One of the things on my to-do list today was to get a Bang & Olufsen Stereopladespiller Type 42 up and running. Unfortunately, I didn’t have a stroboscopic disc for testing the speed. Since a quick search on the Internet didn’t turn up anything I liked, I decided to make my own.If you’d like to download it, it’s available here as a PDF file for A4 paper, and contains the lines for 50 Hz and 60 Hz mains. You can change the magnification to make it fit on different paper sizes, or to increase or decrease the size of the disc. If your magnification is the same in the X and Y axes, then it won’t change anything.
This meant that I had to do a little math, which goes as follows:
mains_frequency = 50 Hz (this is the rate at which the lights blink)
So, here in Denmark where we have 50 Hz mains, I needed to make a disc with a line every 4º. Since I use a Mac, I used graphic.app to do this, but any decent drawing program will do the trick.
If you want to make your own disc, and you don’t want to do the math, here are the results of the possible mains frequencies and revolution speeds
For anyone who knows a thing or two about the Type 42… then I’m already ahead of you. I know that the lines are built into the turntable mat itself. However, I was working in pretty bright daylight, and so I needed more contrast on the lines to be able to see the interference from the lighting. And besides, it was fun as a little light recreational math.
It should not come as a surprise that, when we talk about how a vinyl record works, we can start by looking at the movement of the needle in the groove. If we simplify that connection a little (by reducing the audio signal to one channel, but we’ll come back to that point later), then we can think of this as a needle, sitting on a surface. The needle is at the end of an arm that we call the “cantilever” (because it is fixed on one end and it can move up and down on the other end where the needle is attached) and that cantilever is attached somehow to the tonearm using a springy material of some kind (like rubber, for example).
The simple diagram above shows that arrangement. Of course, I’ve left out a bunch of things, and nothing is to scale, but those details are not important right now.
I’ll make the “spring” in this diagram out of flexible rubber that has some “springiness” or “compliance”. The more compliant the spring, the easier it is to flex. So a stiff spring in not very compliant. (This concept is very important to understand as we go on.)
The audio signal is “encoded” into the surface of the vinyl using bumps and dips that cause the needle to move up and down. I’ve shown this in the simple diagram below.
Notice in that diagram that the needle is in contact with the surface of the vinyl, but the part of the system that connects back to the tonearm has not lifted. This is because the connection between the cantilever and the tonearm assembly is compliant enough to let the cantilever move upwards (or downwards) without moving the rest of the system.
Think of this like driving over a very small bump in the road in your car. The compliance of the tires and the shock absorbers will result in the tire riding over the bump, but the car doesn’t jump as a result.
Remember that the bump in the surface of the vinyl is only passing by, so the needle isn’t raised for long. As a result part of the reason the tonearm doesn’t move upwards (and your car doesn’t jump) is partly because it’s heavy. Its mass results in an inertia that “wants” to stop it from moving up and down. (The other factor that’s involved here is an adjustment in the tonearm called the “tracking force” which is a measurement of how much the tonearm is pushing downwards on the needle.)
Consequently, when that bump comes along, the needle rides on top of it, and the force that is pushing it downwards comes mostly from the “spring” at the other end of the cantilever, as shown below.
If the spring had no compliance (in other words, if it weren’t a spring, and the cantilever were just connected directly to the tonearm) and if the cantilever and needle were strong enough to take the force, then the entire tonearm assembly would jump up and down instead, as shown below. (Imagine riding in a horse-drawn buggy with wooden wheels with steel rims, and no springs on the axles. You’d feel every single rock on the road…)
The tonearm is resting on two points: one is the tip of the needle and the other is at the other end at the pivot point where it also rotates horizontally as you play the album. If we were really dumb turntable designers, then half of the mass of the tonearm would be resting on the needle (and the other half would be resting on the pivot). This would be bad, since your records would wear out very fast. So, a tonearm has some kind of adjustment on it that reduces the amount of weight on the needle. The simplest way to do this is to put a counterweight on the opposite side of the pivot so it’s more like a see-saw at the playground. As you move the counterweight away from the pickup, the downwards force at the needle gets smaller. In fact, you can probably adjust the counterweight so far that the needle-end of the tonearm is lighter, and it is stuck up in the air…
We adjust the amount of downwards force at the needle (called the “tracking force”) to result in a value that is in balance with the compliance of the connection to the cantilever. If the tracking force is too high (or the compliance is too high for the tracking force) then the tonearm will sink like I’ve shown below.
