Phase-locked loops and frequency dividers

[egasimus]

I can understand the concept of a PLL in its own right. However, I've seen it mentioned in the context of synth building several times, but can't make much sense out of it.

For example, PLL is used for shaping a square wave into a triangle and a sine wave, as mentioned in R.G.Keen's post here. Google yields this. Also, I've seen PLL mentioned as a way to divide the frequency of not only square, but also sine and triangle waves.

Anyone care to shed some light on the subject, maybe point out where I can read up on PLLs in music synthesis?

[Tim Servo]

Hi Ega,

In your first link, I can't find any mention of a PLL. What they're talking about is simply converting a guitar signal to a square wave, and then using an integrator to convert that square into a triangle. This would work, but as the messages in the link mention, it will only work well for a narrow range of frquencies. Outside that range, you'll get variations in amplitude.

In the second link, they're using a PLL arrangement to create a control voltage for a sine wave VCO (the 8038 chip). Basically, the loop compares the VCO frequency to the input frequency, and sends a control voltage to the VCO that tries to make the VCO frequency match the input. Any waveshaping or triangle/sine waves come from the VCO portion of the PLL. You might also want to check out the CD4046 data sheet (this is a CMOS PLL chip often used in electronic music), and also take a look at Scot Gravenhorst's PLL frequency tracker / waveshaper which works with square waves:
http://home1.gte.net/res0658s/fatman/4046pll.html

Hope this helps!

Tim (my loop is a little loose) Servo

[egasimus]

[JovianPyx]

You probably already know this - PLL stands for Phase Locked Loop and it's purpose is to lock it's own VCO in phase with the input signal's phase. This actually has nothing to do with frequency and everything to do with phase and phase alone.

Because it is phase and not frequency that is locked, it is possible to make a PLL lock onto a harmonic of a signal and not necessarily to it's fundamental.

Where wide lock range is desired, (assuming a CD4046), Phase Detector 2 should be used and the phase detection happens near one of the input's zero crossings. It is expected that this zero crossing happens at regular intervals (as it would with square, pulse, triangle, sawtooth, sine, etc). If the input signal is not a mixture of more than one unrelated pitch, then the zero crossings will happen at regular intervals and the PLL will lock to it and it's VCO will have a pitch that matches the input signal.

If the input signal is made up of several unrelated frequencies and the component signals are high amplitude with respect to each other and fairly close together in frequency (as would be true in your given example), then the zero crossings will not happen at regular intervals. The PLL circuit, being a closed loop servo, will still try to phase lock to it, but depending on the loop filter, the phase detection circuit will attempt to correct the VCO phase whenever it senses a phase change - which is not happening at regular intervals so the system will "hunt" for it. "Hunt" meaning it will keep moving around attempting to find a regular interval zero crossing. While it does this, the PLL VCO frequency will wander aimlessly and will never lock on one particular pitch. You will get sound, but it will not have a pitch that you would expect, possibly noise or nasty sound.

If the input signal's component frequencies are far enough apart, the PLL may lock to just one of them. For example, given 100 Hz, 223 Hz and 3500 Hz at equal amplitudes where the loop is set to span say 1000 Hz to 4000 Hz, then the PLL will lock to 3500 Hz and remain there, effectively ignoring the 100 Hz and 223 Hz signals. This is used in certain data recovery circuits to exclude unwanted noise.

A PLL is a very versatile circuit and most certainly has musical uses. To be able to use it musically requires understanding how the PLL works as a system. In the case of a CD4046, looking at datasheets from several different manufacturers is suggested because some have better theory explanations than others.

[egasimus]

Thanks for the elaborate post - sure cleared up things a bit for me. So is the 8038 circuit I posted above good for a guitar synth in your opinion? I think its frequency range is a bit limited, and I can't get a 8038 anyway.

I intend to square the guitar's output using something in the lines of a slacktave, and will be eventually going for sine, tri and saw waveforms, although one of them would be perfectly enough for starters. So has anyone come up with a good PLL guitar synth?

[JovianPyx]

On synth-diy (email forum) there is a gentleman named Harry Bissel who has done lots of work with a DIY hex pickup guitar synth. I don't believe he's on electro-music.com. He's posted about using a hex pickup, envelope followers and the like, but I don't know if he's used PLLs for his system. It might be worth it for you to subscribe and ask some questions there. Harry is a wonderfully nice person.

