Behind the scenes of the 'Straight Maths' Virtual Instrument on Pianobook.co.uk

I have been exploring the possibilities of mis-using a sample player recently. Dave Hilowitz's excellent 'Decent Sampler' is, imho, not only much better than merely 'decent', but it has also allowed me to go slightly outside the usual territory of samples and to become an intrepid explorer. Huge thanks also to the team behind Pianobook.co.uk - a great contribution to the world of sampling, created by Christian Henson.

So here is a quick recap of the design thinking behind two of my recent releases on Pianobook.co.uk.

Parallel Inversions

'Parallel Inversions' was my first really developed idea that isn't just a sample replay. It deliberately breaks the rules to produce an 'alien' instrument. In a 5 star review, Michael Milburn said; 

'I don’t understand what these are, but do enjoy the sounds.'

The user interface is the first thing that hits people with this virtual instrument. The top row of controls has 23 vertical faders that look a bit like the 'Mic' sliders that you see in many sample players - except that instead of 2 or 3, or maybe 5, or (extreme) 7 or higher, there are almost two dozen of them! They are split into four sections, and there are some subtleties in the way that these are put together.

The most important section is the one that has the 0 to 4 'Mic' sliders in it. The 'Zero' slider is set at about 75% in the default preset, so that you know it is important. This slider plays the 'fundamental' frequency that is played by Decent Sampler, or rather, it plays that frequency sometimes - the XML code that Decent Sampler uses to specify how samples are played allow all sorts of manipulations, and I'm exploiting this here. So the 'Zero' / '0' slider plays three different octaves, using a random 'Round Robin' assignment. So if you play a C3, then you will actually get a C3, or a C4, or a C5. The ratios are set asymmetrically, with the 'octave down' option half the probability of the others. So for every chord that you play, you may get that chord, or you may get a biased inversion of it instead (a 'bass-light' inversion). This isn't how many instruments work! (But it is an 'alien' instrument...)

The 1 to 4 Mic sliders are actually pitched in semitones up from the 0 (zero), which is why they are arranged in the staggered 'piano keyboard' arrangement. This is immediately obvious if you increase the '1' slider, because you get a C / C# discord! So the 0 to 4 section controls parallel pitches, which (again) isn't how many conventional instruments work - organ drawbars are a bit like this, but...).

The next section to the right is from 5 to 11, and again these are parallel semitones up from the 0 (zero) pitch. The '5' (fifth) slider is set at about 75% in the default preset so that you know it is important (just as with the 0 (zero) slider. So the default preset plays two sine waves, a firth apart, and in both cases, the pitches are inverted (or not) at random, with a preference for one octave up instead of down. All of the inverted pitches are slightly detuned relative to the fundamental pitch, which gives a more interesting tone. All of these 'Parallel & Inverted' sliders are centred in the stereo image.

The combination of fixed parallel intervals (the default 5th is just intended as a hint to get you started) and random inversions kind of breaks 'the rules', and gives this instrument an interesting and unusual character. Have fun breaking all those conventions that you are supposed to follow, and embrace performances that are never the same twice!  

On the far left, there is a single '-12' slider, which was supposed to play a pitch one octave down from the fundamental. Unfortunately, I'm not the world's greatest programmer, and so it actually plays the same pitch as the '0' (zero) slider, except that the random inversions mean that most of the time it plays a different octave. Although Parallel inversions has had 3 versions, I have left this defect in there, because serendipitously, it sounds good. 

The A to K sliders are different again. This time they are panned either hard left or hard right, and they are distorted sine waves, instead of the purity of 0 to 11 and -12. So the A to K sliders add timbre and broaden the stereo image. Again, this isn't how normal instruments tend to work, but...

Finally, the lower row has more 'synthesizer'-type controls than is normal, with a full ADSR 'envelope' control, and I recommend the 'Attack' control for giving gravitas, and the 'Decay' control (with 'Sustain' set to near zero) for adding a 'Radiophonic' or synthetic character that sounds like it is from the 1970s. 

Straight Maths

'Straight Maths' has a busy user interface, but it extends some of the ideas in Parallel Inversions. The left hand side has 48 'Mic' sliders (yep, a lot!), whilst the right hand side has the extended 'synthesizer' controls, but in a more compact vertical format.

The three rows on the left are devoted to three different types of sound source. 

S - Top Row - additive synthesis

The top row (S) is sine waves (with twists) to provide simple Fourier additive synthesis. The '0' (zero) slider is again set as a hint that it is the fundamental in the default preset, but it does tend to get lost with all the other sliders! 

