the following text is a guide to a collection of objects that i call vibrating_rulers. it begins with an overview of the different types of “rulers” (including, in this context, various types of bars and rods), continues with a short discussion of the possible playing setups, and ends with descriptions of several playing techniques. the first stimulus for dealing with these objects was the sound produced by plucking ordinary plastic and metal rulers (like this) (see also the 8th brainstorming session i had for coming up with new ideas for CompositionCloud), and a significant part of the experimentation that led to this text was made in collaboration with percussionist Brian Archinal as part of the preparation for d1-7_sxschVR-Nikel.

 
 

types of rulers

sounds of vibrating rulers range from slow pulses to high-pitched tones. without getting too much into the physics involved, Bart Hopkin gives in his book Musical Instrument Design some helpful insights regarding how the different physical properties of a ruler may affect its sound (which i can generally confirm according to my own experience). broadly speaking, these properties are the mass of the ruler and its stiffness (for the sake of simplicity, the decay time of a ruler is assumed to be reciprocal of its vibrating frequency):

• heavy rulers are loud and vibrate at low frequencies.
• stiff rulers are also loud but vibrate at high frequencies.

both the mass and the stiffness of a ruler are determined by its volume, that is, its length, width, and thickness, and by the material of which it is made:

• heavy rulers are large in volume and are made of dense materials.
• stiffer rulers are short and thick, and are made of rigid materials (materials with high Young’s modulus).

(note that greater thickness means both greater mass and greater stiffness, however, in most situations, the greater stiffness has a stronger effect than the greater mass. in other words, the thicker the ruler, the higher the frequency at which it vibrates.)

to illustrate these principals, let us consider some examples. the photos below (taken at OBI Basel and Coop City Basel) show some promising candidates: aluminum bars, brass bars and rods, PVC bars, galvanized steel threaded rods, and knitting needles made of aluminum and bamboo.

the first difference is between bar-shaped and rod-shaped rulers. because the ratio of mass to thickness, and accordingly to stiffness, is much higher in bars than in rods, rods normally vibrate at higher frequencies than bars (for a bar and a rod of the same thickness, the volume of the bar, and hence its mass, is always greater than the volume of the rod).

the second difference is based on the influence of the different materials of which these rulers are made:

• brass is the densest and the second most rigid of the materials mentioned above.
• stainless steel is the second densest and the most rigid.
• aluminum is in the middle (although the extent to which it is less rigid than brass is smaller than the extent to which it is less dense than stainless steel).
• PVC is the second least dense and the least rigid.
• bamboo is the least dense and the second least rigid (but the extent to which bamboo is more rigid than PVC is greater than the extent to which PVC is denser than bamboo).

assuming identical volumes:

• rulers made of PVC vibrate at the lowest frequencies and are the quietest.
• rulers made of brass vibrate at the second lowest frequencies and are the second loudest.
• rulers made of stainless steel vibrate at higher frequencies than rulers made of brass and are the loudest.
• rulers made of aluminum vibrate at the second highest frequencies and are quieter than rulers made of brass.
• rulers made of bamboo vibrate at the highest frequencies and are the second quietest.

in addition, different materials also tend to produce different timbres, however, these differences are mostly noticed at low frequencies and when at least 2 rulers made of different materials sound one after another.

 
 

playing setups

a basic setup of vibrating_rulers consists of a selection of rulers and a surface, which is required for stabilizing one end of the rulers so that the other end could vibrate freely, as well as for radiating the vibrations of the rulers more effectively to the air. for the maximum loudness, the ideal surface should be both large and light, as well as stiff enough so that the energy of the vibrating rulers will not dissipate too quickly. for a good low-frequency response, some sort of an enclosed air resonator is necessary. a long narrow wooden resonance box is a good option, but more “exotic” surfaces, such as a round glass table (see objects4JamesSaunders-x1), are also possible. as for stabilizing the rulers, they can be pre-mounted on the surface with clamps, resembling an “enlarged kalimba”, or just stopped by hand.

 
 

playing techniques

plucking

while most of a ruler’s properties are fixed according to its type, the length of its vibrating part is determined by the position at which it is mounted or stopped. in the illustration below, one can see that the vibrating part of a ruler is actually made up of 2 different parts:

• the part that vibrates against the surface (vibrating part 1)
• the part that vibrates in the air (vibrating part 2)

both parts affect the frequency of vibration, but the longer vibrating part 1 is, the brighter the sound, and the longer vibrating part 2 is, the longer the duration. if vibrating part 1 is very long but vibrating part 2 is very short, only a brief slap will be heard. if vibrating part 2 is very long but vibrating part 1 is very short, a dull, muted sound will be heard (in fact, for very slow vibrations, a very short vibrating part 1 is very likely to result in silence).

 

2-hand plucking technique

plucking a ruler with one hand and stopping it with the other allows the most control over the vibrating parts of the ruler:

• moving the ruler backward increases vibrating part 1 and decreases vibrating part 2, making the ruler vibrate at a lower frequency, reducing the brightness, and lengthening the duration.
• moving the ruler forward decreases vibrating part 1 and increases vibrating part 2, making the ruler vibrate at a higher frequency, increasing the brightness, and shortening the duration.

to change the length of vibrating part 1 without changing the length of vibrating part 2, one of the following stopping techniques can be used:

• stopping the ruler with the fingers as if the part of the ruler that lies on the surface is the fingerboard of a string instrument
• stopping the ruler with the palm or just with the base of the palm (or anything in-between)
• stopping the ruler with the forearm, varying the amount to which it touches the ruler by using the elbow as an axis

note that the amount of pressure applied when stopping the ruler also affects the sound:

• increasing the pressure makes the ruler vibrate at a higher frequency and lengthen the duration.
• decreasing the pressure makes the ruler vibrate at a lower frequency, shorten the duration, and might even allow its other side (the side behind the stopping position) to vibrate as well.

 

1-hand plucking technique

to pluck a ruler with only one hand, one can either pre-mount the ruler or stop and pluck it with the same hand. for the latter, the ruler is to be stopped with either the palm or the forearm, and the palm should be close enough to the edge of the surface so that one of the fingers could pluck the ruler.

 

using both sides of a ruler

depending on the dimensions of the surface and the length of the ruler, it might be possible to pluck both sides of it and let them vibrate simultaneously. as shown in the illustration below, the stopping position and the position of the ruler on the surface determine the lengths of the vibrating parts of the 2 sides.

 

rubbing techniques

rulers can also be used to rub one another (as well as the surface). generally speaking, in addition to the material of which the rulers are made, which largely determines the timbre of the resulting sound, there are 3 other important parameters:

• the length of vibrating part 2 of the ruler that is being rubbed determines the perceivable pitch.
• the speed of rubbing determines the loudness.
• the point along the rubbing ruler at which it rubs the other ruler determines the brightness.

a rather special case is rubbing a threaded rod or using a threaded rod to rub other rulers (or the surface). in this situation, the speed of rubbing not only determines the volume, but also the pitch of an additional sound produced because of the threaded shape of the rod.

 

other techniques

other possibilities include using auxiliary objects such as bows, hacksaws, friction mallets, electric frothing wands, electric toothbrushes, as well as preparing parts of the surface with various means (nylon bags, aluminum foil, etc.).

 

vibrating_rulers can be heard and seen in d1-7_sxschVR-Nikel, 24d24iS_esO4bsPSpbVRssS2-EPB, and ccloudlab1x2.

an aluminum knitting needle is part of the collection objects4JamesSaunders.