rims.pdf

HDP: 24 – 9

Banjo Rim Sound

David Politzer

∗

California Institute of Technology

(Dated: September 17, 2024)

Four different wood rims are equipped with the same kind of head, tension ring,

hooks, nuts, and shoes. Recordings of taps on the rims and heads yield informa-

tion on their stiffness and impact on head vibration. A single neck with its own

strings, bridge, tailpiece, and co-rod is attached to one rim after the other. The rims

produce different pluck-sound rise times and sustain, in accord with the physics of

weight, stiffness, and coupled oscillators. Timbre variations are complicated. The

combined head-rim systems show slight variations in head resonant frequencies but

clear variations in the amplitudes of head motions in response to excitation by the

bridge at different frequencies. The details presumably depend on the proximity in

frequency of head and rim modes that can couple strongly because of their matching

geometries. Thus, the rims give each banjo a different sound. Sound samples of taps,

plucks, and played tunes are included. Some may conclude that the differences are

not that different from what can be achieved with variations in playing technique

and set-up.

∗

politzer@theory.caltech.edu; http://www.its.caltech.edu/

∼

politzer; 452-48 Caltech, Pasadena CA 91125

Banjo Rim Sound

I. BACKGROUND

Players have opinions about the pro’s and con’s of different rims. But the differences

are subtle and are not totally distinct from what can be achieved by various adjustments

or swappable parts or playing technique. Furthermore, the player is in a privileged position

relative to any other listener. The sound there is most dramatic and is compared immediately

to an intention or expectation. In judging attack or rise time, only the player knows when

the string was plucked. Underscoring the subtlety from a physics perspective, zeroth order

modeling with an assumption of an ideal, perfectly rigid rim does a fair job of accounting for

banjo sound, including adjustments and parts choices that are available post-manufacture.[1]

This study addresses rim sound by comparing four instruments that differ only by the

species and construction of their wooden rims. A single neck, co-rod, set of strings, bridge,

and tailpiece are mounted sequentially on the rims. The complete pots are assembled with

the same kind of hardware (head, tension ring, hooks, nuts, and shoes) with current stock

from Deering Banjos.

II. INTRODUCTION

A technique described in section

VII produces plucks that are highly reproducible in

location and force. This is the sound of a single such pluck on the open 1

string on each

of the assembled banjos:

https://www.its.caltech.edu/

∼

politzer/rims/1st-14th-singles-4sec.mp3

They sound different. (For the hearing challenged, turn up the volume and/or try an up-

grade from laptop speakers.)

The differences are greatest in the earlier part of the sounds. So, the following plucks are

trimmed to the first 0.25 seconds. (That is sedate relative to most banjo playing.) And, to

drive the point home, you will hear a series of ten successive plucks on each banjo:

https://www.its.caltech.edu/

∼

politzer/rims/four-rims-1st-10-TRIMMED.mp3

Just as neck flexing effects the sound by its direct impact on string vibration[2], rim

flexing effects the sound by its direct impact on head vibration. The goal of what follows is

to characterize the sound differences with sufficient precision that some connection can be

made to the physical properties of the rims.

The first three rims are about 9

′′

thick. They are similar but sufficiently different

to sound somewhat different. The fourth rim is about 5

′′

thick and, therefore, much

lighter and much more flexible. Interestingly, those differences are sufficiently large to put

its physics description into a different realm. It works as a banjo, but it works differently.

A. jump to the conclusion?

The pluck sounds linked above aren’t music. And the next two sets of sounds presented

below help characterize the details of the rims, but they are even further from playing the

banjo. If you simply want to hear some music played on the four rims, jump to section

VIII

for some brief samples. However, some of the differences can be hard to discern and might

not seem distinguishable from slight variation in playing technique or recording set-up.

B. outline

In section

IV, the sound of taps on the rims themselves allows a quantitative estimate

of their relative stiffness.

In section

V, mounting the heads and tapping in a reproducible fashion reveals the small

but discernible differences in head vibration produced by the different rims.

