- An Engineering Approach to Wind Chime
Design
- or
- What Makes Toast, Toast
?
- by
Lee Hite
 Clearly
the question should have been, what makes a chime a chime, rather than what
musical notes should be selected when designing a set of wind chimes. I had
originally asked that question and now find that I should have also asked,
"What makes a good chime?" As my good friend Bob pointed out when
faced with the challenge of designing a new state-of-the-art toaster, you first
determine what makes toast, toast; rather than dried bread, before you
design a great toaster. (BTW, a fascinating story)
I used the month of
December, 2000 to study, experiment, build and test nine sets of pentatonic scale
(CDEGA) chimes for fundamental
C2 through C7 for Christmas presents. The gifts were well received
and sounded okay. However, from my research, I found some information good
and some information inaccurate, misleading or wrong. This is not surprising
considering the tremendous breadth of information available on the web. While I would not consider myself an expert by any definition,
and the fact that I have absolutely no musical background, I consider
my findings valuable for the understanding of tubular bells and useful if you
desire to build a great set of wind chimes. My experience with this project is
presented here and may help you in an effort to design a great set of wind
chimes.
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What Is A
Chime?
First, we must ask
what is a chime? Tubular bells (chimes) were developed in the 1880's
when using regular bells became impractical in an orchestra setting because
their sound so closely imitates church bells. Now, that sounds simple
enough but imbedded in that explanation resides two definitions. One
definition is "a chime imitates a bell" and the other definition is that
"a chime does not imitate a bell". While I found those two definitions
to be true I also found that there are about three categories of chimes.
The first category is from about fundamental C6 to C8. Not unlike other percussion instruments
this category is characterized by an audible fundamental pure tone with
overtones mostly absent. Any existing overtones have minimal contribution
to the perceived musical note. The perceived sound is the fundamental
frequency and is not particularly pleasing to the ear. This sound is definitely
a "non-bell" sounding chime, the loudness is low because of the
small radiation surface and the rapid attenuation of high frequencies in the environment.
The second category is from about fundamental C4 to C6. The fundamental is mostly audible
and some overtones contribute to the perceived sound. The perceived sound
is not the fundamental and not the overtones but a combination of both
that produce a perceived musical note. The sound is acceptable but
not great. This has an "almost-bell" sound but not particularly
melodious. The loudness is acceptable but not great.
The third category
is from about fundamental C2 to C4. The fundamental is present but
audibly absent and there are a host of overtones. The perceived sound
is not the fundamental and not overtones but an imaginary tone created
by the combination of the overtones. To the ear this is very
melodious and quite pleasing. This is clearly a "bell-like" melodious sounding chime.
The loudness is quite good because there is adequate
radiation surface for the many overtones.
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The overtone
structure for a chime in not integer harmonics as in string instruments
but instead, inharmonic as in other percussion instrument. Overtones
are multiples of the fundamental by X2.76, X5.40, X8.93, X13.34, X18.64
and X31.87 It was interesting to learn that not all chime frequencies contribute
to the perceived musical note for all notes from C1 through C7. For example,
a
chime cut at
fundamental C2, the fundamental is audibly absent along with little audible
contribution from the first overtone. The remaining overtones combine to
produce a perceived musical note. The perceived note does not
coincide with any specific overtone and is difficult to measure. In contrast to fundamental C2 the perceived musical note from a chime cut at fundamental
C6 and up is mostly the fundamental frequency and overtones are audibly
absent or mostly absent.
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The perceived musical
note
from a chime is more complex and more difficult to determine than
I had originally expected. I had expected this entire project to
be a simple physics exercise, but not true! To gain a better understanding
of the perceived note I examined a set of orchestra chimes manufactured by Premier
of England. The set was 1.5" chrome plated brass with a wall thickness
of .0625 inches and ranged from C5 (523Hz) to G6 (1568Hz). The length
of C5 was 62.625 inches. The fundamental for this length is around
65 Hz, yet the perceived note is C5 at 523Hz. Measurements for other chime
notes in this set of chimes indicated the perceived note to be between
7.8 and 8.3 times the fundamental. I am not certain this is
the correct ratio to multiply for "Premier" chimes but clearly there is
a ratio for each material and configuration involved. In fact, I
believe this style of chime cannot be compared to the traditional chime
tube that is "open-at-both-ends" because the orchestra style of chime is
fitted with an end cap that contains a small hole in the end cap.
