The Degeneracy of Timpani Pitch
An introduction to the degenerate modes of timpani heads and how and why the Paolo Rossini clearing method works.
The objective when clearing timpani heads is to damp the in-harmonic modes as much as possible and project the “harmonic” modes.
How timpani produce a sense of harmonic pitch is quite a complex process. At ground level, you have two competing physical forces (modal sets) working against each other. This is truly the antithesis of how the harmonic series generates pitch on most other instruments. So why are we able to adjust timpani heads so they vibrate with a sense of harmonicity?
Fortunately for the timpanist, one of these modal sets (diametric/asymmetrical modes) is what in the world of physics is called degenerate; doubly degenerate to be exact. The other modal set (concentric/symmetrical) is not. Degeneracy in the case of the diametric/asymmetrical modes is a good thing. When the head is appropriately tensioned, the degeneracy actually helps strengthen the “harmonicity” of the pitch.
Explaining Some Jargon
The scientific world often uses common words and terminology “jargon” to describe functions/processes as well as concepts and ideas. This terminology may have completely different connotations and meanings for those of us in the non-scientific world. This particular article will utilize a couple of terms that may not be familiar to most musicians. Degeneracy is one of those terms.
Degenerate/Degeneracy:
A single preferred diametric mode (e.g. mode 1,1) can correspond to two or more different energy levels (or shapes) when measured. Conversely, a single preferred diametric mode can give the same value of energy (or have the same exact shape) upon measurement.
Degree of Degeneracy: Any deviation from zero.
Thus is the case of the term degenerate and its associates. Because sound is a form of energy, in our case degeneracy will refer to the levels of energy that the pitch producing preferred diametric modes of a vibrating circular membrane can produce. The preferred diametric modes are also referred to as the asymmetric modes, which means that a single mode can correspond to two or more different energy levels (or shapes as we will see) when measured. Conversely, a single mode can give the same value of energy (or have the same exact shape) upon measurement, which is ideally what you want. For the timpanist, zero degeneracy means that the pitch or frequency at all diametric lugs is the same, and the mode (at those diametric points) is balanced. Any deviation from zero is called the degree of degeneracy and we will use the term lift when referring to any deviation from zero, both positive and negative.
For the timpanist, the simplest way to think about lifting the degree of degeneracy is with the actions of the tuning lugs themselves. You have learned over the years that adjusting one lug affects not only the diametrically adjacent lug, but all lugs around the drum to some extent. Let’s say that you have a timpani head where all lugs are balanced, and you consider the head to be completely clear (i.e. zero degeneracy). You are very, very happy…life is good. However, you inadvertently leave the tuning wrench on the lug perpendicular to your playing spot as you leave the room for a well-deserved celebratory cup of coffee. Someone comes along and turns the tuning wrench 360 degrees, but you don’t know in which direction. You strike the drum at a normal playing spot and the head is no longer clear. You hear strange overtones in the sound because the degree of degeneracy of the diametric lug that was disturbed has been lifted. So what just happened? A phenomenon called lifted degeneracy occurred causing a frequency shift between the halves of Mode 1,1 the vibration that creates the drum’s strong sense of pitch .
Figure 1 is a diagram of Mode 1,1 of a vibrating timpani head with zero degeneracy. Mode 1,1 is also called the principal tone because it is what gives the drum its sense of pitch. Mode 1,1 has one nodal diameter (dotted blue line) that bisects the drum. Consequently, half of the head is in a positive cycle while half of the head is in a negative cycle (see figure 1). When the tension of head is evenly balanced at all lug points and the membrane is struck at point X, both halves of the head theoretically will have the same value of energy (shape), and theoretically will vibrate at the same frequency C4 = 262 Hz.