The three blocks of four Mic sliders on the top row have, from left to right:

- a Sine wave (0, 1 or 2 octaves up, shown as 0, 1 or 2), panned to the centre,

- a hollow-sounding, slightly square waveform (-), panned to the centre,

- a slightly bright, slightly sawtooth'y waveform (N (get it?)), panned to the centre, and 

- a detuned stereo 'sweetener' sine waveform (s) that adds a bit of interest and broadens the stereo image. If you want, you can ignore the 's' sliders and add your own preferred chorus effect via VST or outboard...

Yes, there's a bug with the 'S' in the two octaves up section, but that's part of the charm of the user interface, and does not affect the tone! 

The '-2' and '-1' mic sliders are sub-octave sine waves that can add low end to sounds. Use with care! 

As with all additive synthesizers, you mix and match the sliders to give you the combination of harmonics that you want, and then use the ADSR controls to give the sound a bit of shape in time. 

M - Middle Row - Karplus-Strong physical modelling

The second row has 16 different samples of metallic-sounding decaying sounds, derived from the Karplus-Strong hammered/plucked string physical model. '13' is my personal favourite, but it is way too strident for most purposes, and so just the merest hint of it is usually plenty! I resisted the temptation to arrange the sliders in any sort of order (previously I tried a 'tone-to-noise' arrangement), mainly because when I have tried to do this, I have rediscovered just how difficult it is to arrange multi-dimensional differences into a linear order. So I'm afraid that you will just need to play with the sliders until you get used to the sounds. Oh, and 10 and 11 ARE different, but not as different as I wanted! 

The 16 sliders are all tuned slightly differently, and are all stereo. This means that you can use combinations to add harmonics and detuning

My preference is to use the middle row to add a little bit of metallic 'bite' to sounds that are mainly top-row additive at their core. You can completely ignore this and do your own thing, of course!

W- Lower Row - Risset physical modelling

This row mis-uses Risset's work on synthesizing drum sounds, and adapts it to producing 'woody' sounding fast-decaying thumps and clunks to add percussive starts to the higher row sounds. There are four sets of sounds, arranged with the left-most sound in each set being the thickest (three sounds at once) and the others just single sounds. The detuning is toned back for most of these samples. These sounds are in mono, centered in the stereo image. I did play with stereo samples, but at low frequencies there isn't much to gain. To show how self-contradictory I can be, my '9126 Sawtooths' instrument on Pianobook.co.uk has way too much stereo bass!

It is quite fascinating how just a brief 'blip' of woodiness at the start of a sound that is all sine waves can totally change the character and timbre that you perceive. (Oh, and too much reverb is always a good idea!) This low row is influenced by the clicks found in old tone-wheel organs (the idea of adding percussive starts is not 'new' in any way!) and by the rather novel use of samples of the starts of instruments that Roland used in their D-50 synthesizer to augment a simpler digital synthesis technique for the sustained sounds. Roland called this mix of samples and synthesis 'Linear Arithmetic', so 'Straight Maths' is my way of paying homage to a classic 'personal favourite' synthesizer from the 80s. Okay, so now you know where the name comes from!

As before, the lowest row is used to add a little extra bit of character to the sound. The default preset  deliberately adds too much 'W' so that your first experience of 'Straight Maths' is 'Wow!'. Maybe that what the 'W' really stands for? But remember that subtlety is often the best approach, and too much 'W' may take you into cheesy territory...

Trivia

The rows were going to be labelled as: J, AK and C, for Joseph, Alexander, Kevin and Claude, but I thought this might be too obscure. What is interesting is that you now know a famous 'Kevin' - although Karplus still sounds uber-cool to me!

Letiti gave 'Straight Maths' a 5 star review, which is much appreciated, including this comment:

'One of the most innovative and unusual Pianobook entries'

For which I am enormously grateful!

---

If you find my writing helpful, informative or entertaining, then please consider visiting this link:

Synthesizerwriter's Store (New 'Modular thinking' designs now available!)

Buy me a coffee (Encourage me to write more posts like this one!)

Should I look at the Spectrum, or the Waveform? - [Single Cycle Part 4]

One of the 'useful things to remember' that I have always had in my mind is something that I learned reading through a pile of old 'Wireless World' magazines from a cupboard at the back of the Physics Lab at my school:

Spectra can be better diagnostics than waveforms 

(I'm using 'Spectra' here as the plural for 'Spectrum'. You can replace it with 'Spectrums' if you prefer... I won't tell anyone.)