There are lots of things wrong with this. The first is that the needle isn’t at the correct angle to the surface of the vinyl, so it’s not going to move correctly. The second is that the cantilever is at the wrong angle, so it’s not going to move upwards with the same behaviour as it moves downwards, which results in an asymmetrical distortion of the signal. But possibly the most obvious problem is that there’s just too much downwards pressure on the vinyl, so your records will wear out faster.
So, there is a balance between the tracking force and the compliance. That balance ensures that you always have contact between the tip of the needle and the surface of the vinyl as the bumps and dips go by.
Digging into the details
One of the things I do regularly is to measure the magnitude response of a turntable from the surface of the vinyl to the electrical output of the RIAA preamplifier. In order to do this, I play two tracks on a special test record (Brüel & Kjær QR 2010) which has the following audio signals:
20 Hz to 45 kHz sinusoidal tone, log sweep, 5 sec per decade, Right channel
Sometimes (but very rarely), I notice that the needle will skip (or jump) at the transition between the 1 kHz tone and the start of the sine sweep. If this happens, for track 1, the needle will skip forwards into the sweep.
When this happened the first time I thought “Ah hah! The tracking force isn’t high enough, so the needle is being thrown out of the groove. I just need to adjust it.” But after checking the tracking force with my meter (a very small, very precise and accurate scale), I found out that this was not the problem.
Of course, I could make the problem go away by increasing the tracking force, but then it was too high, and my records (and the needle tip) will wear down faster. This would be covering up the symptom, but not correcting the actual problem.
So, what is the problem? It’s that the compliance of the pickup is too low due to an error in the manufacturing process or the fact that it’s just old and the rubber has stiffened over time. In other words it looks more like the system shown in Figure 4, above.
Let’s take a system where the pickup compliance is too low (so the spring is too stiff), so the tonearm can be tossed up off the vinyl surface. We then combine that with the knowledge of how the needle sits in the groove on the vinyl and which channel is on which side of that groove (which I’ve shown below in Figure 6).
Now we can see that, if there’s a bump in the Left channel, it will push the needle on a 45º angle upwards, and if the tracking force and compliance aren’t working together as they should, then the entire tonearm can be pushed hard enough to cause the needle to lift off the surface of the vinyl, heading in towards the centre of the record (towards the left in Figure 6).
What does the signal actually look like?
Let’s go back and look at a recording of that transition between the 1 kHz tone and the start of the 20 Hz sweep, using a pickup that is behaving properly.
The figure above is a screenshot from Audacity that shows the “raw” signal that I recorded at the input of my sound card which is connected to the output of the RIAA preamplifier. I’ve zoomed in to the moment when the track transitions from the 1 kHz tone to the 20 Hz tone at the start of the sweep.
Let’s now use this to go backwards and try to figure out what the surface of the vinyl looks like. I’ll start by re-creating a “perfect” version of that signal in Matlab by joining a 1 kHz cosine wave to a 20 Hz cosine wave.
You might notice that I’ve changed the value a little. I’m simulating one channel of a tone that has a level of at 5 cm/sec, RMS lateral velocity for two channels, instead of the 3.16 cm/sec from the B&K record. But this doesn’t really matter too much – I’ve just done it to make the numbers look nice and be a little easier to talk about.
I’m simulating a system that has a total gain set so that a modulation velocity of 3.54 cm/sec in one channel will produce 354 mV RMS (500 mV peak) at the output of the RIAA at 1 kHz.
Since the lateral velocity of a two-channel tone is 5 cm/sec, then the velocity of one channel will be 1/sqrt(2) of that value because the groove wall is 45º away from the lateral axis and cos(45º) = 1/sqrt(2).
If we take the signal in Figure 8 and filter it with a RIAA pre-emphasis filter (sometime called an “anti-RIAA” or an “inverse RIAA”) and drop the level by 40 dB (a typical gain for a RIAA preamp), then the signal looks like the plot in Figure 9.
As you can see there, the signal much lower in level overall (because of the -40 dB gain) and the 20 Hz tone is much lower in level than the 1 kHz tone (because of the pre-emphasis filter).
The output of the pickup is a current that is proportional to the velocity of the needle. So, we can move farther backwards in the chain and plot the velocity of the needle over time, shown in Figure 10. As you can see, the shape of this plot looks identical to the one in Figure 9. This is because I’m assuming that the current output of the pickup is in phase with the voltage at the input of the RIAA. (This is a safe assumption for the two frequencies we’re looking at here. If you want to pick a fight with me about this, drop by and do it in person. But you’re buying the beer…)
Now comes a jump… the velocity of the needle can be calculated by finding the derivative of the displacement over time, which means that the displacement can be found by integrating the velocity.
