Free-Bar Length Calculator Information Page

Last modified 8-28-2009

Introduction

These notes contain background information, technical notes, helpful hints, and other useful stuff for people using ithe Experimental Musical Instruments Free-Bar Length Calculator.

The purpose of this software is to allow you to calculate how long to cut the bars of a percussion bar instrument in order to get the pitches you want.

To calculate bar lengths from scratch is often problematic, requiring data about the bar material which most casual builders won’t have access to and which is often unreliable anyway.* This calculator takes a different approach. It asks you to cut and tune a sample bar of your chosen material, then uses the pitch and length information from that bar to calculate lengths for the instrument’s remaining bars.

The calculator can also be used in a different way, without requiring you to cut and tune a sample bar. Instead of giving the calculator a sample bar length, you can input a fictitious bar length of 1 (no need to specify a unit of measurement). In that case, the calculator won’t give you specific lengths to cut to, but it will give you relative bar lengths for your desired scale — that is, it will give you a set of multipliers telling you how long each bar of your scale should be relative to the bar represented as “length 1.”

This program actually consists of two calculators: one for those who are tuning for equal temperaments (including the standard western scale known 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.

 *[Footnote from above: For those who wish to calculate specific bar lengths from scratch without making a sample bar, a formula for doing so appears in the article "The Marimba: Scientific Aspects of its Construction and Performance" by Greg Merrill, 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.]


Important Limitations

This software only works for bars, tubes or rods that are regular in shape. If there is irregularity in the shape over the length of the bar, such as hollows, bulges or bends, the calculator’s results won’t be valid. Unfortunately, that means you can’t use it with marimba bars that are tuned by thinning the underside of the center 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.) Bars with such irregular shapes will probably have to be tuned by ear or and/or by prototyping.

Given bars of regular and uniform shape, the calculator’s results will be very accurate for bars made of metal and other materials manufactured to close tolerances. For wood, with its natural variations in density and rigidity, the results will be more approximate, but they can still serve as a useful guideline.

If you plan to use bars or tubes 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 thickness, and run the calculator program separately for each size.

A suggestion for wooden bars or bars of any material that might be slightly irregular in shape: For precise tuning, initially cut the bar a little longer than the calculator recommends, and fine tune by shortening it up to pitch by ear or with an electronic tuner.

 

Two Ways of Using the Calculators (more detail)

As mentioned above, you can use the free bar calculator either of two ways:

Method 1 — the sample bar method. Use this method to get actual lengths for your instrument’s bars. First, cut and tune a sample bar using the same type of material you’ll be using for the bars (details below). Tune this bar to any of the notes in your intended scale. Enter this bar’s length into the calculator’s input fields along with the information requested for the scale you want . (Do this using either the Equal Temperament or the Just Intonation version of the calculator, depending on which kind of scale you want — see details below.) When you click “calculate,” the software will calculate bar lengths for the remaining notes of your scale.

Method 2 — the relative lengths method; no sample bar needed: 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 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. Example: 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.


One Sample Bar, or More than One Sample Bar?

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. Typically it is enough to work from a single sample bar. However, cutting more than one sample bar from different parts of the range may give more accurate results if you are working with non-uniform materials such as woods or with instruments having a large range.


Preparing the Sample Bar

You can tune the sample bar to any of the pitches of your intended scale. It’s usually best to tune it to a pitch somewhere near the middle of the intended range.

To cut and tune a sample bar, begin by cutting a bar a little longer than you’ll need for the note you intend to tune to. But, of course, you don’t know what that length is yet, so you’ll be guessing — maybe a wild guess, maybe an educated guess. Just cut to what seems like a reasonable length. Then test the bar by comparing its sounding note to the intended note. You can do this comparison by ear, using an electronic keyboard or other source for the reference tone, or you can do it using an electronic tuner. If the bar’s 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 for the material in hand. Check the pitch after each shortening 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.

When you’ve cut and tuned your sample bar, carefully measure its length. This is the information you will input into the calculator.


Equal Temperament vs. Just Intonation

When it comes to thinking about musical scales in a mathematical way, there are two common approaches: equal temperament and just intonation.

  • 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, and this is the scale most instrument builders will want to use. However, other divisions of the octave can also be used. For instance, 19 tones per octave and 31 tones per octave both are popular.

  • In just intonation, the musical intervals that make up the scale are defined by specifying ratios between the frequencies of the pitches. Many people believe that just tuning produces musical intervals that are more natural and sweeter sounding than those of equal temperament. Just intonation’s basis in ratios more closely reflects the way our ears and brains process pitch information.

The next several paragraphs will give you more information on these two scale types.


Inputting Information to Specify your Scale

In the equal temperament version of the calculater, all you need to do to specify what sort of scale you want is to tell the calculator how many tones per octave — 12 for the standard twelve-tone equal temperament; 19, 31, or some other number for other equal temperaments.

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. There is an infinity of possible just scales that could be specified in this way; if you are a practitioner of just intonation then you may already have plenty of ideas about the ratios you’d like to work with and how to manage them. In case not, here’s a little more information to help you in working with the calculator, using the most common form of a just major scale as an example:

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, 5/1, 16/3, and so forth.

Ratios for other just scales you might want to try can be found below.


OK, So What Scale Do You Want to Tune To?

To tune to the standard western chromatic scale including all twelve notes per octave, use the equal temperament version of the calculator, and specify 12 tones per octave.

Alternatively, you may want to use a standard western tuning, but don’t want all twelve notes per octave of the chromatic scale. For instance, you might 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. Other subsets of the chromatic scale include the standard western minor scales, various pentatonic scales and so forth. In this case, do as follows …

To tune to a subset of the western chromatic scale such as a standard major, minor or pentatonic scale, then once again, use the equal temperament calculator and specify 12 tones per octave. Then use the results for the notes you want and ignore the rest.

To tune to an equal temperament other than twelve-tone, use the equal temperament calculator and input a different number of tones per octave. As mentioned above, 19 and 31-tone equal temperament are often favored because the resulting scales happen to do a good job of approximating the most important just intonation intervals. If you want something more strange and unsettling sounding, try 11 or 13. For something exotic yet peaceful, try 5.

To tune to a just intonation scale, the possibilities are endless. Here are a few potentially useful ones.

Basic just major scale: 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1

Basic just minor: 1/1, 9/8, 6/5, 4/3, 3/2, 8/5, 9/5, 2/1

Just pentatonics: select five tones from among the minor or major scales above

A nicely exotic, slightly bluesy scale: 1/1, 11/9, 4/3, 3/2, 7/4, 2/1

A Note Concerning Dual Pitch Effects in Imperfectly Round or Square Bars

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 pitches: 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 rods 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.

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.

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