By taping thirteen rulers to a desk, each with a different amount of overhang, we have a rudimentary musical (?!?!?!?) instrument. So, how much overhang is necessary, how do we do it, and why? Well...
If there's one piece of science that every kid knows, it's that twanging a ruler on the edge of the desk makes a noise, and by changing the amount of overhang will change the pitch. Give anyone a new ruler, and the first thing they'll do is twang it, to see how it sounds. This is an interesting fact of life.
To intellectualise this pastime we can say that we are discovering the relationship between wavelength and frequency (the longer the ruler overhang, the lowest the frequency of the note), and that we're listening for the timbre (pronounced tam-ber), which indicates the character of an individual sound, and is why a violin and piano sound different when playing the same note. This is less interesting fact of life.
Step 1: Prepare the Rules
We start by determining the length of each overhang. By knowing how much is needed for the lowest note, C in our case, we can calculate the others mathematically. Finding the lowest note is done in typical school kid fashion by experimenting, unless you have a keyboard, guitar, or other musical instrument and an extra pair of hands, er, to hand.
You will notice that some lengths do not produce notes at all. Very short distances just produce a click, while very long ones make no sound at all. To make a complete octave, the overhang of the lowest note will need to be twice as long as the shortest (highest note), so if the rules you're using only make sounds between 5cm and 8cm you won't get a full octave.
TIP: Hold the rule to the desk as tightly as possible to produce the best audio fidelity (read: twang) possible.
Step 2: Setting the Rules Lengths
Taking the lowest note as a guide, measure the distance from the edge of the table to the tip of the rule. This is not, alas, the distance marked on the rule as most have a gap at each end. You will need to measure this. Taking this total distance, divide it by 1.0594630943592952645618252949463 to compute the overhang of the next rule. Oh, you want that number explaining? Ok!
Musical frequency is a logarithmic scale. The frequency of each note on an instrument is always half the frequency of the note one octave above it. That is, a=2f. So, with 12 notes to an octave, we raise 2 to the power of 1/12 to get our multiplier, 1.0594630943592952645618252949463.You can create very interesting, weird, and ethereal music by using different scales, but rules are not good enough to reproduce it, but it can be done effectively with soft synths. But I digress...
So, if your first rule (as the lowest note, on the left) overhangs by 104mm, the next must overhang by 98mm. Line this up with the edge of the rule - remembering that there's probably a 7mm gap between the edge of the rule and the numbers, so measure off 91mm and tape this rule to the desk.
Note - Number - Length (with 7mm gap considered)
C (low) - 0 - 9.700000
C#/Db - 1 - 9.116292
D - 2 - 8.565346
D#/Eb - 3 - 8.045321
E - 4 - 7.554484
F - 5 - 7.091195
F#/Gb - 6 - 6.653909
G - 7 - 6.241165
G#/Ab - 8 - 5.851588
A - 9 - 5.483875
A#/Bb - 10 - 5.136801
B - 11 - 4.809206
C (hi) - 12 - 4.499997
You can make a smaller C-scale rule organ, by only using the lengths highlighted in bold.
The method I found best for lining up the rules was to place tape on the edge of the desk, so that the base of the rule stuck to it at the correct distance. Then, once all the rules where laid, I securely fastened them from above, removed this strip, and inserted double-sided tape underneath. It's also easiest to work from the left, as this means the first edge you fix to the tape has the millimetre scale on it.
NOTE: Never tape to varnished desk, as the tape will remove the varnish.
Step 3: Weighting
Taken from Rule organ details