Building a Monochord:
experiencing the history and science behind musical scales

The companion article in this column, Building the WHAP!-a-phone, seemed to generate a lot of interest among readers. Several wanted more information on its construction so they could create replicas while others just wanted to experience its sound.1 With that curiosity in mind, I thought it might be interesting to detail a second project that our two junior high acoustic engineers, Julie and Barbara, completed for their science fair competition. This one was a monochord. While the monochord doesn’t have a cool name like the Whap!-a-phone, it does have advantages in being smaller and less expensive to build. It also has centuries of science and history behind it stretching all the way back to the ancient Greek civilization. Creating a monochord is a wonderful way to explore how musical scales and tuning systems work.

As its name implies, a monochord is a simple instrument consisting of a single string stretched tightly between two posts on top of a resonant box. It also features a moveable bridge with which the user can shorten or lengthen the vibrating part of the string. The string is usually plucked to create the tone; however, it can also be bowed if a sustained sound is desired. The monochord can be used to play music, but it is more useful as tool to study the science behind how musical scale used in music is derived.

(A monochord showing its moveable bridge.)

Some ancient theory on how to create a musical scale

The Greek mathemetician, Pythagoras, used the monochord-- some say he actually invented it-- to demonstrate the numerical relationship of the scale notes used in music. To Greek philosophers, music was a manifestation of universal perfection, and Pythagoras sought to demonstrate this by showing that scale intervals could be created out of the simplist, whole-number ratios. Pluck any tight string and it produces a musical tone because of its regular vibrations. Divide this string perfectly in half (a ratio of 2:1) and it produces a tone a perfect octave higher than the original-- the purest sounding interval. That’s because the string is vibrating twice as fast as the original.

Divide the same string perfectly into thirds (a ratio of 3:1) and it produces a tone a perfect fifth above the octave-- the next purest sounding interval. That’s because it’s vibrating 3 times as fast. In order to bring the perfect fifth down into the same octave as the original tone, simply invert the string length ratio (3:2 rather than 3:1).

We can keep generating progressively more complex ratios to produce progressively more complex sounding musical intervals. Here is a chart of these ratios, the musical intervals they produce, and how most listeners would classify their consonance/dissonance:

But remember, Pythagoras was a philosopher and mathemetician, not a musician. He was primarily interested in proving mathematical perfection in music. He reasoned that the entire musical scale could-- and therefore should-- be generated from just the two simplist ratios, 2:1 and 3:2. Following Pythagoras’s “simplicity equals perfection” philosophy, musicians created scales based on his reasoning for the next thousand years. In fact this method is still called Pythagorean tuning. Here’s how it works:

Start with any tone-- let’s say the note A, vibrating at a frequency of 110 cycles per second.2 If we multiply this number by 3 and divide by 2 (3:2), we get the note E (165 cps).

Multiply 165 by 3 and divide by 2 (and again by 2 to bring it into the first octave) and we get B 124. If we keep doing this process on each new number generated, we will end up with all twelve notes in a musical scale:

Enough theory, lets build the monochord

Julie and Barbara wanted to use Pythagoras’s concepts to build a scale they could use to make music, but first they needed to build the basic instrument that they could put the scale on. Their first step was to create a frame using 1 x 8 boards as shown here:

The frame was constructed as a trapezoid to help it resonate at a wide range of frequencies. The size was roughly 48 inches long by 24 inches wide at the widest end and 8 inches wide at the narrow end. Screws and Elmers glue were used to hold the frame together.

Because the string needed to be held taut, Julie constructed a peg box for the narrow end of the body. This was similar in size to what would be found on a string bass. However, It would have been difficult to create a conical peg that held the string taut by friction alone, so she used an alternate method of attaching a bar that could be screwed down tightly against the string after it had been stretched. Notice that the raised “nut” (actually a piece of aluminum bar) has already been placed at the top the pegbox. It will hold the string about 1/4 inch above the instrument’s body.

