Like most bells, Hand Bells have a rich set of vibration modes/overtones/harmonics. The overtones/harmonics are different from stringed instruments. That is because strings vibrate largely in 2 dimensions, i.e., in a plane, and are constrained, i.e., attached at each end. In contrast, a bell’s ringing is created via a flexing of the bell material in 3 dimensions, without any boundary constraints.
In the so‐called English tuning of Hand Bells (made of bronze), which is followed by most Hand Bell makers in England and the United States, the principal modes are the [2,0] mode and the [3,0] mode. The strike tone of a Hand Bell is determined by tuning of the fundamental [2,0] mode, or what Pitit & Fritsen call the prime.(1) The fundamental (2,0) mode radiates a rather strong second harmonic partial. So the sound spectrum has prominent partials at the first three harmonics. Additionally, the (3,0) mode is tuned to three times the frequency of the (2,0) mode. The difference of these two principal modes in a bronze bell is exactly a perfect twelfth (one octave and a perfect fifth).
The Petit & Fritsen website refers to there being five partials they focus on in their bronze bells. They give the example of a C2 bell, whose five tones for those partials are: C1 (hum-tone); C2 Prime/fundamental; C3 Nominal; E#2 (tierce); and G2 (quint).
Hand Bells are roughly scaled in size so that the diameter (d ) is inversely proportional to the square root of frequency, except in the smallest bells, in which it varies inversely with the cube root of frequency. The thickness of the Hand Bell (t ) is then adjusted so that t/d *2 is nearly proportional to the frequency.(2) The fundamental is largely determined by the mouth diameter and wall thickness, while other partials are controlled by the profile.(3)
For economies, foundries commonly provide one bell size casting for the manufacturers to make two (or more) consecutive pitched bells. The pitch tuning of the bells by the manufacturer is done principally by removing metal from the inside, i.e., the parameter t (thickness) of the bell is decreased while the outside d (diameter) is kept the same (invariant.)
Starting from a particular size cast bell, the manufacturer tunes the bells by removing the approporiate amount of material from the inside of the bell. The higher pitched bell made from the same bell casting have more material left inside the bell. As a result, the lower pitched bell made from the same bell casting size will have a thinner wall and less weight than the higher pitched bell(s) made from the same cast size. As a result, a lower pitched bell can be lighter than the higher pitched bell made from the same size bell casting. This seems to violate the assumed “principle” that “lower bells are heavier.”(1)
FYI - Both Malmark (Plumsteadville, PA 18949) and Schulmerich (Hatfield, PA 19440) use the same foundary (Bridesburg Foundry, Whitehall, PA 18052) to cast their bells for them, which they then manufacture to create their finished products. All three businesses are located in southeastern PA.
Bridesbury Foundry does casting for a number of industies in addition to Hand Bells.
Another feature of bells is that their physics allows them to be scaled. For example, Hand Bells are scaled down from significantly larger church bells, but can sound the same fundamental pitches. The same concept of scaling is also applied in making very low pitches of bass Hand Bells, which otherwise would become even larger and heavier than they are. For example, all the Malmark aluminum Hand Bells with decreasing pitches below F#2 have almost exactly the same outside diameter, and have decreasing or about the same weight. Similar is true for Malmark’s bronze bells below F#2.
For those who may be interested, the theories of acoustic physics for bells is fairly well developed. A mathmatical overview of those theories can be reviewed in chapter two of Michael Jedamzik's book.(1)
Below the pitch of F#2, the bass bells of both Bronze and Aluminum have been scaled to avoid increasing weight. However, the aluminum bells are considerably larger in diameter than those made of bronze.(4) (A bronze Hand Bell has the same fundamental frequency as an aluminium bell whose diameter is approximately 124% larger.)(1) Another trade off with aluminum is that in addition to being much larger than a comparable bronze bell, alumininum is also are a more fragile material.
The accuracy of pitch perception by the average human at the extreme low frequencies in octave 0 is vague. Piano composers have usually only used such very low notes near A0 for tonal colors and special effects. That is because due to the incapability of most humans' hearing to clearly distinguish between the notes at very low frequencies, it is not possible to make a consistent harmonic and melodic use out of such low notes. Such notes tend to be heard as something approaching noise. Although they may be different in pitch, in practice, one very low pitch sounds similar to the next. That same limitation of accuracy of note perception for very low note Hand Bells is also certainly true, and the "muddiness," especially of the bronze bells overtones, makes it more so.(6)
[FYI - The highest pitch on a standard full piano is matched by the C8 Hand Bell. Malmark makes bells that go an octave higher than that, i.e., up to C9.]
Dominant harmonics for bronze bells are:
Dominant harmonics for aluminum bells are:
In low bass bronze bells, the multiple overtones can be perceived by listeners as producing a more “muddy” sound. In contrast, in the higher pitched bells the additional overtones of bronze bells are perceived as producing a more mellow sound.
Another factor contributing to this “muddiness of sound” at very low pitches is that large bass bronze bells radiate sound waves inefficiently. This is because the speed of bending waves in these bells is considerably lower than the speed of sound in air, a condition known as “being below the coincidence frequency.” (3-4)
An example of a very low frequency bronze Hand Bell is Mr. Sue ringing the bronze G0 [G zero, i.e., octave 0, (fundamental frequency of 24.5 Hz)] at Malmark. Click here to see and hear the G0 on YouTube Hear the G0 Hand Bell.
