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 neighbor, 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.

It took a month 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 internet 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.

Another engineer, Chuck's Chimes, has made a significant contribution to the understanding of chimes and I encourage to look over his site too.

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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 ranges 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 ranges 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 ranges 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 instruments. 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.
Some sources are:

http://hyperphysics.phy-astr.gsu.edu/hbase/sound/subton.html#c2

http://ccms.ntu.edu.tw/~karchung/Phonetics II page thirteen.htm

http://en.wikipedia.org/wiki/Missing_fundamental

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 through 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 of the older version you can download from my site Chime.exe. 

If you have trouble unzipping Greg's new version here are the two files you need. Chime32A.exe  and TUNING.DAT
Place both files in the same folder and run the exe file.

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Note Selection:

Note selection is mostly a personal choice.  I selected 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 a "non-bell sounding" chime and attempting to excite exact notes, exact tuning is required. On the other hand, if you desire a "bell-sounding" chime then cutting a tube to the length suggested by the formula listed at the bottom of this page should be adequate. For your convenience, pre-calculated lengths for various materials and sizes are listed in the table below in addition to calculate your own dimensions in these Excel sheets. Note that the pre-calculated table uses a correction factor or 10.3% while the DIY Excel Sheet suggest a correction factor of 15%.

Calculating exact lengths appropriate for use in an orchestra setting is complicated and uses parameters not available to most people. Most of us have selected an approach that simplifies the process and simplifies the formula.

With that in mind, I decided to provide an approach that offers the best of both worlds. This method uses a correction factor that you can adjust to suit the needs of the your metal of choice. There is a wide range of hardness for metals.  Since most "wind chime" builders will not have the resources to determine the hardness of their metal, the formula provides you the choice to get it "exact" or to get it "about right".

To get it "Exact" is fairly easy. Take any tube of any metal, cut it to some random length and measure the exact frequency of resonance. Go to the DIY Excel Sheet and adjust the "correction factor" until the sheet provides an exact match for the length calculation compared to the length of the measured tube. This process calibrates the Excel formula to your specific metal.

To get it "about right" you can select a correction factor of about 10 to 15%. I suggest 15% for an all-around number for any material. This method works because the table will place each length in the correct relationship to each other. i.e. A2 will be in the correct relationship to A3 and so on. It may not be an exact A2 for an orchestra setting, it is close enough for good sounding "wind chimes".

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 better understand the difficulty in selecting a chime note to match a selected 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 chime tube mechanical support:
Chime support is at a node point 22.42 % from the either end. I found it 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 best and the line is available in a variety of colors.  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.

A variety of methods can be used for mechanical support

Method 1	Method 2	Method 3	Method 4
Most typical and the most stable in high winds.	One support point and somewhat less stable than method 1. More difficult to de-burr the inside support hole.	May be more pleasing to the eye with less visible string. Less stable that method 1. More difficult to de-burr the inside support hole.	A  mounting pin eliminates the string knot and provides a cleaner look.

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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 view 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 select 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 "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 Wind Chime 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 considerable 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 from high winds.

If you live in a area with little wind then eliminating the 1oz weight may be desirable.  See the next section on striker location for more info on striker weight and location.

The small diameter disk was used to minimize the striker from contacting more than one chime at a time.  Attempting to strike several chimes simultaneously 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 the striker is, at best, disappointing. A larger diameter disk may be preferred as it is less likely to be blown to the outside of the chime circular mounting profile during high winds.

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. Striking the center assures excitation of the second harmonic.

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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A Better "Ringer Dinger"
Update 2009, The traditional striker describe above works well, however, I was looking for an approach that would produce a more consistent sound.  To my surprise, I believe there is a better way. Striking the chime at the very end consistently produced a much better sound that striking near the end or in the center. Try it for your self. On most chimes it made a difference except on the high notes.

A tapered striker was used to accomplish this goal and to assure the striking of all chimes at the very end.  If you have access to a wood lathe, it's an easy task to machine a tapered striker from wood.  I used pressure treated stock but any hardwood would work well too. It is important to under-cut the inside to minimize excessive weight which will inhibit striker resonance. 

This heavier striker made it more difficult to resonate. (see Striker Location, below). As described in "Chuck's Chimes" the choice of material depends on the note selection. For lower frequency chimes a soft, heavy striker is best and for higher frequency chimes a harder, lighter striker works better.

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The Wind Chime Striker Location a new approach "The Resonate Striker"

Update 2009, I happen to live in a wooded area with little wind and have struggled to achieve good strike energy with low winds. With that in mind, I set out to improve the low wind performance of the striker.

The objective is to maximize striker movement with little input energy. The easy solution was to resonate the line holding both the striker and the catcher using the second mode bending principle. This resonance will help to amplify and sustain the motion of the striker with little input energy from the wind.

You can easily recognize this movement by using both hands to hold a string vertically and have a second person pluck the center of the string. The natural resonance of the string will cause the center to move back and forth horizontally. If you position the striker at the exact center between the top support and the catcher you can achieve this resonance.

Jack Maegli, on Chuck's Chimes page mentions that the catcher/sail should have a weight equal to about 25% of the striker. For medium to high winds and for a non-resonate mounting,  I completely agree with that suggestion. In this instance when you attempt to resonate the support line,  I suggest a light weight striker to assist with good resonance. A heavy striker would not resonate.

