Cartridge / Arm Matching....or things you didn't really want to 
know about resonant frequency of a system tonearm/cartridge which is coupled to 
the record by the compliance of the cantilever suspension. ... 
1) 
above photo #1: showing the cantilever and location of suspension.  
The suspension element may be as simple as a tight fitting elastic grommet that 
grips both the shaft and the cartridge body.
 
2) 
above photo #2: Showing stylus mounting in the cantilever.  This is a 
nude mounting that features a precise 'force fit' between stylus body and the 
bore in the cantilever.  Additionally, a small bit of adhesive is applied over 
the joint to ensure a lasting fit. 
side note: Whoever said "diamonds are forever" was not referring 
to phono cartridge stylii. They wear out. Yet, like their jewelry counterparts, 
they are expensive nonetheless.
Matching a cartridge to a tonearm (pivoting tone 
arms)
The focus here is upon elements of the cartridge design, and elements of the 
tonearm design which affect how a vinyl addict might go about choosing a given 
cartridge to fit a given tonearm.  First, some definitions.
Effective mass versus compliance results in a Resonant 
Frequency (of the cantilever)
Compliance: Let's take a look at a spring (any spring) designed to 
carry a given load.  Place too much weight on this spring and it collapses.  
Place not enough weight on the spring and the spring won't compress at all and 
remains rigid.  Think of the cantilever as a spring. 
In the case of a phonograph cartridge, the cantilever is a rigid arm 
connected to a springing medium mounted up within the body of the cartridge.  
This springing element may be as simple as a rubber donut that holds enough 
tension against the cantilever mounting to maintain relative position of the 
cantilever.  Most importantly, the suspension must control the attitude of the 
stylus fitted at the cantilever's end.   
The amount of distance that a cantilever deflects under a given force 
load is referred to as cantilever compliance.  Higher compliance cantilevers 
deflect a greater distance when a given amount of force is applied.  Lower 
compliance cantilevers deflect lesser distances when the same force is applied.  
In other words; high compliance = softer, low compliance = 
stiffer.
Vertical Tracking Force (VTF): The amount of vertical force placed 
upon the cantilever is controlled by a careful balance between the weights of 
the arm on either end of the pivot.  Typically, the tonearm will have a long 
tube leading to a head shell with a phonograph cartridge and stylus at that 
end.  The other end of the tonearm will have weights intended to balance the 
mass of the long end.  By careful manipulation of the counterweight position, 
relative to the distance from the pivot bearing, a precisely measured amount of 
vertical force may be applied at the stylus end.
Effective Mass: The amount of force felt at the 
stylus under dynamic conditions in any (xyz) arcing vector about the pivot. This 
differs from VTF which is set static and remains constant only under 'peaceful' 
conditions while the record is in play.  Effective mass is influenced by the 
weight of the various appendages of the tonearm assembly in ratio to the 
distance from the pivot.  Weight that is further from the pivot center will 
account for higher effective mass than the same weight if positioned closer to 
the pivot.  Said slightly differently,  the heavy bits on the tonearm need to be 
closer to the tonearm pivot or excessive effective mass will be the 
result.  
Phonograph cartridges have different weights from one product to the next, 
therefore tonearm makers rate their arms in terms of effective mass before a 
cartridge is mounted.  In the case of the SME 3009 Improved with fixed head 
shell, the arm is rated to have an effective mass of 6.5 grams.  Compare this to 
the Thorens TP16 Mk 1 tonearm  which has a rated effective mass of 
16.5grams.
Resonant frequency of the cantilever is used as a 
guide to match suspension stiffness (compliance) of the cartridge to the tone 
arm's (effective) mass.
Resonant Frequency (of the cantilever) The acoustic frequency at which 
the cantilever will become excited and vibrate out of control.  :)))  This 
frequency is measured in cycles per second.  Also referred to as 'hz'.  Resonant 
frequency of a cantilever is regarded as inescapable and the effect is 
controlled by manipulating this frequency to exist in a range below human 
hearing but not so low that it will become excited by external vibrations such 
as foot fall disturbance or that of a warped record.  This ideal frequency range 
is 8 to 12 hz.  The lowest of low organ notes rarely go below 20 hz.  Footfall 
and record warps happen below 6 hz.  
The effective mass of a tonearm in combination with the compliance of the 
cartridge cantilever serves to determine where the resonant frequency of a given 
tonearm/cartridge match up will be.  In general terms, arms with high 
effective mass fitted with cartridges of high compliance result in resonant 
frequencies that fall below the ideal range.  At the opposite end, arms with low 
effective mass mated to cartridges of low compliance result in resonant 
frequencies above the desired range.  Both extremes are to be avoided.  