What I can tell you from reading his posts is that guitar synths are not easy to make work correctly. For starters, the waveforms coming out of the pickups are not easy to shape. Ideally, one would want the number of zero crossings per second to be exactly double the frequency of the string. The problem is that there are many strong harmonics that cause the number of zero crossings to be a number other than double the string's frequency (pitch, if you will). Systems that can work with that effectively can be very complex. Envelope followers are frustrating because you want them fast (to follow correctly). But when made fast enough to do envelopes for high notes, they follow the waveform too much at low pitches. When made slow enough for low pitches, they tend to be noticably sluggish for high notes.

[egasimus]

Well, winding a hex pickup is in my plans, but, if I build one, I'd probably end up just connecting it to a microcontroller and using it to control VSTi synths - although connecting it to an analog synth would also be interesting. That's all fine and dandy, but it's a quite distant goal for me, as it would require an investment of time and money which I currently cannot afford to make.

Some quick thoughts:
Here's a version of the Slacktave which I'm gonna build.

Any PLL+VCO circuit that doesn't involve the expensive and virtually unobtainable where I live ICL8038?

On a side note, http://proguitarshop.com/store/effects-synthesizer-pedals-c-602_14/pigtronix-mothership-analog-synthesizer-p-701 - here's what inspired me in the first place and showed me that an analog guitar synth is possible. I'm really aiming for something much simpler, though.

[JovianPyx]

The 8038 VCO is used to get better waveforms. The 4046 already has a VCO built into it, but it's got only one waveform where the 8038 has several. So it's possible to use the 4046 as it is, but you'd have to give up the other waveforms unless you want to figure out ways to waveshape the 4046. Too bad about the 8038...

[egasimus]

What waveform does the VCO output, though? Judging from the several datasheets I've looked at, it depends on the inputs. Can a 4046 produce a sine/tri/saw from a square input?

[JovianPyx]

This subject came up several years ago on the Synth-DIY email forum. The output (pin 4) is a square wave.

The VCO schematic can be found in one or more of the datasheets out there, Texas Instruments has a nice explanation with schematic. Looking at the schematic, the core is something I would call "flying capacitor" and results in a squarewave main and intended output, but nothing else useful. Another person on the list came up with a circuit using JFET input opamps TL074 (high input impedance) across the timing capacitor to extract waveforms on either side of the capacitor. They look like a sawtooth that is missing every other tooth (where it is simply a constant voltage [which may or may not be zero volts]). This person combined those waveforms with opamps to get a sawtooth. I found the drawing, which was actually a VCO design that didn't use the PLL as a PLL, just as a VCO. So ignore the bit about "Linear CV" and PWM and concentrate on the opamp arrangement connecting to C1.

Attribution to this schematic goes to Bjorn Nillson, and I don't know if that's his real name. He'd been on the synth-DIY list for awhile, made some very odd comments that offended some people (I thought they were just poorly worded jokes) and had left and rejoined under different names a couple of times.

Anyway, as you can see just getting saw is a bit tricky, but you can get saw and you can get square (pin 4 output). I believe I remember reading someone say that it looked possible to derive triangle from the A and B waveforms (I also believe that is true). Once you have triangle, there are several ways to make sine, but we're talking a few more ICs per PLL. I would imagine that triangle derived this way is not glitch free and if sine is taken from that, it would be far from pure.

Also note that the sawtooth would be an octave above the square wave output.

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[egasimus]

Great stuff! I think I've found my way of resynthesizing a square wave from an earlier topic I posted. Read someone's post that went along the lines of "pll multiplication->binary counter->lookup table". And although what I did doesn't look like a lookup table to me, it seems to work nicely.

Beyond this very long link lies a rough model of my concept. (Warning - might load down your CPU a bit). Square the guitar using a slacktave, multiply square wave frequency using PLL, divide using binary counter IC, mix through resistor ladder. Result: instant ramp wave. And, given that there are 14-bit counter ICs available, it doesn't seem like it's going to suffer from aliasing, either. My main concern would be finding accurate resistors, 1% ones are hard to come by where I live.

