Advanced Subtractive Synthesis: Tonewheel Organs

I titled this “Advanced Subtractive Synthesis” but I don’t like the term “subtractive synthesis;” I think that the segregation of subtractive and additive synthesis is a false dichotomy. Using phase differences, additive synthesizers can subtract, and, as I hope to show you, adding is an essential part of advanced subtractive synthesis. I’ll be using Reason’s Subtractor, one of my favorite synthesizers, but you can follow along with any synthesizer that has two or more audio oscillators and two or more low pass filters (gain compensated and mild slopes, preferred).

First, let me explain how tonewheel organs work, for those who don’t know. Tonewheel organs were an early attempt at additive synthesis. Being that certain models are now coveted for their unique sound, I’d say that they were (conceptually) a miserable failure but you could do worse than being so highly regarded as the Hammond B3, the most coveted of them all. Tonewheel organs work by spinning tonewheels (sprockets, in essence) in front of a pickup (like an electric guitar’s) to produce a close approximation of a sine wave. A small number of tonewheels are then mixed as harmonics, using eight-position “drawbars,” to create a complex sound. In the case of the B3 and many others, these tones are, in ascending order and according to Hammond’s very strange naming convention that has everything in the wrong octave (less strange, from Hammond’s point of view – they were following pipe organ tradition), the octave below the fundamental, the fifth above the fundamental, the fundamental, the second harmonic, the third harmonic, the fourth harmonic, the fifth harmonic, the sixth harmonic, and the eighth harmonic.

But that’s not all there is to it. Not only do the tonewheels not produce pure sine waves but their signal gets distorted multiple ways en route to the output. And there’s leakage between tonewheels. Also, because the sound would otherwise be static and, therefore, musically uninteresting, tonewheel organs are almost exclusively played through rotating speakers (a particular make – Leslie – too) which further shapes the sound in ways beyond the scope of this not-as-short-as-any-of-us-would-like explanation. We won’t be synthesizing that, anyway. We will, however, have to incorporate an electronic key click and “percussion” effect.

So, let’s get started synthesizing a commonly used and thankfully simple Hammond registration (as each unique combination of drawbar settings is called), the first three drawbars (suboctave, fundamental, and fifth) at equal volume (or, 888000000). On synths that have three or more audio oscillators, this is naturally quite easy to do, but we’ll have to be clever if we want use Subtractor.

First off, we have to set the oscillators to produce the right harmonics. No one waveform will work; a saw is the only one that produces each harmonic but they’re proportions are too far off and I’m not confident in our ability to filter out the fourth. We can, however, manage this sound, if we start with two waveforms: a square, which produces the first (suboctave) and third (fifth above) harmonics and leaves a gap where the fourth should be, and a sine wave, an octave above, taking the place of the missing second harmonic (fundamental). I won’t be pitching them, an octave down but it doesn’t really matter.

We now have two of the harmonics in place, the first and second at equal volume, but the third’s still to low and there’s many more odd harmonics than there should be. Luckily, we can solve both of those problems with the filter. You can use filter resonance to pick out and reinforce harmonics. All we have to do is turn on keyboard tracking, set the 24 db/octave low pass filter cutoff to the third harmonic, and turn up the resonance.

(Subtractor’s filter cutoff increases and decreases in increments of slightly less than a semitone; with keyboard tracking at 100%, a cutoff value of 59 is approximately at the triggered key.)

Well, there’s still one problem: the second harmonic is within the resonance band and will also be boosted. We could try using the oscillator mix control to compensate for this but that will also affect the fundamental. As with elsewhere, I’ll opt for the “does it sound good?” method of calibration. It will undoubtedly be wrong (though, after comparison, I got pleasingly close) but it will at least sound good.

Having set the resonance to a value of 73, we’re left with just the key click and percussion to synthesize. The key click is a simple, bright “clicky” noise and the percussion is a short emphasis of the second or third (forth or sixth) harmonic. A crude but usable approximation of these is easily created by quickly sweeping down the filter, at the start of each note. To do this, I’ve set the filter envelope sliders to 0 and the amount to 8. Our work is now done but there’s still the matter of the rotary speaker, which is almost as much of the sound as the organ itself. There really isn’t much that Subtractor can do to synthesize its sound but there are many emulations. For a free, Reason based solution, I recommend Jiggery Pokery’s combinators.