Mr. Sue referred to the sound of that bell in the following way –
You probably noticed that the G0 had a rather muddy sound due to the mess of its overtones.(5)
The bronze G0 is an example of several one-of-a-kind very large low pitch bells produced by Malmark that are somewhere between research tools (for testing the harmonics of really large, low pitched bronze bells) and a “because we could do it” demo. A couple other such very large research bells made by Malmark include an A0, and B0.(5)
The physics of a Hand Bell’s ringing characteristics are also influenced by the material they are made from. Metals other than bronze have different ringing properties for various reasons, including different masses and stiffness. Aluminum is an example of another metal that has noticelably different Hand Bell harmonics, and sound radiation efficiency.
In order to take advantage of aluminum's higher sound radiation efficiency and less harmonics, and thereby enhance the sound of very low pitches for bass bells, in 1991 the Malmark Company created a new bell design using aluminum rather than bronze.
Another feature of aluminum bells in contrast to bronze, is the different harmonics that aluminum bells have. They do not have as many harmonics. Their dominate harmonics are limited to:
This means ringing of aluminum bells produces much less complex overtone harmonics than produced by bronze bells. As a result, bass aluminum Hand Bells are perceived to have a much purer fundamental pitch.
As a result of the simpler harmonics and the greater radiation efficiency, the sound of low bass alumimum bells cut well through the sound of even a huge number of bronze bells. For example, the Bay View Week of Hand Bells can balance the sound well between ten bronze choirs versus only a double set of aluminum octave 2s, plus G1-B1.(4)
In addition to having lower coincidence frequencies that lead to more efficient radiation of bass notes, aluminum bells are considerably lighter in weight, and thus more easily handled by bell ringers, who even with the scaling of bronze bells, have to deal with increasingly heavier bass bronze bells as the pitch goes lower, down to about F#2.
Currently the lowest pitch bass aluminum bell in Malmark’s production range is G1, and the highest pitch production bell cast in aluminum is F#3.(3,7,8)
Mr. Sue said: Just speculating, but they (Malmark) might attempt to make aluminum bells below G1 someday. But, if they do, they’ll be really expensive; I’d guess $7,500-$10,000 each.(5)
Reasons they don’t make aluminum bells for pitches higher than F#3 likely include:(5)
- The lack of the multiple overtone harmonics in aluminum bells means the perceived timbre or mellowness of the bell is perceived to become weaker as the pitch increases. Malmark has chosen not to produce aluminum bells above the F#3 pitch, 4 steps below C4 (middle C).
- Also, the physical advantage of having a lighter aluminum bell above F#3 becomes physically less important. The next bell up from F#3 is G3, which in bronze weighs approximately 4.25 pounds. A reasonably fit person can play such bells for an entire rehearsal or concert without too much strain.
- An examination of the weights of the F#3 shows the bronze bell is about a pound heavier than the aluminum bell at 6.15 pounds.The bronze G3 at 4.25 pounds is almost two pounds lighter than the bronze F#3. The G3 is clearly manufactured from a smaller foundary bell casting than the F#3.
- Additionally, aluminum is a softer metal than bronze, and so manufacture and durability possibly are also important considerations.
Larry Sue holding an aluminum C2 Hand Bell at Bay View Week of Handbells
In the above photo, Mr. Sue is holding a nine and a half pound aluminum C2 Hand Bell manufactured by Malmark from a bell cast by the Bridesburg foundry. [C2 is in the second octave down - or to the left on a standard piano keyboard - from the piano's middle C, which is a C4.] (A bronze C2 bell weighs just over 50% more at a bit over fourteen pounds, but is only about half as large.) Another advantage of an aluminum bell is cheaper price. An aluminum C2 bell is about $5,000.00, while a bronze C2 bell is about $6,400.00.
1 Handbell_Compendium-1, Michael Jedamzik, can be read online or saved as a pdf, http://www.handglockenchor-wiedensahl.de/wp-content/uploads/2017/03/Handbell_Compendium-1.pdf
2 Acoustics of handbells. The Journal of the Acoustical Society of America 91, 2450 (1992) https://asa.scitation.org/doi/abs/10.1121/1.403097
3 Craftsmanship, https://www.malmark.com/html/craftsmanship.php
4Bass Handbells of Aluminum, Thomas D. Rossing, Deepak Gangadharan, Edward R. Mansell and Jacob H. Malta, Published online by Cambridge University Press: 29 November 2013, https://www.cambridge.org/core/journals/mrs-bulletin/article/bass-handbells-of-aluminum/A359EAC77D300AD5C274D9B6A33BA091.
5 Email from Mr. Sue, 4/18/2020.
6 Why is the lowest note on the piano an A?, Music: Practice & Theory Stack Exchange, Feb., 10, 2015, https://music.stackexchange.com/questions/29675/why-is-the-lowest-note-on-the-piano-an-a
7 To date, only Malmark produces aluminum Hand Bells. Whether that might be associated with a copyright or not is unknown to the author.
8 Malmark has a history of developing new products. A list of new products, and when they were introduced to the public, is found at Malmark Inovations.
9Handbell, Wikipedia, https://en.wikipedia.org/wiki/Handbell