Banjo Physics 411

by David Politzer



The banjo clearly has many features in common with other stringed instruments. It certainly has plenty of unique features, beloved of players and luthiers, alike. And the physics of much of this is well-understood. This would be the content of "Banjo Physics 101." E.g., what determines the pitch of a plucked string? If you're curious about that, you'll have to look elsewhere. What I hope to do here is report on my research efforts to explore issues of banjo acoustics, banjo construction, banjo mechanics, and banjo science more generally, addressing things I don't understand but believe I could. Why does a banjo sound different from a guitar? Or, even more to the point, why does one banjo sound different from another? It should be unnecessary to mention but unfortunately it needs repeating: the biggest differences one hears between performances come from the musician and not the hardware!

There are always some projects in the works. So, if you find anything here of interest, you might want to check back to see what's new since.


Here is something S.S. Stewart wrote about professional acousticians of musical instruments.



NEW IN AUGUST 2017:

Banjo Drum Physics -- theoretical preliminaries is 30-plus pages thick with differential equations, Fourier series, and complex numbers (but no sound files). It is, of course, aimed at helping understand how banjos work. In particular, it presents the simplest way I could figure out to do the physics of the effect of the air inside the pot on the produced sound -- via its interaction with the head.

The goal is to gain some overall perspective. Correct, formal solutions of the equations have been known for over a hundred years, but to figure out what they're telling me, particularly as relates to the banjo, required a simpler approach than any I'd found in the literature.



MARCH 2017:

NOTE ADDED: A flange of this sort is a key aspect of the Indian tabla and the Remo line of "Powerstroke" drum heads.

DYI Mylar Flange for a More Mellow Banjo Head describes how you can cut a second mylar head and install it under your regular head to reduce some of the sharpest "ping" sounds --- fast, inexpensive, and totally reversible.

Here are the sounds of head taps before and after. For links to actual plucked strings (and instructions), click here for the write-up.



SEPTEMBER 2016:

A Hoseus Banjo Restoration describes some simple, amateur efforts to bring a bottom-of-the-line 19th Century banjo back to life. It features an 1886 patented head support system that never really caught on. There's no physics, just banjo tomfoolery.

The accompanying sound files (also linked in the pdf) are: ("steel" refers to steel strings and a 2 oz. bridge; "fish" refers to fishing line strings and a 1 oz. bridge)

steel-HM.mp3

steel-JH.mp3

fish-HM.mp3

fish-JH.mp3



JULY 2016:

(The file now linked is an improvement over earlier versions. Added are two ways to look more carefully at the two resonances below 800 Hz.)

Banjo Rim Height and Sound in the Pot is a follow-up to the first endeavor in this banjo physics series. (See DECEMBER 2013, below.) That study focused on the two lowest frequency modes. The totally new material here is an iquiry into the effect of rim height on the whole spectrum. Measurements on the three different height rims bear out the standard calculation of sound resonances inside cylindrical containers. While this might not sound too impressive, just looking at the results tells you something very significant about the role of rim height in the transformation of string pluck to radiated sound. For the acousticians of musical instruments, a careful repeat of the 2013 measurements establishes a stark contrast between the banjo and the guitar (an other wood-topped instruments) regarding the Helmholtz resonance.



JUNE 2016:

A variety of acoustics experiments are presented in Physics of the Bacon Internal Resonator Banjo in an effort to capture what is going on inside the pot, its relation to the design, and its effect on the sound. The right experiments coupled with some simple ideas yield a fair number of qualitative and quantitative insights.

(That's an 8 MB file. If you're going to print it out on a B-and-W printer, here is a 6 MB version, already in grey-scale.)

Three sound files are linked in the write-up but are also available here:

pluck comparisons

frailed sample A

frailed sample B

(8/2/16: The discussion of FIG. 14 misidentifies which pot is which. If you're serious, you can get it right; you've just got to picture the actual pot geometries. Why didn't I fix it? This same manuscript is in two other "permanent" archives. It's hard to say whether it's worth messing with them.)

Air modes of the Bacon internal resonator banjo is a terser, more "professional" write up of the same investigation (14 pages of text versus 22 -- but you'd miss some great photos). It was summarily rejected by the Journal of the Acoustical Society of America. Apparently, "the conclusions are very weak and simple." It was faulted for not explicitly referencing relevant previous work. However, the only relevant previous work is over 150 years old and a standard part of the sophomore physics curriculum where I work. There also is no section titled "Methods," as well as other violations of the Scientific Method as articulated by Francis Bacon (perhaps no relation) in the 16th Century and taught to STEM junior high students everywhere.



APRIL 2016:

A Bacon Tone Ring on an Open-Back Banjo is an investigation into the sound of a reproduction Bacon tone ring. This ring is an integral part of Bacon's internal resonator design. However, while investigating the internal resonator, I was surprised by what the tone ring does on its own.

There are sound files linked in the write-up, but I include the first one here as a teaser. What you hear are the taps of a piano hammer on four Deering Goodtime pots, tapping from center to rim. The pots, in order, are 1) an old wood rim, 2) an old wood rim fitted with a 1/4" diameter brass ring, 3) a new wood rim, and 4) a new wood rim fitted with a Bacon tone ring.

I found the differences to be noteworthy and something of a challenge to explain. The write-up is long but has no equations. It also has no triumphal confirmation of theory by experiment, but I believe that the suggested perspectives are enlightening in this context and more broadly.



DECEMBER 2015:

Banjo Bridge Wood Comparisons compares bridges made of different kinds of wood. (!) Together with luthier/designer Ken LeVan, we compared bridges of the same design and weight but very different species, i.e., bamboo versus mahogany and spruce versus walnut. As judged by spectral analysis, the differences were discernable but very slight.