Now let’s ask ourselves a question: can this work with more complicated registrations? I think this is a question worth answering and, to that end, we’ll try synthesizing a favorite registration of mine, 006646440, which was used by one of my favorite keyboardists, Tony Banks. (He’s a very, very clever musician; I highly recommend his work.) Just to recap, this registration will have the fundamental, second harmonic, and fourth harmonic at equal volume and the third, fifth, and sixth harmonics at roughly two thirds that volume. As with before, a sawtooth wave would produce all of the harmonics we need but in grossly wrong proportions and I’m not confident in our ability to filter out the ones we don’t want, most notably, the seventh.

But I have an idea. We’ll again use oscillator one, for odd harmonics, and oscillator two, for evens but, this time, with a square and a pitched up sawtooth, respectively. The square produces its usual odd harmonics and the sawtooth, because it’s pitched up an octave, produces only evens (one becomes two, two becomes four, etc.). We now have first and second harmonics of equal volume, too weak third, forth, and sixth harmonics, and a much too weak fifth. And too strong seventh and above harmonics. And two resonant filters.

Choosing which combination of harmonics to reinforce is tricky and I wouldn’t say that there’s a wrong answer. One thing to keep in mind is that, because Subtractor’s filters are (hardwired) in series, filter one will reduce the amplitude of the harmonics passed to filter two. The first harmonic I’ll reinforce is the third (using filter one in 12 db/octave low pass mode). It needs to be reinforced more than the fourth and I generally find that fifths are more important to Hammond sounds than octaves. Additionally, it’s the percussion sound at the beginning of the note. The second and fourth harmonics are within the filter’s resonant band, which is again bad, in the case of the of the second, but good, in the case of the fourth, which needs a boost.

The second harmonic I’ll reinforce is the fifth (note: Subtractor’s second filter is always a 12 db/octave lpf). It needs to be reinforced far more than the sixth and reinforcing it, instead of the sixth, keeps the seventh harmonic out of the resonant band, if only just. The sixth harmonic (and the fourth, too) are also reinforced somewhat by the filter. Resonance is again calibrated using the “does it sound good?” method. Again, the relative amplitudes of the harmonics are surely wrong but when comparing my patch to a The Colony of Slippermen, I’m pleasently surprised by how similar it sounds (perhaps Banks used a slightly different registration than I had thought).

So, can we use subtractive synthesis to emulate tonewheel organs? Well, kind of. The key click/percussion sound in the first patch isn’t very satisfying at all. It could be improved by adding filter two but that would be wrong for the body of the note. (The key click/percussion sound for the second patch isn’t perfect either but I’ll take it.) Our organ may also be a bit too clean sounding. We can solve this by mixing in some dark noise at a just-noticeable level. For the Leslie sim I used, color and level values of 21 and 66, respectively, worked. This makes it sound a bit more organ-y. (The organ in question may need servicing but beggars can’t be choosers.) This also improves the click/percussion sound, by giving the filter more partials to sweep through.

One thing I didn’t mention is that organ harmonics use equal temperament. This is of little consequence for our 888000000 sound, as equal temperament fifths are quite close to just, but it may be holding back our 006646440 sound, which includes a third. And then there are the countless odd but important behaviors of organs that these patches don’t begin to incorporate. (For instance, percussion should only be present on the first note triggered, after all previous notes have been released.)

But, in a much less academic look at usability, how do these patches fair? Well, if I were hired as the keyboardist in a jazz B3 combo, this is NOT how I’d get my sounds. But, with a good Leslie sim (or a real Leslie) and loud bandmates, I think I might get away with using my Slippermen patch at a Genesis tribute show. I’d certainly have no qualms about using it in a demo.

But, in that case, I wouldn’t go to the trouble of trying to program an organ sound using just one device, I’d put multiple tone generators and a mixer in a Combinator and give myself proper control over the individual harmonics. Though, as of this writing, there actually isn’t a way to get a proper Hammond organ sound within Reason (no, not even Revival).

So why did I go through all the trouble to teach you this method, if I’m just going to tell you to not bother with it? Because I thought it would be an educational bit of synthesizer programming. :)

Frequency Modulation – It’s Not So Bad Part III: Synthesizing Brass

FM can produce quite nice brass sounds and, in this tutorial, we’ll be synthesizing a very 80s trumpet.

