Bells, Gongs and Triangles:

Math and Design of Simple Vibrators

In Progress


Twelth Root of two  Binary series.  Graphic Copyright (c) 2001 Jock Dempsey
The 12th Root of Two, Binary series:

The Piano does not use a "pure" scale. Pure notes have frequencies that are whole numbers and the spacing is not equal. This was found to not work well on early keyboard instruments. The mathematics of the "tempered scale" was invented in the 17th century by mathematician, Andreas Werckmeister. This has even spaces in a 12 note octave (8 full notes and 4 half notes - the black keys on the piano). J.S. Bach popularized it in his "The Well Tempered piano" (Das wohltemperirte Clavier).

All Western music now uses this "compromise" scale. The series of numbers generated by the formula is used to generate tables of frequencies using what ever standard you want to start with. Today this is A4=440. That results in a C4 that equals 261.63 instead of the old Pythagorean standard of 256 (a binary number).

The same formula also generates the spacing of frets on guitars and can be used for calculating lengths of other vibrators (I was sure Pete F would know this one). The large 7 pedal orchastral harp also uses this system.

Early stringed instruments used a system of fractional proportions that resulted in frets of unequal spacing (half, thirds, fourths and fifths). Some books on guitars and banjos give the value 1/18th for dividing the fingerboard but this results in progressive errors that throws the 6th fret off by 1/8" (3mm) or so.

Many other cultures still use pure scales. The use of tempered scales is what makes non-western music sound so strange to to those of us that are used to it.

Although all our Western music is based on this mathematics it is very difficult to find information about it in most references on music and it is not taught because most musicologists do not write on the technical aspects of musical instrument construction where the mathematics is very important. When the tempered scale is mentioned it is in musical terms of cents (1/1200th Octove) or flats and sharps but NOT the mathematics. A few modern books on guitar construction discuss the mathematics but this is the exception.

This is typical of ALL references regarding musical instruments. They are writen about in musical terms and their place in music but never in any technical terms. The simplest facts such as length and width of an instument are complete mysteries in modern references. That is why I wanted to write a book on musical instruments from a mechanical engineering and construction viewpoint. I spent 18 months searching for such a reference to proove it does not exist (it does not). While doing some of this research on the internet I found the Blacksmiths Junkyard and stated answering questions there. . . and that brings us back to anvilfire. Now you know the rest of the story.

I'd still like to write that book. It was planned to include detailed CAD drawings of all the major orchastral instruments and construction technology. It would require a LOT of research (3 to 4 years) including dismanteling and reverse engineering many instuments. NO, virtualy none of this information is available anywhere. Much of it is still handed down from one maker to another (father to son so to speak). A real paradox in our "information age".

I produced a 3 volume bibliography in my search and collected several dozen references (some quite rare) while doing my research. I estimated a cost of $300,000 to do the job at the time but it is more likely a half million dollar job.

The problem is it is NOT a job for a musicologist OR oganologist (Organology is the study of the construction of musical instruments). Most of them have forgotten more about music than I will ever know. On the other hand they know virtually nothing about the physical constuction or manufacturing methods of musical instruments much less the design and engineering.

It is a book that NEEDS writing. People ask me why don't I write a book about blacksmithg. . . Why? There are more than enough and most are VERY good. The world does not need another one. My book on details of musical instrument construction does not exist. And the world needs it. It is a reference that would find its way to EVERY library on the planet. Now THAT is a book worth writing!

Ah, why is it a binary series? Waves go UP and DOWN. Standing waves (pure tones) in strings and other vibrators (including air) have pairs of opposite wave crests. The tempered scale has few pure tones but the series still applies.
   - guru - Monday, 09/24/01 18:40:30 GMT
A friend of mine (an experienced blacksmith), has been asked to make a set of pentatonic scale chimes for his daughter's schoolyard. He'll be using oxy bottles and is wondering if there is a formula for cutting the lengths for the various notes. We are wondering if slicing into them lengthwise effects the notes? Also any general tips or common mistakes to avoid would be helpful. Thanks so much.
   Christine - Monday, 11/19/01 20:47:38 GMT
Tones Christine, Yes there is (a formula). I'll have to dig into my old music research stuff. I THINK you use the formulae for a simple virbrator. The trick to it however is that there are several variables. Length and density (mass per unit length) resulting in weight. But I also think that there is a secondary vibration that has to do with diameter and mass in a circular direction. Then there is the fudge factor due to the hemispherical top end. Its not all math, there is some art involved.

