Bell tone

Robert wrote,

>Bell makers use the sequence
>1/12, 1/6 1/5 1/4 1/3.
> <http://www.oakcroft13.fsnet.co.uk/lehr.htm>
http://www.oakcroft13.fsnet.co.uk/lehr.htm

Looking at this, Robert, I see the following:

>Considering that the striking point for a bell is nearly fixed, then one
can quite reliably say that partials with >nodal circles near the strike
point would be heard weakly. Partials with nodal circles far from the
striking >point will be heard loudly. In this way one can observe that the
hum tone, prime, tierce, nominal, twelfth, >double-octave, etc. will be
dominant in the spectrum. Interesting is that this sequence gives a relative
tone >scale c - c1 - es1 - c2 - g2 - c3 (C3 - C4 - E4b - C5 - G5 - C6),
which is very acceptable.

In ratios, this might be 5:10:12:20:30:40, or 1/24:1/12:1/10:1/6:1/4:1/3

So it looks like you did some octave fudging.

But what's the perceived main pitch of the bell? It's C! This shows that the
timbre is not heard as an undertone series but simply as

1:2:4:6:8

with an "extraneous" Eb that adds inharmonic interest to the tone, and,
according to the article, is a consequence of the bell's general shape:

"One can vary the slope of the bell profile, the height or diameter of the
bell, and so forth. This means that the designer can do something about
internal relationships within a group. But one cannot expect too much, for
experience shows us that to conserve the harmonic structure of the lowest
partials, a variation of 4% in frequency is maximum. As a consequence, the
major third bell, with partial I-3 a major third above the prime, has not
yet been realized, in spite of the claims of certain bell founders. Also it
is not possible to make the interval between I-3 and I-4 a perfect octave"

Furthermore, the article states:

"The essentially nonharmonic spectrum of the bell has been made quite
acceptable in this way, for it is the dominant overtones that reinforce the
tuned pitch of the bell."

This agrees with what I'm saying -- we essentially hear otonally, so the
more otonal characteristics can be tuned into the bell's spectrum, the
better.

Finally, Robert, here are the conclusions from the article you cited:

"many musicians have correctly observed: that the bell, with its conspicuous
minor third, gives rise to confusion in carillon playing . . . Correctly
then, the musicians state that the minor third should be as weak as
possible."

"Apart from the question of whether this is possible or not, one has to
consider seriously that the nodal circles of the other partials of group I
would be shifted down by approximately the same degree. However, it is
exactly in this group that the bell sound finds a strong basis for
unambiguous tonal height."

Unambiguous tonal height is what I've been talking about, Robert, and it's a
sensation that can only be given by a harmonic series.

"Thus, a weaker third opposes the production of a strong tonal height
sensation for the bell; it is clear in such a choice what we would prefer."

What the author is saying here is that, due to the physics of bell
vibrations, weakening the minor third would necessarily weaken other
frequencies which are, in fact, at or near harmonic overtones of the
perceived main pitch. So as much as musicians might wish to eliminate the
"extraneous" minor third from the bell's spectrum, bell-makers can't do so
without adversely affecting many of the bell's "correct" partials.

Hi Paul,

Yes, looking at the article, it seems that bell tuners are actually trying to
approximate the otonal sequence as closely as possible.

However, at least in the 17th century, they ended up instead with
1/12: 1/6: l/5: 1/4: 1/3

and the 1/5 is there because they can't get rid of it!

I'm not clear from the article about whether modern bell
tuners also target this sequence.

I wonder if a bell with a major instead of a minor third
would still sound like a bell? An experiment one could
do with partials synthesis of sounds.

Seems bell tuners would make such a bell if they could,
especially for use in carillions.

Robert

Robert wrote,

>I'm not clear from the article about whether modern bell
>tuners also target this sequence.

From the frequency analysis, it looks like a slightly flattened 6:5 is
targeted, perhaps to result in slightly less beating when two bells an
equal-tempered minor third apart are played together.

>I wonder if a bell with a major instead of a minor third
>would still sound like a bell?

It wouldn't sound like a carillon bell, but it would still sound very
bell-like, due to the amplitude envelopes of the various partials. It would,
of course, have a remarkably clear pitch-height.

>Seems bell tuners would make such a bell if they could,
>especially for use in carillions.

Yes, though some music has been written specifically for carillons with the
usual minor third partials. This music tends to avoid major tenths in the
harmonies -- so major triads are usually in first inversion, etc.

Robert wrote:

> However, at least in the 17th century, they ended up instead with
> 1/12: 1/6: l/5: 1/4: 1/3
>
> and the 1/5 is there because they can't get rid of it!
>
> I'm not clear from the article about whether modern bell
> tuners also target this sequence.

This is discussed in Sethares' "Tuning, Timbre, Spectrum, Scale"
(essential reference, etc) pp107-110. The ratio is given as
0.5:1:1.2:1.5:2 which is equivalent to yours. So it still seems to be the
target.

> I wonder if a bell with a major instead of a minor third
> would still sound like a bell? An experiment one could
> do with partials synthesis of sounds.
>
> Seems bell tuners would make such a bell if they could,
> especially for use in carillions.

p.108: "Bell makers have recently found out how to shape a bell in which
the tierce becomes 1.25, which is the just major third 5/4. These are
called `major third' bells." I remember a TV program about these, when
they were a new invention. They have a bulge at the top instead of the
usual sloping shoulders, and I think they used a computer to work out this
shape. I don't have a phonographic memory of this, but they did have a
brighter sound, not the mournful sound you expect of a bell.

Graham

Interesting web site with book by a carillioneur.
http://www.msu.edu/~carillon/batmbook/index.htm
Old bell that somehow achieves a major third partial
without using the modern design
http://www.oakcroft13.fsnet.co.uk/collect/dorkold.htm

Robert