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The physics of a perceived
note:
To make a great set
of wind chimes it is not necessary to understand why a chime note behaves
as it does, but in my case I find it necessary. It turns out that several
other people have spent time investigating the "missing fundamental"
and the "perceived note' from a chime. One such source is
HERE,
another
HERE,
and
HERE
I spoke with the folks at Musser Chimes and confirmed that indeed the process
of tuning a chime is a complex process of accounting for all frequencies
from the fundamental to the many overtones.
An integral
part of the "perceived note" effect is the sensitivity of the human ear to loudness
and to frequency. You can see the loudness sensitivity range
and frequency sensitivity range of the ear by viewing the Fletcher/Munson
"Equal Loudness Curves" found
HERE
Clearly
the ear has more sensitivity in the range from about 300 Hz to about 4 KHz than
at other frequencies.
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Chime
Emulation:
There is a terrific piece of shareware software "Windchime Designer V1.0" by Greg Phillips
that will emulate a chime
for any note in many different scales. It will help you determine
what notes sound good on chimes and what scale to use. In addition, there
is a comparison calculator that can determine the length of tube once
you have a measured length as a reference. An updated version (2006) is
available
HERE. The older version requires a sound card, is for older computers
and is no longer available on the web. If you want a copy you can download from my site Chime.exe.
In case you have trouble unzipping Greg's new
version here are the two files you need.
Chime32A.exe and
TUNING.DAT
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- Note selection is mostly a
personal choice. I chose the pentatonic scale to build the nine sets
of chimes; however, I discovered later in the year that there is probably
a better choice. The pentatonic scale was a safe choice and sounds
very good close to the chime set but not so good at a distance. A
set of chimes designed for the C2 or the C3 octave have very good acoustic
radiation properties and can easily be heard at a distance of 150
feet. The problem is that at that distance the ear has difficulty
detecting the separate notes of the pentatonic scale (CDEGA). All
notes (CDEGA) had a strong tendency to sound alike at a distance of 150
feet. The next set of chimes will be designed for notes that have
considerable separation but maintain an overall coordinated sound.
(More work is required to determine the correct notes for this approach.)
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Tuning:
If you are building "non-bell sounding" chimes and attempting to excite exact notes, exact tuning
is required. If you desire "bell-sounding" chimes, cutting a
tube to the length suggested by the formula listed below and used in the
attached Excel work sheets should be adequate.1"Brass
3/4"Copper
3/4"Alumnium
1"Alumnium
Attempting to tune a low frequency tube to the exact frequency for fundamental
C2 through C4 is largely a waste of time because the perceived sound is
dependent on the many overtones and not the fundamental. Having said that,
I want to emphasize that good tuning will certainly help to accurately produce
the appropriate overtones for the selected note, particularly for the higher
notes.
To get an idea about the difficulty
in choosing a chime note to match a chosen musical note
see the Excel sheet
ChimeFreq.
This
will give you a colored picture of the many overtones present for each note and on how any
specific frequency is created by more than one chime. You can
see the wide range of notes present in a single chime by observing the
horizontal axis. The diagonal axis represents the many different opportunities
for a specific frequency to be generated.
In addition
to the many overtones present for each chime we have the difficulty of
knowing which overtones are prominent for each note because of the ear's sensitivity
as represented by "The Equal Loudness Curves". As you might suspect,
the prominence of a particular overtone changes as we move up the scale.
For a typical ear sensitivity range of 300 Hz to 3 KHz, see the sheet named
300Hz-3KHz.
Obviously this is not the entire
audible range of the ear but is presented
as a simple example of the limited ability of the ear to hear all the
frequencies generated by the overtone structure. In particular, the range of
C2 to C3 contain a large number of audible overtones while the range of
C5 to C7 contains very few. The range of C2 to C4 produces the most
melodious sound and is the easiest set of chimes to build. Precise
tuning (+ or - .1Hz) is not required.