If you don’t like calculus, then you can think of it this way: In the old days, if you drove from Struer to Copenhagen, you had to take a ferry to get from the island of Fyn to the island of Zealand. Every once in a while, there would be a policeperson, walking around the parking lot as people waited to board the ferry, handing out speeding tickets to some of the people there. What happened was that the licence plates were recorded with time stamps as they crossed the bridge to Fyn from Jutland – which is about 75 km away from the parking lot. If you arrive at the ferry too early, you must have been speeding, and you get rewarded with an earlier ferry, and an extra charge…
In other words, you can calculate your speed (velocity) by your change (difference) of distance (displacement) over time.
You can also do this backwards: if you know how fast you’re going, you can calculate your displacement over time (you’ll be 100 km away in an hour if you’re driving 100 km/h the whole time, for example). If your velocity changes over time (say you drive a different speed every hour for 10 hours), then you can still calculate your displacement by dividing time into slices (in this case, 1 hour per “slice”) and adding up the individual displacements for the velocity you had during each slice of time. If you divide time into infinitely short slices, then you are integrating instead of adding, but the process is essentially the same.
Back to the story: if we take the signal in Figure 10 and integrate it (and scale it – which isn’t really important for this discussion), we get the curve in Figure 11.
This gives us a good idea of the actual shape of the left wall of the groove in the vinyl for that particular signal.
So, as you can see there, if the connection between the cantilever and the pickup doesn’t have a high enough compliance, it’s no wonder that the needle gets thrown out of the record groove. That’s a heck of a bump to deal with! To be honest, it’s also a little amazing to me that the needle that’s behaving (like the one that produced the output shown in Figure 7) can actually put up with that kind of abuse.
(Special thanks to Jakob Dyreby for helping me to wrap my head around the simulation part of this posting. I did the math, but only after he pointed me in the right direction.)
Every once in a while, someone will send me a link to a YouTube page that shows an electron microscope “video” of a needle tracking a groove in a vinyl record. If you listen to the explanation of that video, he explains that it’s not really a video. It’s a series of photographs that he took, one by one, and then assembled into a video.
This means that, in that video, the needle isn’t really behaving like it does in real life when the vinyl is moving underneath it.
Imagine setting up a video camera on the side of the road, next to a small speed bump, and making a video of a car driving over it. You’d see that, as the car drives by, the wheels move up into the wheel wells and the car doesn’t get pushed upwards as much, since some of the vertical movement caused by the speed bump is “taken up” by the car’s springs and shock absorbers.
If, instead, you set up a camera, and got the car to move forwards 5 cm and stop – and you take a photo, then the car moves forwards another 5 cm and stops – and you take another photo, and the you repeat this until the car is out of the frame – and then you assemble all of those photos into a video, it would look very different. The car would not remain horizontal when the wheels are on the speed bump because the springs and shock absorbers wouldn’t be compressed at all.
That video is like the second “video” of the car. Of course, it’s still interesting, and it’s well-explained, so no one is playing any tricks on you. But it’s not a video of what actually happens…
This article, from The Gramophone magazine, August 1932 foretells the future of turntables with platters driven by electric motors. Note that, to test this particular one, they increased what we would today call the “tracking force” to 3.5 pounds (about 1.6 kg) on the outside groove of a 10″ record without reducing the speed. Try that on a turntable today…
Sad to see a familiar mantra here though: “the motor is remarkably efficient, very well made and ridiculously expensive.”
What caught my eye was the discussion of gramophone needles made of “hard wood”, and also the prediction that “the growth of electrical recording steps … to grapple with that problem of wear and tear.”
The fact that electrical (instead of mechanical) recording and playback was seen as a solution to “wear and tear” reminded me of my first textbook in Sound Recording where “Digital Audio” was introduced only within the chapter on Noise Reduction.
Later in that same issue, there is a little explanation of the “Electrocolor” and “Burmese” needles.
The March 1935 issue raises the point of wear vs. fidelity in the Editorial (which starts by comparing players with over-sized horns).
I like the comment about having to be in the “right mood” for Ravel. Some things never change.
What’s funny is that, now that I’ve seen this, I can’t NOT see it. There are advertisements for fibre, thorn, and wood needles all over the place in 1930s audio magazines.
