The Basic Plucked String
In these pages, we’ll build a model of a bowed cello, complete with a resonating wooden body. We’ll then modify that model to imitate a violin, a viola, and a contrabass.
This will be, by far, the most complicated model in these pages. In fact, we’ll need all four DSPs in a G2 to play a single note.
Getting started
Before we bow a string, we’ll pluck it. Plucking is easier to patch, and easier to understand, so we’ll begin there.
Of course, the G2 already has a String Oscillator module, which simulates a plucked string. Unfortunately, we’ll have to bypass it, and make our own from scratch. We’ll eventually see why.
Below is a plucked string patch. It’s the standard waveguide-type physical model of a plucked string.
How does it work?
The string itself is made up of the two yellow sections labeled “Pluck To Nut” and “Pluck To Bridge”. If it looks familiar, it’s because it’s the same resonator used by our reed woodwind model. That model was called a “blown string”. Well, here again is the string part of the model, in its basic form.
Let’s trace the signal flow again. We’ll begin with the “Pluck To Nut” section. This section contains a delay line and a gain control. The output then goes to the “Pluck To Bridge” section, and is inverted as it enters. The “Pluck To Bridge” section is identical. It contains another delay line, another gain control, and a lowpass filter. The output then returns to the “Pluck To Nut” section, and is again inverted as it enters.
The delay in the “Pluck To Nut” section represents the length of string from the pick to the nut, and the other delay represents the length of string from the pick to the bridge. The lowpass filter and the inverting gain controls represent the reflections and energy losses at the bridge and nut. Some waveguide models of strings have an additional lowpass filter at the nut, but we’ve left that out to save a module.
To pluck the string, we’ll inject a short noise pulse into both delays simultaneously. That’s the job of the “Pluck Excitation” circuit in red. The pulse will travel through the loop made by the delay lines again and again. With each trip it will get a little softer, because the gain controls are less than unity. It will also get a little duller, because the lowpass filter will trim off some high-frequency energy.
This is actually quite near to how a real string works. When you pluck a string, the pulse travels in both directions toward the bridge and nut. When it reaches the end of the string, it is reflected back. Its polarity changes and it loses some energy. High frequencies experience more loss than low frequencies. This is all modeled by the delays, the inverters, the VCAs, and the lowpass filters.
Some odds and ends.
The position of the pluck point depends on the ratio of the two delay lengths, and the pitch of the string is related to the sum of the two delay lengths. Here’s some boilerplate copied from the reed woodwind model:
- The delays are tuned using an old trick from the original Nord Modular: the Level Scaler Module. This module can be linked to the keyboard and adjusted to output a -6dB/octave slope. Since -6dB represents a 50% reduction, and since we want the delay line to become 50% shorter with each increasing octave, this module is perfect for the task.
- We’ll control the position of the pluck point by controlling the ratio of the delay line lengths, via a Pan module. If the Pan module is centered, the two lengths are equal, and our string is plucked in the center. But if one length is longer than the other, the string begins to include even harmonics. Regardless of the position of the Pan module, the total length of the loop remains the same, and this is why the pitch doesn’t change when the Pan module is changed.
- The need for controlling the position of the pluck point is the big reason why we can’t take advantage of the String Oscillator module. We’ll need access to the pluck point, because it will soon become the position of our bow.