The Sound of Bells

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Bell pitch and tuning of nominals

For bells of a great range of weights - from several kilogrammes to a couple of tonnes - with a conventional profile or cross section, the nominal and a few upper partials including the superquint and octave nominal determine the pitch which is assigned to the bell by a listener, if the bell is struck hard with an iron clapper on the soundbow. The reasons for this are subtle, as explained in the section on strike notes. The ear seizes on a set of partials with a rough harmonic relationship and produces a sensation of pitch an octave below the lowest, an effect known as the 'missing fundamental'. In many cases, the pitch perceived by the listener will not be found as a frequency among the bell's partials. The pitch heard is called the strike note or tap tone.

As the profile and tuning of a bell departs from the conventional, the pitch can move away from an octave below the nominal, and it can sometimes be difficult to assign a unique pitch to a bell. For some bells, especially those in continental carillons, struck gently, the pitch assigned by the ear corresponds to the prime, not an octave below the nominal. In very small bells, the pitch assigned is that of the hum, because the ear does not hear the high partials well and the 'missing fundamental' effect breaks down. In large bells, the pitch an octave below the nominal is so low that the ear seizes on an alternate set of harmonically related partials with higher frequency, assigning a higher pitch (the 'secondary strike') to the bell than would be expected. In heavy UK bells, the secondary strike is usually about a fourth above the octave below the nominal, because of the frequencies of the other partials found in bells of normal profile. In heavy continental bells, the secondary strike is flatter - nearer a major third - which to my ears is much more harmonious. As can be seen from this brief description, the pitch of bells is a complex and interesting subject explored in more detail elsewhere on this site.

The ringing of bells in changes throws special emphasis on virtual as opposed to spectral pitch effects.

For most bells, the assumption that the pitch is about an octave below the nominal is a good one. Therefore, tuning of the nominals across a peal of bells has a big part to play in whether they sound in tune. However, quality or timbre of each bell is also important, and sometimes precise tuning of the nominal of a bell is sacrificed in favour of a better overall tone, especially when an old peal of bells is retuned.

Bells used for change ringing are tuned to a diatonic scale - the scale used for the vast majority of Western music. There are three effects which are common in church bells which allow or require the nominal tuning to diverge from the 'ideal' diatonic scale to give acceptable results. These effects are differences in temperament, virtual pitch effects including octave stretch, and 'shading' of tuning errors.

Temperaments

There are many different ways or 'temperaments' to tune a diatonic scale such that it is acceptable to the ear. The origins and reasons for the different temperaments are described in An Introduction to Historical Tunings or Bells and their music. Three tuning styles commonly met in church bells are equal temperament, where all the semitones in the scale are exactly the same interval, just tuning, which has rather flatter thirds, sixths and sevenths and where all the frequency ratios are simple whole numbers, and the meantone temperaments, which are a compromise between the two. Most musical instruments were tuned in variants of meantone for a couple of centuries up to the middle of the 19th century - almost all are now tuned equal. Well temperaments (such as Werkmeister's Temperament III) represent stages in the transition towards equal temperament. Pythagorean tuning is a very old tuning style, suited to melodic music but not very pleasant for harmonised music because of its sharp thirds and sevenths. The following list of intervals in cents for a peal of eight shows the difference in these tunings:

Bell Equal
temperament
Just
tuning
Meantone
temperament
Werkmeister
temperament
Pythagorean
tuning
Treble 1200 1200 1200 1200 1200
second 1100 1088 1083 1092 1110
third 900 884 890 888 906
fourth 700 702 697 696 702
fifth 500 498 503 498 498
sixth 400 386 386 390 408
seventh 200 204 193 192 204
tenor 0 0 0 0 0

The normal tuning tolerance for the partials of a church bell is 10 cents. Some of the differences between the temperaments exceed this tolerance, and can be clearly seen when analysing bells. For example, the difference between the sixth in Just / Meantone and Pythagorean is 22 cents - almost a quarter of a semitone.

In most musical instruments, the chief argument for just tuning rather than equal temperament is that it avoids beats between higher overtones of the notes in common chords, producing a more consonant effect. The argument for equal temperament is that all keys sound the same, allowing for arbitrary modulation. As bells do not have partials which fit a harmonic series, and modulation between keys is not an issue in change ringing, neither argument justifies a choice between the tuning styles. In fact there are three factors which come into play. First, non-equal temperaments or tunings were the standard for all instruments until the 19th century. Second, some people would say that the flatter thirds, sixths and leading notes in a just-tuned peal are easier on the ear, giving a softer, warmer effect. Finally, equal temperament has become more common in the last hundred years because of the requirements of the carillon market, and because it is the tuning to which modern ears are accustomed.