Figure 1: Mode 1,1 with Zero Degeneracy
Figure 2 is a diagram of the same Mode 1,1 the principal tone where lug e has been altered by one full turn (-1) lifting the degeneracy of the mode by creating a lowered energy value (slightly different shape) in the half of the membrane at lug e. This result is a noticeable Frequency Shift in this half of the head. The mode is no longer equally balanced, and you may hear strange overtones and even beats, and the pitch is not “clear.” Half of Mode 1,1 is vibrating at C4 = 262 Hz and the other half is vibrating at C4′ = 254.83 Hz, which is -48¢ flat of the intended 262 Hz. The mode is truly asymmetrical. Even though the perceived pitch of one-half of the head has been lowered, the original zero degeneracy of the mode is said to have been lifted. Remember, any deviation from zero is called the degree of degeneracy and the term lift refers to this degree of deviation from zero, both positive and negative.
Figure 2: Mode 1,1 with Lifted Degeneracy
For the sake of simplicity, a mode that is doubly degenerate means that the energy level of that mode (and its complement) can correspond to two or more different measurable states. This is why when you adjust the tension of the head at one lug, the tension of the head at the lug directly across from it is affected as well (see the figures above). This also includes the complementary orthogonal points at 90º; this is the double degeneracy part. Ideally you want to have the tension (measurable state) the same at all four locations. This is one reason why timpani with eight lugs are purported to be easier to clear than timpani with six lugs. When you use a cross-lug tuning sequence, it helps balance the energy levels of that mode set at all four locations.
Complementary Degenerates of Mode 1,1
Because the modal sets that produce a sense of pitch on timpani are doubly degenerate, there are an infinite combination of energy levels that we have to get to the same state, yet we only have six or eight tuning lugs which we can alter to make that happen. Hmmm….no wonder it is so hard to clear timpani heads 🙂
On the flip side, the modes that detract from the sense of pitch are not degenerate, nor are they doubly degenerate; their energy levels are considered fixed or constant. This is also a good thing because they tend to be static, which makes them very efficient at radiating their energy, so their sound dies away quickly. However, they will always be present in the sound.
Since the mode that gives the drum its “fundamental” sense of pitch (mode 1,1) bisects the drum, both halves of the head need to have the same energy level when excited in order to produce a single frequency. Creating a mode 1,1 can happen anywhere you strike the head around the circumference of the drum at the normal striking position, so there are an infinite number of possible modes 1,1 that can be generated. Because mode 1,1 is doubly degenerate, when the head is “clear” or balanced, a single iteration of mode 1,1 can excite other areas of the head that have the same energy level because of their double degeneracy; remember the orthogonal complement at 90º? This can only happen when the tension at the lug points is such that it creates the same or very similar energy states when the head is excited. The process becomes even more complex as we add the other “harmonic” modes into the equation (modal entanglement). That will be for another time.
The more of these “balanced” mode 1,1 energy states that can be excited both orthogonally and sympathetically, the more the non-degenerate mode 0,1 and its subsets are suppressed resulting in more “harmonicity” and clarity of the pitch.
The short video below shows the energy state and the shape of mode 0,1 (the dissonant fundamental) as it peaks and troughs at the center of the drum. It also shows the energy states and how the shape of a balanced mode 1,1 bisects the shape of the drum. Notice how the energy states are even. When the two are combined, the energy states of a single balanced mode 1,1 becomes unequal, which will result in a loss of pitch. As more of the “harmonic” modes are added (mode 1,1 and its subsets) the energy state of mode 0,1 (and its subsets) is damped. Remember the objective when clearing a head is to damp the in-harmonic modes as much as possible and project the “harmonic” modes.
Why will Paolo Rossini’s clearing method work? Running your finger diametrically across the head as you tap the head creates a nodal point that bisects the drum similar to what a mode 1,1 does. This lets you feel the amount of energy that mode 0,1 and its subsets have as you cross the head and reach dead center. Dead center is where mode 0,1 has the most measurable energy. When your finger is dead center, you want to feel very little energy from mode 0,1 and its subsets. If you do, adjust the tension at opposing and orthogonal lug points of the drum until there is minimal or no energy perceived. Instead of listening for pitch as you clear, you are sensing the vibrations with your fingers that you want to eliminate. As the various iterations of mode 1,1 and its complement become balanced, the vibrations from the in-harmonic concentric modes are reduced and the vibrations that create a sense of “harmonic” pitch are projected.