A peal of bells can be assessed by recording them individually, digitising the sound into a PC, and using Wavanal to determine and compare the frequencies of the various partials. The procedures are straightforward and are described in the following sections.
The section of this website on Wavanal, and in particular the Wavanal documentation, gives practical tips on taking recordings, reading them into a PC, downloading the Wavanal software, and running it. I find it only takes me an hour or two to record, digitise and analyse eight bells - plus travel time from tower to home, of course.
Using Wavanal, load each bell sound in turn and use Wavanal to analyse it using these instructions. If the recordings are sufficiently good quality Wavanal should find the partials automatically. Make a note of the frequencies of the main (named) partials of each bell. The quint is often quiet and therefore hard to find, it's not really an issue if you fail to pick it up. To save writing all the frequencies down, Wavanal can save them to a file which you can then load into a spreadsheet.
Cents are a way of measuring intervals independent of frequency. 100 cents make one semitone, 1200 cents make an octave. Negative values indicate an interval down, positive an interval up. (If two tones have frequencies f1 and f2, the interval between them in cents is 1731.234 * ln {f2/f1} where ln is the log to base 'e'.)
I use a spreadsheet to automate calculation of cents from the partial frequencies and analysis of tuning in a number of different temperaments. The spreadsheet, including instructions for its use, can be downloaded here in Excel 97 format. The spreadsheet includes some macros, open it with them enabled.
The intensities as well as the frequencies of the various partials are of considerable interest. Excel or another spreadsheet package can be used to plot and print graphs of partial intensity, by saving the transform from Wavanal, loading it as a comma-separated-value file (Excel opens these automatically as a spreadsheet) and plotting the result. However, for most purposes displaying the transform in Wavanal will suffice.
Here are the results of analysis of a peal of bells (a peal of six at Chapel-en-le-Frith in Derbyshire, cast by Rudhalls in 1733). If you are unfamiliar with the names and significance of the partials, see the explanation here. To interpret the results, go to the section on the quality of a peal of bells.
Here is a list of the five lowest partials and intervals between them. The intervals of all the partials are given in cents from the nominal of that bell, except for the nominals themselves, which are given in cents from the nominal of the tenor.
Bell |
Hum |
Hum cents |
Prime |
Prime cents |
Tierce |
Tierce cents |
Quint |
Quint cents |
Nominal |
Nom. cents |
1st |
365 |
-2247 |
605.5 |
-1371 |
800.5 |
-887 |
1083 |
-364 |
1336.5 |
910 |
2nd |
312.5 |
-2297 |
539 |
-1354 |
696 |
-911 |
943 |
-385 |
1178 |
692 |
3rd |
269 |
-2350 |
511.5 |
-1238 |
620.5 |
-903 |
867.5 |
-323 |
1045.5 |
485 |
4th |
259 |
-2339 |
498.5 |
-1205 |
602.5 |
-877 |
706.5 |
-601 |
1000 |
408 |
5th |
242.5 |
-2250 |
434.5 |
-1240 |
533 |
-887 |
724.5 |
-355 |
889.5 |
205 |
6th |
212 |
-2277 |
390.5 |
-1220 |
473.5 |
-886 |
649.5 |
-339 |
790 |
0 |
Here is a plot showing the partial intensities of the third bell against frequency (given as cents from the bell's nominal).
Last updated January 3, 2002. Site created by Bill Hibbert, Great Bookham, Surrey