Just as in tutorial two, I’ll have you set your amp and filter envelopes first, so that we don’t have to interupt any FM-y goodness for them. For the amp envelope, use a ~15ms attack, very snappy decay, and very high sustain. The filter envelope should have a ~20ms attack and a medium sustain. You can leave the decay and release stages at their defaults.

Now, for the FM-y goodness:

You’ve probably read that trumpets produce a complete harmonic series (if you hadn’t, you have now!), so we’ll create an FM pair oscillator and… leave the ratio at 1:1. To give the sound shape, we’ll turn the FM amount down and assign the filter envelope to it in the matrix. I found that a value in the 60-70 range is about right.

If you play a few notes in the trumpet’s register, you should hear the beginnings of a trumpet sound. It’s a bit dull and weak, though. We can beef it up with a second FM pair using a ratio that also produces a complete harmonic series, I went with 2:1. Connect it to the filter envelope (I used a mod amount of 50) and it sounds pretty trumpet-y.

FM really can be quite simple.

The boot leather of the sound is there, so we’ll just add some transient detail and vibrato, before I leave you.

A few things could add a nice ‘prp!!!’ to the start of a note. My first thought is to try a detuned modulator. More modulation will sharpen the attack and, because it’s detuned, there will be a cluster of sidebands at each harmonic.

Create a third FM pair, drop the modulation, and detune it a bit (I used -5 cents). Route it to the first two pairs in the mod matrix, scaled by both the mod envelope and the key note, then set the mod envelope attack and decay times to 15-20 milliseconds. Raise the FM amounts (I used values between 60 and 70) and play.

Did it sound good? It worked for me. Trumpet vibrato is between five and six hertz; I set a delay and scaled mine by the mod wheel. Add a bit of reverb and play some chords… very 80s. I’ll leave you to do the velocity scaling.

FM – it’s really not so bad.

Part I: Introduction

Part II: Synthesizing Bells

Frequency Modulation – It’s Not So Bad Part II: Synthesizing Bells

It’s not for nothing that FM is famous for bell sounds. In this tutorial, we’ll create a sound along the lines of wind chime

We’ll create and initialize a Thor and remove the analog oscillator and filter. For a bell, we’ll want to turn down the sustain levels in the envelopes and raise the release times to around the same value as the decay times.

For flexibility, we’ll apply FM via the matrix. Create two FM pair oscillators and route the second to the first (don’t forget to turn down the FM amount knob). To find the right ratio, just keep hitting the up button until you find one that sounds bell like. Yes, that’s really all you need to do!

I like 1:7. For the purposes of following along, I recommend you use it.

So that the timbre changes over time, we can scale the FM by the filter envelope. Once we find an appropriate amount of FM (I used 50), we have a basic bell patch.

But we can do better.

Unlike a real bell, the harmonics of ours are all in tune. But, remembering the previous tutorial, it’s easy to change that by detuning the modulator. Playing with it a bit, I like -15 cents. This is an improvement but it’s still a bit to stable – a second modulator can fix that. We’ll stick with the 1:7 ratio but tune a few cents sharp (I chose five).

With FM amount values in the 15-20 range, we can introduce a nice ringing dissonance, much more bell like than the perfect 1:7 sound. And it wasn’t very difficult; all we did was use trial and error to find a bell-like ratio and detune the modulation. Why was anyone ever afraid of FM?

To polish off the patch, we’ll add a bit of flavor to the transient. FM is quite good at this because of a little bit of magic that happens when you modulate a modulator: the carrier is modulated by every frequency in the first modulator.

We can just use the second FM pair for this. Set the mod envelope’s decay very short and assign it to the second pair’s FM amount. We don’t want too many partials created, so we’ll try a ratio for the second pair that has lots of gaps in its spectrum, 7:14 (the same as 1:2).

I’m pleased with the sound and declare our work here done.

A few notes:

With all the detuning, we’ve dragged our patch off-pitch. This can easily be fixed by shifting each oscillator’s pitch by the same amount. How much will depend on how much detuning you did.

