I was leafing through some old editions of Wireless World magazine this week and came across an article in the July, 1968 issue called “Computing Distortion: Method for low-power transistor amplifiers” by L. B. Arguimbau and D.M. Fanger.
I was immediately intrigued by the first sentence, which read:
Unlike those of thermionic valves, the non-linearities in junction transistors for low collector currents are highly uniform and predictable, hardly differing from one transistor to another.
Now, as an “audio professional”, I’m very used to seeing the “±” sign in data sheets. Any production line of anything has some tolerance limits within which the product will fall.
For example, the (on-axis, where applicable) magnitude response of a loudspeaker or headphone is typically spec’ed something like ± 3 dB within some frequency range. This would mean that, at some frequency within that range, when measured under identical conditions, two “identical” products (e.g. with the same brand and model name) might be as much as 6 dB apart.
For different devices and components inside those devices, the tolerance values are different.
This is why, for example, when I read that someone says “headphone model A has more bass than headphone model B”, I know that if you included the missing information, it would actually read “my sample of headphone model A has more bass than my sample of headphone model B”.
However, when it comes down to the component level, I’m used to seeing tighter tolerances. Of course, if you save money on resistors, they might be within 20% of the stated value. However, if I look at the specs of a decent DAC (which, in my case, is a chip that would be used inside a product – not a big DAC-in-a-box that sits on your desk), I’m used to seeing numbers like < ±1 dB within pragmatically usable frequency ranges.
Since I’m only a young person, I’ve only really worked with transistor-based equipment, both when I worked in studios and also since I started working in home audio. So, I’ve always taken it for granted, and never even considered that the distortion characteristics of a transistor would vary from one to another. This is because, as the article from 1968 states: they don’t… much…
However, I’ve never thought about the (now obvious) possibility that two “identical” tubes/valves will have different distortion behaviour, even at low levels, due to manufacturing differences.
So, the next time someone tells you that this tube amp is better than that tube amp (which I translate in my head to actually mean “I prefer the sound of this tube amp over the sound of that tube amp” since “better” is multi-dimensional with different weightings of the different dimensions by person), remind them that the full sentence should be:
“I prefer My sample of this tube amp with the tubes that are currently in it to that tube amp with the tubes that are currently in it.”
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.”
This episode of 99 Percent Invisible tells the story of the Recording Ban of 1942, the impact on the rise of modern jazz music, and the parallels with the debates between artists and today’s streaming services. It’s worth the 50 minutes and 58 seconds it takes to listen to this!
At the end of that episode, the ban on record manufacture is mentioned, almost as an epilogue. This page from the January, 1949 issue of RCA’s “Radio Age” magazine discusses the end of that ban.
Interestingly, that same issue of the magazine has an article that introduces a new recording format: 7-inch records operating at 45 revolutions per minute! The article claims that the new format is “distortion free” and “noise-free”, stating that this “new record and record player climax more than 10 years of research and refinement in this field by RCA.”
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.
Glossary
Cantilever
The rod or arm that connects the stylus on one end to the “motor” on the other.
Effective Length
The straight-line distance between the pivot point of the tonearm and the top of the stylus
Equivalent Mass
definition to come
Flutter
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).
Frequency Drift
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”
Groove
The v-shaped track pressed into the surface of the vinyl record, in which the stylus sits
Linear Tracking
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.
Modulation Width
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.
Mounting Distance
The distance between the spindle and the pivot point of the tonearm.
Needle
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).
Null Radius
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).
Offset Angle
The angle between the axis of the stylus and a line drawn between the tonearm pivot and the stylus. See the line diagram below.
Overhang
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.
Pickup, Electromagnetic
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.
Pickup, Piezoelectic
Produces an output proportional to the displacement of the stylus.
Pitch
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.
Radial Tracking
A tonearm that rotates on a pivot point with the stylus tracing a circular path around that pivot.
Radius
The distance between the centre of the vinyl disc and the pickup stylus.
RIAA
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.
Side Thrust
definition to come
Skating Force
definition to come
Spindle
The centre of the platter around which the record rotates
Stylus
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.
Tracking Error
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.
Warp Wow
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”.