This is just a collection of information about turntables and vinyl for anyone wanting to dig deeper into It (which might mean that it’s just for me…). I’ll keep adding to this (and completing it) as time goes by.
The rod or arm that connects the stylus on one end to the “motor” on the other.
The straight-line distance between the pivot point of the tonearm and the top of the stylus
definition to come
Higher-frequency modulation of the audio frequency caused by changes in the groove speed. These may be the result of changes in problems such as unstable motor speed, variable compliance on a belt, issues with a spindle bearing, drive wheel eccentricity, and other issues.
Flutter describes a modulation in the groove speed ranging from 6 to 100 times a second (6 Hz to 100 Hz).
Very long-term (or low-frequency) changes in the audio frequency, typically caused by slow changes in the platter rotation speed.
Typically, changes with a modulation frequency of less than 0.5 Hz (a period of no less than 2 seconds) are considered to be frequency drift. Faster changes are labelled “Wow”
The v-shaped track pressed into the surface of the vinyl record, in which the stylus sits
A tonearm that moves linearly, following a path that is parallel to the radius line traced by the stylus. This (in theory) ensures that the tracking error is always 0º, however, in practice this error is merely small.
The distance measured on a line through the spindle from the start of the modulated groove to the end of the modulated groove. This is approximately 3″ or 76 mm.
The distance between the spindle and the pivot point of the tonearm.
Also known as the stylus. The point that is placed in the groove of the vinyl record. Some persons distinguish between the “stylus” (to indicate the chisel on the mastering lathe that creates the groove in the master record), and the “needle” (to indicate the portion of the pickup on a turntable that plays the signal).
The radius (distance between the spindle and the stylus) where the tracking error is 0º. A typically-designed and correctly installed radial tracking tonearm has two null radii (see this posting).
The angle between the axis of the stylus and a line drawn between the tonearm pivot and the stylus. See the line diagram below.
The difference between the Effective Length of the tonearm and the Mounting Distance. This value is used in some equations for calculating the Tracking Error.
Includes three general types: Moving Coil, Moving Magnet, and Variable Reluctance (aka Moving Iron). These produce an output proportional to the velocity of the stylus movement.
Produces an output proportional to the displacement of the stylus.
The density of the groove count per distance in lines per inch or lines per mm. The pitch can vary from disc to disc, or even within a single track, according to the requirements of the mastering.
A tonearm that rotates on a pivot point with the stylus tracing a circular path around that pivot.
The distance between the centre of the vinyl disc and the pickup stylus.
A pre-emphasis / de-emphasis filter designed to fill two functions.
The first is a high-frequency attenuation de-emphasis that reduces the playback system’s sensitivity to surface noise. This requires a reciprocal high-frequency pre-emphasis boost.
The second is a low-frequency attenuation pre-emphasis that maintains a constant modulation amplitude at lower frequencies to avoid over-excursion of the playback stylus. This requires a reciprocal low-frequency de-emphasis boost.
The first of the two plots below, show the theoretical (black lines) and typical (red) response of the pre-emphasis filter. The second of the two plots shows the de-emphasis filter response.
definition to come
definition to come
The centre of the platter around which the record rotates
Also known as the needle. The point that is placed in the groove of the vinyl record. Some persons distinguish between the “stylus” (to indicate the chisel on the mastering lathe that creates the groove in the master record), and the “needle” (to indicate the portion of the pickup on a turntable that plays the signal).
Stylus, Bonded vs. Nude
Although the tip of the stylus is typically made of diamond today, in lower-cost units, that diamond tip is mounted or bonded to a metal pin (typically steel, aluminium, or titanium) which is, in turn, connected to the cantilever (the long “arm” that connects back to the cartridge housing). This bonded design is cheaper to manufacture, but it results in a high mass at the stylus tip, which means that it will not move easily at high frequencies.
In order to reduce mass, the metal pin is eliminated, and the entire stylus is made of diamond instead. This makes things more costly, but reduces the mass dramatically, so it is preferred if the goal is higher sound performance. This design is known as a nude stylus.
The angle between the tangent to the groove and the alignment of the stylus. In a perfect system, the stylus would align with the tangent to the groove at all radii (distances from the spindle), since this matches the angular rotation of the cutting head when the master was made on a lathe. A linear tracking arm minimises this error. A radial tracking arm can be designed to have two radii with no tracking error (each called a “Null Radius”) but will have some measurable tracking error at all other locations on the disk.
One side-effect of tracking error is distortion of the audio signal, typically calculated and expressed as a 2nd-harmonic distortion on a sinusoidal audio signal. However, higher order distortion and intermodulation artefacts also exist.