Banjos are assembled for section

VI. An impulse hammer gives reproducible taps on

the bridge with strings both damped and open. The recordings simply sound like taps,

albeit slightly different from each other. The damped string sounds are a reflection of the

bridge admittance, a function that encodes the bridge motion that would be produced by

the various frequency components of a properly plucked string. The open string sounds give

a picture of the inharmonic partials that accompany every string pluck.[3]

The 1

string pluck sounds linked above are examined in detail in section

VII. Rise time

and sustain are gross features rather than being frequency specific. The differences between

the first three thick rims behave in these respects in ways that follow directly from the

weights and stiffnesses. However, understanding the sound of the thin rim requires reference

to its frequency spectrum, which is substantially lower than the corresponding resonances

of the other rims because of its relative flexibility.

The spectrograms of the 1

string pluck sounds reveal many slight differences in frequency

component intensity and time dependence. There are no simple patterns. Rather, they

must be the consequence of the interplay of the head and rim resonances. Brief qualitative

explanations are presented.

Played music samples are presented in section

VIII. There is also a set of alternate takes,

presented without the rim identity as a test of whether one can actually identify the rims

by their sound.

III. THE RIMS

In the figures and sound samples, the fours rims are always presented in the same order.

It’s actually by age, oldest first. The first is from a year 2000 Deering Goodtime and has 11

plies of maple. So, it is designated “11 ply.” The second is from an example of the upgraded

Goodtime, i.e., 3-ply maple — and labeled “3 ply.” The third is from the limited edition

2023 “Cherry” Goodtime. It has three plies: cherry - maple - cherry — and labeled simply

“cherry.” These rims are all of the same dimensions. In particular, they are

∼

′′

thick. The

fourth is fashioned from a 12-ply cherry drum shell (from https://nordicshells.com/) that is

∼

8mm (

∼

′′

) thick — and labeled “8mm.” TABLE I lists the labels and descriptions for

convenient reference.

label

description – see p. 1 photo

11 ply

laminated maple

3 ply

bent maple

cherry

3-ply bent cherry-maple-cherry

8mm

12-ply laminated cherry

TABLE I: identifying rim labels

IV. BANJO RIM TAPS

The rims are suspended from one of their hanger bolt holes and tapped at 45

◦

from the

suspension point. A directional mic is positioned close to the outer surface and at 45

◦

the other side of the suspension point. These locations emphasize exciting and recording

the lowest mode. (See Appendix B regarding cylindrical shell normal modes.) For these

recordings, the same current stock shoes (the things that hold the tension hooks & nuts)

are attached to each rim. The 16 shoes and their screws together weigh 5 oz.

This is the actual sound, and FIG.s 1 & 2 are waveforms, spectrograms, and spectra of

those sounds:

https://www.its.caltech.edu/

∼

politzer/rims/rim-taps-one-each.mp3

FIG. 1: taps on rims

A. lessons from a bit of physics

The actual sounds of the rims themselves have no direct relation to the sounds produced

by the assembled banjos, where the rims are constrained to move with the tension ring and

edge of the head. However, a bit of physics brings some order to the analysis of these rim

sounds and produces some useful results. In particular, we can deduce the rims’ relative

stiffnesses.

First, recall the formula for the frequency

of the ideal mass on a spring:

r

FIG. 2: rim tap spectra

which applies to any oscillation at small enough amplitude.

The resonant frequencies of a particular rim are given by a series of dimensionless (pure)

numbers times a single frequency, which is obviously proportional to the square root of some

measure of its stiffness divided by its total mass.

The four rims’ observed fundamental frequencies and measured weights are displayed

in TABLE II. The third column is the stiffness deduced from the frequencies and weights

relative to the results for 3-ply maple. As long as the fundamental modes’ motions are

perpendicular to the thickness, this is legitimate because all four rims’ dimensions are the

same in directions perpendicular to the thickness. The rank order of stiffnesses are what

many people might have expected.

Furthermore, the dimensionless numbers that give the series of resonant frequencies for

a particular rim should be (approximately) the same for all rims that are sufficiently similar

in their geometry and differ most in their weight and stiffness. For thin, squat cylinders,

this will work best for the first few resonances, i.e., before the rank order of different types

of motions gets scrambled by the differences in physical parameters. (See APPENDIX B.)

