Tonearm resonance theory – or how I learned to stop worrying and love the heavy Japanese arms.

So you have a turntable with a tonearm and a cartridge and you are wondering if the combination of tonearm and cartridge are performing as they should. You might be able to enjoy your music, but you wonder if there an option to tweak the system by either upgrading the cartridge, tonearm or the entire turntable system. You’ maybe started researching and stumbled on the phrase Tonearm Resonance frequency and calculators that can tell you if it works or doesn’t. But cartridges are listed from the manufacturer with a lot of numbers and data, the same goes for your tonearm. Maybe the producer of your tonearm and system doesn’t list any data and stats for your tonearm, in which case you are even harder pressed. What should you buy?

You could go out and buy that expensive Moving Coil cartridge and step-up, and fingers crossed it might be the upgrade you want it to be. There is a lot of different cartridges out there and you might want to invest a lot of money in your hobby. But how do we actually use the specified data to make a decision into what should be the best combination? Even more important, how do we use the data we know to tell if it would be a problem?

A good place to start is by looking at the final combination of cartridge and tonearm. That is specifically tonearm/cartridge resonance frequency. This article will highlight all the relevant theory of tonearm resonance, and with this article you’ll be better suited for making a good decision. I will not be making a calculator or quick-fix tool, but instead explain the theory involved and the math behind a usable and precise method. But please note: as with a lot of theory, there is a gap between theory and practice when it comes to applying your findings to your real-world system. That is just the nature of audio-systems. What you think sounds good in the end IS a subjective matter and a lot of different aspects factor in to the listening experience. Tonearm resonance is just one part of a good analog system.

What is tonearm resonance?

When you are playing a record, the cantilever on your cartridge is both responsible for moving the cantilever in accordance to the grooves in the record producing sound and at the same time moving the entire arm across the record surface. This means your cartridge is doing two things at the same time:

1. being stationary enough for the cantilever to move freely on 2 axis producing sound. Since the signal is generated in the cartridge from the cantilever movement relative to the cartridge body.

2. Being able to move freely so as when the record grooves are moving across the record surface towards the spindle the while tonearm follows, positioning itself relative to the groove.

This is quite the task to ask for from a mechanical system, but it works and it is the exact task our cartridges and tonearms are designed to do. To allow this to happen a cartridge has a “springiness” to it – it has a totally rigid body, but the cantilever can move in 2 axis. How easily this movement is done, ie. the springiness of the cartridge is called the compliance. Different cartridges have different values for compliance, as the suspension, holding the cantilever has different stiffness.

The resonance of the tonearm and cartridge combination is describing both the frequency and the amplitude, where vibrations starting at the cantilever will propagate throughout the cartridge and tonearm. The frequency denotes when and the amplitude denotes how much.

It is important to separate the two terms frequency and amplitude here. look at the illustration below, the red arrow highlights the amplitude – measured in voltage and experienced as increased or decreased volume. The green arrow highlights one cycle of a tone, how many of these cycles in a second is the frequency. it is measured in cycles per second or hertz and experienced as pitch (a high note will have a high frequency and a low note a low frequency).

The resonance frequency is a direct product of the cartridge compliance and the effective weight of the tonearm – ie, how much force the needle “sees” in order to move the tonearm around the spindle. The amplitude of the resonance is a product of the tonearm dampening. IE dampening an arm-tube, using an oil-damper and so on will not affect the resonance frequency, but could dampen the severity of the resonance and the impact on playback.

How do we measure tonearm resonance?

Using a test record providing a tonearm resonance test, you can hear or see or measure the resonance frequency. I usually use an Ortofon Test record, but a lot of test-records are available, and they would provide you with the same result. I use the lateral test, as my cartridge Compliance is provided from the manufacturer as a dynamic lateral value.

The way the test works, is that the frequency in the track is also cut with a 1000hz tone, and when the cartridge starts to resonate the fluctuations are clearly audible in the playback of the record. The Ortofon test record goes from 20hz to 5hz in 1hz steps, and at the resonance frequency I can clearly hear the fluctuations in the test-tone, but also see that the entire cartridge is visibly vibrating.

The test will highlight the resonance frequency of the combination, but if you record the signal digitally you can also pinpoint the resonance-frequency by looking at an increase in amplitude on the waveform.

By measuring my tonearm and cartridge combination – the Fidelity Research FR-64S with FR-S3 headshell and Ortofon Quintet Black S cartridge i measure the resonance to somewhere between 6 and 7hz, those two tests are easily the most severe, and it is had to tell them apart in terms of severity.

How do we calculate tonearm resonance?

Resonance of a tonearm is given by the below formula:

In order to calculate tonearm resonance we need to know the compliance of the cartridge and the effective mass of the tonearm. As mentioned before the compliance is a value denoting the “springiness” of the cantilever. The effective arm mass is the mass in relation to the spindle that the cartridge sees.