A modulation of the frequency of the audio signal caused by vertical changes in the vinyl surface (a warped record). This typically happens at a lower frequency, which is why it is “warp wow” and not “warp flutter”.
Low-frequency modulation of the audio frequency caused by changes in the groove speed. These may be the result of changes in problems such as rotation speed of the platter, discs with an incorrectly-placed centre hole, or vertical changes in the surface of the vinyl, and other issues.
Wow is a modulation in the groove speed ranging from once every 2 seconds to 6 times a second (0.5 Hz to 6 Hz). Note that, for a turntable, the rotational speed of the disc is within this range. (At 33 1/3 RPM: 1 revolution every 1.8 seconds is equal to approximately 0.556 Hz.)
Disk size limits
Outside starting diameter
6.78″, +0.06″, -0.00″
172.2 mm, +1.524 mm, – 0.0 mm
9.72″, +0.06″, -0.00″
246.9 mm, +1.524 mm, – 0.0 mm
11.72″, +0.06″, -0.00″
297.7 mm, +1.524 mm, – 0.0 mm
Start of modulated pitch
6.63″, +0.00″, -0.03″
168.4 mm, +0.0 mm, – 0.762 mm
9.50″, +0.00″, -0.03″
241.3 mm, +0.0 mm, – 0.762 mm
11.50″, +0.00″, -0.03″
292.1 mm, +0.0 mm, – 0.762 mm
Minimum inside diameter
Lockout Groove diameter
3.88″, +0.00, -0.08
98.552 mm, +0.0 mm, -2.032 mm
4.19″, +0.00, -0.08
106.426 mm, +0.0 mm, -0.762 mm
4.19″, +0.00, -0.08
106.426 mm, +0.0 mm, -0.762 mm
Unmodulated (silent) groove width
2 mil minimum, 4 mil maximum
0.0508 mm minimum, 0.1016 mm maximum
Modulated groove depth
1 mil minimum, 5 mil maximum
0.0254 mm minimum, 0.127 mm maximum
The figure below shows the typical, minimum, and maximum groove depths, drawn to scale (with a 13 µm spherical stylus)
A typical standard reference level is a velocity of 35.4 mm/sec on one channel.
This means that a monophonic signal (identical signal in both channels) with that modulation will have a lateral (side-to-side) velocity of 50 mm/sec.
Typically measured with a 3150 Hz sinusoidal tone, played from the vinyl surface
This signal is then de-modulated to determine its change over time. That modulation is then filtered through the response shown below which approximates human sensitivity to frequency modulation of an audio signal. More detailed information is given below
The AES6-2008 standard, which is the currently accepted method of measuring and expressing the wow and flutter specification, uses a “2σ” or “2-Sigma” method, which is a way of looking at the peak deviation to give a kind of “worst-case” scenario. In this method, the tone is played from a disc and captured for as long a time as is possible (or feasible). Firstly, the average value of the actual frequency of the output is found (in theory, it’s fixed at 3,150 Hz, but this is never true). Next, the short-term variation of the actual frequency over time is compared to the average, and weighted using the filter shown above. The result shows the instantaneous frequency variations over the length of the captured signal, relative to the average frequency (however, the effect of very slow and very fast changes have been reduced by the filter). Finally, the standard deviation of the variation from the average is calculated, and multiplied by 2 (hence “2-Sigma”, or “two times the standard deviation”), resulting in the value that is shown as the specification. The reason two standard deviations is chosen is that (in the typical case where the deviation has a Gaussian distribution) the actual Wow & Flutter value should exceed this value no more than 5% of the time.
All of these are available online. Some of them require you to purchase them (or be a member of an organisation).
“Tracking Angle in Phonograph Pickups” B. B. Bauer. Electronics magazine, March 1945
“Minimising Pickup Tracking Error” Dr. John D. Seagrave, Audiocraft Magazine, December 1956, January 1957, and August 1957
“Understanding Phono Cartridges” S.K. Pramanik, Audio magazine, March 1979
“Tonearm Geometry and Setup Demystified” Martin D. Kessler and B.V.Pisha, Audio magazine, January 1980
“Understanding Tonearms” S.K. Pramanik, Audio magazine, June 1980
“Analytic Treatment of Tracking Error and Notes on Optimal Pick-up Design” H.G.Baerwald, Journal of the Society of Motion Picture Engineers, December 1941
“Pickup Arm Design” J.K. Stevenson, Wireless World magazine, May 1966, and June 1966
“The Optimum Pivot Position on Tonearm” S. Takahashi et. al., Audio Engineering Society Preprint no. 1390 (61st Convention, November 1978)
“Audible Effects of Mechanical Resonances in Turntables” Brüel and Kjær Application Note (1977)
“Basic Disc Mastering”; “ Larry Boden (1981)
“Cartridge / Arm / Turntable Followup: Loose Ends and New Developments” The Audio Critic, 1:43 (Spring/Fall, 1978)
“Have Tone Arm Designers Forgotten Their High-School Geometry?” The Audio Critic, 1:31 (Jan./Feb. 1977).
