Understanding the Vectorscope or Audio Goniometer
This scope based meter can be used to get an idea of general stereo spread and phasing differences between left and right channel audio mixes.
Apple Logic Pro has one as does FLS via Wave Candy, when the Vectorscope tab is clicked (Rclick, Settings):
So what does it represent and how does it function?
The basic idea comes from a comparison of vector functions of two waveforms that are displayed simultaneously on an oscilloscope, using the X/Y function control, that most dual beam oscilloscopes should have – usually by an “X/Y” button, or a position on the time base selector switch.
So, first – what is vector? There are MANY types across many scientific disciplines, but for this article’s purpose, I’m interested in a very simple type in 2D Euclidean, flat plane geometry.
Google states:
Noun

Being familiar with an oscilloscope in normal operation, where the Y axis represents amplitude (magnitude in vector terms) and the X axis represents time, we know that the simplest cyclical event or oscillation based on a constant circular rotation will be a sine wave, which you should be very familiar with by now:
If we set up Synthmaker as usual with a Test Tone circuit with 2 oscillators – one for Left and one for Right stereo, something like:
The dual beam oscilloscope can be set to view both at the same time – via the X and Y channel inputs by taking the LR outputs from a stereo cable, where the tip should be Left, the centre ring Right and the Earth is common to both channels.
To change the frequencies, we add a vector knob:
And use its output to controls both oscillators. I add one each later. I have a 1Hz minimum to 1kH max:
Using mini croc clips you should be able to see the same “In Phase” waveforms on the real Scope as Wave Candy displays:
Now watch when the XY button is pressed:
So what does this indicate?
If you think of the sine wave of each channel in normal view, it is varying with time across the horizontal X axis. If you imagine looking at one wave from the side, all you would see is the dot moving up and down only, with a complete cycle being completed when the dot has travelled from any start point you like – the maximum height say (peak) – moving down through the centre line position, to the lowest negative point of the wave (trough) and back to maximum peak to complete the cycle.
When the XY button is pressed, the waves are set relative to each other, but with the Y axis moving the dot up and down in a vertical plane, and the X axis moving the dot to and fro in the horizontal plane. The combination of the two directions being superimposed at the same time causes the dot to move overall in both the X and Y directions equally between both – at 45 degrees anti clockwise from the (easterly convention) X axis.
The displacement of the dot is the combination of each relative X and Y magnitude (distance), and their different directions (90 degree separation) – its Vector. Below, the maths is for the condition when both planes are moving from 0 to maximum only.
Mathematically, this direction is 45 degrees from horizontal, and its magnitude can be worked out by Pythagoras’ Theorem or trigonometry.
http://en.wikipedia.org/wiki/Trigonometric_functions
http://en.wikipedia.org/wiki/Pythagorean_theorem
As both X and Y are equal in this case – say 1 unit measurement (whether Volts, cm etc.) The dot moves 1 unit up and 1 across, with its resultant in red, which denotes the hypotenuse (long side) of a right angled triangle:
From Pythagoras, 1 squared + 1 squared = 2 squared so hypotenuse = 1.4142135623730950488016887242097 units. The square root of 2.
Or, from Trigonometry of a Right triangle:
As the sin 45 degrees is the opposite/hypotenuse, or 1/sin45 = hypotenuse = 1/0.70710678118654752440084436210485 = 1.4142135623730950488016887242097 units also.
The Vector of the dot has a magnitude of 1.414 units in a 45 degree angle direction above easterly horizontal – the red arrow above.
To account for full travel of the dot for both positive AND negative half cycles of the waves, we would have a negative component of the same value, in the opposite direction so overall, the dot cycles between the two extremes:
Also, this exercise tells me that my oscilloscope needs calibrating, as when no signals are present in X/Y mode the dot should be stationary right in the centre of the grid – or with signals, the line should be central – which it is not, as you see in the photo above.
All good so far, but so what!?
Well, at this stage it infers that any dual signals – stereo left and right – if in phase with each other and of equal amplitude, will tend upwards toward the 45 degrees anticlockwise from an Easterly horizontal, and downwards 225 degrees anticlockwise from an Easterly horizontal.
Taking the logic further, if one signal was 180 out of phase (due to a phase button being pressed on equipment, microphone placement etc.) we would expect to see the vector move in a different direction. If the Y component is at maximum when the X component is at minimum and vice versa, in XY mode, we would have:
This may infer a phase issue between right and left channels.
There is an interesting visual display condition that arrives out of phase differences that are called Lissajou figures, when displayed on a scope.
http://en.wikipedia.org/wiki/Lissajou
As you know, a sine wave is created from circular motion, so the opposite is true, and when 2 equal sine signals are set 90 out of phase, a perfect circle can be observed. To achieve this, we can add another vector knob to the Synthmaker circuit attached to the Phase input of one oscillator, and set its minimum movement to 1 and maximum to 2, so at ½ way, the Phase will be 180 out, and ¼ way round it should be 90 degrees out. You can experiment with values to get what control you require with your kit:
Anything more or less than 90 degrees will result in the circle becoming more elliptical in shape – you may be able to move in steps, so you go from a straight angled line (full in or out of phase) to a perfect circle.