It was from an article where they described how a project to recreate the sound of a church organ by reproducing the waveform failed because the result sounded totally different. From the first part of this series then you may not be suspecting that they probably only matched up the 'top' 30 to 40 dB of the sound (the visible bit on a 'scope) - the '30 dB Rule' as I call it. When I've experimented with A/S (Analysis/Synthesis), the iterative synthesis technique where you analyse the target sound/timbre, get a reasonably close synthesised version of it, then subtract the two to get a 'residual', and then synthesize that, and so on, then I wondered if you could use this to keep removing layers of 40 dB or so of visibility, getting a better approximation each time...

Anyway, a reasonably good spectrum analyser is going to show you a lot about the spectrum of a sound - and the harmonics that it shows will give you detail well below 40 dB down. But the spectrum isn't perfect either, because it shows the magnitude of the harmonics in a sound, but generally, not the phase relationships. As was shown in part two of this series (Single Cycle 2), then phase relates to the tiny timing difference between the same point on two waveforms - it could be zero crossings, or positive peaks: anywhere that is easy to compare. Although the horizontal axis is the 'time' axis, many people think of the phase more in terms of the shape of the waveform 'sliding' horizontally - which kind of removes the link that is implicit in a 'time waveform'! But this 'sliding' approach does explain how it is possible to have phase differences that are not directly related to time - if you take a waveform and invert it, the two waveforms are then 'out of phase' even though neither of them has moved in time (although it might take a finite amount of time for the inversion to happen, of course!)

Where this gets interesting is when the waveform is not symmetric. If you invert a sawtooth, then what does 'phase' mean? The zero crossing position gives a reasonably neat alignment of the sawtooth waves, but using the positive peak is confusing, and it would be better to use the fast 'edge' between the positive and negative peaks - but is this then ignoring the time for that fast edge. So should the zero crossing in the middle of the fast edge be used? 

When the waveform is even less symmetric, then neither peaks nor zero crossings may be a viable choice for a reference point. In the example above, inverting the waveform means that the positive peaks are different, and there are two candidate zero crossings. When waveforms are this different, then phase starts to lose any meaning or value for me... Of course, you could use the fundamental of the two waveforms, in which case the inverted waveform would be seen as out-of-phase or inverted.

Phase is important in filter design (like in loudspeaker crossovers, for example), in noise cancellation (two anti-phase signals will cancel out to give silence, although getting two precisely out-of-phase signals is not very easy in a large volume in the real world), and in creating waveforms (in additive synthesis, for example). It turns out that the phase can be very important as a diagnostic tool: so a visually smooth filter cut-off might well be hiding a phase response that goes all over the place. 

Why is Phase Important?

The standard example to show why 'phase is important' is to take a 'square'-ish waveform made from a few odd harmonics, and to change the phase of one of them. Suddenly the square wave isn't square any longer... 

What has always fascinated me is the number of harmonics that are required to get waveforms that are close to the mathematically perfect, sharp, linear wave shapes that you see in text books. In the example above, then 23 harmonics are used to make a 'wobbly' square wave - actually, of course, then because a square wave is made up of odd harmonics, then there are not 23 actual sine waves used to make up the square wave, since just under half of them have zero amplitude. 

So when the phase of the third harmonic (three times the frequency of the fundamental) changes, then two things happen. Most text books will show the changed waveform, and will note that it still sounds like a square wave (the harmonics are the same...). But it is more unusual for there to be any mention of the change in the peak amplitude - the 'F3 out of phase' waveform on the right hand side is about 50% bigger, peak-to-peak, than the 'conventional' square wave approximation on the left hand side. It turns out that changes in the phase of harmonics can affect the shape and the peak-to-peak amplitude, and more: the phase of the harmonics can be used to optimise a waveshape for some types of processing, although this is normally used in applications like high power, high voltage electricity distribution rather than audio.

But this 'phase is important to the shape of the waveform' principle applies to any waveform, and this can give surprising results. Take a triangle wave: it has only odd harmonics, and they drop off rapidly with increasing frequency, so the triangle really is what it sounds like: a sine wave with a few harmonics on top. Now you are probably intrigued by this, and rady to explore it yourself, so there's a very useful online resource at: http://www.mjtruiz.com/ped/fourier/ It is an additive synthesizer that lets you explore the amplitude (volume/size/value) of harmonics, as well as their phase! (This is called a Fourier Synthesizer, after the Fourier series, which is the mathematics behind adding different sine waves together to give waveforms...)