In the previous tutorial, I told you that we’d be using key-scaling to compensate for the deficiencies of the matrix’s algorithm. I left that out of this tutorial, as I couldn’t get a result I was happy with. Even with a compensating algorithm, FM often requires more sophisticated key-scaling abilities than subtractive synthesis, so Thor isn’t up to the task.

Part I: Introduction

Part III: Synthesizing Brass

Frequency Modulation – It’s Not So Bad Part I: Introduction

I don’t claim to be an expert in FM; I don’t claim to be especially good at it, for that matter. I do claim, however, that FM is not as hard or scary as you might have heard and I hope to prove this over the course of three tutorials. From my tinkering with FM, I have come to the conclusion that it is difficult to learn because of the dirth of good user interfaces, rather than any property of the method, itself.

The trouble is that FM synthesizers is that they, by necessity, have more parameters that need to be set than most others; presenting them all intuitively is difficult. Conceptually, FM is actually simpler than subtractive, in my opinion. Whereas with subtractive, you build a complex sound to then take away from, with FM, you just build a complex sound.

For these tutorials, we’ll be using Thor as our FM sythesizer (it’s not ideal but it’ll do) and in this first tutorial, we’ll be getting ourselves acquainted with some basic FM concepts. Create a Thor, initialize the patch, remove the filter, set the amp sustain to full, and replace the analog oscillator with an FM pair oscillator.

Observations on Basic FM Sounds

The math averse among us need not fear; we’ll be using our ears. For the following exercises, it would be best to stick around middle c.

The first observation I’d like to make is that raising and lowering the modulation amount produces very different changes in timbre than raising or lowering a filter cutoff frequency. If we were to perform a filter sweep, only the harmonics above the cutoff frequency would be affected and they would be affected linearly, that is, their amplitudes would change at a steady rate and in just one direction (resonance notwithstanding). If we sweep the FM amount knob on the oscillator, we’ll find that this is not the case for FM. All harmonics are affected and the effect on their respective amplitudes is non-linear.

Those with an ear for this sort of thing will have noticed that the default ratio (1:1) produces the full harmonic spectrum. Now let’s observe what happens when the frequency of the modulator is raised. You should hear that gaps in the spectrum appear. With ratios above 1:10, or so, sounds can take on an enharmonic quality, though you’re actually just hearing higher harmonics than you’re used to (observation 3).

For observation 4, we’ll compare two ratios, 1:2 and 1:4. (We’ll be sticking with 1:n ratios, as they’re the most musical.) If you listen closely, you may be able to tell that they produce the same harmonics (odds) but at different levels. If you can’t hear it, try comparing 1:2 with an FM amount of ’44’ to 1:4 with an FM amount of ’19;’ the sounds should be quite similar. Different ratios can produce the same harmonics, they just won’t be at the the same amplitudes.

Next, we’ll turn the FM amount knob back to ‘0’ and create a second FM pair and connect it as a modulator in the mod matrix, with an amount of ~35. You should hear something similar to a filtered sawtooth. So far we’ve only used perfect 1:n ratios. Listen to what happens when you detune the modulator (+/-10 cents). You should hear beating (observation 5).

Now disconnect the second pair from the first and connect it to the output. Set the two pairs to different 1:n ratios and turn up the FM amount. Make careful note of the sound. Disconnect the second pair, turn the FM amount down, and set the carrier frequency to that of the first pair’s modulator. Create a third FM pair and set the carrier frequency to that of the second’s modulator. If you connect the second and third pairs as modulators to the first pair in the mod matrix, you should be able to produce the same sound as you just heard, because two modulators modulating a carrier is the same as two modulators modulating separate carriers of the same frequency (observation six).

To recap:

1. Amplitude changes of a harmonic are nonlinear.

2. 1:1 produces the full harmonic series; other 1:n ratios above have gaps.

3. High 1:n ratios can sound enharmonic, though they aren’t.

4. Different ratios can produce the same harmonics but at different amplitudes.

5. Detuning the operators results in beating but only of the overtones.

6. Two or more modulators modulating a single carrier is the same as two modulators modulating separate carriers of the same frequency.

All FM Is Not the Same

This observation has a big effect on FM in Thor, so I’ve given it its own heading: there are different kinds of FM. The FM pair oscillators and the mod matrix use different algorithms and so produce different sounds. The oscillators use the same technology as the DX synthesizers, whereas the matrix is based on analog technology. The analog based algorithm has two key deficiencies: artifacts at low FM amounts and a dulling of the sound as you play up the keyboard. There’s no avoiding the former but the latter can be compensated for, somewhat, with key-scaling.