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Location
for mechanical support:
Chime support is
at a node point which is 22.42 % from the either end. I did find it very easy to destroy
the Q (hang time) of a hi-Q chime by improper support. Thin wire,
rubber grommets and plastic inserts all worked but contributed to a lowering
of the Q. Nylon plumb line with no inserts worked the best. Of course
it is necessary to de-burr and burnish the drilled support holes to minimize
wear and tear of the line. Sources for rubber or plastic grommets include Radio Shack, Home Depot, Lowes and your local model airplane&
hobby store.
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Frequency measurement:
We can measure the
fundamental frequency and the overtone content with DSP (digital signal processing) via
FFT (Fast Fourier Transform Analysis), but from a practical
stand point I found it of little value in determining the actual musical
note.
An octave band filter
between the accelerometer and a frequency counter made it easy to choose
the overtone the counter would measure. Using a period measurement
and converting to frequency solved the issue of a short hang time.
A commercial electronic
music tuner by Krog worked well for fundamental measurements but can be
tricky because of the short hang-time from the chime.
A software solution
is to use a good piano tuning program. I found a shareware program
"Tune Lab 97 worked well once you were close to the desired frequency. TuneLab 97" is available HERE
.
This is particularly good if you are attempting a very exact tune because
it also compares the phase of the chime to the internal clock of the computer.
If you need to tune the phase between many chimes then "TuneLab 97" makes
it easy.
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Mechanical support
for frequency measurement:
I had good success
supporting the chime horizontally at both nodes, one by a rubber band,
and at the other node by a thin wire attached to an accelerometer.
The accelerometer eliminated the annoying background noise when using a
microphone.
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The Striker:
 Orchestra chimes,
of course, need a human to strike the chime and a rawhide-covered rubber
mallet works well for that application. However, for wind chimes
there is little strike energy available from the wind catcher so preserving
and applying that energy is the challenge. I tested a number of strikers
and found that maximum strike energy can be applied by using a 2" diameter
oak disk machined to a knife-edge and loaded with about 1oz of weight.
I also used a small 1/16-inch brass tube about 5 inches long as an axle
for the disk. The axle keeps the disk horizontal during rapid and
sudden movements. The small diameter disk was used to prevent the
striker from striking more than one chime at a time. Attempting to
strike several chimes together to produce a chord was a waste of strike
energy and ineffective. There is little strike energy available to
start with and attempting to strike a musical cord with a chime is, at best, disappointing.
There are two locations
on the chime that work well for striking. If you are building
a "non-bell sounding chime" for fundamental C6 and up, striking at the center
or the end works equally well. On the other hand, if you are building
a "bell sounding" chime it is important to excite all possible modes for
good overtone representation. This is easily accomplished by striking at the very end of the chime. Striking
at the end will assure the excitation of all modes since all modes exhibit
high impedance at the end of the chime.
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Choice of Material:
As I am sure many
already know, the choice of material makes a considerable difference in
the timbre of the chime sound. I tuned three tubes of the same diameter
and wall thickness made from brass, copper and aluminum to the same fundamental
frequency of C2. The resulting timbre was as different as day and
night. I suspect the difference in perceived sound is because
of the varying ability of the material to support overtones in varying
degrees of loudness. Researching this effect was way beyond the scope
of the project. Aluminum had the very best overall sound for fundamental
C2.
-
For the "non-bell sounding"
chime tuned to fundamental C6 there was little noticeable difference among
the three materials. This is not surprising because there is a lack
of overtones at these frequencies and the chime approaches a pure tone
at the higher frequency. Good tubing sources seem to becoming
more scarce at time goes on. Here are a few I found but I have no personal
experience with them.
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Source
of Tubing:
I
have had considerable success in locating brass and aluminum tubing at my local
metal recycler.
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Measuring Tape:
I found it much easier
to work in millimeters rather
than inches. The problem was finding a tape
measure that uses mm here in the USA. I found one made by The L.S.