[JovianPyx]

I don't want to rain on the parade, but I looked at the slacktave page and the caveats there seemed to indicate some problems with range of playing. Something about "works best on the inner 4 strings" and "works from 3rd to 17th fret". He also notes that it's best to use the neck pickup with the tone control to bass. I would thus wonder if guitar model and manufacturer makes a difference as well. Noisey pickups could play hell. Exceptionally bright pickups could play hell.

If you can work with that, you'll be fine - but as a musician, I like to be able to play the entire instrument's range.

You might want to first try building one slacktave channel and see how well it works in reality - to your own ears. As a guitar player myself, I've messed with similar cicruits and they do have these limtations. And the limitations can be exacerbated by playing style. Problems that appear at the slacktave output will be carried forward into the rest of the circuitry that follows.

[egasimus]

Indeed... They also mention cutting off everything above 700Hz as an alternative to the neck pickup + tone control thing, though. I'll do what you say - thanks for keeping me aware that there's more research to be done

[JovianPyx]

One other thing I noticed - slacktave is targeted as a tool for use with a 6 string guitar and standard pickups. As such, the designer tried to find a "happy medium" between the highest pitch and lowest pitch playable - and the result is a circuit that works best in the middle.

The reason for the caveats is that a lowpass filter that is tuned to a frequency best suited for the lowest string and get a fundamental might have such low output on a high string that it can't make a reliable square wave from the output. Put the frequency up too high and low strings start to have too many harmonics and you get a crappy "square" wave.

There could be an upside if you use a hex pickup and 6 slacktave circuits. However, they can't be identical slacktaves - I would consider that since the slacktave is really a lowpass filter, you might think about building 6 of them, but use pots for the frequency setting resistors. That is, you could optimize each slacktave for the range of the string it will process. If the slacktave doesn't have to worry about dealing with 4+ octaves of range, it should cover it's one string much better than the generic. You will need a hex pickup that isolates it's strings from the others rather well. Hex pickups that are commercially available are usually meant to be mounted near the bridge. Given the slacktave caveats, you might consider the hex pick up as a replacement for the neck pickup since a bridge pickup will give more harmonic content than a neck pickup.

If you go with that plan, you can build a single slacktave with frequency pots and try to opmtimize it for each string. Optimization should hopefully give you playability starting at open string. If the optimization works well enough, this might work. Once you have a string optimized, measure the pots so that you can build slacktaves that are optimized using fixed resistors.

Just thinkin' out loud here...

[egasimus]

I'm thinking, I'm thinking! Maybe a VCF at the input, the CV for which is roughly derived from the frequency of the input signal... I've come across several F to V circuits, although I can't find one when I need it...

P.S. I have thought of the hex pickup idea myself - it's definitely doable and could eventually become an entire guitar-controlled synth string ensemble... This is why I'm trying not to think about it, lest it consume me

[JovianPyx]

Ok - I'll make it worse -

I have a Roland GR-50 guitar synth. It has a GK-2 hex pickup. It's a pretty nice deal, but it isn't perfect. The problem I and others have with it is that it takes a longer time for a note to start up when playing a low string than when playing a high string. For strings 1, 2, 3 and 4, it seems responsive enough, but 5 and 6 have this short but annoyingly noticable delay.

What I did was to replace strings 5 and 6 with strings that are exactly 1/2 the diameter. This allowed me to tune these strings one octave higher than normal. Note that the fingering for chords is the same - but the low 2 strings no longer cause trouble. Yes, it's a bit different, but I can play it that way. This is called "Nashville Tuning".

Perhaps the same thing can be done with a single slacktave and make it more playable? Changes in optimization might likely be required.

More thinking out loud...

[egasimus]

But isn't the GR-50 digital? Having come out in 1988, its sample rate is probably quite limited by the technology available back then, and it would therefore take more time to reliably detect the pitch of a lower note. (Or am I wrong?) The fairly recent Boss SYB-5 also seems to suffer from some lag, though, from what I hear. One cycle of a bass guitar's E string would be about 25ms, and I don't know how can the frequency be calculated from anything less than a cycle (maybe abandon fundamentals altogether and read harmonics?) Well, it seems we can't do it better than Roland. Or at least I can't.

[JovianPyx]

Yes, the GR-50 is digital. It's engine is based on the D-110 (or D-10). It's sample rate is the standard audio 44.1 kHz.