These deficiencies will become important in tutorials II and III.

Some Math, for the Curious

This is the equation for the frequencies generated:

Equation Image 2


In English, there are sidebands (new partials) of frequencies (‘w’) equal the carrier plus and minus integer multiples (‘n‘) of the modulator. So, for 1:1, there are positive sidebands of 2c (c+1m), 3c (c+2m), and so on. Negative – yes, negative – sidebands are 0c (c-1m), -1c (c-2m), and so on. The negative sidebands are “reflected” back at the reverse polarity.

The reason different ratios can produce the same harmonics but at different amplitudes is that the harmonics will be generated as different sidebands. In the case of 1:2 and 1:4, the first positive sidebands will be the third and fifth harmonics, respectively; the amplitudes of the sidebands will be the same but they’re representing different harmonics.

A detuned modulator will create beating because they generate detuned sidebands. Let’s take a ratio of 1:99, as an example: positive sidebands will be 1.99c (c+1m), 2.98c (c+2m), 3.97c (c+3m), and so on; reflected sidebands will be .98c (c-2m), 1.97c (c-3m), 2.96c (c-4m), and so on. Obviously, this is to the detriment of intonation. Happily, though, a detuned carrier is mathematically the same and mitigates the flattening effect.

The equation below determines the amplitude of sidebands:

Equation Image 3

In English, the modulation index (‘β‘), i.e., brightness, is equal to the change in the carrier’s frequency divided by the modulator’s frequency. Using this equation to actually calculate the amplitude of a given sideband requires can’t really be done with pen and paper but, in practice, it’s not necessary. However, I would like to make note of the denomitator, the modulator’s frequency. As it increases, i.e., as you play up the keyboard), the modulation index (‘β‘), i.e., the number and aplitude of sidebands, decreases. Dedicated FM synthesizers compensate for this; analog and analog based synthesizers do not.

Part II: Synthesizing Bells

Part III: Synthesizing Brass

Equation images courtesy of Gordon Reid.

Three kHs ONE Tips

I’ve had the ONE Re since before it was released (I was a beta tester) but I’ve never said much about it, here or elsewhere. I thought I’d make up for lost time and share a few tips that are off the beaten path.

1. The sub oscillator doesn’t have to be a sub oscillator.

The initial patch sets the sub oscillator’s octave to -1. This is is very sensible but it will lead some to overlook that it can be set all the way up to +5. Using the sub oscillator’s unique tones in higher octaves greatly expands the range of sounds that you can create. (I’ve found it very helpful when creating pipe organ patches, for example.) Be warned, though, the sub oscillator’s anti-aliasing isn’t very good (in normal use, of course, it doesn’t need to be).

2. The oscillators can be freerunning.

Another feature you won’t find in normal use. When an oscillator’s gain is increased from 0% AFTER a note has been triggered, there will be a phase difference between it and the others. I’m not sure if this is a feature or a bug (I can’t confirm it’ll work in other formats) but there are some cool modulation effects to be had by modulating an oscillator’s gain and its shape and/or sync parameters simultaneously.

3. Simulating a different filter.

This one works on other synths, as well. With most resonant filters, increasing the resonance to high levels will attenuate the pass band quite noticeably. Others, like sallen-keys, Subtractor’s, and ONE’s, are gain compensated – pass band attenuation is never more than a few dbs. What to do if you want to emulate a patch that used a non-gain-compensated filter with a high level of resonance? Use both of ONE’s filters. A high pass filter between two and three octaves below the low pass filter will get you in the ballpark. Tweak to taste. (Sorry, 12db/oct cutoffs only.)

Note by Note: The Making of Steinway L1037

I don’t watch many documentaries but I did watch this one, last nIt wasn’t wasn’t as educational as i hoped it would be but I enjoyed it. It follows the making of a Steinway concert grand – a year long process – and includes interviews with Steinway workers and pianists and happenings in the world od Steinway. There’s a great scene with jazz greats Bill Charlap and Kenny Barron selecting pianos. Barron is having a great time, playing beautifully on each of the pianos, while Charlap just doesn’t fit with any of them, playing many abortive attempts at Gershwin’s second prelude. Sadly, the audio recording of the whole film was quite poor. Very frustrating.

