the first (and left)is the 13 o break angle; the second (and right) is 0 o .Listen to the comparison cut off at 2.5 seconds after the pluck: Click here for a comparisonof 4 th ~ politzer/zero-break/4th-comparison-2.5sec.mp3.Here are the same two plucks, this time cut off after 0.5 seconds: Click here for a com-parison of 4 th ~ politzer/zero-break/4th-comparison-0.5sec.mp3.Spectrograms of these two versions are shown in FIG.s 6 and 7. (The pile-up in FIG. 7at the end of the 0.5 second selections is an artifact of the math algorithm and the sharpcut-off of the sound file.)FIG. 6: High frequency resolution spectrograms of single plucks of the 4 th string — 13 o and then 0 o , both cut off after 2.5 seconds.FIG. 7: High frequency resolution spectrograms of single plucks of the 4 th string — 13 o and then 0 o , both cut off after 0.5 seconds.Again, there is little difference below 1500 Hz.The effect seems strongest with the 4th (lowest) string. This agrees with the “theory.”The stretching is proportional to bridge motion, and that is greatest for the lowest the notes.The dramatic differences between the two set-ups are confined to the first 0.2 - 0.3 seconds0of the pluck. The longer sustain is dominated by the common sound of lower harmonics.On the other hand, in normal playing, notes typically come at 4 per second or faster, oftenmore than one at a time. So there are lots of things are going on at once. TAILPIECE DOWN PRESSURE & STRING DRIVING EFFICIENCY
Some people believe that the down pressure due to break angle contributes to effectivetransmission of string motion to the head, and the more the better — at least up to a point.However, the zero break configuration was perfectly loud. Furthermore, a simple physicsanalysis suggests just the opposite: an increase in equilibrium down pressure reduces headmotion in response to a given string motion. And that’s what ultimately chokes out thesound at extreme break angles.At equilibrium, when the strings are at rest, their downward force on the bridge is exactlycanceled by the upward force from the deformed head. That is a stable equilibrium. If thebridge is displaced upward, the combined forces of the strings (still not vibrating themselves)and the head, push it backward. The string downward force increases for two reasons. Thetension increases because the strings are stretched, and the break angle becomes steeper (sothe downward component of tension is greater). Conversely, the head upward force decreaseswhen the bridge is displaced up (for exactly the analogous reasons). So an upward bridgedisplacement results in a greater net downward force on the bridge from strings and head.The important point is that the restoring force (i.e., the force tending to return thebridge to its equilibrium position) for a given displacement is greater for a larger equilibriumbreak angle. The string stretching and increase in tension are greater, and the downwardcomponent of the tension (even for a given value of tension) is greater.In the context of normal playing, the head moves because string vibrations make smallchanges in the down force of the strings on the bridge. However, the bridge moves underthis force and the net force from its being displaced. Hence, increased break angle reduces the net motion of the bridge. Ultimately, it can be enough to produced a discernibly chokedresponse.At the other end, a zero break angle minimizes the return force of quiet strings and headon the bridge. For small amplitudes, bridge motion is solely determined by the vibratingstring forces. Consequently, the head moves more air for a given amplitude pluck. However,the perception of loudness is not purely an issue of sonic power. There is a frequency1dependence, too. I believe that we are more likely to take note of higher frequencies. Sotiny increases in the energy of high frequencies will produce a perception of louder sound.
CONTRIBUTORS TO BANJO SOUND
Virtually every design aspect contributes in some way to the sound of a banjo. Andsome design choices emphasize what some people consider “banjoness.” Head density andtension are obvious examples. I’ve written previously about a few not-so-obvious examples.Bridge mass has a huge impact.[5] Extreme added-on mass is one common form of mute. Itcomes with dramatic changes in timbre. In particular, the sustain is increased, and higherovertones are suppressed relative to lower ones. But these effects are clearly discernible evenwithin what is considered the normal range for bridge masses. The geometry at the edgeinside the pot and just below the head also can be chosen to impact the the amount of“ping” associated with each struck note.[6] And, of course, resonator backs emphasize theattack and very high overtones.[7]But there’s something that makes all (real) banjos recognizable as banjos. And “all”includes low-tuned, 13 (cid:48)(cid:48) , fretless, skin-head open-backs constructed as they were in the mid-19 th Century. The combination of first order string stretching and consequent frequencymodulation is one of the few (though not only) things they have in common. So maybe itreally is the key. [1] D. Politzer,
Banjo timbre from string stretching and frequency modulation ~ politzer — or directly as String Stretching,Frequency Modulation, and Banjo Clang .[2] J. Chowning,
The Synthesis of Complex Audio Spectra by a Means of Frequency Modulation ,J. Audio Eng. Soc. 21 (7) 526 (1973).[3] The Karplus–Strong algorithm[4] can be used as the basis of an electronic analog computerto simulate plucked string sounds. Its crucial parts parallel the salient physics features of thetime evolution of a plucked string. There is an initial disturbance. On a real string, this is Digital Synthesis of Plucked String and Drum Timbres , ComputerMusic Journal (MIT Press) 7(2), 1983.[5]
Banjo Bridge Mutes,” ~ politzer/inertial-mute.pdf.I suspect that wood species has little to do with it, except in as much asit determines the density; see: “Banjo Bridge Wood Comparisons,” ~ politzer/bridge-wood/LeVan-bridges.pdf.[6] Rim top geometry is known to astute builders as having important consequences. But unrav-eling the underlying physics seems too formidable a task — except to note that the radiationof high frequency sound from the head is dominated by motion in that region. The importanceof that region for head motion and sound radiation figures in two other related design choices. “DIY Mylar Flange for a More Mellow Banjo Head,” ~ politzer/mylar-flange/mylar-flange.pdf, describes something any-one can do to reduce the “ping” of a very tight head. Bacon and Dobson tone rings alsoperform this function; see “A Bacon Tone Ring on an Open-Back Banjo”, ~ politzer/bacon/bacon-ring.pdf.[7] “The Resonator Banjo Resonator, parts 1 and 2,” ~~