6 June 2018

Low-frequency Arm and Cartridge Interaction, Part I

We have to slightly postpone further investigation of flexure bearings as neither the reference playback system nor the laser displacement meter are functional. Hope to get both sorted out before the holiday season.

Not to waste my (and your) time waiting, I have decided to publish what we know about a rather controversial topic — cartridge compliance.

Not to waste my (and your) time waiting, I have decided to publish what we know about a rather controversial topic — cartridge compliance.

If you open AES Disk Recording Anthology, a good third of the articles in Tome 1 would touch the subject one way or the other. And yet, few other ideas in LP reproduction are as misunderstood as a relationship between cartridge compliance, tonearm effective mass and the low-frequency resonance.

On the surface, it's all quite simple. There's a formula quoted on literally thousands of sites:

$$f={1\over{2\pi\sqrt{m \cdot c}}}$$

You plug in the cartridge compliance**c**, tonearm's effective mass **m**, and you get your LF resonant frequency *f*. Too high, you'll get bass distortion, too low and you'll get mistracking. Aim for 8-10 Hz, and you should be all right. Easy, isn't it?

$$f={1\over{2\pi\sqrt{m \cdot c}}}$$

You plug in the cartridge compliance

There's one problem with this calculation. It doesn't seem to be very well supported by both measurements and listening experience.

There's one problem with this calculation. It doesn't seem to be very well supported by both measurements and listening experience. Have a look at our armtube experiment's summary of findings. The widest, in tens of grams, variation in armtube weight results in single Hz difference in main LF resonance.

Sonically, there are numerous examples of relatively "stiff" low compliance moving coil cartridges performing wonderfully in very light arms. And, while a lot fewer people do this, high and medium compliance MM cartridges (except Grado) work very well when mounted on heavy tonearms. I vividly remember a Shure V15-IV, beautifully singing its little heart out on a monster Gray 208s tonearm. Gray's effective mass isn't published, but is clearly above 50 grams.

I am not the only one to think that there's something wrong with this formula. Origin Live, who obviously know a whole lot about tonearms, tell their customers to simply ignore the calculations.

Sonically, there are numerous examples of relatively "stiff" low compliance moving coil cartridges performing wonderfully in very light arms. And, while a lot fewer people do this, high and medium compliance MM cartridges (except Grado) work very well when mounted on heavy tonearms. I vividly remember a Shure V15-IV, beautifully singing its little heart out on a monster Gray 208s tonearm. Gray's effective mass isn't published, but is clearly above 50 grams.

I am not the only one to think that there's something wrong with this formula. Origin Live, who obviously know a whole lot about tonearms, tell their customers to simply ignore the calculations.

Well, the formula isn't derived there. It's referenced from an AES publication, which in turn sends us to another paper, and another…

in the end, we find the original in a 1954 article, "Resonance, Tracking and Distortion: An Analysis of Phonograph Pickup Arms" by R.E.Carlson of Fairchild, back then a big maker of both tonearms and cartridges. Here's that hand-drawn piece of audio history for your enjoyment.

Mr Carlson assumes a simple elastic motion in translation and rotation, and applies the basic harmonic oscillator formula, where **k** is a spring constant and *m* is effective mass:

$$\omega= \sqrt{k\over m}$$

Converting angular frequency to Hz, and substituting spring constant*k* for compliance *c* (compliance is just *k* written backwards), we get our formula. Hurray, we found it!

$$f= {1\over {2\pi}}\sqrt{k\over m} = {1\over {2\pi}}\sqrt{1\over{m \cdot c}} = {1\over {2\pi\sqrt{m \cdot c}}}$$

$$\omega= \sqrt{k\over m}$$

Converting angular frequency to Hz, and substituting spring constant

$$f= {1\over {2\pi}}\sqrt{k\over m} = {1\over {2\pi}}\sqrt{1\over{m \cdot c}} = {1\over {2\pi\sqrt{m \cdot c}}}$$

Let's have a look at the typical 1950s cartridge and compare it to the modern one.

The modern stereo cartridge, however, uses the elastomer "donut" as the main component of its suspension, providing both damping and location. I've tried to highlight the differences in an illustration on the right.

So naturally, our first question can be:

In the next issue, I will try to go throught the most important differences between springs and elastomers as used in the cartridge suspension, and what it means for Mr Carlson's formula. Then we'll see if other differences play a perceptible role too.

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