So....wouldn't it be useful if you could calculate a 
resonant frequency between a given arm and cart...?   You can,  just 
plug in some vital statistics into the short formula below.  Keep in mind the 
formula is intended as a rule of thumb.  The test record will be the final say 
on what a given arm cart combo can give in terms of cantilever resonant 
frequency.  Think of the formula as a 'on paper test' and the test record as a 
'real world actual test'.
rf = 159 / sqrt ((eff. mass + cart weight + fastener weight) * 
(compliance)) 





rf: resonant frequency in hz



eff. mass: rated by tonearm 
manufacturer



cart weight: rated by cartridge 
manufacturer, but if accurate scales exist, an actual weight value could be 
used



fastener weight: screws, nuts, 
spacers, washers, shims.  They have weight and add to the mass over the 
stylus



compliance: rated by cartridge 
manufacturer
  

 
Example No. 1: Now let's think about 
matching a Shure V15VxMR to the above pictured TP16 Mk 1 tonearm.
 





effective mass rating: 16.5 
grams



cartridge weight: 6.6 grams



fastener weight: .5 grams



compliance: 25

(16.5 + 6.6 + .5) * 25 = 590
sqrt 590 = 24.2899
159 / 24.2899 = 6.5459 hz calculated
The above figure was approximately verified with the HFNRR test record 
getting a test result value of 6 hz
I'll admit that I have lived with the above combination for a time.  I found 
the setup susceptible to footfall but it seemed to track record warps just 
fine.  A marginal matching.  It is the wrong side of marginal, too.  Another 
visual note about this match-up is that the high compliance of the Shure 
cantilever  was quite obvious when dropping the stylus into the lead in groove.  
Considerable sideways deflection was evident.  This is not the same case with 
the above match up between the Shure and the much lighter SME tonearm.  For more 
notes about The Shure cartridge and it's dynamic stabilizer  when fitted to the 
TP16 mk 1 see the ** below.


 
Example No. 2: with the TP16 Mk 1 tonearm.  
This time with the above pictured Blue Point Special cartridge...?  Let's crunch 
the numbers and see.





effective mass rating: 16.5 
grams



cartridge weight: 9.0 grams



fastener weight: .5 grams



compliance: 12

(16.5 + 9.0 + .5) * 12 = 312
sqrt 312 = 17.6635
159 / 17.6635 = 9.0016 hz.


Example No. 3: let's calculate the rf for 
my SME 3009 Improved Fixed Head shell tonearm and Shure V15VxMR cartridge.  
 





The effective mass rating for the arm is 6.5 
grams



The weight of the cartridge is rated at 6.6grams



fastener weight: .5 grams 



Compliance is rated at 25 (x 10¯6   cm/dyne)*

(6.5 + 6.6 + .5) * 25 = 340 
sqrt 340 = 18.4391
159 / 18.4391 = 8.6230 hz calculated
HFNRR test result: side 2 band 2 low freq. horizontal resonance 
test......10 hz 
Notes about the discrepancy between the test record figure and the 
calculation.  The test record value is considered to be the valid reference.  
One of the variables in the calculation must be incorrect.  Perhaps effective 
mass rating of the tonearm may actually be lighter in the real world situation.  
The finger lift was not used and so does not contribute to the effective mass.  
I have no accurate means to measure the weight of the finger lift.  I used the 
shortest possible screws in the cartridge mounting.  Said screws and nuts are 
aluminum.


 
*Shure does not disclose compliance figures for this cartridge.  This value 
was found elsewhere on the web. The 'actual' HFNRR test result reads better this 
value.  Another note: when fitted to the SME, the 'dynamic stabilizer' of the 
Shure cartridge is not required.
 
 
**  :  The Shure V15VxMR phono cartridge uses a device that Shure calls a 
"dynamic stabilizer".  They (Shure) describe this device as a shock absorber for 
their cantilever.  If this device is put to use, it is claimed that a much wider 
range of effective mass tone arms may be used with this cartridge.  I have used 
this 'dynamic stabilizer' when the cartridge was fitted to the heavy TP16 mk1 
and found that the cartridge tracked all records without any apparent fault.  
When the device was parked in it's 'up' position (taken out of use) a much 
greater amount of cantilever deflection could be witnessed and the tonearm was 
more susceptible to external disturbance.  The 'device' was not put to use with 
the SME tonearm as the resonant frequency of the cantilever falls within the 
optimal range when fitted to this arm.