I'll look it up. Remind me if I don't get back to you.
   - guru - Monday, 11/19/01 21:07:43 GMT
All,

While we're talking about re-cycling tanks, let me give a quick method of hanging them so they "ring free".

Pick up a pipe cap the right size to fit the threaded opening at the top. Drill and tap it. (I use 3/8" X 16 TPI) Run a safety nut onto the eye bolt, then thread the ey e bolt into the tapped hole in the pipe cap. Add a second safety nut on to the eye bolt from the bottom, and "trap" the pipe cap between the safety nuts. Now thread the pipe cap into the top of the cylinder.

They ring best is hung from an organic fiber. I use leather boot laces, doubled as necessary for strength. But if they are going to be outside, you can use a light chain, and the eye bolt, pipe cap combination seems to permit the bell to ring clearly.
   Paw Paw Wilson - Monday, 11/19/01 22:53:13 GMT
Guru, Paw Paw, Pete F. and Matt: Thank you all for the chime tips. Guru, you said to remind you to look for the "formula" for rough cutting the chime lengths. Please consider this a gentle nudge.
Thanks again.
   Christine - Monday, 11/26/01 19:48:09 GMT
Chimes Christine, I found SOME info but not exactly what I wanted. Roughly.

1) A beginning note in a series can start at almost ANY random frequency. That is unless you plan on playing with other instruments tuned to a standard. In that case you need a darn good ear OR a sophisticated oscilloscope to hit the right frequency(s).

2) Technicaly a gong made from a welding cylinder is a bell. Bells have numerous overtones and the classic bell shape is designed to balance these into a pleasant sound. There is no one true frequency for a bell so tuning is tricky. Most bells are tuned to the first major overtone.

3) In cylinder bells the top hemisphere is a null (it doesn't add to the sound). When comparing the sounding length of two bells the null part is discounted. This is approximately 3/4 of the hemisphere. So you measure length from the lower part of the bell's "shoulder". You estimate this on your first bell, tune the second exactly an octave higher then itterate the exact point.

4) Once you have your first bell the others can be made in proportionate lengths based on the "standard". In just intonation (pure notes) these lengths are fractional based on whole numbers. This series is 17/18, 15/16, 7/8, 3/5, 3/4, 5/8, 1/2 (see series below). In standard western tempered notation (like a piano and modern guitar) the steps are even multiples of the 12th root of 2 (1.059463094359...) or its reciprical (0.94387431268..)

Where I am having trouble is finding the correct way to apply the fractions. In the case of strings it is straight length but in tubes I THINK a square root is stuck in the ratio/formula.

What this means is that in a string 1/2 length = next octave. But in tubes and bars you use the square root of the ratio. . . . I think. I'm still looking and actual research does me no good because I am tone deaf. . Or at least I can't tell one note from another. :(

The best I can tell is that
L2 = SQR(a/b) * L1

L1 is your standard or starting point.
L2 is your goal.

a/b is your ratio of notes. Using this series where L1 (your longest gong/bell = 2/1)

15/8, 16/9, 5/3, 8/5, 3/2, 4/3, 5/4, 6/5, 9/8, 16/15, 1/1

That is an octave set.

As Pete mentioned, cut long and then trim to tune. The shorter the tube the higher the note. If you go too short save the piece for the next note and start again.

I'm sorry I have not had time to prove the above math. Four years ago when I was doing research on musical instruments I could have spit it out and sworn to its accuracy.

   - guru - Monday, 11/26/01 21:10:44 GMT
More on Bells: Christine, if your friend makes the set we would like to know the dimensions and how it came out.