The two main foundries in the UK, Taylors and Whitechapel, had rather different policies on nominal tuning throughout the 20th century. Taylors produced some peals with flattened thirds in the early part of the century but the bulk of their output from the mid 1920's onwards was equal temperament, driven by their business in carillons. Whitechapel, on the other hand, almost always used Just tuning up to the early 1960's and still often tune in temperaments other than equal.

In an attempt to find out if differences in temperament can be heard in a complete peal, some while ago I put together an on-line experiment which I invite you to try. The conclusion, based on results from about 25 listeners, is that the majority cannot distinguish the various tunings. Subsequent work has convinced me that virtual pitch effects can be as important as differences in temperament in judging bell pitch.

Virtual pitch effects and octave stretch

Peals of twelve, ten and sometimes eight produced in the 19th century and the middle part of the 20th century commonly have sharp treble nominals, by up to 25 cents per octave. Here are several examples of the nominals of peals with sharp trebles:

Peal 1 2 3 / 1 4 / 2 5 / 3 6 / 4 7 / 5 8 / 6 9 / 7 10 / 8 11 / 9 12 / 10 founder / date
Cambridge
Great St. Mary
?? 1734 1637 1438 1223 1124 912 708 489 412 221 0 Phelps 1722/3
Tuned Taylor 1952
Painswick (10) - - 1625 1421 1219 1120 893 700 496 403 197 0 Rudhall 1731/2
St. Paulís
Cathedral
1972 1728 1632 1424 1239 1107 898 710 538 413 215 0 Taylor 1878
Stockport
St. George (10)
- - 1624 1422 1217 1112 897 693 490 404 208 0 Mears 1896
Tewkesbury 1939 1735 1632 1423 1222 1122 917 708 507 402 203 0 Taylor 1962

This effect is known as octave stretch. Stretch is very familiar to piano tuners, who routinely tune sharp octaves. In pianos, the stiffness of the string causes its first overtone to be slightly sharp, so that tuning for zero beats between octaves gives the stretch. Some handbells intended for tune ringing, especially those from Malmark, are tuned with stretch so as to be in tune with a piano. Prof. Terhardt suggests that stretched tuning is often the more natural style than tuning in strict octaves. However, the explanation which stands good for pianos does not explain the usage in bells, because of the inharmonic bell partials. As explained in the section on strike notes, the correct explanation in bells is a virtual pitch effect due to the fact that the trebles of higher numbers usually have flat upper partials, probably as a consequence of their mechanical design. Work in the 1980s by Terhardt and Eggen and Houtsma demonstrates the effect. I finally concluded the reality of the effect after considerable research and experiment, prompted by hearing peals of bells with treble nominals exactly in tune which to me sounded flat.

Prior to the introduction of the tuning fork in UK bellfounding in the late 19th century, bell pitch was presumably tuned by ear, incorporating the effects of virtual pitch. The resurgence of stretch tuning in the 1950s was to achieve an aesthetic effect of 'brighter' trebles.

My research work on virtual pitch has incidentally disproved a theory suggested on a previous version of this page, that flat primes have a significant effect on pitch. Further work is needed to determine numerically how bell pitch changes with upper partial tuning.

Shading of tuning errors

The ear is quite tolerant of absolute errors in the pitch of bells. A peal containing a bell whose nominal is badly out of tune which for some reason (for example, due to the age of the bell) cannot be tuned, can be made to sound more pleasant by ensuring that the error between adjacent bells is not too great. Bells either side of one which cannot be tuned can be detuned to disguise the problem. For example, a peal of eight tuned as follows:

Bell 1st 2nd 3rd 4th 5th 6th 7th 8th
Cents 1210 1120 930 740 530 420 210 0

will sound well enough in tune. A real example of this effect can be seen in the tuning of St. Paul's Cathedral given above, where the tuning shades towards the seventh from either side. The seventh is the flattest bell in the peal. As sharpening bells is hard to do successfully without spoiling the tone, I surmise the tuner judged it better to flatten the bells on either side.

My experience of analysing many hundreds of peals of bells suggests that discrepancies of up to 40 cents in nominal tuning can be tolerated, though they can be heard.


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Last updated May 1, 2004. Site created by Bill Hibbert, Great Bookham, Surrey