The girls then traced the outline of the frame onto two 2’ x 4’ sheets of Luan plywood. This material is thin, strong, and available inexpensively at any wood store. After the top and bottom were cut to the outlined size, a six inch diameter tone hole was cut in the top piece with its center about 10 inches above the wide end. Julie and her mom then glued and nailed the top and bottom pieces onto the frame.

To complete the basic instrument, the girls cut a 2 inch bridge from 1/4 inch angle aluminum and glued it in place 1100 millimeters from the nut. This number was chosen mostly for convenience in generating all the other numbers which would be used to mark the exact note places under the string. It was the only round number that would work between the tone hole and the large end of the body . They then pulled a 5 foot piece of thick nylon string from a fastener at the large end of the body over the bridge and nut and fastened to a hole drilled through the peg on the pegbox. This string is easy to find and inexpensive in the lawn-trimmer section of any hardware store. Finally they cut another 2 inch piece of 1/4 inch angle aluminum to act as the moveable bridge.

Marking the note places

Now it was time to mark in the “frets” where the moveable bridge would be placed to play each note of the musical scale. The girls did the math using the ratios that Pythagoras said would make the “perfect” scale.

First, they measured the total length of the string to be sure it was 1100 millimeters. Having assured themselves that it was, they put a fret mark half of that distance-- 550 millimeters from the bridge-- on a strip of masking tape running right next to the string. Using their moveable bridge, they tested the pitch and found that indeed it produced a perfect octave.

Since they knew that they could derive the perfect fifth by shortening the vibrating portion of the string to 2/3 of its original length, they multiplied 1100 by 2 and divided by 3 to get 733 millimeters. That meant they needed to place the fret mark for the perfect fifth 733 millimeters from the bridge or 367 millimeters from the nut (1100 - 733 = 367). They followed similar calculations to find all the other fret marks:
1. Multiply the current length by 2 and divide by 3.
2. If the number is less than 550, multiply it by 2 to get it into the first octave.
3. Mark that distance from the fixed bridge.
4. If we want to also mark an octave above this note, we divide the number by 2.

Here’s a list of all the fret distances Julie and Barbara marked from the fixed bridge for the first octave. We’ve attached note names here based on the string being tuned to A as we used in the frequency list above.

The problem with ratio-derived scales

But wait: isn’t that last fret place in the list above, the A an octave above the whole string A, supposed to be 550 millimeters from the bridge, not 542 mm. The answer is yes, it is. Pythagoras’s method of deriving the musical scale had a glaring problem: the octave resulting from the cumulation of 12 perfect 2:3 relationships didn’t match the octave built from a perfect 2:1 relationship. This difference caused a big tuning problem and came to be know as the Pythagorean “comma.” So much for mathematical perfection in music. Pythagoras’s scale worked well enough for the thousand years that people sang simple melodies with little or no accompaniment, but as instruments and harmony became more prominent in music, the tuning problems caused by the comma became more and more annoying. Over the years musicians tried many complex methods to “temper” the intervals so that the out-of-tuneness caused by the comma would be distributed among many notes rather than a few. These methods included just tuning, meantone tuning, Werckmeister tuning, Vellotti and Young tuning just to name a few of the more important ones. Still they all had problems because they all tended to work well in certain keys but not others.

The method of deriving scales that we use today was developed at around the time of Johann Sebastion Bach (1685 - 1750) and it’s actually the simplest. It has come to be known as equal temperament. To use it, simply multiply the frequency of any scale note by the 12th root of 2 (1.059463094) to derive the next scale note. Conversely, to place fret marks, simply divide the length of string at any fret mark by 1.059463094 to derive the next higher fret mark.

In equal temperament there are no perfect intervals except the octave. Everything is a little out of tune but nothing is terribly out of tune. What it sacrifices in these small tuning discrepencies, it makes up for in flexibility. Equal temperament can be used in any key with no glaring tuning problems. Here is a list of fret mark positions for an equal tempered scale compared to those for Pythagoras’s ratio-derived scale.

Building a monochord is a great way to experience the science and history behind our musical scales. For those who can’t afford the time to actually build the instrument but who still want to try out the scales, there is a version of this article along with a playable virtual monochord online at http://multimedia.utsa.edu.