There is a lot of online tools and quick formulas for calculating tonearm resonance¹, unfortunately these are mostly not correct – or they are correct enough, as they are based on estimates and approximations. The reason for this, is that effective mass is not one number, and you can’t simply add the weight of the cartridge and screws to the manufacturers given number.

Effective mass is the moment of inertia divided with distance to spindle (or Effective Length) to the power of 2:

EM = MI / EL²

Where MI is Moment of Inertia and EL is effective length. ie. distance between needle tip and spindle axis.

The moment of inertia of a compound shape as a tonearm is the same as the moment of inertia of all the parts of the tonearm. so for ease of calculation i’ve broken my tonearm into 4 shapes that i can more easily calculate. What i need is the mass of the part and the parts center of gravity distance to the spindle. A straight tonearm here is much easier that an S-shaped arm, and if your tonearm has a taper like the SME 309 or V tonearms, it makes it a lot harder.

So while you can’t add the individual parts mass together, you can add the individual moment of inertia. as mentioned above the moment of Enertia is the relation between mass and distance between spindle and center of gravity, and the total moment of inertia of the combination is the sum of all parts individual moment of inertia.

MI = M * L²

where M is mass and L is between the center of gravity and spindle axis.

Let’s take my Fidelity Research FR-64s with the FR-S3 headshell and an Ortofon Quintet Black S cartridge – all measurements of distance is made as the tonearm is setup with the cartridge having 2gr of tracking force.

Counterweight:

M = 244gr, L = 3,6cm, MI = 3162,24 gr*cm²

Armtube between spindle and headshell:

M = 23,07gr, L = 12,5cm, MI =3604,69 gr*cm²

Armtube behind the spindle:

M = 5gr, L = 2,65cm, MI = 35,11 gr*cm²

headshell with mounted cartridge:

M = 29,37gr, L = 24,9 cm, MI = 18209 gr*cm²

The total MI (ie. the sum of all the moment of inertia) is 25011,7 gr*cm² and the effective length is 24,9 cm so the total effective mass of this combination is 40,34gr.

By using this value in the formula for calculating resonance frequency, well get a resonance frequency at 6,46hz which is well below the optimal range

The consequences

From the measurements and the calcuation we reach the same conclusion – a resonance frequency at around 6 to 7hz. This is not an ideal or optimal situation. an ideal range would be between 8 and 12 hz, where someone even limits this to between 9 and 11. But what does an out-of-interval measurement actually mean in terms of your turntable performance?

Whatever the resonance frequency is, it would mean your system is susceptible to resonate if exposed to those frequencies. For high resonance combinations this could happen by the music you play, ie. bass-notes in the recording could be affected and it could even make your cartridge skip. However for low resonance frequencies, the system is susceptible to footsteps or rumble or vibrations in your record player.

I am lucky, that my turntable setup is very well isolated from any outside noise, as well as very good tonearm dampening meaning the a very low risk of external forces, making the system resonate. But another effect could be unwanted dampening of the high-end.

As you can read, suddenly you have to take your system and room into account as well.

Should I care?

Well perhaps you should – it is a good idea to have a rough knowledge of the limitations of your system and in what direction is should be improved. Take my experience, the Quintet Black S performs fine despite the very low resonance frequency, but that is also due to my entire setup being as isolated as it is. However I have plans for purchasing a Fidelity Research headshell that weighs considerably less for sporting this cartridge. I have also fitted another tonearm to my turntable, the SAEC WE-407/23, which is lighter while still being a heavy tonearm.

The thing is – I love my FR-64S tonearm. It is beautiful and a very good sounding tonearm – despite being so heavy that it shouldn’t sound good. So in my system, with my gear i’ve found that I really don’t care because it sounds amazing when it plays, and as soon as the needle drops I forgot al about it not being optimal, and just listens to music.

If however I want to try another tonearm or another cartridge, i would go for a lighter arm and a lower compliance cartridge to have the flexibility in my system to match cartridges and tonearms in other ways and go up in resonance frequency instead of down. If you own a vintage system, note that the market for low-mass vs high-mass tonearms shifted in the 80’s with low-mass arms and high-compliance cartridges being popular but today’s high-end cartridges are better suited in medium to high-mass tonearms.

See the SME Series-3000 article for an overview of the evolution of that tonearm

The best thing I can tell you to do is to listen, borrow and test equipment before you buy it – you’ll need to try tonearms and catridges out on your own system before you purchase them – get social with other audiophiles and hifi-enthusiasts, and share experiences.

1. A list of tools on VinylEngine: https://www.vinylengine.com/cartridge_database_tools.php