Temperament and bell tuning

[oakcroft13]

Richard Offen:
> Where's Bill Hibbert when you need him?!!!!
Following the tuning discussion with great interest!

Roderic Bickerton: 
> This sounds like a very good case for modern computer modelling

I have created simulations of two peals of twelve, one with nominals tuned to the Lucy temperament and the other in equal temperament, which you can listen to and compare for yourself at http://www.hibberts.org.uk/lucy.

Of course we are concerned about both outer tuning (the relationship between nominals in different bells) and inner tuning (the relationship between partials within each bell. I encountered a good example of the trade-off between these while comparing two peals of six bells the other day. In one, at Old Wolverton near Milton Keynes, the individual bells are quite good toned (very good for their date) but the nominals are rather wild. The fourth is sharp by 44 cents but masked by several other bells also being sharp. In the other, at Chapel-en-le-Frith, the nominals of the bells are pretty good (the worst is only 15 cents out), but tonally the individual bells are poor. Old Wolverton sound much better to my ears - though these things are a matter of personal taste.

The issues with temperaments in bell tuning have already been well explained by other posters; what follows is an attempt to put the arguments in more detail.

In most musical instruments, the individal notes have overtones which are very close to harmonic - so for a fundamental frequency f, the frequencies present are f, 2f, 3f, 4f, 5f etc. This is not the case in bells. The partials which determine the pitch (nominal, superquint, octave nominal etc.) can never all be harmonic in bells of normal shape, and the lowest one is missing. We still hear the dominant pitch or strike note about an octave below the nominal but it is generated in the head. Of course the partial called prime, second or fundamental can be placed an octave below the nominal by the founder and tuner but experiments have shown that it makes little contribution to pitch except in small bells.

The overtones in a harmonic tone are as follows, with figures given to the nearest cent.
Harmonic: 0, 1200, 1902, 2400, 2786, 3102, 3369, 3600, 3804, 3986

If we reduce these to intervals of an octave or less we get:
0, 1200, 702, 1200, 386, 702, 969, 1200, 204, 386.

When we tune harmonic tones against each other for zero beats or least harshness between the overtones (i.e. to sound 'in tune') we get the Just intervals: octave of 1200 cents, fifth of 702 cents, major third of 386 cents, etc. However, it is not possible to modulate to different keys in a Just tuning without encountering intervals that sound badly out of tune. This, and the arithmetic problems of the cycle of fifths (12 pure fifths exceed the corresponding number of octaves by over 23 cents) motivates the introduction of temperaments - where intervals are changed a little from their Just values to allow modulation. These changes introduce beats or harshness between overtones when notes are played together that give different audible effects or 'colours' to different keys in different temperaments . In equal temperament, all the intervals in the scale are detuned to make all the semitones equal in size (which means all the keys sound the same). Purists would say they they all sound equally out of tune.

If we compare the overtones of some sample bells with a harmonic tone the differences become obvious:

Harmonic tone: 0, 1200, 1902, 2400, 2786, 3102, 3369, 3600, 3804, 3986
Bullring treble: -, 1200, 1815, 2281, 2649, 2954, 3213
Great Tom (Oxford): -, 1200, 1835, 2348, 2274, 3138, 3453
Laira (Plymouth) treble: -, 1200, 1860, 2375, 2791, 3137, 3434
Dorking tenor: -, 1200, 1915, 2482, 2954, 3350, 3687
Little Somerford 2nd: -, 1200, 2050, 2610, 3074, 3448, 3806.

Remember that the lowest overtone is missing in bells. The samples were chosen to give a range from the most squashed up to the most stretched out, corresponding to relative thickness, especially of the soundbow. I hope that it is clear from the figures that any theory about the sound of different temperaments based on harmonic tones breaks down irretrievably in bells because of the wildly inharmonic partials. If we are to judge whether intervals between the nominals of bells are in tune, we cannot rely on audible clues from interaction of overtones. Instead, we rely on our memory of 'in tune' intervals, and our taste and discretion. Experiments show that most people, even skilled musicians, are not very good at judging intervals in the absence of clues from interaction of overtones - discrepancies of 10, 20 or 30 cents are hard to hear.

In this connection, it has been suggested that very good choirs singing unaccompanied will settle into Just tuning. However, actual measurements I have made show that unaccompanied choirs singing 'in tune' tend to sharpen their thirds - even sharper than Equal. I can suggest two reasons for this: first, we are surrounded by equal tempered music in our daily lives, and are attuned to it. Also, it lends a brightness to the sound.

So what are the issues in choosing a temperament for bell tuning? I can think of several - and I am sure there are more:
* In the days of hand calculation of frequencies (i.e. in the 19th and early 20th centuries), Just tuning had the great advantage that the sums could be done by simple multiplication and division
* If casting bells for stock, equal temperament has the advantage that a stock bell can be used for any degree in the musical scale. I guess (without any evidence) that this influenced the Rudhalls
* In carillons, equal temperament has the advantage that modulation between keys is painless. However the nominals in a carillon are tuned, the tierces need to follow the same tuning to avoid unpleasant effects
* Equal temperament in bells has the disadvantage that the third and leading note of the scale are only a semitone away from the next bell up - which can sound harsh or overly bright. It has been suggested to me that a motivation in choosing a temperament for a peal of bells is to lessen the difference between the tones and semitones
* In higher numbers, where the trebles are sometimes stretched or tuned sharp to offset the effect of their flatter upper partials, tuning in a temperament with flattened thirds moves the nominal of the bell affected the wrong way - its strike note is already flat because of its shape.

There's much more that could be said but I think this is enough!

Bill H