The Sound of Bells

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Tewkesbury, Gloucestershire, The Shed, 10, 0-0-16 in F#

Recorded: WAH 29/5/99
Analysed: WAH 15/6/01

These bells are a mini-ring of ten cast by Matthew Higby and Richard Bowditch, tuned by David Bagley and Matthew Higby, and hung in David Bagley's garden shed. My thanks are due to David for allowing me to analyse his bells, and for much of the background information below. The acoustics of such tiny bells differs enormously from church bells of normal size, as will become clear from the analysis. I do not yet have a recording of the bells rung in changes, though some of the individual bell recordings appear below.

Bell Founder Tuning
1, 2 Higby and Bowditch 1996 Matthew Higby
3, 6, 10 Higby and Bowditch 1994 David Bagley
4, 5, 7, 8, 9 Higby and Bowditch 1993 David Bagley

Tuning of main partials

These bells, being quite small, have quite different acoustics to larger bells. As explained below, their pitch derives from the lowest frequency partial. Note therefore that the left-most column below shows the relative pitch of each bell, not the 'nominals' column as usual. In this column, the doublet frequency given first is the strongest. Rather than name the various partials, I have referred to them as 1st, 2nd, etc. I have tried below to identify some of the partials with their equivalents in larger bells but without checking the modes of vibration, this is speculation.

Tenor pitch: 732Hz (with a doublet at 735Hz).
Due to the doublets on the first partials of each bell, an average value for the 1st partial is given in the second column. This probably gives a better view of the overall tuning of the peal.

Bell 1st 1st av. 2nd 3rd 4th 5th
1 1607
1622
1611 1195 1534
1535
1989 2442
2 1403
1405
1401 1151
1156
1429 1973 2309
3 1208
1202
1202 1153 1597
1599
1896 2163
4 1107
1103
1101 1213 1735
1745
2028
2029
2210
5 910
926
914 1222
1229
1955
1973
2104
2113
2218
6 721
700
707 1199
1211
1370
1378
- 2171
7 505
492
495 1310
1322
- 1790 2201
8 404
423
410 1288
1291
1312
1314
1366
1368
2112
9 231
215
220 1118 1284
1292
1835
1887
2527
10 0
7
0 1447
1454
1583
1587
1856 2472

(The figures in this table are all given in cents. For all columns except '1st' and '1st av.', the intervals are given from the stronger doublet of the 1st partial of the bell. Cents of the 1st partials are relative to the tenor. Pairs of values indicate a doublet.)

Intensity plots

Here are intensity plots of the treble, the sixth and the eighth, each of which demonstrate points of interest:

The Shed treble

The Shed 6th

The Shed 8th

Commentary

The first question to settle about these bells is which partial determines the pitch. Comparison with a pure sine tone adjusted to sound the same note shows that it is the lowest partial which determines the pitch of these bells. This identification is made much easier by the fact that few of the partials form perfect octaves, so it is perfectly clear which partial matches the pitch. (In bigger bells it is the nominal - the 5th partial - which determines the pitch.) Confirmation of the pitch partial for these bells can be had if we notice that many of the partials are doubletted. The sound of the sixth, for instance, has a definite wobble at about 10 beats a second (more easily heard after the first second or so). The wobbling pitch is clearly the initial pitch of the bell. The doublets on the various partials are: 10Hz on the first, 3 Hz on the second, 2Hz on the third, and 2.5Hz on the 4th. So, the first partial gives the pitch of these bells.

There are two probable reasons why the 1st, not the 5th, partial is heard as the pitch. First, the pitch frequency of these bells is very high. I believe that the effect of the missing fundamental, which fills in the strike note for bells of lower frequency, tries to find a lower strike pitch than is possible from the 5th partial of these bells. Secondly, the 1st partials of these bells are almost all the strongest partials, often by a considerable degree.

Having decided how to identify the pitches, we can look at their tuning across the peal. All the 1st partials have doublets. There are two ways of dealing with this: take the loudest of each pair as the frequency to determine the pitch, or (since all the doublets are relatively equal in intensity) take the average value for each bell. Choosing which half of the doublet to pick is sometimes not obvious. In the fifth, both halves of the doublet are the same intensity. In the seventh, the 1st partial has frequencies of 972.5Hz and 980Hz. Using Wavanal over one second, I picked 980 as the stronger. David Bagley, using a microphone connected to a DSP over a longer period, picked 972.5Hz as the stronger. The plot below of the intensities of these two over time, shows that in changes, the partial at 980 is more important:

The Shed - 1st partials of the 7th

(I had a similar discussion with Whitechapel about the tuning of the sixth at St John, Lambeth. We disagreed about the nominal of the sixth, in fact this partial is doubletted. The frequency they had in their tuning books was much the weaker of the pair.)

Theory suggests that the actual pitch of a doublet is related in a complicated way to the two frequencies. Given the relatively equal intensity, I took the average value as a reasonable compromise. In fact, the table above of tuning figures shows that either approach give similar results for these bells. The fifth, sixth and ninth are a little sharp, and the remainder are pretty well in tune, bells two, three and four especially so.

David Bagley sent me the following description of the tuning of these bells:
"The bells were tuned on the inside with an angle grinder while rotating in a lathe. The first step in tuning the new bells was to pitch them all. This was done on the original 6 bells, and it turned out that the 4th of 6 was the flattest. The other 5 bells were tuned down to match this bell using 2^(1/12) to calculate the frequencies required. However, when I augmented by adding the treble and tenor, the 3rd was replaced (the original bell was destroyed during tuning, and was replaced on a temporary basis with a gunmetal bell) and the previous 4/6 was tuned down a semitone anyway to become 5/8. To tune them, I used an audio signal generator and a small loudspeaker. When the sig gen is tuned to the same pitch as a resonance, you can quite easily hear the difference in amplitude, and you can also hear the beats if you tune off slightly."

It's not worth going into a lot of detail on the exact tuning of the other partials of the bells, as I suspect that due to the small size of the bells it is not possible to tune them with any precision. All the bells have different arrangements and intensities of the partials, due no doubt to differences in profile. Take first the sixth, and look at its intensity profile above. This bell is clearly thick (i.e. not profiled like a handbell) because it is possible to identify the 'thick profile' partials. That at 0 cents is clearly the hum, that at 1200 cents a very accurate prime, that at 1370 cents a rather flat tierce, and that at 2170 cents a very flat nominal. The octave prime, taken with the low intensity of the other partials, gives this bell a clear sound.

The profile of the treble is nearly all hum, with just a little tierce, again giving this bell quite a pure sound. At the other extreme, however, we have the eighth. The partials I speculatively identify with prime (at 1312 / 1314 cents) and tierce (at 1366 / 1368 cents) are very close together, which gives the bell a somewhat boingy tone. This degree of closeness between prime and tierce would be a problem in a big bell also. Last but not least, here is the tenor, a nice sounding bell but with a prominent doublet which beats at about 3 cycles a second.

A final note on the doublets. Given the small size of the bells and their method of tuning, I can understand how hard it must be to get the bells exactly round. David Bagley points out that had he had a big enough lathe he could have substantially eliminated the doublets by turning them with a lathe tool rather than an angle grinder. However, given the way these bells are rung, i.e. very quickly (recent quarters have averaged 195 mS per bell-strike, or about 5 bells per second) it is not very likely that the doublets much affect the sound in changes.


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Last updated June 24, 2001. Site created by Bill Hibbert, Great Bookham, Surrey