FIG.s 1 & 2 bear some resemblance to this expectation. In FIG. 1, the linear frequency

axis of the spectrograms shows similar ratios of the spacing of first to second and second

to third resonances. In Fig. 2 with a logarithmic frequency scale, this shows up as roughly

equal spacing of peak 1 to 2 and 2 to 3.

rim

fundamental frequency – Hz

rim & shoe weight – oz.

relative stiffness

11 ply

316

0.83

3 ply

354

cherry

335

0.90

8mm

133

0.10

TABLE II: wood rims & shoes

The second relevant bit of physics is that, for a thin, uniform material, the stiffness to

bending perpendicular to the thin direction is proportional to the cube of its thickness:

stiffness

∝

thickness

Wood references give standard approximate values for stiffness. In particular, the ratio of

Young’s modulus for cherry to maple is 0.83 (accidentally approximately equal but totally

unrelated to the relative stiffness of the 11-ply rim, as given in TABLE II). This allows a

reasonable comparison of the two laminated rims, i.e. 8mm cherry to 11-ply maple. The

stiffness ratio of the 8mm to 11 ply deduced from weight and fundamental frequency as in

TABLE II is 0.12. This is to be compared to the ratio of Young’s moduli times the cube of

the ratio of thicknesses: 0

= 0.14.

(In the realm of banjo physics, I say that’s

a triumph. Were those laminated with different glue? Is that relevant? If nothing else, the

closeness of the Young’s modulus-based estimate to the measured/deduced value demonstrates

that the discussion above is not obviously wrong.

Also, the photo on page 1 suggests that the 11 ply and 8mm rims are “plywood,” i.e., with

grain going in two different directions. That makes them more stable but inherently less stiff

than were all layers with grain going around. Even though the bending moduli are different in

different directions, the two laminated rims have roughly the same mix of the two directions.

So, the ratio of rim stiffness may well be about right using just the along-the-grain value.)

B. shoes vs. no shoes

In retrospect, recording rim taps of the wood alone, i.e., without the shoes, would have

been clearer in its interpretation — even if a bit further from the action of the assembled

instruments.

The scaling of the resonant frequencies of each rim in proportion to, say, each lowest

frequency, is only true if the densities of all rims are distributed in the same proportions.

Inclusion of the 5 oz. of shoes & screws violates this. (That is over 30% of the weight listed

for the 8mm rim and

∼

23% for the others.) For these rims’ profiles, the lowest mode

presumably has four straight node lines spaced 90

◦

apart around the circle and no circular

node line going all the way around. (See Appendix B.) The next mode higher in frequency

is likely one with six equally spaced straight lines. The next one might be the one with

eight. Where the modes with circle node lines enter the series is difficult to say without a

detailed numerical calculation. Their frequencies depend on how the mass is distributed in

the vertical direction, i.e., perpendicular to the plane of the head. Because the same type

of shoes were used on all rims, the

relative

mass distributions were different in the vertical

direction.

In any case, deducing relative stiffness using the lowest mode frequency works just as

well with shoes as it would without as long as the lowest mode for each rim has the same

spatial shape, i.e., node line pattern suggested above. If so, the inertia of that mode is

simply proportional to the total wood and shoe weight for each particular rim.

V. BANJO POT DRUM TAPS

Each rim was supplied with head, tension ring, hooks, and nuts from current Deering

factory stock and tightened to 88 on a DrumDial. (The tensions were left to stabilize and

readjusted over a period of a week.)

The next set of sounds are well-controlled taps on the head. Without neck, bridge, &

strings, these are drums. The taps were performed by dropping a pencil from a position

and height using a rigid frame pictured in FIG. 3 as a guide. I analyzed a set of ten clean

ones for each rim. The differences from tap to tap were much smaller than the differences

between rims. One consistency check was to repeat a set of taps on the first rim (the 11

ply) after doing all the others – to confirm that it actually was just like the first round and

obviously different from the others.

