We conclude our analysis of the fabulously complex beast that is the Leslie rotary speaker.
Of course, these days, there are plenty of available digital rotary speaker simulators, but as with previous instalments of this series, I'm going to describe the process using analogue principles, as it's easier that way to relate the constituent parts to conventional synthesizer components, and understand how everything works.
Let's start by returning to what this series was examining way back in SOS August 2000. That month, I showed how the concepts behind Sample and Hold (or S&H) synth modules are related to those behind analogue-to-digital converters, and thus to all of digital audio. Today, I find myself at the same starting point, and, although it may not be obvious how discussions of S&H circuits and Leslie speakers should be so closely linked, I'll ask that you bear with me because — as always — all should soon become clear.
A Quick Recap
To understand S&H and how it leads to the technology of modulated effects, I'm going to review some of the ground that we covered back in 2000, starting with Figure 1 (below), which I've copied from the previous article. As you can see, this is a remarkably simple circuit, comprising just two components: a capacitor and a switch.
Imagine that, just for an instant, the switch in the diagram closes. If the capacitor can react quickly enough, it then charges up (or discharges down) to the voltage at the input, thus 'sampling' that voltage. Then, once the switch has opened again, the voltage across the capacitor cannot change. This is because, on the left-hand side, there is no circuit and, on the right-hand side, the impedance — which is represented by the mathematical symbol 'z' — is infinite (which means that no current can flow). However, although no current flows, you can still measure the voltage across the output.
That's all there is to it... when the switch is closed, the capacitor 'samples' the input voltage. When the switch is open, the capacitor 'holds' that voltage, allowing other circuits to respond to it as appropriate.
Figure 4 (below) explains the nature of the output. Each time the S&H module receives a trigger, it measures (or 'samples') the voltage of the input signal (shown in red). It then holds this voltage (the blue line) until it receives the next trigger, at which point it repeats the operation. It 'samples' and then 'holds', just as I've described.
As I suggested last time, this result would not be very interesting if a sine wave was the only signal you could present to the module's input. Fortunately, the input signal can be anything: a synthesized audio waveform, an external signal such as the output from a turntable or CD player, or even the 'live' sound of an instrument being played. And this is where we begin to diverge from my previous discussion. Whereas traditional synth S&H effects are derived mostly from using a random 'noise' signal as the input, and directing the output to the control inputs of other synthesizer modules, we are now going to concentrate on affecting the actual sound of an instrument being played. But before we do so, we have to convert the S&H circuit into a delay line...
The Bucket Brigade Device
Now imagine that this sequence of events repeats, but that this time it is Switch 2 which closes for a moment, and then re-opens. When Switch 2 closes, the second S&H circuit takes the sample held in the first as its input, samples it, and holds it. In other words, the sample is passed down the line!
Before moving on, we need to eliminate two problems encountered when sampling and reconstructing a continuous waveform. Just as when sampling digitally, all that stuff about keeping the maximum frequency at less than the sampling rate holds true here, too (for more on this, look back at Part 17 of this series, in SOS September 2000).
Because of this, we need to ensure that the highest frequency presented to the delay line is less than half the sampling frequency. In this case, the sampling frequency is half of the clock frequency, because, as illustrated in Figure 5 earlier, a new sample is taken every two trigger pulses. Anyway, to ensure that the input is suitably band-limited, we need to add a low-pass filter before the signal input. Secondly, we want to eliminate the 'blockiness' from the output waveform shown in Figure 3, and we do so by smoothing the output using a second low-pass filter.
Putting all of this together, we now have a circuit description for an analogue 'bucket-brigade device' (or BBD) delay line, so-called because its operation is analogous to handing buckets of water, each filled to a different depth, along a line of people (see Figure 7).
By the way, the low-pass filters I have drawn — simple 1-pole devices — are much less powerful than one would normally use for these purposes, so please treat them as representative rather than an exact circuit description. The first of these is called an 'anti-aliasing filter' because it removes high frequencies that lead to aliasing. The second is known as a 'reconstruction filter' because it reconstructs the smooth waveform from the 'blocky' one at the output.
Clock Modulation & Waveshaping
But what happens if we modulate the clock so that adjacent samples measured at one rate are presented to the reconstruction filter at a different rate? For the purpose of this discussion, you can think of the delay line as a tape recorder, with a record head at one end and a playback head at the other, and an infinitely long strip of tape running past them. If the speed of the tape is, say, 15ips when a middle 'C' (C3) is recorded at the start, but just 7.5ips when that part of the tape passes the playback head, the note will be replayed as C2, an octave lower. Conversely, if the tape speeds up to 30ips, the note will be raised an octave, and reappear as C4.
