Stretch tuning and my research

[Bill Hibbert]

A bit more unfinished business (from over two weeks ago - sorry!).

Drew Douglas:
> how much is the flattening of the trebles subjective and likely to 
different interpretation by different ears?

Here's an attempt to quickly summarise what I found or confirmed 
(some of it was known before) in my research into strike notes and 
stretch tuning.

1) Strike notes are generated within the ear or brain of the listener 
by sets of partials with frequencies in the approximate ratio 1 : 
1.5 : 2 and so on. The note or pitch we hear is roughly that of the 
lowest partial in the set, though usually in a lower octave. The 
effect in the ear generating the strike note is very powerful - 
greater than the effect of any individual partial - and is well 
established by research, though the cause is not fully understood.

2) In bells, there are two sets of partials that have the right 
frequency ratios to generate strike notes: the nominal, superquint, 
octave nominal etc.; and I-7, I-9, I-11 etc. These are all partials 
which have an anti-node at the rim of the bell (i.e. are stimulated 
by the clapper blow). The I-n notation, due to André Lehr, is 
explained in my thesis for those not familiar with it.

3) The lowest of the set of partials with ratios 1 : 1.5 : 2 has to 
lie broadly in the range 500Hz to 1500Hz for a strike note to he 
heard: the exact range differs from person to person. In bells whose 
nominals lie in this range, the nominal determines the strike note. 
In bigger bells, the higher series of partials based on I-7 comes 
into play, giving rise to the secondary strike about a fourth above. 
In small bells, the nominal is too high to form a strike note and the 
pitch of the bell is determined by the prime or hum. In small true-
harmonic bells nominal, prime and hum will all give the same note 
(albeit in different octaves). If the bell is old-style this is not 
so, hence the discordant sound and ambiguous note of smaller old-
style trebles.

4) Now for the really interesting bit. When a strike note is formed 
by a set of partials with ratios approximately as 1 : 1.5 : 2 etc., 
if the partials are squeezed together (as generally happens in 
thicker bells) the strike note flattens. If they are stretched apart 
(as is often the case in thinner bells) the strike note sharpens. The 
variation can be considerable: 1/4 or 1/2 of a semitone. The extent 
of the shift depends both on the listener and the position of the 
bell's nominal in the audible spectrum: but all the dozens of people 
who have completed the experiments experience the effect to a greater 
or lesser degree. A subsequent experiment has shown that the shift is 
fairly independent of partial amplitude.

In passing, I should say that it is difficult to measure strike 
notes, because one is measuring the reaction of people rather than 
anything physical. I had to devise a new way of measurement, which I 
discover is also now being used by a researcher in Finland to measure 
similar effects in pianos. 

I have had two opposite reactions to the experiment results showing 
shifts in strike note. Those familiar with the practice of bell 
tuning find it hard to believe that the strike note can move away 
from the half-nominal in this way. The academics are surprised that 
anyone *doesn't* accept that strike notes shift with partial spacing. 
Perhaps the effect in bells has remained in doubt until now because 
the shifts, though big enough to be musically significant, are not so 
big as to be impossible to ignore. 

I used the experiment results from 26 people, plus analysis of the 
tuning of about 2,000 bells, to develop a way to predict the average 
shift in strike note from the frequencies of the nominal and octave 
nominal partials. In peals of twelve, because the trebles are cast to 
a thicker scale than the tenors, they have relatively flatter strike 
notes. I applied the predictions to half a dozen stretched twelves: 
Melbourne (Australia), Tewkesbury, Cambridge, Cornhill, St Paul's, 
and Preston Minster; and in each case the experimental results 
predicted well the actual stretch in the trebles.

The experiments were deliberately designed to emphasise listening to 
strike notes rather than individual partials. In practical 
situations, in trebles of twelve, ambiguity exists, and different 
listeners hear the flattened strike pitch, but also the hum, prime or 
nominal, in different circumstances, so that there is not one right 
answer to how to tune trebles. Certainly, to sharpen trebles to the 
full extent required to give strike notes that are in tune means that 
the nominals are audibly and unpleasantly sharp. On the other hand, 
trebles (such as those at the Bullring) which are very thick but are 
tuned without stretch sound odd rung with the thinner tenors, because 
the ear hears both the in-tune nominals and the flat strike notes.

The effect of pitch shift is not restricted to peals of twelve: 
Southwold (where the two trebles have a thicker profile than the back 
bells) is a vivid example, covered in an RW article in June 2003.

Bill H

PS My thesis is available online at 
http://www.hibberts.co.uk/phd/phd.htm; it is a long read and covers 
many other aspects of bell acoustics as well as stretch tuning. There 
are some introductory articles intended to provide an easy way in.