When resonance is working well you will notice that as the catcher comes to rest, the striker will continue to bounce off the chimes for a few more strikes, an indication of dissipating the stored energy from resonance. See this VIDEO for a demo of the "resonance striker." Notice the large movement of the striker compared with little movement from the catcher.

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The Wind Chime Wind Catcher, a new approach. "The Unstable Catcher"

Update 2009, Traditional wind catchers have worked well for years, hang vertically, and can be configured in a variety of materials, sizes and shapes. The objective of a better catcher is to cause the catcher to become unstable and to move the striker in a more circular motion from the force of single-direction winds and to better capture the effect of turbulence.

First, my dissatisfaction with the traditional wind catcher is that in single-direction winds it has a tendency to cause the striker to swing to and from the direction of the wind and not strike adjacent chimes. Much like the periodic motion from the pendulum effect and sounding like ding dong, ding dong, etc.

Secondly, with turbulent winds the traditional catcher does not do a good job of contacting all the chimes. Certainly, the wind is random in direction and the catcher will eventually cause the striker to contact all chimes, but I wondered if there might be a better way?

Thirdly, as you know, winds vary considerably across the country from low speed to high speed and from predominately single-direction winds to mostly turbulence winds, and in any combination. The best solution for you will depend on your type of wind.  You may need to try a few catchers.

From some windmill research I did several years ago I recall that wind, close to the ground, can behave quite differently than winds aloft. As you know, wind can be quite turbulent and often do not blow horizontally as intuition would suggest. Instead, it is a two dimensional force simultaneously blowing in both the horizontal direction and the vertical direction. Swirling and blowing uphill, downhill, and horizontally. Wind sheer is a common occurrence close to the ground and perhaps we can exploit that force to make a better wind catcher.

To better understand wind turbulence mixed with single-direction winds watch this 20 sec VIDEO showing a bi-directional wind vein mounted on my deck. You probably noticed the swirling motion mixed with single-direction winds and the random uphill and downhill movement, pitch & yaw. Let's take advantage of this movement to create a striker movement that is somewhat rotational in nature and does a better job of striking all the chimes.

To better capture turbulence the solution is quite simple. Mount the catcher at 45^O to the horizontal so as to catch the pitch and yaw forces. See the picture to the left and right for an easy  solution. Thread the support line through two small holes next to the center of an old CD disk and tie the knot slightly off-center to create the 45^O slope. You may need to glue the line in place for the long term.

A second approach to capturing turbulence and causing the striker to move in a more circular motion is to hang the catcher perfectly horizontal.  This may seem counter intuitive but depending on your particular type of wind it can work surprising well.

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Choice of Material for Wind Chime Tubes:
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 of a lack of overtones at these frequencies and the chime approaches a pure tone at the higher frequency. Tubing sources seems to becoming more scarce at time goes on.  Here are a few I found but I have no personal experience with them. I have had considerable success in locating brass and aluminum tubing at my local metals recycler.

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Source of Tubing:

Aluminum and brass tubing tend to exactly follow their stated ID and OD dimensions, however, copper tubing does not. See the table below for actual dimensions for Type L copper tubing.  More on type K, L, and M  here

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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, 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  C5 and up, tuning is required. 

Having said that, I  am not convinced that selecting a musical note for the range of 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.

Your particular wind type (single-direction or turbulent) and wind speed will determine the best choice for both the wind catcher and for the chime striker.

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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, and so on, I selected a single correction factor E (based on actual measurement) to incorporate into the traditional formula.  This correction factor allowed me to move easily among materials.

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

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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

1 in (inch) = 25.4 mm

Calculate your own dimensions two methods available

Method 1 "Excel Sheet"-  All dimensions calculated and based on OD, ID and Material type*.

Method 2 "Excel Sheet"** - Single dimension calculated and based on Note selection, OD, ID and Material type*.

* OD = Outside dimension of tubing (inches), ID = inside dimension of tubing (inches), Material type =  Aluminum, Brass, Cast Iron, Copper, Pyrex^TM  and  Steel.  Note selection by frequency in Hz.

**Method 2 courtesy of Ken Petrocelly

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Pre-calculated lengths for some common materials used in chimes are in the table below. For a different material or size use the formula above, or method 1 or 2 Excel sheet above.

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Suggested Reading

An interesting physics class, student project,  authored by Professor G. William Baxter and Assistant  Professor Keith M. Hagenbuch, both from Penn State, Erie

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.  
The missing fundamental effect
The missing fundamental (Hanover College)
Fletcher/Munson Curves   Fletcher/Munson Curves with ISO 
 

Software:

Windchime Designer V1.0" by Greg Phillips
If you have trouble unzipping Greg's new version here are the two files you need. Chime32A.exe  and TUNING.DAT
Place both files in the same folder and run the exe file.

Tune Lab 97 software

Some good links:

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.

Not exactly related but an interesting video Steam Driven Chimes   

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A Few Sources for Chimes

SQUIDOO

Wind Chimes & Gongs

The Wind Chime Page

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Last updated on 10/28/2009