# (0,1), (0,2) and (0,3) modes of vibration for a timpani

For a timpani, when hitting it in the center, how do you produce these three modes of vibration? Second mode has a higher frequency than the first, does that imply do you hit the timpani harder?

Thank you!

Some information for this answer taken from The Well Tempered Timpani.

Since a real world timpani is not perfectly shaped and can never be perfectly struck in the center, any time you strike a timpani, you will always activate multiple vibration modes.

Also, timpani membranes are two dimensional, so the numbering of the modes is more complicated. I'm going to assume the modes you are talking about are the (0,1) mode ("first"), the (1,1) mode ("second"), and the (2,1) mode ("third"). When a timpani is actually played, the preferred modes that are emphasized are actually (1,1), (2,1), (3,1), (4,1), and (5,1), and sometimes (6,1). The (0,1) mode is inharmonic with the other modes, so it is generally avoided to make sure the timpani is playing a note.

By striking a timpani as close as possible to the exact center, you will activate many vibration modes, but the "first" mode, (0,1) will be the loudest. So striking firmly in the exact center is how to hear the first mode. This might not sound like normal timpani playing, since normally they are played closer to the edge, and should come across a bit boomy. More like a bass drum.

To hear the "second" mode, (1,1) best, you have to both strike the timpani in a way to make the second mode louder and also damp the timpani to make the first mode more quiet. To emphasize the second mode, strike the timpani about 1/3 of the way along a diameter of the timpani. Another way to say that is to strike it about 2/3 of the way from the edge to the exact center. Also while you are striking the timpani, gently place at least one finger in the exact center of the timpani. You may get better results by trying to make a line with your fingers that goes through the middle of the timpani and make sure the line is perpendicular to the line from the edge to the center that you are striking along. You might have to adjust the strike point slightly somewhere between 1/2 to 2/3 the distance from the edge to the center (closer to the center). I may edit in a picture or diagram later if I can, since that would be clearer.

If you succeed in activating the second mode, it should sound like it's an octave higher than the first mode. Resist the temptation to strike a lot harder, a firm strike is what you want. It's the position and damping that controls the vibration, not the hardness of the strike (too soft or too hard will change the vibration modes, but in a more uncontrolled way).

Each higher mode is harder to isolate, so the "third", (2,1) might require a lot more practice. I'm actually not sure how to activate the third, but I have an educated guess. Again you want to place your finger in the center of the timpani, and ideally you would damp in two perpendicular lines going through the center of the head. Then you would hit inside one of the four quarters of the head created by the two lines, close to the middle of the quarter, and slightly towards the overall center of the head.

• Thanks a lot! This is very useful information. However, sorry I didn't specify but when referring to the different modes of vibration for a hit in the center, I actually meant (0,1), (0,2) and (0,3). I apologise for not specifying. So, the timpani is still struckin the same spot at the centre but there are different modes of vibration. I'm still unsure how to achieve the different modes. This is still very insightful though! Also, I think I might try to propose a theoretical situation where everything is perfect. Apr 18 '18 at 15:42
• @KevinNguyen Since the (0,2) and (0,3) modes have concentric nodes, it's next to impossible to isolate them. You would have to somehow create a very thin damping ring of the right circumference and place it in the right place on the head and then strike the head in the middle to get those. Apr 18 '18 at 15:56
• @KevinNguyen Also, (0,1), (0,2) and (0,3) modes are generally called the "first", "fourth", and "ninth" modes. You might edit your question to use those words or the specific numbers if you think someone else might have better information than mine about those modes. Apr 18 '18 at 16:06
• In summary, :-) , strike the surface at a point which will be a maximum for the desired mode, and preferably also a minimum for the undesired modes. If it were possible to adjust the Fourier composition of the "impulse" blow to match the desired modes' frequencies, that would be great -- not practical in the real world. Apr 19 '18 at 13:21
• @CarlWitthoft Sort of, but for all modes where the first number is 0 beyond (0,1), that is, for modes (0,2), (0,3)... (0,n) with n>1, it's essentially impossible to make that the loudest mode by striking alone. Damping at nodes would also be required, but those modes all have concentric circular nodes, so that's a real damping challenge. Apr 19 '18 at 14:58

Without being a timpanist, I will hazard a guess.

First, you will know when you have hit the higher modes because the tone changes as you would expect.

I suggest gently touching the surface of the timpani with one finger as you play it. The exact location where you touch it will determine whether your finger simply dampens the sound or dampens only some modes of vibration.

The round surface is symmetric in rotation (even if the modes of vibration aren't), so you can find out where to touch the surface just by testing from the outer rim to the center (like a needle following the groove on a record) and by testing sets of points ( for ( >0 , _ ) modes) that start at any arbitrary "cardinal direction" ...you don't need to explore the entire surface.

• Do you think changing the modes have anything to do with the pedal? Since the pedal changes the tone, allowing it to play higher or lower notes. Apr 18 '18 at 15:18
• If I understand the instrument right, the pedal increases the tension on the skin, changing the fundamental frequency but not switching the mode of vibration. I think if you find a place to touch the instrument and change the mode of vibration, that place will be the same however you adjust the pedal. Apr 18 '18 at 15:20
• After some research, apparently if the fundamental frequency is changed then mode of vibration should also change. Refer to the frequency differences between (0,1) and (0,2) acs.psu.edu/drussell/demos/membranecircle/circle.html Apr 18 '18 at 15:25
• Great link! I don't think you're following me regarding the pedal, though. The different modes all have their associated frequency, but what I call the "fundamental frequency" is a function of the tension of the skin, its weight, and its size: not the mode of vibration. Apr 18 '18 at 15:29
• Not all modes are circumferentially symmetric (nor do all modes have an axis of symmetry either!) . Over in optics-land, we often use the Zernicke decomposition in a circular aperture -- which is what the drum head is. Apr 19 '18 at 13:22