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INSTRUCTIONS AND USER NOTES
Many users, especially those with prior knowledge of the subject area, will be able to use this bar-length calculator program without need for additional information. Other users will benefit from the notes that follow here. The program actually has two calculators:
one for those who are tuning for equal temperaments
(including the standard western scale known technically as
12-tone equal temperament) and one for those who are tuning
to just scales. You can use whichever is appropriate for
your purposes. Fuller information on just
intonation and equal temperament
appears later in this page. Calculating Bar Lengths Without Preparing a Sample Bar This program will not give you specific bar lengths unless you prepare a sample bar as described below, and input its pitch and length information. Without cutting a sample bar, you can still use this program to calculate how long the bars should be relative to one another for a particular scale, but without specific lengths. To do this, enter a value of 1 as your sample bar length (in the just intonation calculator, you'll typically enter it alongside the ratio for your first, lowest, or 1/1 tone, but it you could enter it alongside another tone as well). The calculator will then calculate what the other bar lengths would be relative to the sample bar, e.g.: if the calculator comes back with a length for the second bar of 0.9632, that means the second bar should be 0.9632 times as long as the bar for which the sample bar length value of 1 was given. [For those who wish to calculate specific bar lengths without having to cut sample bars, a formula for doing so appears in the article "The Marimba: Scientific Aspects of its Construction and Performance" by Greg Merrill, available on the web at http://faculty.smu.edu/ttunks/projects/merrill/MarimbaH.html. The formula is taken from The Musician's Guide to Acoustics by Murray Campbell and Clive Greated. (New York: Schirmer Books, 1987). It requires values for the density and Young's modulus of the bar material. (Young's modulus is a measure of rigidity.) The article provides these values for a few woods commonly used in marimba making. For a more comprehensive and more technical treatment of the topic, see The Physics of Musical Instruments by Neville H. Fletcher and Thomas D. Rossing (New York: Springer Verlag, 1991), pages 54-60.] The formulas that this program uses to make its calculations are valid only for bars that are uniform in shape over their length. Thus, it will work for hollow tubes, be they circular or rectangular in cross section, as well as solid cylindrical or rectangular bars, I-beams, and other forms for which the cross sectional shape is the same over the length of the bar. It won't work for bars that are scalloped underneath, as wooden marimba bars often are, or for other shapes for which the cross section is not the same over the length of the bar. (To come up with formulas for irregularly shaped bars would be an unrealistic task because of the infinity of different possible shapes, all showing different acoustic properties, that the user might come up with.) Bars with such irregular shapes will probably have to be tuned by ear or by prototyping. For this program to return valid results the sample bar should be the same thickness or diameter as the new bars to be calculated and made. If you plan to use bars of different thickness or diameter for different parts of your instrument's range, you will need to cut and measure a new sample bar for each bar size, and run the program separately for each size. For instance, if you plan to make a 2-octave tubulon (that's a steel-conduit marimba) using 1" tubing in the upper octave and 1.5" tubing in the lower octave, follow these steps: 1A) Cut a 1" sample bar and tune it to some note in the upper octave. 1B) Run the program to calculate your bar lengths for the upper octave based on the 1" sample bar. 2A) Cut a 1.5" sample bar and tune it to some note in the lower octave. 2B) Run the program to calculate your bar lengths for the lower octave based on the 1.5" sample bar. This program will return theoretically perfect results for theoretically perfect bar materials. In reality, whatever material you use will be imperfect, having some degree of irregularity in shape and rigidity. For metal tubings and bars manufactured to very close tolerances, you may find that you can get good results simply by cutting the bars to the lengths that the program calculates for you. For woods, and especially woods with irregular grain or knotholes, the real-world results tend to be variable enough that you will probably want to take the more cautious approach: cut each bar a little longer than the calculated length, and tune it up to pitch by gradually shortening with frequent listening tests, using your ear or an electronic tuner. For metals not manufactured to close tolerances the results will generally be more dependable than with wood, but you may still choose to cut a little long and tune up. Which approach you take in any given case will depend in part on how picky you are about accuracy in tuning. Well made bars that are perfectly round in cross section will produce a single fundamental pitch. Bars that are rectangular in cross section will in theory produce two: a higher one when vibrating sideways, as from a side-strike on the narrow side surface, and a lower one when vibrating up-and-down, as from a strike on the broad surface from above. In practice, it's not likely that anyone will strike a flat, rectangular bar from the side, and the effect of the sideways fundamental will be negligible in the tone. However, in bars that are not square but nearly so, both fundamentals may come into play, producing a dual pitch, with one or the other predominating to varying degrees depending on the direction of the strike. The same can happen with imperfectly circular bars or tubes. The dual-pitch effect is especially common in tubes having a seam running lengthwise inside. To avoid potential problems in these nearly round or nearly square cases, be consistent in how you orient the bars or tubes when playing them. In tuning the sample bar, be consistent in striking it from the same direction relative to any cross-sectional irregularities, and try to position all the bars on the finished instrument in such a way as to make it natural to consistently strike from that same angle. Additional note: With nearly circular or nearly square bars, the two fundamentals may be very close in pitch. In that case, the bar or tube may produce what seems like a single pitch but with a "beating" -- a sort of tremolo or wah-wah effect, when both sound together. If you like this effect, you can try to bring it out by striking from a direction that brings both into play. If you don't like it, try to set up the instrument so that the striking direction will consistently be one that favors one vibrational mode over the other.