Here's a screenshot of a triangle wave produced using the Fourier Synthesizer from M J Ruiz:

I have edited the colours of the sliders to emphasize the harmonics which have zero amplitude (black), the harmonics which are 'in phase' with the fundamental (blue), and the harmonics which are 'out of phase' with the fundamental (orange). In-phase is shown as a value of 0 in the screenshot - meaning zero degrees of phase, where a complete cycle would be 360 degrees. Out-of-phase is shown as 180 degrees - half way round a cycle of 360 degrees.

 The screenshot above shows an unedited view of the same triangle wave, but with the phases changed so that all of the harmonics are in-phase. The result is more like a slightly altered sine wave than a triangle wave - but it sounds like a triangle wave...

Earlier I pointed out that the sound of a square wave with the third harmonic changed in phase was the same as a square wave with no phase change on the third harmonic. It turns out that your ears are not sensitive to phase relationships of this type, and so the square waves, and the triangle waves, all sound the same regardless of the phase relationships of the harmonics. BUT if you change the phase of a harmonic in real-time, then your ear  WILL hear it. Static phase relationships between harmonics are not heard, but changes in phase are...

If you think about it, then this is not as surprising as it might at first sound. Your ears are very good at detecting changes of phase, because that's how they know what frequency they are hearing! But fixed differences in phase just change the shape, and your ears don't pick that up. One possible explanation for this is that your ears evolved as they did because the harmonic content of sounds was important for survival (maybe locating sources of food, or danger!), but the shape of the waveform was not. Discovering that the human hearing system is not optimised for sound synthesis may be a disappointment for some readers...

One other thing that you may have noticed in the Fourier Synthesizer screen-shots is the small amplitudes of the harmonics for the triangle wave. The sliders used to control the amplitudes are linear, whereas the way that harmonics are typically shown in a spectrum analyser is on a log scale: as dBs. 

The spectrum above shows this quite nicely (plus some other interesting things as well, most of which are because this isn't a 'real' triangle wave, but one that I constructed inside Audacity). The fundamental frequency of the waveform is 50Hz and goes higher than the 0dB reference level (maybe +5dB), and the 3rd harmonic at 150 Hz is at about -24 dB which translates to -19 dB when you add that +5dB. But the Fourier Synthesizer showed this as 0.11 on the linear scale. It turns out that -19dB is a voltage ratio of about 0.11. Thhe 5th harmonic is at -33dB, which is -28dB when you add the +5 dB, and this is a voltage ratio of 0.04, which matches the Fourier Synthesizer value of 0.04. The 7th harmonic is -41dB, which becomes -36 dB which is  0.016, and the Fourier Synthesizer has 0.02. 

So the spectrum analyser harmonic levels are nice numbers, whereas the Fourier Synthesizer harmonics amplitudes are small numbers. I much prefer the spectrum analyser log scale, and here's a chart that shows how the spectrum analyser dBs down relates to the Fourier Synthesizer slider value:

Note that 0.5, which means that you set the Fourier Synthesizer slider to the half way point, corresponds to -6dB. Anything below -40dB is a slider value of below 0.01, which means moving that slider by 1 hundredth of its travel, which is a small distance. This table kind of reinforces why the 40dB Rule mentioned in part 1 of this series exists - the slider values are just tiny, and this means that the harmoncs are going to be tiny too. Probably too small to be seen on a screen!

So if I was going to be designing a Fourier Synthesizer, or an Additive Synthesizer, then I wouldn't use the slider values, because they are typically small and are going to be hard to set easily. Instead I would use the dB values, which are simple numbers and are going to be much easier to set correctly.

Conclusions (so far)

From this series so far, there's quite a lot of things that we now know about waveforms and spectrums:

- Anything below -40dB is going to be difficult to see on a screen

- A single cycle waveform might have unusual harmonics in it

- The waveform does not always tell you what harmonics are present

- The spectrum always tells you what harmonics are present

- Spectra can be better diagnostics than waveforms 

- Phase is important for waveshapes, but not what they sound like

- Your ears can only hear changes of phase

- Controlling the level of harmonics should use a log (dB) scale

In the next part, I'm going to talk about noise in single cycle waveforms, and why it doesn't do what you might expect.

---

If you find my writing helpful, informative or entertaining, then please consider visiting this link:

Buy me a coffee (Encourage me to write more posts like this one!)

Synthesizerwriter's Store (New 'Modular thinking' designs now available!)