HARMONIC BELLS:DESIGN, CARILLONS, ORCHESTRAL BELLS and HANDBELLS

Australian Bell has invented a series of bells with harmonic overtones. These are currently being patented in Europe and the USA. Harmonic bells have the clearest pitch salience yet achieved in any bell or other tuned percussion instrument, and can be played in any musical key without dissonance caused by inharmonic partials.

The largest harmonic bell yet cast is a little over one tonne, with a diameter of 1.2 meters and a fundamental frequency of 73 Hz (D2, almost 2 octaves below middle C). The smallest is a handbell just 80mm in diameter with a fundamental frequency of 2488 Hz (D#7, just over 3 octaves above middle C).

Three Harmonic bell shapes

HARMONIC BELL DESIGN

Harmonic bells have been designed for use in four contexts:

carillons, large bell installations, orchestral bells, and handbells.

The harmonic bell was designed using Finite Element Analysis (FEA) with shape optimisation. The first stage of the design process was to find bell shapes that vibrated with only circumferential modes within the range of partials to be tuned. Circumferential mode frequencies remain approximately evenly spaced for a wide range of shapes, and so could be tuned to a harmonic series. If a bell had any mixed modes interspersed amongst the circumferential modes it can only be tuned harmonicially by tuning the mixed mode to the same frequency as one of the circumferential modes. This leads to one partial sounding much louder than the surrounding partials which disturbs the pitch salience. Such a bell is also very difficult to tune.

Bell shapes with only circumferental modes in the first 7 partials were found by increasing the stiffness of the form to axial vibration through increased cone angles and increased wall thickness toward the crown of the bell. Typically, harmonic bell walls taper to thinner walls at the mouth of the bell. Once the approximate shape of the bell has been found, shape optimisation was used to fine tune the frequency of each mode to the harmonic series. The bell can then be scaled to tune given that frequency is inversly proportional to size (eg. twice the size vibrates at exactly half the frequency).

However one shape cannot be used for the entire frequency range described above. For example, scaling the wider angle bell to C6 (two octaves above middle C) would result in impossibly thin walls. Since the frequency of circumferential modes can be increased by reducing the circumference, the cone angle is decreased and the wall taper increased to arrive at the generally thicker and higher pitched harmonic bell shown below. For larger bells a convex curve can also be used to stiffen the axial modes.

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Two Harmonic bell profiles
	Spectrum of an harmonic bell with a fundamental of 220 Hz.
	This graph shows the modelled pitch perception for an harmonic bell with a fundamental frequency of 220 Hz (see spectrum above). Notice the pitch sensation at 220Hz is twice as strong as any other calculated percept. The only perceivable aspects to the sound above 220 Hz are the second partial and third partials as much weaker virtual pitch sensations. The lower virtual pitch percepts are subharmonics of 220 Hz and are very weak, partly spurious results. The average pure tonalness of this bell is 1.16 as compared to the european bell shown elsewhere at 0.81 and the 8/5 polytone at 0.46.

Harmonic bells have been used in all three of the Federation Bell projects and are expected to have a wide range of musical applications as discussed below.

CARILLONS A set of 32 harmonic bells is part of the Federation Field of Bells to be installed in the new riverside park in Melbourne. Since these bells are intended to be heard at very close range they could be designed with relatively thin walls compared to tower bells. The thicker bell profile shown above was specifically designed for installation in carillons.
ORCHESTRAL BELLS Harmonic bells are ideal for inclusion in instrumental ensembles. They have perfect pitch salience and no inharmonic partials to create dissonance amongst standard instrumentation and tuning. A two octave, chromatically tuned harmonic bell set has been presented to the Melbourne Symphony Orchestra. Harmonic bells can be tuned to the same pitch range as sets of tubular bells currently found in most symphony orchestras. They will be able to be used in repertoir that includes tubular bells with far greater certainty of pitch and consonance. The thin walled profile shown above is prefered in this context as it produces lower pitches at less wieght than thicker profiles suggested for carillons. Thinner profiles are also louder for the same energy of impact.
HANDBELLS Australian Bell has manufactured 2001 harmonic handbells as part of the celebrations of the Centenary of Federation of Australia. These bells are chromatically tuned over 2 octaves starting at E5. A special set of 1/4 tone bells has also been made for a piece for choir and bells by Constantine Koukias.