Arc Angles: Optimizing Tonearm Geometry
If you've been reading audio magazines for the past 30 years or more, you may recall that in the late 1970s there was a short-lived furor, first in the US and then in the UK, about forgotten lessons of tonearm/cartridge alignment. It turned out that seminal work on the subject by H.G. Baerwald, published in 1941(footnote 1) and refreshed by J.K. Stevenson in 1966 (footnote 2), had been disregarded by many pickup-arm designers, presumably because they'd never been acquainted with it. (It has since come to light, principally through the work of Graeme Dennes (footnote 3), that Erik Löfgren first published the optimum alignment equations more generally credited to Baerwald over three years earlier, in 1938. Perhaps because Löfgren's paper was in German and Baerwald's in English, the latter's became the recognized reference. Löfgren's paper has since been translated by Klaus Rampelmann (footnote 4)).
At the time this all blew up, LP was thriving, of course. Some would argue that it is beginning to do so again, but either way, we're talking 30 years ago. In the interim, the same process of forgetfulness and obfuscation has inevitably occurred, so that today, once again, despite the recent efforts of Michael Fremer and some fine Web resources (footnote 5), many audiophilesand perhaps even some new-generation turntable designerswill be unclear about what's involved in optimally aligning a pickup cartridge. Fewer still will know about the history and theory of the subject, and the important ramifications that theory has for the design of pickup arms and their use. If you're interested in the more complete picture, then this introductory article will give you a good grounding, and point you in the direction of further reading.
To do the history part right, we have to go back before Löfgren and Baerwald, and turn our gaze to England in 1924, the second year of publication of The Gramophone, cofounded and edited by novelist (later Sir) Compton Mackenzie. The magazine's technical editor was the remarkable Percy Wilson, a larger-than-life character whose mathematical background was used to good effect at the time, not only in modifying the flare of the exponential horn on the more realistic assumption of spherical rather than planar wavefronts, but also by being applied to what, in an article published in the September and October 1924 issues (footnote 6), he termed "needle track alignment."
What was then already understood was that (using modern terminology) because tonearms or pickup arms were pivoted, but cutter heads ran on a linear drive mechanism, angular errors were inherent in record replayand exactly the same remains true today (footnote 7). The cutting stylus traverses a radius of the disc, so that the lateral groove modulation is nominally at right angles to a tangent to the groove at all timeswhereas the replay stylus traverses an arc, so that the modulation is generally read with some angular disparity, which generates distortion (footnote 8). Today we know that angular disparity as lateral tracking error (LTE), and the resulting nonlinearity as lateral tracking error distortion (LTED).
Wilson calculated what alignment of the replay components would minimize LTE across the disc, basing his figuring on the assumptionwrong, as we will seethat this would also minimize the resulting distortion. To achieve this, he found, the radius of the stylus (needle) arc had to be greater than the distance from the arm pivot to the center of the disc, by a distance that today we term the overhang; and the pickup cartridge had to be twisted relative to a line joining stylus to arm pivot by an angle termed the offset. This overhang-plus-offset geometry, Wilson's great contribution to arm/cartridge alignment, is still used today, even though his equations for calculating the two parameters have been superseded.
If we assume an arm effective length (that is, the horizontal distance from stylus to arm pivot axis) of 230mm, which is about the 9" of many modern pickup arms, and maximum and minimum modulated groove radii of 146.3 and 56mm (more on this later), then Wilson's alignment gives the result shown in fig.1. Actually, this is two graphs superimposed: the red trace, which is partly overlaid by the blue trace, shows LTE vs groove radius, and the blue trace the absolute value of LTE vs groove radius, which results in a W shape that will become more familiar as we continue. There are two important features to note about this graph. First, the alignment minimizes the maximum LTE by setting it equal at three points across the disc: the innermost modulated groove radius, the outermost modulated groove radius, and a point in between represented by the peak of the curve. (The sign of the LTE is irrelevant; a positive angle gives rise to the same distortion as an equivalent negative angle.) Second, the LTE is zero at two points across the disc termed the zero tracking error radii, here 64.3 and 127.4mm.
Footnote 1: H.G. Baerwald, "Analytic Treatment of Tracking Error and Notes on Optimal Pick-Up Design," Journal of the Society of Motion Picture Engineers, December 1941.
Footnote 2: J.K. Stevenson, "Pickup Arm Design," Wireless World, May and June 1966.
Footnote 3: Graeme Dennes, "An Analysis of Six Major Articles on Tonearm Alignment Optimisation," 1983, updated 2009, downloadable here.
Footnote 4: Erik Löfgren, "On the Non-Linear Distortion in the Reproduction of Phonograph Records Caused by Angular Deviation of the Pickup Needle," originally published in German in Akustische Zeitschrift, November 1938, Vol.3, pp.350362; downloadable in English translation here.
Footnote 5: Such as www.vinylengine.com and www.audioasylum.com/audio/vinyl.
Footnote 6: Percy Wilson, "Needle Track Alignment," The Gramophone, September (pp.129131) and October (pp.167169) 1924. Available from Gramophone's online archive, these articles can be downloaded, a bit laboriously, as PDFs. Only the September article contains alignment equations.
Footnote 7: I ignore here radial-tracking (aka linear-tracking or parallel-tracking or tangential-tracking) arms since they remain rarities, and those pivoted arms with headshell-rotation mechanismssuch as that on the Garrard Zero 100 of yesteryear because they are rarer still.
Footnote 8: I was once upbraided by a reader for forgetting, along with everyone else who had ever written about arm/cartridge alignment, that the record groove is a spiral, and therefore its tangent is never truly at a right angle to a radial line drawn from the center of the disc through the stylus. While true, this is irrelevant to the alignment issue, since the aim is not to align the cartridge front-back axis as closely as possible to a true tangent to the groove, but to align it as closely as possible to the front-back axis of the cutting head, which is aligned (nominally, at least) at a right angle to the disc radius. Even if it were relevant, the angular error is tiny. If we assume a groove pitch of 0.135mm, then even at the inner modulated groove radius of an LP, the angular difference between a tangent to the groove spiral and a tangent to a circle of the same radius is just 0.022°. For comparison, the LTE of an optimally aligned 9" arm at this radius is 0.85°; ie, almost 40 times greater.