It was interesting to see how much of the work was done by hand. According to the film, this is because the work required too much precision. That’s untrue, though; Taylor Guitars, an industry pioneer in the use of CNC (Computer Numeric Control) machines, is able to make instruments the way they do because of the precision afforded by machine cutting (their neck joints are designed with such precision that the thickness of the finish has to be factored in.

Not to discount the quality of their pianos but it’s worth noting that Steinway’s success doesn’t stem from that alone. Historically, they have very aggressively pursued name recognition and familiarity among young musicians. This, among other business practices, is what makes them so dominant in the American piano market, today. I felt strange, hearing a Steinway executive lament other makers going out of business.

Merry Christmas

Merry Christmas. I like Christmas. It’s hard not to.

This Christmas brings a gift for the Reason users among us. A free crowd-sourced refill was created for the occasion and released several days ago. I intended to have my piano combis included but there was a miscommunication between myself and the organizers (my fault). Oh well; the patches readily available on my Free Combies Page.

It’s a 326 mb refill with patches for most Reason devices and a fair number of Rack Extensions. The total patch count is 1200. I’ve looked through it and there’s some good stuff. Kudos and thanks to all involved.

Review: LOXX Strap Locks

LOXX strap locks

Strap locks are a more secure way of connecting straps to guitars, basses, and the like than the factory supplied hardware. While there are low-tech options, the term most often applies to mechanical varieties. They come in two main parts, a replacement strap button and a device that catches on an inner lip when inserted  into the strap button (this is to be attached to the strap). While these are in some ways superior to the standard option (a conical piece of metal on the instrument and an expanding hole on the strap), they do have drawbacks.

They can outright fail. They can break free of the strap (in most designs, they attach to the side of the strap towards the instrument), they can slip loose of their mate, they can unscrew themselves from the instrument. Because the strap now attaches at the end of the strap button (unless the button is countersunk into the instrument), there is increased leverage and therefore increased strain on the wood which can rip the whole device free of the guitar. Some even creak and rattle and the sharp edges of the devices can damage the instrument’s finish (though this is rare, thankfully).

But there is one variety that solves most of these problems, albeit with a drawback that put me off at first. LOXX strap locks have a pin installed in place of the strap button which is then grasped by the piece attached to the strap. This has the obvious fault of keeping it from being used with an unmodified strap but not for no reason; this way, the strap is kept almost as near to the instrument as it would with a standard strap button. And since I really only ever use one strap with each instrument (as I expect is the case with most people), this drawback may be overlooked.

The LOXX design is new to guitars but has been used in the automotive and marine industries for decades, securing soft tops and the like. The design seems perfectly robust and reportedly will require 220 lbs of force to be exerted on it to fail. Because the pin is spherical, the lock pivots smoothly rather than requiring the strap to flex as the instrument’s position on one’s body shifts (not strictly necessary but nice – this reduces strain on the strap that might cause it itself to fail). Even the variety of finishes is impressive (I chose black).

Installation is simple: screw the pins into place and attach the threaded strap insert with the supplied purpose built washer. It even comes with a multitool. There was one hiccup (well, two – I accidentally cut myself) in my installation but it wasn’t a problem with the product; one of the pins went in crooked. It could be that I made a mistake screwing it in or it could be that the whole was already messed up (the guitar’s previous owner took it to a tech I used once). I’m pretty sure it was my fault but I’m going to blame that tech (shame on him).

Once the strap locks were installed, I was entirely pleased. They look far better in person and they seem very secure. The mechanics feel smooth while anything can break, I’m not concerned about accidents. Not only must you pull up on the mechanism with even pressure on each side (i.e., even if it got caught on something, which it won’t, nothing would happen) but the device resists disengaging when carrying the instrument’s weight. Best of all, the mechanics attach through the strap, from the side away from the instrument; the washer is just there for when the strap isn’t in use.

All in all, I have to give this a big thumbs up. I’m very happy with the product and would gladly receive samples for more exhaustive examination and review (another electric set and an acoustic set, nickel thanks *wink wink*). Much more information is available at their website, including an informative, albeit biased, comparison of their design and two common alternatives.