Wow
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
7″ discs
6.78″, +0.06″, -0.00″
172.2 mm, +1.524 mm, – 0.0 mm
10″ discs
9.72″, +0.06″, -0.00″
246.9 mm, +1.524 mm, – 0.0 mm
12″ discs
11.72″, +0.06″, -0.00″
297.7 mm, +1.524 mm, – 0.0 mm
Start of modulated pitch diameter
7″ discs
6.63″, +0.00″, -0.03″
168.4 mm, +0.0 mm, – 0.762 mm
10″ discs
9.50″, +0.00″, -0.03″
241.3 mm, +0.0 mm, – 0.762 mm
12″ discs
11.50″, +0.00″, -0.03″
292.1 mm, +0.0 mm, – 0.762 mm
Minimum inside diameter
7″ discs
4.25″
107.95 mm
10″ discs
4.75″
120.65 mm
12″ discs
4.75″
120.65 mm
Lockout Groove diameter
7″ discs
3.88″, +0.00, -0.08
98.552 mm, +0.0 mm, -2.032 mm
10″ discs
4.19″, +0.00, -0.08
106.426 mm, +0.0 mm, -0.762 mm
12″ discs
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)
Signal levels
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.
References
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)
45 500: Hi-Fi Technics: Requirements for Disk Recording Reproducing Equipment
45 507: Measuring Apparatus for Frequency Variations in Sound Recording Equipment
45 538: Definitions for Disk Record Reproducing Equipment
45 539: Disk Record Reproducing Equipment: Directives for Measurements, Markings, and Audio Frequency, Connections, Dimensions of Interchangeable Pickups, Requirements of Playback Amplifiers
45 541: Frequency Test Record St 33 and M 33 (33 1/3 rev/min; Stereo and Mono)
45 542: Distortion Test Record St 33 and St 45 (33 1/3 or 45 rev/min; Stereo)
45 543: Frequency Response and Crosstalk Test Record
45 544: Rumble Measurement Test Record St 33 and M 33 (33 1/3 rev/min; Stereo and Mono)
45 545: Wow and Flutter Test Records, 33 1/3 and 45 rev/min
45 546: Stereophonic Disk Record St 45 (45 rpm)
45 547: Stereophonic Disk Record St 33 (33 1/3 rpm)
45 548 Aptitude for Performance of Disk Record Reproducing Equipment
45 549: Tracking Ability Test Record
IEC Publications
98: Recommendations for Lateral-Cut Commercial and Transcription Disk Recordings
98: Processed Disk Records and Reproducing Equipment
386: Method of Measurement of Speed Fluctuations in Sound Recording and Reproducing Equipment
Late-night free-floating anxiety is not a modern phenomenon. One of my favourite descriptions of it is Dorothy Parker’s 1933 short story called “The Little Hours”, published in The New Yorker.
However, more than 30 years before this, The Phono Gram magazine published the following, which a Dr. J. Leonard Corning proposed to cure the problem of late-night melancholy with what we would, today, called a “pair of headphones” (although the design that he describes probably wouldn’t sell well to anyone who is not interested in S&M…) playing Wagnerian arpeggios and minor chords. (Personally, I just put the timer on my iPhone to turn off after 30 minutes, and turn on an old episode of “QI” or “8 Out of 10 Cats Does Countdown” – Sean Lock’s voice drowns out my own internal ones.)
I live in Denmark where people speak Danish. One interesting word that I use every day is “højtaler” which is the Danish word for “loudspeaker”. I say that this word is “interesting” because, just like “loudspeaker” it is actually two words glued together. “Høj” means “high” or “loud” and “taler” means “talker” or “speaker” (as in “the person who is doing the talking”).
Sometimes, when I have a couple of minutes to spare, instead of looking at cat videos on YouTube, I sift through old audio and electronics magazines for fun. One really good source for these is the collection at worldradiohistory.com. (archive.org is also good!). Today I stumbled across the December, 1921 edition of Practical Electrics Magazine (which changed its name to The Experimenter in November of 1924 and then to Amazing Stories in April 1926*) It had a short description of a stage trick called the “Haunted Violin”, an excerpt from which is shown below.
The trick was that a violin, held by a woman walking around the aisle of the theatre would appear to play itself. In fact, as you can see above, there was an incognito violinist with a “detectophone” that was transmitting through wires connected to metal plates under the carpet in the theatre. The woman was wearing shoes with heels pointy enough to pierce the carpet and make contact with the plates. The heels were then connected with wires running through her dress to a “loud talker” hidden inside the violin.
Seems that, in 1921, it would have been easier to learn at least one word in Danish…
Side note: This is why, when I’m writing about audio systems I try as hard as possible to always use the word “loudspeaker” instead of “speaker”. To me, a “speaker” is a person giving a speech. A “loudspeaker” is a thing I complain about every day at work.
* That April 1926 edition of Amazing Stories had short stories by Jules Verne, H.G. Wells, and Edgar Allan Poe!
Post Script: My wife reminded me that it’s the same in French: “haut-parleur”. It’s a reminder that the original loudspeakers were never intended for music, I suppose…
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)
where
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.)
Important notes:
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)
where
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
Final Post-Script:
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.