“How the Stereo Disc Works” Radio-Electronics, (July 1958)
“Manual of Analogue Sound Restoration Techniques” Peter Copeland (2008)
“On the Mechanics of Tonearms” Dick Pierce (2005)
“Reproduction of Sound in High-Fidelity and Stereo Phonographs” Edgar Villchur (1966)
Journal of the Audio Engineering Society (www.aes.org)
“Centennial Issue: The Phonograph and Sound Recording After One-Hundred Years” Vol. 25, No. 10/11 (Oct./Nov. 1977)
“Factors Affecting the Stylus / Groove Relationship in Phonograph Playback Systems” C.R. Bastiaans; Vol. 15 Issue 4 (Oct. 1967)
“Further Thoughts on Geometric Conditions in the Cutting and Playing of Stereo Disk” C.R. Bastiaans; Vol. 11 Issue 1 (Jan. 1963)
“Record Changers, Turntables, and Tone Arms-A Brief Technical History” James H. Kogen; Vol. 25 (Oct./Nov. 1977)
“Some Thoughts on Geometric Conditions in the Cutting and Playing of Stereodiscs and Their Influence on the Final Sound Picture” Ooms, Johan L., Bastiaans, C. R.; Vol. 7 Issue 3 (Jul. 1959)
In the last posting, I reviewed the math for calculating the tracking error for a radial tonearm. The question associated with this is “who cares?”
In the March, 1945 issue of Electronics Magazine, Benjamin Bauer supplied the answer. An error in the tracking angle results in a distortion of the audio signal. (This was also discussed in a 3-part article by Dr. John D. Seagrave in Audiocraft Magazine in December 1956, January 1957, and August 1957)
If the signal is a sine wave, then the distortion is almost entirely 2nd-order (meaning that you get the sine wave fundamental, plus one octave above it). If the signal is not a sine wave, then things are more complicated, so I will not discuss this.
Let’s take a quick look at how the signal is distorted. An example of this is shown below.
In that plot, you can see that the actual output from the stylus with a tracking error (the black curve) precedes the theoretical output that’s actually on the vinyl surface (the red curve) when the signal is positive, and lags when it’s negative. An intuitive way of thinking of this to consider the tracking error as an angular rotation, so the stylus “reads” the signal in the groove at the wrong place. This is shown below, which is merely zooming in on the figure above.
Here, you can see that the rotation (tracking error) of the stylus is getting its output from the wrong place in the groove and therefore has the wrong output at any given moment. However, the amount by which it’s wrong is dependent not only on the tracking error but the amplitude of the signal. When the signal is at 0, then the error is also 0. This is not only the reason why the distortion creates a harmonic of the sine wave, but it also explains why (as we’ll see below) the level of distortion is dependent on the level of the signal.
This intuitive explanation is helpful, but life is unfortunately, more complicated. This is because (as we saw in the previous posting), the tracking error is not constant; it changes according to where the stylus is on the surface of the vinyl.
If you dig into Bauer’s article, you’ll find a bunch of equations to help you calculate how bad things get. There are some minor hurdles to overcome, however. Since he was writing in the USA in 1945, his reference was 78 RPM records and his examples are all in inches. However, if you spend some time, you can convert this to something more useful. Or, you could just trust me and use the information below.
In the case of a sinusoidal signal the level of the 2nd harmonic distortion (in percent) can be calculated with the following equation:
PercentDistortion = 100 * (ω Αpeak α) / (ωr r)
ω is 2 * pi * the audio frequency in Hz
Apeak is the peak amplitude of the modulation (the “height” of the groove) in mm
α is the tracking error in radians
ωr is the rotational speed of the record in radians per second, calculated using 2 * pi * (RPM / 60)
r is the radius of the groove; the distance from the centre spindle to the stylus in mm
Let’s invent a case where you have a constant tracking error of 1º, with a rotational speed of 33 1/3 RPM, and a frequency of 1 kHz. Even though the tracking error remains constant, the signal’s distortion will change as the needle moves across the surface of the record because the wavelength of the signal on the vinyl surface changes (the rotational speed is the same, but the circumference is bigger at the outside edge of the record than the inside edge). The amount of error increases as the wavelength gets smaller, so the distortion is worse as you get closer to the centre of the record. This can be see in the shapes of the curves in the plot below. (Remember that, as you play the record, the needle is moving from right to left in those plots.)
