Bowing as Periodic Plucking

The ``leaning sawtooth'' waveforms observed by Helmholtz for steady state bowed strings can be obtained by periodically ``plucking'' the string in only one direction along the string [404]. In principle, a traveling impulsive excitation is introduced into the string in the right-going direction for a ``down bow'' and in the left-going direction for an ``up bow.'' This simplified bowing simulation works best for smooth bowing styles in which the notes have slow attacks. More varied types of attack can be achieved using the more physically accurate McIntyre-Woodhouse theory [287,407].

Commuting the string and resonator means that the string is now plucked by a periodically repeated resonator impulse response. A nice simplified vibrato implementation is available by varying the impulse-response retriggering period, i.e., the vibrato is implemented in the excitation oscillator and not in the delay loop. The string loop delay need not be modulated at all. While this departs from being a physical model, the vibrato quality is satisfying and qualitatively similar to that obtained by a rigorous physical model. Figure 7.5 illustrates the overall block diagram of the simplified bowed string and its commuted and response-excited versions.

Figure 7.5: a) The simplified bowed string, including amplitude, pitch, and vibrato controls. The frequency control is also used by the string. b) Equivalent diagram with resonator and string commuted. c) Equivalent diagram in which the resonator impulse response is played into the string each pitch period.

In current technology, it is reasonable to store one recording of the resonator impulse response in digital memory as one of many possible string excitation tables. The excitation can contribute to many aspects of the tone to be synthesized, such as whether it is a violin or a cello, the force of the bow, and where the bow is playing on the string. Also, graphical equalization and other time-invariant filtering can be provided in the form of alternate excitation-table choices.

During the synthesis of a single bowed-string tone, the excitation signal is played into the string quasi-periodically. Since the excitation signal is typically longer than one period of the tone, it is necessary to either (1) interrupt the excitation playback to replay it from the beginning, or (2) start a new playback which overlaps with the playback in progress. Variant (2) requires a separate incrementing pointer and addition for each instance of the excitation playback; thus it is more expensive, but it is preferred from a quality standpoint. This same issue also arises in the Chant synthesis technique for the singing voice [362]. In the Chant technique, a sum of three or more enveloped sinusoids (called FOFs) is periodically played out to synthesize a sung vowel tone. In the unlikely event that the excitation table is less than one period long, it is of course extended by zeros, as is done in the VOSIM voice synthesis technique [203] which can be considered a simplified forerunner of Chant.

Sound examples for linear commuted bowed-string synthesis may be heard here: (WAV) (MP3) .

Of course, ordinary wavetable synthesis [280,185,306] or any other type of synthesis can also be used as an excitation signal in which case the string loop behaves as a pitch-synchronous comb filter following the wavetable oscillator. Interesting effects can be obtained by slightly detuning the wavetable oscillator and delay loop; tuning the wavetable oscillator to a harmonic of the delay loop can also produce an ethereal effect.

The externally excited, filtered delay loop can be used also to simulate wind and other musical instruments. In fact, any quasi-periodic tone can be approximated using an appropriate excitation signal (which may be varied over time) together with some loop filter (which also may be varied over time). The fact that the delay line is approximately one period in length restricts application of this type of structure to quasi-periodic tones. However, aperiodic tones which can be well approximated by a superposition of a few quasi-periodic tones can be synthesized using multiple delay loops added together in parallel and excited by common or separate excitations. Thus, piano, marimba, and glockenspiel can be approximated, for example. For wind instruments, a filtered, enveloped noise excitation is needed. In summary, the externally excited, filtered delay loop can be viewed as an efficient compression technique for arbitrary quasi-periodic signals with musically desirable parameters.

Figure 7.6: Generalized table-excited, filtered delay loop synthesis system.

Figure 7.6 illustrates a more general version of the table-excited, filtered delay loop synthesis system. The generalizations help to obtain a wider class of timbres. The multiple excitations summed together through time-varying gains provide for timbral evolution of the tone. For example, a violin can transform smoothly into a cello, or the bow can move smoothly toward the bridge by interpolating among two or more tables. Alternatively, the tables may contain ``principal components'' which can be scaled and added together to approximate a wider variety of excitation timbres. An excellent review of multiple wavetable synthesis appears in [185]. The nonlinearity is useful for obtaining distortion guitar sounds and other interesting evolving timbres.

Finally, the ``attack signal'' path around the string has been found to be useful for reducing the cost of implementation: the highest frequency components of a struck string, say, tend to emanate immediately from the string to the resonator with very little reflection back into the string (or pipe, in the case of wind instrument simulation). Injecting them into the delay loop increases the burden on the loop filter to quickly filter them out. Bypassing the delay loop altogether alleviates requirements on the loop filter and even allows the filtered delay loop to operate at a lower sampling rate; in this case, a signal interpolator would appear between the string output and the summer which adds in the scaled attack signal in Fig. 7.6. For example, it was found that the low E of an electric guitar (Gibson Les Paul) can be synthesized quite well using a filtered delay loop running at a sampling rate of 3 kHz. (The pickups do not pick up much energy above 1.5 kHz.) Similar savings can be obtained for any instrument having a high-frequency content which decays much more quickly than its low-frequency content.

Figure 7.7: Example of a filtered noise excitation implementation.

For good generality, at least one of the excitation signals should be a filtered noise signal. An example implementation is shown in Fig. 7.7. In this example, there is a free running bandlimited noise generator which is filtered by a finite impulse response (FIR) digital filter. The filter coefficients are computed in real time as a linear combination of a set of fixed FIR coefficient sets stored in ROM. A recursive filter may also be used, in which case ladder/lattice forms can be used so that the coefficients can be interpolated without stability problems. In a simple implementation, only two gains might be used, allowing simple interpolation from one filter to the next, and providing an overall amplitude control for the noise component of the excitation signal.