Output sample rate, however, says nothing about internal computation and input sample rate. But 44.1 kHz ought to be fast enough.

The problem is in fast pitch correction. I wish I could find the article, but I read somewhere that Roland used a system that looks at dv/dt (or rate of rise) of the first 1/4 cycle of the input to determine a rough estimate of the pitch. It uses that estimate of pitch when it tells the synth to "go". As the note plays, the GR-50 is using other methods (which I don't know) to determine pitch accurately and sends information similar to a pitchwheel message to correct the pitch. By the time the synth is out of it's "attack" state, the pitch is already fully corrected and nobody is the wiser.

Still, even with this technology, it is slower (noticably so) on lower notes. And this is affected by playing style - heavy handed playing messes it up causing it to take longer.

You're correct that what Roland did was digital and what you want to do is analog. But it's interesting that I was able to "fix" the Roland problem by raising the two lower strings by one octave. It isn't the same, obviously, but what I read about slacktave indicates a spectrum width problem. If the lower notes weren't lower anymore, then the range is now compressed and it just might work a lot better if the lowpass filter doesn't have to deal with such a wide spectrum.

Ultimately, I think the hex pickup method is the best. In fact, Roland attacked that also - they made the GR-300 and possibly other synths - which is an analog guitar synth. You had to buy both the guitar and the synth from Roland (which is probably why it didn't sell well), but it play very smoothly. It had a hex pickup as well as 2 audio pickups. A friend had one that he loaned me for a few months to see what I could do with it. The problem for me was that I was already spoiled by the sounds I could get out of MIDI synths and the GR-300 sounded sort of "plain" to me. It was a very simple thing, 2 VCO, 1 VCF, ringmod, EG, etc. per string. Also, a hex pickup system allows you to play power chords, full chords, or whatever - and you can even be a little sloppy - like if you accidentally hit two strings when you wanted one, the single slacktave system can't work with chords at all and will make some kind of "blup" "florp" "crack" or other unwanted noise that would probably be unnoticed with the hex system and 6 channels of slacktave driven analog synth.

[egasimus]

Well, yeah, I meant the input sample rate, of course.

[JovianPyx]

"Pigtronix Mothership" interesting name, but sorry - I have no idea.

[egasimus]

Do you think that using a Schmitt trigger instead of a comparator would improve tracking somewhat?

[JovianPyx]

Comparator versus schmitt trigger in the slacktave? - I think that it will be different, but not better and depending on the schmitt trigger circuit. A CMOS schmitt trigger gate will have hysterisis that you can't change... You can build a custom schmitt trigger with two inverting CMOS gates and a couple of resistors and you can control the hysterisis, but even then, there will be problems that just won't go away. Under some conditions, one might be better than the other, but those conditions are fluidly dynamic and change as you play the instrument.

As you probably know, the reason for the poor "tracking" is that the guitar waveform has many harmonics and doesn't by nature produce zero crossings at numbers exactly double the fundamental's frequency. The slacktave attempts to remedy this by reducing the harmonics of the string using lowpass filtering. But reducing is not the same as eliminating. This lowpass filter cannot possibly work the same with the B string as it does with the low E, and therein lies the flaw. This is why I would suspect a hex pickup with 6 pitch range optimized slacktaves would work much better than a single slacktave on a standard pickup - and even then I doubt it would be perfect.

[egasimus]

Well, I was hoping of using an opamp I mean, a Schmitt trigger would probably remove some of the extraneous zero crossings from higher harmonics, wouldn't it?

If I feed a square wave into a PLL, in order to multiply it N times, I need a divide by N counter in the feedback loop. Does that mean a lag of at least (1/f)*N between input and output? Bad news for bass... Or maybe just use XOR gates to multiply the frequency?

[JovianPyx]

A schmitt trigger *might* do that, but it will also cause the signal to drop out earlier when the amplitude gets very small. It's worth a try.

As for frequency multiplication adding delay, yes, I believe it does, at least there is a delay in attaining lock which could become a problem adn there may also be a delay in how quickly it can change pitch. My circuit multiplies the input by only 12 times. Delay wasn't apparent to me when playing a keyboard using it. However 12 is much smaller than 256.

[egasimus]

Well, I'll try out what 8x or 16x sounds like - although that's only 3/4 bits, maybe with some filtering it could come out pretty decent.