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POLYTONE & OTHER BELLS  

Bells have many more timbral possibilities than most other instruments. Bells vibrate flexurally in three dimensions and so have inharmonic partials. Therefore it is possible to design bells so that their partial frequencies actually contain sets of frequencies in more than one harmonic series, and produce more than one pitch percept.

Australian Bell has designed a series of bells where each bell has two or three perceived pitches. They sound like musical chords.

The European carillon bell has more than one perceived pitch. A strong pitch percept is based at the fundamental frequency of the bell and is supported by the partials with frequencies at 2, 3 and 4 times the fundamental frequency. Another weaker pitch percept is based on the partial at 2.4 times the frequency of the fundamental (a minor third interval). Australian Bell has designed a minor third polytone bell in which partials at 3.6 and 4.8 times the fundamental frequency strengthen the minor third percept to about the same strength as the fundamental (depending on how the bell is struck).

Australian bell has designed and cast eight just tuned polytone bells with a fundamental pitch of D4 (293Hz) for the Federation Bell project (as shown in the table below). The 6/5 cylinder was replaced by the European 6/5 bell as a reference to this bell tradition. Some of the bells are shown here before a patina is applied to their surfaces.

BELL

TUNING RATIOS

SPECTRA

SIZE AND APPEARANCE(HEIGHT, DIAMETER, MASS)

6/5 CYL.

1

2.0

2.4

3.0

3.6

4.0

4.4

4.8

5.0

820mm, 510mm, 205kg

6/5 EUR.

1

2.0

2.4

3.0

4.0

510mm, 600mm, 220kg

5/4

1

2.0

2.5

3.0

3.75

4.0

4.4

5.0

5.25

820mm, 510mm, 205kg

7/5

1

2.8

4.0

4.2

5.0

5.6

7.0

7.5

8.0

440mm, 300mm, 32kg

8/5

1

1.6

2.0

3.0

3.2

4.0

4.4

4.8

825mm, 1060mm, 782kg

7/4

1

1.75

3.0

3.5

4.0

4.4

5.25

6.0

520mm, 570mm, 132kg

9/5

1

1.8

3.0

3.6

4.0

4.4

5.0

5.4

520mm, 570mm, 132kg

4/3, 5/3

1

1.33

1.66

2.0

3.0

3.33

4.0

5.0

775mm, 1220mm, 335kg

 

With close inspection of these tunings and spectra some features become evident. Most importantly not all the harmonics of each pitch percept are present. For example, the 6/5 and 5/4 bells do not have the fundamental present for these tones. This is because to tune the second partial to just 1.2 times the fundamental would have resulted in many other partials occurring at frequency ratios too low to tune. The 4/3 & 5/3 bell is an example of a very compressed timbre that results from having a small interval to the second partial.

Keen observers will notice the tuning ratio 4.4 appearing in the higher partials of a number of bells. This is simply a partial that could not be tuned to one of the harmonic series and so was tuned to a common value for all the bells so that it did not produce dissonances or random timbral variations. This partial will not contribute significantly to the timbre of any of the bells in which it occurs. In fact careful listening to the bells during tuning revealed little contribution to pitch perception of more than about the first five partials (when the bells were struck in a manner consistent with how they will be used).

Analysis of the pitch perception of these bells is still be undertaken as the preferred striker design is being developed. This graph shows the pitch percepts calculated for the 8/5 bell with a fundamental at 293 Hz and the next partial at a ratio of 8/5 (or 1.6 times the fundamental frequency). The spectral and virtual pitch percepts for the fundamental calculated by the analysis alternate in weight producing the two results near 293 Hz (the green and light blue lines). This is not a readily perceivable effect and the pitch weight could be considered to sum with a very slow (around 1 second) frequency vibrato around 293 Hz. This is due to the higher partials at harmonics of the fundamental being slightly flat in the cast bell.

The 8/5 pitch percept can be easily seen (a dark blue line) to be around an equal weight as the fundamental after about 1 second. There are many weaker virtual pitches calculated by the model at subharmonics of the stronger percepts described above. The lower their frequency the less important they are to the perception of the bell.