TT speed

What you’re describing is tangency, I don’t know why you’re not comfortable with the word commonly used to describe this aspect of pickup arm geometry. Tangency has zero effect on freqeuncy (pitch). But I’m done with this conversation - the question you’ve raised is easily answered several ways, as I’ve already pointed out.

@cleeds I am not comfortable with the word, because

1. It is the wrong word.
2. Tangency does not affect pitch.

According to audio technical what we are talking about is not tangency:

The term “translocation” was used earlier, and that at least is not able to be easily confused with the stylus being parallel to the track, or tangent with the track… which is what “tangency” commonly refers to.

The OP is correct and his numbers are spot on target. The math to prove it is not difficult.

If the position of the cartridge changes by 20° from beginning to end of the LP, it will lengthen the time of the play back by 100mS: 20°/360°=1/18th of a rev; at exactly 33 1/3 RPM, 1 rev=1.8seconds x 1/18=100mS. The 100mS change in play back takes place over 500 revs (15 min) or 200µS/rev. So each and every revolution will experience a longer playback time of 200µS. 200µS/1.8S (expected time per rev @ 33 1/3)=0.00011111 or 0.0111% longer. Another way to look at it, the playback of each revolution will take 1.800200 seconds instead of 1.8S so the effective speed would be 33.3296 RPM (60Sec/1.80020). This will change a 1kHz tone to 999.888 Hz, assuming the record was cut with a linear tracking cutting head and the speed was held constant at 33 1/3 RPM.

The shift in frequency is insignificant and not audible even to someone with perfect pitch. It is not immeasurable or beyond calculation. It has nothing to do with tangency or FFT algorithms. It is purely a timing issue.

Another way to conceptualize this: Imagine a stylus is playing a 1kHz tone in a locked groove for 15 minutes (500 revs). Over the course of that time, the tone arm is moved in such a way that the tracking error stays within the expected range, but the stylus location finishes at a point that is 20° CW from its starting point in the locked groove. The effect would be the same as what is described by the OP.

My tonearm is 10" effective length. I placed a piece of masking tape on the platter to form a radial "spoke" on the platter.  If I align the stylus tip with the tape at the outer groove radius (146mm) then rotate the arm to the inner groove radius (60mm), it is slightly more than 3/4" CW of the stationary spoke of tape.  At 60 mm radius, 20° represents 0.824" of circumference.  So a 20° shift in location is a reasonable amount. 

I thought this through over the last few days, and I agree, but I would explain it differently. I’ve done a 180, reversal of my prior opinion.There’s more than one path to the same conclusion. Still, I’d like to see a demonstration.

Perform the demonstration in the last paragraph of my previous post.  Keep the platter stationary and move the stylus from the outer groove to the inner groove area and measure the distance from the stylus to the tape when located at the inner groove.  This is the linear distance the stylus will advance over the span of 15 minutes. At 60mm radius inner groove diameter, the circumference is 14.84".  Divide the distance from the tape by 14.84 for the portion of the total circumference and multiply by 360 to give you degrees.

The fact the stylus is CW of the tape proves the length of the groove used for playback increases as the stylus travels towards the dead wax area.  It's fairly easy to crunch the numbers and determine how much it advances on each rev. and therefore the change in pitch that will occur.

@phoenixengr 

Thank you.

Needless to say that a linear tracking arm is the exception here.

Thanks for the clearer explanation @phoenixengr
It is indeed a SFA* amount of shift.

* SFA is a UK/Au expanding to “Sweet F__ All”, which generally means is to the point of not mattering.

@holmz Yeah, that was weirdly interesting...

I have to credit Dover with causing me to re-think my prior position.  He simply mentioned that the cutting lathe is linear and the tonearm is pivoted (assuming a pivoted tonearm).  Rather than quantify the error, my simplistic approach is to start with the premise that the accuracy of the pitch (1000Hz) on this perfect LP being played on a perfectly speed stable TT depends upon the stylus remaining at a fixed point on the radius of the LP.  That, of course, doesn't happen with a pivoted tonearm; it touches the radius only at each of the two null points, assuming proper alignment.  When the stylus tip is off the radius, that represents relative movement of the stylus tip with respect to the recorded signal, which movement must change the pitch. Good thinking, Holmz.  Interesting discussion, too.

I’ve done a 180, reversal of my prior opinion.

Real or imaginary part?

I have to credit Dover with causing me to re-think my prior position.

Thanks @lewm It looks like we can credit @larryi , @dover  and others for helping me explain it properly. Many thanks fellows.

At first I was “… like… WTF.”
But then the math resulted in, “<and>… like… who cares…”

It is prolly just of minor intellectual curiosity, but does put “another one” in the clip of the digital crowd’s ammo against analogue. 😋 

@holmz

but does put “another one” in the clip of the digital crowd’s ammo against analogue.

I have a friend who designs DAC’s way beyond what is available publicly.

His description of red book digital is "it is only a little bit out ALL of the time".

Without going into the weeds truncation errors due to the sine x/x calculation are built into the red book standards. Digital data transfer also contains errors in transmission - it is possible to fix these as each packet has a check digit, but you would need a Fugaku supercomputer to calculate the corrections in real time.