Starrett Company and it is model # CS1-8ME12. Lowe's Home Improvement
carries the item but only at their web site. It cost about $10-. Another
possibility is L.S. Starrett model # CH12-10DME
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Conclusions:
Clearly there is
more to a chime than I had anticipated and I am sure I did not learn all that there is to know about the physics of a chime. This was
a Christmas present project and not a focused research project. I am convinced that it is not necessary to tune a set of "bell-like" chimes designed for a
musical note from fundamental C2
through C4 because the formula achieved the desired frequency
within 2 Hz. Tuning to achieve an accuracy closer than 2 Hz was a waste of time. However, for a fundamental
note from about C5 and
up, tuning is required. Having said that, I am not convinced
that choosing a musical note for the range from C2 through C4 by choosing
the fundamental frequency is the correct approach. The actual musical
note depends upon the configuration of the overtones and they are dependent
on the choice of metal used to manufacture the tube. Therefore, the
correct length is not the length for the fundamental note but a length longer
than the fundamental. I leave the determination for the correct tube
length to achieve an exact musical note for another time.
However, building a set of chimes for fundamental C2 or C3 sounds very melodious
and
is definitely worth the effort. Also, for or a chime set between C2 and about C4
I believe it is necessary to spread the notes
apart so they maintain their
individuality at a distance.
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Calculations:
Here it was necessary
to keep my eye focused on the goal of building wind chimes rather than
pursuing an occupation for true calculations, so I cheated a little.
Rather than be faithful to all the physical constants of density, Young's
modulus, material temperature, speed of sound, and so on, I chose to use
a single correction factor E (based on actual measurement) into the traditional
formula. This correction factor allowed me to move easily among materials.
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|
Update
December 2002 - 2007 |
|
Another
engineer (Chuck's Chimes) has posted a very good site on this subject and I would
encourage you to look over his site. He addresses an issue that we
looked in to but never completed and that is the resonance of the air
column matching the resonance of the tubing. The site contains a
nice calculator for determining length based on his approach.
Please
see
http://mysite.verizon.net/cllsj/windchimes/home.htm |
Pre-calculated lengths for tubing can be
found at these links
For
a different
material size use the formula below.
Formula for
the length of an open
end tube at a specific frequency.
L (mm) = (√
((E*3.14159*K*V)/F))*10
L (inches) = (√
((E*3.14159*K*V)/F))/2.54
Top
L= Length of chime
K= Tubing size and
wall thickness constant
ID = Tubing inside
diameter (inches)
OD = Tubing outside
diameter (inches)
E = Correction factor
determined from measured data. Suggest 1.15
V= Velocity of sound
(cm/s)
F = Frequency (Hz)
K = (√
((ID*2.54*0.5)^2+(OD*2.54*0.5)^2))/2
|
Approximate
Speed of Sound cm/s
|
|
Brass
|
Aluminum
|
Copper
|
Pyrex
TM
|
Lucite™
|
Steel
|
Cast Iron
|
|
359,000
|
500,000
|
375,000
|
517,000
|
184,000
|
500,000
|
448,000
|
|
|
|
|
|
|
|
|
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Suggested
Reading
Thanks to the suggestion of a visitor (Larry) here is a
book "The Physics of Musical Instruments by
Neville
H. Fletcher,
Thomas
D. Rossing" available at eBay
HERE
that has a great chapter on chimes and bells.
and
some very good links on the subject:
http://www.hibberts.co.uk/index.htm
This site has not only nice pages on bell sounds and tuning but offers free
software that lets you listen to the effects of overtones
and allows you to tune your bell or chime using a sound card and microphone.
Really nice.
http://www.msu.edu/~carillon/batmbook/index.htm
Chapter 5: The Acoustics of Bells is a nice introduction to bell physics.
http://www.mmk.ei.tum.de/persons/ter/top/pitch.html
Psychoacoustics of pitch perception.
http://www.mmk.ei.tum.de/persons/ter/top/strikenote.html
The strike note of bells.
Thanks Larry.....
Additional Reading
The missing
fundamental effect
The
missing fundamental (Hanover College)
Fletcher/Munson
Curves
Tune Lab 97
software
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A
Few Sources for Chimes
Wind
Chimes Free Shipping
Wind
Chimes & Gongs
The Wind Chime Page
Top
Last updated on
05/19/2008
|