You can listen to sound files of actual playing with those bridges on the same banjo here, in this directory. The bridges are identified there only by number. You can and should do a "blind" listening test and only consult the Banjo Bridge Wood Comparisons manuscript key (at the end of the paper) after forming your own opinions.

Design and weight certainly impact the sound. However, by matching bridges as we did, there's not much left to make a difference. The speeds of sound through the bridges may be different but are too high to matter acoustically. And the differences in hardness, which would impact flexibility, are largely compensated by the matching of weight. (I.e., the softer woods are less dense; so to make up the weight, there's more of it [in the "thickness" dimension].)



JUNE 2015:

The Resonator Banjo Resonator, part 2: What makes 'em "really crack"? explores a way to get a dramatic quantification in the differences in sound between different backs -- in particular, resonators versus open backs. The big differences are all above 4500 Hz, with sound energy dissipation as the primary source of difference and the nature of sound hole projection secondary. A digital multichannel equalizer defined by the measured difference on one banjo can be applied to any openback recording and give it the sound of a resonator banjo.

The relevant sound sample files are ppA.mp3, ppB.mp3, ppC.mp3, and ppD.mp3. Good speakers help. You'll have to read to find of what they are and which is which.

I came across a very relevant tidbit from Jim Rae. (See p. 5.) He writes, "At least 99% of a banjo's sound power occurs below 5000 Hz,..." Jim has done extensive measurements on resonator banjos, some in collaboration with Tom Rossing (e.g., see The Science of String Instruments) and some as consultant to Steve Huber in the development of the Huber "Truetone" rim and ring. That's why I initially only looked below 5000 Hz. Noting the measurements presented in FIG.s 2 and 3 in part 2:... (above), that's why overall loudness is hardly effected by the resonator but the resonator is a significant factor in tone determined by yet higher frequencies.



MAY 2015:

The Resonator Banjo Resonator, part 1: Overall Loudness presents the recordings and measurements that convinced me that the resonator on a resonator banjo does not make the sound any louder. It was a surprise to me, though it may be well-known to others. I stumbled on it while trying to learn about sound holes. My modest conclusions there (and perhaps a couple of new perspectives) are described in the next item, Sound Hole Sound.

The accompanying Resonator Banjo sound file is here, as well as being linked within the article.

AND:

Helmholtz and Rayleigh did the classic work on the flow through a sound hole, which determines how much the hole contributes to the volume of the produced sound. In Sound Hole Sound, I describe their work and do some measurements relevant to real instruments' geometries. In particular, at least a qualitative account is given of the the effects of the side walls of the sound box and the possibility of multiple holes. In the end, this discussion has essentially no relevance to banjos (except, perhaps, to cigar box instruments).

Also, this article is heavy on the physics (albeit fairly elementary) and very light on the music.



MAY 2015:

Writer and filmmaker Jim Carrier interviewed me about Banjo Physics for a half-hour science show on WORT-FM out of Madison, Wisconsin. They archive their shows for later listening.

A couple of years ago, Jim made a wonderful documentary, The Librarian and the Banjo about a very important book, Sinful Tunes and Spirituals: Black Folk Music to the Civil War, and its amazing author, Dena Epstein. It is a major, pioneering piece of historical scholarship about an era and subjects previously largely ignored. Banjo history is really just only one small part of that story.



MARCH 2015:

Banjo Bridge Mutes begins with a demonstration and simple explanation of how and why mutes that clamp to the bridge increase sustain and mellow the sound --- in addition to making things quieter. Not for the faint of heart is the ensuing, only partly successful attempt to make that discusion more precise and quantitative.

mute sound file



Simple Anti-Tone Rings for the Banjo: Do-it-yourself Electric, Surgical, and McGhee tone rings that can be easily and inexpensively installed to make modern banjos sound old.



JULY 2014:

Zany strings and finicky banjo bridges explains how tiny details of the string-bridge connection dramatically effect how the strings vibrate and the sound they produce. This version is equation-free; the math details are in The plucked string: an example of non-normal dynamics, which provides an elementary introduction to the more general subject of transient growth and "non-normal" equations -- published in the American Journal of Physics, Am. J. Phys. 83, 403 (2015). (This is a slightly expanded and improved version than the original.)



APRIL 2014:

String Stretching, Frequency Modulation, and Banjo Clang That's a pdf file. If you read it on a Web-enabled device, the accompanying sound files may appear with live links, in blue boxes. If the blue boxes aren't apparent, you should retrieve those files. The URL's are listed as you read along, but they're all collected in a single directory here. (The text and sound files are revised as of 4/22/14; the changes from the originals are minor -- unless you caught some mistakes, in which case you might check whether I caught the same ones.)


Here is a very short version of that same idea, Banjo timbre from string stretching and frequency modulation, published in Acta Acoustica (united with Acoustica), AAuA 101(1) 1, January 2015. However, the language is considerably more technical. It's been slightly revised since my first posting here in response to a certain amount of professional acoustician refereeing.



DECEMBER 2013:

NOTE ADDED in 2016: The JULY 2016 entry extends and improves on this study. The write-up lacks much of the folksy charm of this first banjo report, but, for banjo physics understanding, it's an essential companion piece on this subject.

This was my first Banjo Physics project, written up in December 2013: The Open Back of the Open-Back Banjo. There are four essential, accompanying sound files which you'll need to follow along with the manuscript. (You might want to save them on the side, to listen as you read.) They are:

head taps, while opening up

the sound of Banjo A

the sound of Banjo B

the sound of Banjo C




glass banjo by M. Desy; soft banjo by Sally Suzuki



middle

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