You can also see in those plots that the percentage of distortion changes significantly with the amplitude of the signal. In this case, I’ve calculated using three different modulation velocities. The middle plot is 35.4 mm / sec, which is a typical accepted standard reference level, which we’ll call 0 dB. The other two plots have modulation velocities of -3 dB (25 mm / sec) and + 3 dB (50 mm / sec).
Sidebar: If you want to calculate the Amplitude of the modulation
Note that this simplifies the equation for calculating the distortion somewhat.
Also, if you need to convert radians to degrees, then you can multiply by 180/pi. (about 57.3)
Of course, unless you have a very badly-constructed linear tracking turntable, you will never have a constant tracking error. The tracking error of a radial tonearm is a little more complicated. Using the recommended values for the “well known tonearm” that I used in the last posting:
Effective Length (l) : 233.20 mm
Mounting Distance (d) : 215.50 mm
Offset angle (y) : 23.63º
and assuming that this was done perfectly, we get the following result for a 33 1/3 RPM album.
You can see here that the distortion drops to 0% when the tracking error is 0º, which (in this case) happens at two radii (distances between the centre spindle and the stylus).
If we do exactly the same calculation at 45 RPM, you’ll see that the distortion level drops (because the value of ωr increases), as shown below. (But good luck finding a 12″ 45 RPM record… I only have two in my collection, and one of those is a test record.)
Everything I’ve shown above is not to be used as proof of anything. It’s merely to provide some intuitive understanding of the relationship between radial tracking tonearms, tracking error, and the resulting distortion. There is one additional important reason why all this should be taken with a grain of salt. Remember that the math that I’ve given above is for 78 RPM records in 1945. This means that they were for laterally-modulated monophonic grooves; not modern two-channel stereophonic grooves. This means that the math above isn’t accurate for a modern turntable, since the tracking error will be 45º off-axis to the axis of modulation of the groove wall. This rotation can be built into the math as a modification applied to the variable α, however, I’m not going to complicate things further today…
In addition, the RIAA equalisation curve didn’t get standardised until 1954 (although other pre-emphasis curves were being used in the 1940s). Strictly speaking, the inclusion of a pre-emphasis curve doesn’t really affect the math above, however, in real life, this equalisation makes it a little more complicated to find out what the modulation velocity (and therefore the amplitude) of the signal is, since it adds a frequency-dependent scaling factor on things. On the down-side, RIAA pre-emphasis will increase the modulation velocity of the signal on the vinyl, resulting in an increase in the distortion effects caused by tracking error. On the up-side, the RIAA de-emphasis filtering is applied not only to the fundamentals, but the distortion components as well, so the higher the order of the unwanted harmonics, the more they’ll be attenuated by the RIAA filtering. How much these two effects negate each other could be the subject of a future posting; if I can wrap my own head around the problem…
One extra comment for the truly geeky:
You may be looking at the last two plots above and being confused in the same way that I was when I made them the first time. If you look at the equation, you can see that the PercentDistortion is related to α: the tracking error. However, if you look at the plots, you’ll see that I’ve shown it as being related to | α |: the absolute value of the tracking error instead. This took me a while to deal with, since my first versions of the plots were showing a negative value for the distortion. “How can a negative tracking result in distortion being removed?” I asked myself. The answer is that it doesn’t. When the tracking error is negative, then the angle shown in the second figure above rotates counter-clockwise to the left of the vertical line. In this case, then the output of the stylus lags for positive values and precedes for negative values (opposite to the example I gave above), meaning that the 2nd-order harmonic flips in polarity. SINCE you cannot compare the phase of two sine tones that do not have the same frequency, and SINCE (for these small levels of distortion) it’ll sound the same regardless of the polarity of the 2nd-order harmonic, and SINCE (in real-life) we don’t listen to sine tones so we get higher-order THN and IMD artefacts, not just a frequency doubling, THEN I chose to simplify things and use the absolute value. Post Script to the comment for geeks: This conclusion was confirmed by J.K. Stevenson’s article called “Pickup Arm Design” in the May, 1966 edition of Wireless World where he states “The sign of φ (positive or negative) is ignored as it has no effect on the distortion.” (He uses φ to denote the tracking error angle.)