Figure 9 shows approximately 24 cycles of a sine wave that, for the sake of argument, I have presented to the input of our delay line. I shall now modulate the clock frequency to obtain Figure 10, which shows that I have increased the wavelength of some cycles, thus lowering the frequency, and decreased the wavelength of others, thus raising the frequency. It should be obvious from this somewhat exaggerated example that this is an extreme example of pitch modulation.
What I have described here is, of course, the basis of the frequency-modulation synthesis (or FM) used in Yamaha's DX series of synthesizers, and it is very similar to the 'Phase Distortion' (or PD) synthesis used in the Casio CZ series of keyboards. But instead of modulating an oscillator, as we did when investigating FM synthesis earlier in this series (see SOS April 2000 and SOS May 2000), we are now frequency-modulating any sound.
It's possible to build a mathematical model of the 'clock distortion' FM synthesis implied by Figures 9, 10 and 11 using a sine-wave oscillator to modulate the frequency of the clock (see Figure 12). There's nothing special about sine-wave modulation in this context — I could use any waveform — it's just that it's simple to implement a sine wave in a model of this nature. Using this, you can generate vibrato when the modulation oscillator is running in the LFO range, and many recognisable 'DX' and 'CZ' waveforms when it runs at audio frequencies.
Synthesizing The Leslie
Audio-frequency FM and PD synthesis are fascinating topics, but they are not the purpose of this month's Synth Secrets, so we have to leave them behind, return to the Leslie, and now ask what its modulation depth and frequency might be. Surprisingly, the modulation depth created by the doppler effect in a Leslie speaker is quite small — around ±1 percent, which would be no problem for the mechanism in Figure 12 (see earlier).
To make this model accurate, we must split the audio signal into two bands — a treble band above 800Hz and a bass band below 800Hz — just as in a real, dual-rotor Leslie speaker. The easiest way to do this is to split the audio into two signal paths and apply appropriate band-splitting EQs to each. We can then duplicate the modules in Figure 16, defining independent 'rotation' speeds and transition speeds within each path, as shown in Figure 17 (which, for pertinence, I have drawn with a keyboard rather than a microphone as the signal source).
Now all we need to do is add the amplitude and tonal modulations discussed last month (see Figure 18) with each 90 degrees out of phase with respect to the LFO 'rotation' rate. I have added small delay lines in each of the control signal paths to generate this delay, but it is far from a complete description because, as the rotation rate changes, the lengths of these delays also need to change. This can be achieved by adjusting the clocks driving the secondary delay lines, but I suspect that you'll forgive me if I don't plumb the details of this.
Anyway, with all the delay lines, filters, amplifiers, LFOs, EQs, CVs and Slew Generators in place, we now have the glorious, analogue... argghh!! Figure 18 shows just one direct signal path for each rotor, without any of the reflections that occur within or outside the Leslie cabinet. To re-use last month's analogy, we have two roundabouts but no office blocks. Fortunately, a BBD is an appropriate device for creating simple reverberant effects so, in theory, the addition of another couple of delay lines (the fifth and sixth) might help to overcome this. But given the difficulties in getting this far, and the complexities I've just sidestepped regarding the phase relationships of the various modulations, I imagine that it's becoming clear why no analogue emulation of the rotary speaker cabinet was ever fully successful. To be fair, there was one — the Dynacord CLS222 — that was pretty damn good, and the effect on the Korg BX3 organ was useable if you were prepared to open the instrument up and tweak the internal trimmers.
The Digital Leslie
For most people, the dream of a light, portable, inexpensive and authentic-sounding Leslie effect became a reality only with the advent of digital electronics, and the development of algorithms capable of modelling all the above factors successfully. These algorithms can calculate thousands of signal paths, each exhibiting different pitch shifts, different phases and different amplitudes. Sure, it takes a lot of processing power to implement them but, nowadays, that's not a problem.
Of these, my favourite remains the Korg ToneWorks G4, a combined 'valve overdrive and rotary speaker' emulator. If you hook one of these up to a Juno 60 or the Kawai K3 I discussed a few months ago, the results are magic. The G4's overdrive is more realistic than the distortion imparted by the Juno's VCA, the rotary effect is remarkably authentic, and its speaker simulation gives it, in my opinion, just the right amount of dull woodiness. Connecting everything together, we obtain Figure 19.
Of course, the algorithm in the Korg G4 is synthesizable using analogue electronics, and with a wall of filters, clocks, modulation oscillators, delay lines and amplifiers, you could create a convincing electronic recreation of the rotary speaker effect. You would be mad to try, but you could do it.
Epilogue
We have achieved a huge amount this month, learning how closely linked the seemingly disparate technologies of S&H, delay lines, phase-distortion synthesis, and digital converters prove to be. Moreover, armed with our new understanding of BBD delay lines, we could continue to develop our analogue 'Leslie' effect. Alternatively, we could extend some of this month's ideas to create all manner of effects, such as echo, flanging, chorus and ensemble. And that's what we're going to look at next month.