Another interesting view is if you change one of the oscillators to be slightly higher in frequency than the other whilst 90 degrees out – say 200Hz an 201Hz – this causes one wave to drift in and out of sync with the other which, can be seen in normal dual scope view, but when viewed in XY mode causes the circle to look like it is spinning at an angle.
If one waveform is twice the frequency of the other, 2 facets or a figure 8 is shown, and this can be made to look like it rotates also if one frequency is slightly more than a multiple of the other as before.
If you choose say, 100 and 300Hz you get a 3 face shape etc. Try with different frequency multiples like 400:300, 500:400 etc. To make them spin, add 1 Hz, to say, 501Hz.
3 Rotating Rings video
How the circle is produced from 90 degree phased sines
Synthmaker Setup
I found a setup of 3 vector knobs worked well. I set up each that is connected to each oscillator’s frequency connections at 200Hz to start experimenting. The 3^{rd} knob is for changing the phase angle of one waveform relative to the other. This was set to min1, max 2, and I could get a 90 degree shift with this. For the shape rotation I made one frequency 201Hz and multiples thereof – 301, 401, 601 etc.
There is a fundamental difference between how Wave Candy and a real oscilloscope handle the displayed images and I don’t know why that is at this stage. A stereo In Phase pair gets displayed as a vertical bar in Vectorscope – not at a 45 degree angle as on real scope, as below.
It also shows a 180 degree phase shift as a horizontal bar, not a 45 degree slope as a real slope:
It does, however, show a 90 shift as a circle – the same as a real scope!
I don’t know whether (or why) the FLS guys would program the software differently from the way a real scope works but it makes me glad I spent the research time to go from first principles for this topic, using a real scope first. It really messed up my initial attempts to do a summary for the overall purpose of understanding this tool though, when I found the Wave Candy view did not conform with my expectations!
This means that there is a 45 degree offset slope difference between views on a real scope and WC. Again, if one channel is turned down, on a real scope this would produce either a fully horizontal or vertical line, as there would be full amplitude in either the X or Y direction only, but in WC, it produces a positive or negative 45 degree slope:
With a real scope, a mainly In Phase piece of music would lean toward 45 degrees right of vertical, not at the vertical as WC does. A mainly out of phase piece would lean toward 45 degrees left of vertical, not toward the horizontal as WC does.
To find out what the effect of a difference in amplitude makes between the two waves and the Vectorscope view, I added a variable amp to each side, and changed the frequencies to 200Hz in line with the setup I described above that worked well visually for a real scope:
So, changing one of the channels to half the amplitude of the other:
The line shifts to about 60 degrees for a decrease of 50% volume of the left side, and slopes negative for a drop in the other channel instead.
For a real oscilloscope, I would expect an angle of about 30 degrees above horizontal if the left channel was the Y axis, from drawing the vector manually and calculating as above:
Calculate the inverse tangent:
Opp/Adj = tanA = tan a/b = 0.5/1, X = 0.5/tan = 26.565051177077989351572193720453:
I verified this angle visually with a real scope at 50% volume drop of 1 channel. If the other (X) channel is dropped 50% instead, the slope goes to about 30 degrees right of vertical as you would expect.
I’ll leave you to do the maths for this one.
The Lissajou figures
Can we replicate the same rotating patterns seen on the scope in Vectorscope? Yes!
Putting the amplitudes back to equal and setting one frequency to 201Hz we get an oscillation between horizontal and vertical lines that oscillates through ellipses and a circle:
Doubling the frequency of one to a double and a bit to 401Hz, we get 2 rotating loops, and changing view settings for clarity:
At 3 times right channel frequency (200Hz) we get 3 joined rotating loops– 601Hz
So what is the final point?
If you take a piece of music and play it you will see the complex interplay between all the instantaneous waveforms parameters of frequency, amplitude and phase between left and right channels. This generally shows the amount of phase and relative L/R amplitude (stereo dynamics) there is in a mix.
As I stated above, my initial attempts to summarise in this section were thwarted between the conflicting behaviour of a real scope and Vectorscope, which confused me, but as I explained it up there, it is the same concept just shown displaced at varying angles axis in Vectorscope, so a distorted shape in some views compared to a real scope, but still an analogue of the relative movements result.
“In” phase signals tend toward the vertical Y axis in Vectorscope and out of phase signals tend to the horizontal X axis, but at one positive and one negative 45 degree slopes for a real scope. 90 degree phase shifts tend toward circular motion, as can be seen in both Wave Candy and a real scope:
Amplitude differences between channels show as angles between horizontal and vertical, both between 45 degrees of positive and negative slopes for WC and a real scope.
If you set the left channel (cable tip) to the Y axis, then the more the signal tends to the 45 positive slope on a real scope, and the higher the volume of the right channel, the more to 45 negative slope for a real scope. Now you know all this, play some music through FLS and put Wave Candy on the Master channel, and see what you make of it.
The overall 45 degree offset direction between WC and a real scope when playing music.
There is a really well explained YouTube video explaining Lissajou concepts, with a real scope and digital sig gens here:
How Lissajous are created with mechanical pendulums