More on Tuning: As I mentioned an oscilloscope is about the only way to match an exact frequency. However almost any other standard instrument can be used to help tune another. As mentioned above, make bell #1 then make one an octave higher. This will prove your calculations and the null fudge factor.

I have an old HS band Glockenspiel that needs rebuilding. The plan was to take it apart, carefully weigh and measure each bar and apply the standard math to it comparing the theoretical to the as-built. I got as far as testing several bars to see if the theoretical math fit (IT DID). I also applied the theoretical to tuning forks and it worked but the NULL is a factor that needs iteration as above. What all this proved is that for simple vibrators you could predict the exact frequency from the theoretical.

Sadly all this is burried in my old PC and in note books and slips of paper. . . Most of it was not formalized so it is difficult to dig out today. If I had thought it would apply to blacksmithing I would have taken better care to file it.
   - guru - Monday, 11/26/01 21:22:57 GMT >
I have been making triangle dinner bells from coil springs[small car strut shock absorbers]straightened with a vise and a torch.I forge both ends to a point, scroll one end and forge an eye/hook on the other end.I bend the corners with a torch,and heat up the three side to a cherry red,wrap in durablanket and leave it over night.\ When hit lightly the bell rings clearly, but not loudly.When hit hard the clarity goes south.The harder its hit the worse it sounds.What am I doing wrong??? Summer is hanging in in the north country,Tomatoes ripening on the vine, unheard of.
dimag -- Thursday, 08/27/98 02:25:27 GMT


TRIANGLES (musical or dinner bells - dimag):

Although small musical triangles SEEM to make a sweet clear tone all triangles make a dissonant tone (many overlaping frequencies not necessarily in sync).

Hard triangles work better than soft but I found all I needed to do was quench the mild HR bar immediately after bending. Hard corners seem to do the trick and 20-30 point carbon steel is good enough.

Shape and proportion makes a HUGE difference in triangles. I always made a tuning fork variety (three bends, open ends facing each other). After making a triangle from 1" wrought iron that was a complete failure I made a short study of the subject. I experimented with a LOT of proportions. In the end I found a 30" length of 1/2" HR made my "Best Ringer Ever". Bends were at center and 5" from the ends with about 1/2" betw. the ends. If you make this one I'd appreciate credit for the pattern!

-- guru Thursday, 08/27/98 22:20:52 GMT

Hi Guru,
I purchased a wind chime last month, and it is made of small metal figures and designs (each about 6-8 inches tall) tied together with string. I would like to make other wind chimes like it, but do not know where I can puchase the small metal figures. I've gone to several craft stores but can't find anything like them. The metal looks like it will rust over time (that's the desired look). The figures/designs are pretty rough in the way they are cut and I image wouldn't be too expensive. Do you know of any website or stores that would sell such things? Any info would be appreciated. thanks, Dave
Dave   - Wednesday, 02/09/00 21:01:50 GMT
WIND CHIME: Dave, Those figures are probably cut with a computer operated cutting torch (REAL hand craft) and are the product of the person making the wind chimes. Craft stores around tourist attractions are full of this stuff that is supposed to look like "primitive" art. Some is, but most is cranked out by the thousands so that it can be sold in catalogs.

There are THREE ways to do this. One is to purchase a machine, make your own CAD templates (HO HO! Its not even LOW TECH primitive art), and cut the pieces as needed.

The second method is to make CAD templates and have your local steel service center cut them for you.

The third method is to pay someone to produce the CAD templates (I charge $50/hr) and take it to the steel service center. . . Ah. . . Carolina Steel has a $100 minimum. . . Then there is shipping.

OR, you could take a cutting torch and cut them one at time like the tourists think it is done. . . The steel plate figurines I've seen in Asheville, and Black Mt, NC were made from CAD templates that even had the "shakes" of an inexperianced welder. . But they were all EXACTLY alike.
- guru - Wednesday, 02/09/00 21:44:23 GMT

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