FIG. 3: pencil/pot tap rig and mic

The following is the sound of one representative tap for each sequence of ten. Note that

the 11 ply is repeated, the second version having been recorded after the other three to

demonstrate the consistency of the tap mechanism. However, it is played here immediately

after the first one — to demonstrate that they sound the same. So, there are five head taps:

11 ply, 11ply, 3 ply, cherry, and 8mm.

https://www.its.caltech.edu/

∼

politzer/rims/drum-taps-one-each.mp3

Some of the complexity and subtlety of rim physics is already apparent in these sounds.

Rayleigh, who was seriously interested in the physics of drums, made the distinction of

those with

definite

pitch to those of

indefinite

pitch. “Definite” required that the low lying

resonant frequencies be integer multiples or nearly so. The discerned pitch frequency is that

common factor. Examples include the kettledrum, which Rayleigh studied and the Indian

tabla, whose heads sport a patch of flexible iron-weighted clay that C.V. Raman showed

produced the magic ratios. More generally, a series of strong frequency components whose

spacings are integer multiples of a single frequency can be heard as that spacing frequency

even if they are all offset from that one as a base.[4]

It is more common for drums to have frequencies in rough agreement with the Bessel

function zeros analysis of the “ideal” drumhead. Most human brains do not recognize that

series as perfectly well-ordered. It is certainly not “musical.”

Nevertheless, human brains struggle mightily to discern a particular pitch wherever pos-

sible — even when technically “indefinite.” The wood rim taps in the previous sound sample

likely seem to have pitches. Likewise, the indefinite pitch drums in the immediately preced-

ing sound sample also likely have associated pitches. However, people do not all agree on

which is higher and which is lower![5] For example, does the lowest pitch of each set belong

to the same rim? And which in the sequence is, indeed, the lowest? People answer with

confidence, but their answers might be different.

FIG. 4 shows the waveforms and spectrograms of those chosen representative pot drum

taps. And FIG. 5 shows computed spectra for each rim, averaged over the set of 10 each.

FIG. 4: spectrograms of the representative tap for each rim, with 11 ply repeated

So, the sounds, spectrograms, and spectra of the four pots are definitely different.

There is some qualitative physics in FIG.s 4 & 5 that will contribute to understanding

some of what happens with the fully assembled banjos.

FIG. 5: pot drum tap spectra – average of 10 for each pot

The brain has the knack of listening to and analyzing many different aspects of heard

sound before deciding and reporting “what it sounds like.” No one spectrogram can capture

that. Rather, the brain considers a great many analyses and criteria simultaneously. It

also compares what it is hearing to what it heard in the past on all time scales, i.e., from

immediately before to long ago.

FIG. 6 displays spectrograms of the same representative plucks as used to produce FIG. 4.

However, the spectrogram analysis parameters are quite different. The frequency range goes

to 6000 Hz, and the frequency resolution is much lower, which allows a much higher time

resolution. For all string instruments, on average, the spectrum decreases substantially

with increasing frequency. Audacity offers a feature that allows viewing high frequency

ranges along with low. In particular, their “High Boost” amplifies the frequency analyzed

signal by some number of decibels per decade of frequency. That is used in FIG. 6 to get

meaningful comparisons up to 6 kHz of differences that certainly are apparent with normal,

good hearing.

FIG. 6: spectrograms of the representative tap for each rim, with 11 ply repeated

A. coupling of the rim to the head

We can think of any head motion as a superposition of its normal modes, each with its

own frequency and spatial pattern along the top of the rim. Considered individually, these

each tug on the rim with their own frequency and pattern. Similarly, rim motion can be

decomposed into normal modes. And we can imagine the total interaction as a pairwise sum.

(This is an application of the approximate linearity of the physics for small amplitudes.)

It is natural to think of the head as a driver of the rim. The effect of a particular head

mode on a particular rim mode depends on four things. 1) There’s the tension of the head

and the amplitude of the particular head mode. 2) The rim response is inversely proportional

to a measure of its mode inertia. Those are equal to the total rim mass times a series of

dimensionless numbers that depend on the mode number. As discussed in section

III (and

Appendix B), that series depends on the spatial distribution of the mass. The four rims are

sufficiently similar that these numbers are expected to be about equal for the lowest few

modes. 3) There is an effective coupling which depends on the degree of spatial matching.