No Sample Bar, One Sample Bar, or More than One Sample Bar? If you do not cut a sample bar, you can still use this program to calculate how long your bars should be relative to one another. Without the information from a sample bar, however, the program will not be able to calculate actual lengths in inches or centimeters. If you do cut sample bars, you may choose to cut one sample bar, or more than one. If you cut just one, then all the bar lengths for the instrument will be calculated based on that bar. If you cut more than one, then the length calculation for each new bar will be based on the information from the sample bar closest in pitch to the bar being calculated. Using more than one sample bar will usually give more accurate results, especially if you are working with non-uniform materials such as woods or with instruments having a large range.
You will tune each sample bar to one of the pitches of your intended scale. The calculator program is flexible; it can base its calculations on sample bar information for any degree in your scale. If you will be preparing just one sample bar, it will be best to set it to a pitch somewhere near the middle of the intended range. If more than one, try to space them out across the range -- e.g., if you're making a three-octave instrument, you could tune one sample bar to a pitch near the middle of the upper octave, one near the middle of the middle octave, and one near the middle of the lower octave. To cut and tune a sample bar, begin by cutting a bar a little longer than you'll need for the scale degree you intend to tune to. But, of course, you don't know what that length is yet, so you'll be guessing -- maybe it will be a wild guess, maybe an educated guess. Just cut to what seems like a reasonable length. If the resulting pitch is below your intended pitch, that's good. You can tune it up to the desired pitch by shortening. Shorten it a little bit at a time, using a saw, a grinder, a tubing cutter or whatever else works. Check the pitch after each shortening, either by ear or with an electronic tuner, until you arrive at the intended pitch. Remember, the sample bar pitch can be any pitch in your scale (although a central pitch is preferable), so if you find yourself conveniently landing on some scale pitch or other, you have the option to stop there and use that as your sample bar pitch. Be sure to tune carefully, as the accuracy of the subsequent calculations depends on the sample bar information you input. When you've cut and tuned your sample bars, note their pitches and carefully measure their lengths. This is the information you will input into the program.
When it comes to thinking about musical scales in a precise and mathematical way, there are two widely-used approaches: just intonation and equal temperament. In musical scales using just intonation, people define the musical intervals that make up the scale by specifying ratios between the frequencies of the pitches. In equal temperament, the scale is defined by breaking the octave into some number of equally spaced steps. 12-tone equal temperament is the standard western scale, but other numbers of divisions per octave can also be used. Because the two approaches call for different sorts of user input as well as different underlying mathematics, this software presents two separate calculator interfaces -- one for ET and one for JI. You can use whichever suits your needs. If you want to work with the standard western scale, go for equal temperament, and specify 12 tones per octave. If you want to stay in the standard western scale but won't be using all 12 chromatic tones of the octave then go ahead and have the program calculate the full 12-equal set of bar lengths. You can then use the results for the notes you want and ignore the rest. This is what you'd do, for instance, if you want a simple C-major scale matching the white notes of the piano. (The piano is normally tuned to 12-equal, and the C-major scale is a subset of this 12-equal scale.)
For equal temperaments, all you need to do to specify your scale is tell the program which equal temperament you want -- that is, how many tones per octave. Based on this and your sample bar information, the program will then do the calculations. For just intonations, more input will be needed. Just intonation scales are specified by giving a frequency ratio for each step of your desired scale. The frequency ratios are relative to the fundamental pitch for the scale. This fundamental pitch is given by the ratio 1/1. The musical interval of an octave corresponds to a doubling of frequency, so the pitch an octave above 1/1 will have a frequency of 2/1. Between 1/1 and 2/1 will be a set of ratios serving to specify the pitches for one octave of your scale. For instance, a standard just major scale over one octave would have the following ratios: 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1. (This scale will sound much like a major scale in the standard 12-tone equal temperament, but to the ear of a just intonation theorist will be more perfectly in tune.) To continue this scale up through a second octave, each of the ratios from the lower octave will be doubled. After doubling the ratios from the first octave and reducing them, you get a second octave that looks like this: 2/1, 9/4, 5/2, 8/3, 3/1, 10/3, 15/4, 4/1. The next octave would involve doubling again: 4/1, 9/2, 10/2, 16/3, and so forth. It is a series of ratios, such as those described above, that you'll input to specify your scale. You can specify any just intonation scale this way (an infinite number of different just scales are theoretically possible). Based on this and your sample bar information, the program will then do the calculations. Just intonation is a big topic, and lots of composers and theorists are active in scholarly exploration of it. Our book, Musical Instrument Design, has a clear, concise and practical treatment of the subject in Chapter Three. (This book also has a lot more information on free bar instruments, both practical and theoretical.) Click on the "Catalog" button above for information on the book and how to get it.
Back to ExMI Free-bar Calculator page .
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