Inducing Unusual Dynamics in Acoustic Musical Instruments

Sound Examples and Additional Info

Edgar Berdahl
September, 2007

This is the website corresponding to the paper:

Brief note on resonant ring modulation:

Websites for related conference papers:

I. Examples With Multiple Equilibria

Figure 1. Ten plucks with linearly increasing magnitude (above), Corresponding loop gain L(t) (below)

Fig. 1 shows the results from the simulation where the string is virtually plucked ten times with initial conditions corresponding to increasing magnitudes. Fig. 1 (top) demonstrates that the final equilibrium RMS state depends on the initial condition. Four different stable equilibria in xp are evident. Fig. 1 (bottom) reveals that for each pluck, the loop gain L first wraps around a number of times while the energy outside the main resonances decays, and then L converges to the value inducing marginal stability as desired. Because velocity feedback is used instead of the integral of displacement feedback, the balance between the energy in the various harmonics does not change much even as the level is driven to a target.

The RMS level is quickly driven to a neighboring target. Some slight distortion may be heard directly following the attacks. This corresponds to L(t) varying wildly as the level detector first starts seeing the transient.

II. Band-Pass Filter Effect

The bandpass filter effect may also be used advantageously to control the RMS level of a single mode without affecting others. This involves tuning a high-Q band-pass filter (two-pole, no zeros) to the mode in question.

Here the band-pass filter is tuned to the second harmonic. Positive feedback is applied in order to damp the second harmonic quickly. The other harmonics remain unchanged.
This is the same as above, except that negative feedback is applied to increase the decay time constant of the second harmonic.

The block diagram in Fig. 2 shows in particular how multiple channels may be connected together to control multiple harmonics simultaneously.

Figure 2. System block diagram for implementing
multiple level-regulating controllers concurrently

The 1st, 2nd, 3rd, 4th, 5th, 6th, 9th, and 10th harmonics were controlled simultaneously in order to make the vibrating string model sing. To this end, they were driven to the spectral envelope outlined by the three formants f1, f2, and f3 for the the short u and a vowels. Since the sensor and actuator were placed realistically in the model accoring to real physical characterstics, both the 7th and 8th harmonics are both harder to sense and actuate. This is a controllability/observability problem due to the sensor and actuator being placed near a node for the 7th and 8th harmonics. Thus, we did not attempt to control them due to signal to noise ratio considerations (see discussion regarding Fig. 5). This means that the formants are not perfectly modeled, but it still should be possible to hear which vowel is being "sung." Fig. 3 shows the performance of the controller for the second harmonic. The initial pluck at t=0s causes some overshoot of the target and the loop gain L2 to be underestimated. However, soon the disturbance subsides and the controller adapts to the correct loop gain using the adaptive controller. Then at t=5s, the target vowel switches from u to a. After a few seconds, the RMS level is again driven to the correct target. Of course in both cases the same loop gain is used to induce marginal stability as soon as the target RMS level is achieved.

Figure 3. Driving the RMS of the second harmonic to 135 until t=5, at which point the RMS is driven to 24

The match of the spectra was quite good. Fig. 4 shows the spectrum in blue of x at the end of the u vowel after the system has nearly converged to the steady-state solution. The red circles are the targets as specified by the formants of the vowel.

Figure 4. Match between the desired levels and the attained levels (left, zoomed out; right, zoomed in)

The match is quite good (within +/-1.5dB) given the controller is non-ideal in some ways. For example, band-pass filters that are not perfectly tuned to the harmonic will impart a gain error. This means that, even though the adaptive controller for a given harmonic believes it has achieved the target, it may be slightly off-target because the band-pass filter is mistuned. Another non-ideality is that dividing signals by RMS level estimates leads to some harmonic distortion. Fig. 5 shows the spectrum of the nearly steady-state signal u1, which is the control signal for the first harmonic, as estimated from the same time frame as Fig. 4.

Figure 5. Spectrum of steady-state signal u1

This is the sound of the eight harmonics being controlled as explained above to synthesize a u and then an a vowel.
This is essentially the same example except that some modulation has been added because this aids in perceiving vowels. In particular, the error signals were slightly amplitude modulated at 1Hz.

III. Resonant Ring Modulation

Figure 6. RMS levels for resonant ring modulation

Resonant ring modulation involves amplitude modulating the feedback signal by a sinusoid with carrier frequency fc. For more information, see an Analysis Of Resonant Ring Modulation. One difference here, which holds for all of the following examples except for the last two, is that an additional first-order low-pass filter is applied in the feedback loop preceeding the modulation to form the effect. This helps improve the sound quality by preventing too much energy from being stored in the higher harmonics.

Here fc is swept linearly from f0 to 2.05f0:
In this case, the loop gain L is held constant. This corresponds to the blue line in Fig. 6. Note how the quieter sections sound somewhat hollow because the controller is not driving the plant hard enough during these sections. This example has a noticeable pluck at the beginning because the loop gain L begins at the constant.
This time, the adaptive controller is applied to drive the RMS level to a target. This not only reduces the dynamic range of the instrument's output, but it also helps shift the applied parameters closer to the space of interesting-sounding ones. For instance, this sound does not contain any "hollow"-sounding sections. Note that this example does not have a noticeable pluck at the beginning because the loop gain L begins at zero.

Here fc is swept linearly from f0 to 4.05f0:
Now fc is swept over a much larger range revealing more parts of the space with unique dynamics. The adaptive controller is applied here.
This is the same as above, except that we are listening to a limited version of the feedback signal rather than the output from the string itself. Note that as a result, it sounds less resonant because we do not tend to hear the natural string harmonics ringing out because they are instead modulated.

For the last two examples, the simplest form of resonant ring modulation is implemented. That is, there is no low-pass filter in the feedback loop, and so the sound examples may sound tinny due to the additional higher-frequency content:
Here we have another example of a three octave sweep using the adaptive controller.
This is the example from the note on resonant ring modulation. We can hear the partials converge to the target spectrum.

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