This is greatest when the head and rim modes have the same number of nodes around the

edge of the head and the top of the rim. There can be nearly complete cancellation if they do

not. The situation is simple if both systems have exact circular symmetry. On real banjos,

both are more complex. And 4) there is the resonant response as suggested by the curves in

FIG.s 7 & 8. A given rim mode responds most to head modes of nearby frequencies. The

effect of higher head frequencies goes to zero, while all lower head frequencies contribute to

the rim motion.

FIG. 7: generic shape of resonant amplitude response; note the behavior away from the

peak. The horizontal axis is the driving frequency, and the peak is at the oscillator’s

natural frequency.

FIG. 8: amplitude for sinusoidal forcing of a harmonic oscillator, with and without

damping, as a function of forcing frequency

FIG. 8 is a reminder of the effect of damping in the driven rim mode. The green curve is

for zero damping and goes to infinity on resonance. The blue curve has twice the damping

as the yellow one. When it is said that damping increases the width in frequency of the

resonance, that is with respect to FWHM – full width at half maximum – as a measure of

width. It is nevertheless true that damping reduces the maximum response amplitude but

has little to no effect away from the resonant frequency.

Of course, the rim pushes back on the head, establishing joint modes of the coupled

system. The sound of tapping on the head or plucking the strings is produced overwhelmingly

by the head because the head is a much more efficient radiator of sound than the rim or

strings. (Hunting down errant string buzz is an amusing example. Wherever its origin, the

sound of the buzz comes off the head.)

In FIG.s 4 & 5, we see peaks that are generally similar from rim to rim. The strongest

ones are presumably descendants of modes of an ideal head with a perfectly rigid rim. There

are differences in frequency and intensity due to the way in which the complete set of rim

modes alter the particular head mode due to their coupling. For the comparisons in the

figures to be relevant in terms of intensities and not just frequencies, it is important that

the excitation was in the same place and with the same impulse in each case. (That was the

purpose of the pencil-drop guide structure pictured in FIG. 3.)

One interesting mode is the lowest one, i.e., just below 300 Hz. The lowest pure-head

mode is simply up and down simultaneously over the whole head, with no node lines. This

mode is expected not to couple to the rim in any appreciable way. Any rim mode for

which the whole top edge goes in simultaneously and then goes out is what Rayleigh termed

“extensional.” The whole rim has to stretch and then compress. “Inextensional” modes

involve bending. Their restoring force and, hence, frequency, is generally substantially lower.

This, of course, depends on the actual dimension ratios. (See Appendix B.) In FIG.s. 4 & 5

the lowest frequencies of the four rims are essentially equal, reflecting the further fact that

they were the same type and size of head, tensioned to the same DrumDial reading. FIG. 5

reveals that they are not quite the same intensity, which reflects the fact that the same total

impulse gets divided among the modes as they see fit, and the higher rim modes do, indeed,

couple noticeably to the head.

VI. BRIDGE TAPS

The next round involves fully assembled banjos. As stressed before, the same one neck

(Goodtime-CNCed white oak), tension ring, bridge, string set, and tailpiece were used on

all four rims. Still not normally playing, the protocol was to tap on the bridge, first with all

strings damped and then with all strings open. The rig shown in FIG. 9 produced nearly

identical taps — in the same place and with the same impulse.

Below are sounds, waveforms, and spectrograms of one representative tap for each rim.

FIG. 9: impulse hammer for bridge taps

https://www.its.caltech.edu/

∼

politzer/rims/hammer-strings-damped-one-each.mp3

https://www.its.caltech.edu/

∼

politzer/rims/hammer-strings-open-one-each.mp3

The sounds are different for each rim. Even the string component of the taps with open

strings are impacted by the differing rim interactions with the head. While the dominant

physics is the same for all rims, as in the rim drum discussion, head modes get combined

with rim modes in ways that depend both on the details of the frequency spectra and on

the spatial geometry of the modes. The result is variation from rim to rim that cannot be

described as a single, simple trend.

Adding the strings has added further physics. To be sure, the static down-pressure of the

bridge on the head has altered what might be thought of as the pure head modes. A big

change due to the strings is that they significantly damp the head vibration, i.e., take away

energy, compared to the “drum taps” without strings – whether the strings themselves are

damped or not.