Penultimate Post Script:
J.K. Stevenson’s article gives an alternative way of calculating the 2nd order harmonic distortion that gives the same results. However, if you are like me, then you think in modulation velocity instead of amplitude, so it’s easier to not convert on the way through. This version of the equation is
PercentDistortion = 100 * (Vpeak tan(α)) / (μ)
Vpeak is the peak modulation velocity in mm/sec
α is the tracking error in radians
μ is the groove speed of the record in mm/sec calculated using 2*pi*(rpm/60)*r
r is the radius of the groove; the distance from the centre spindle to the stylus in mm
I’ve given this a lot of thought over the past couple of days and I’m pretty convinced that, since the tracking error is a rotation angle on an axis that is 45º away from the axis of modulation of the stylus (unlike the assumption that we’re dealing with a monophonic laterally-modulated groove in all of the above math), then, to find the distortion for a single channel of a stereophonic groove, you should multiply the results above by cos(45º) or 1/sqrt(2) or 0.707 – whichever you prefer. If you are convinced that this was the wrong thing to do, and you can convince me that you’re right, I’ll be happy to change it to something else.
The June 1980 issue of Audio Magazine contains an article written by Subir K. Pramanik called “Understanding Tonearms”. This is a must-read tutorial for anyone who is interested in the design and behaviour of radial tonearms.
One of the things Pram talked about in that article concerned the already well-known relationship between tonearm geometry, its mounting position on the turntable, and the tracking error (the angular difference between the tangent to the groove and the cantilever axis – or the rotation of the stylus with respect to the groove). Since the tracking error is partly responsible for distortion of the audio signal, the goal is to minimise it as much as possible. However, without a linear-tracking system (or an infinitely long tonearm), it’s impossible to have a tracking error of 0º across the entire surface of a vinyl record.
One thing that is mentioned in the article is that “Small errors in the mounting distance from the centre of the platter … can make comparatively large differences in angular error” So I thought that I’d do a little math to find out this relationship.
The article contains the diagram shown below, showing the information required to do the calculations we’re interested in. In a high-end turntable, the Mounting Distance (d) can be varied, since the location of the tonearm’s bearing (the location of the pivot point) is adjustable, as can be seen in the photo above of an SME tonearm on a Micro Seiki turntable.
The tonearm’s Effective Length (l) and Offset Angle (y) are decided by the manufacturer (assuming that the pickup cartridge is mounted correctly). The Minimum and Maximum groove radius are set by international standards (I’ve rounded these to 60 mm and 149 mm respectively). The Radius (r) is the distance from the centre of the LP (the spindle) to the stylus at any given moment when playing the record.
In a perfect world, the tracking error would be 0º at all locations on the record (for all values of r from the Maximum to the Minimum groove radii) which would make the cantilever align with the tangent to the groove. However, since the tonearm rotates around the bearing, the tracking error is actually the angle x (in the diagram above) subtracted from the offset angle. “X” can be calculated using the equation:
x = asin ((l2 + r2 – d2) / (2 l r))
So the tracking error is
Tracking Error = y – asin ((l2 + r2 – d2) / (2 l r))
Just as one example, I used the dimensions of a well-known tonearm as follows:
Effective Length (l) : 233.20 mm
Mounting Distance (d) : 215.50 mm
Offset angle (y) : 23.63º
Then the question is, if I make an error in the Mounting Distance, what is the effect on the Tracking Error? The result is below.
If we take the manufacturer’s recommendation of d = 215.4 mm as the reference, and then look at the change in that Tracking Error by mounting the bearing at the incorrect distance in increments of 0.2 mm, then we get the plot below.
So, as you can see there, a 0.2 mm error in the location of the tonearm bearing (which, in my opinion, is a very small error…) results in a tracking error difference of about 0.2º at the minimum groove radius.
If I increase the error to increments of 1 mm (± 5mm) then we get similar plots, but with correspondingly increased tracking error.
If you go back and take a look at the equation above, you can see that the change in the tracking error is constant with the Offset Angle (unlike its relationship with an error in the location of the tonearm bearing, which results in a tracking error that is NOT constant). This means that if you mount your pickup on the tonearm head shell with a slight error in its angle, then this angular error is added to the tracking error as a constant value, regardless of the location of the stylus on the surface of the vinyl, as shown below.